Fundamentals of Spectrum Analysis

Fundamentals of Spectrum Analysis
Fundamentals of
Spectrum Analysis
Christoph Rauscher
ISBN 978-3-939837-01-5
Fundamentals of Spectrum Analysis
Christoph Rauscher
Christoph Rauscher
Volker Janssen, Roland Minihold
Fundamentals of Spectrum Analysis
© Rohde & Schwarz GmbH & Co. KG, 2001
Mühldorfstrasse 15
81671 München
Germany
www.rohde-schwarz.com
Sixth edition 2008
Printed in Germany
ISBN 978-3-939837-01-5
PW 0002.6635.00
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Table
of
Contents
Table of Contents
1
Introduction
7
2
Signals
8
2.1
Signals displayed in time domain
8
2.2
Relationship between time and frequency domain
9
3Configuration and Control Elements of a
Spectrum Analyzer
17
3.1
17
Fourier analyzer (FFT analyzer)
3.2Analyzers operating in accordance
with the heterodyne principle
27
3.3
30
Main setting parameters
4Practical Realization of an Analyzer Operating on the
­Heterodyne Principle
32
4.1
RF input section (frontend)
32
4.2
IF signal processing
44
4.3
Determination of video voltage and video filters
55
4.4
Detectors
61
4.5
Trace processing
73
4.6
4.6.1
4.6.2
4.6.3
Parameter dependencies
Sweep time, span, resolution and video bandwidths
Reference level and RF attenuation
Overdriving
76
76
80
86
Fundamentals
of
S p e ct r u m A n a l y s i s
5
Performance Features of Spectrum Analyzers
95
5.1
Inherent noise
95
5.2
Nonlinearities
102
5.3
Phase noise (spectral purity)
114
5.4 1 dB compression point and maximum input level
120
5.5
Dynamic range
125
5.6
Immunity to interference
135
5.7
LO feedthrough
138
5.8
Filter characteristics
139
5.9
Frequency accuracy
140
5.10
5.10.1
5.10.2
5.10.3
Level measurement accuracy
Uncertainty components
Calculation of total measurement uncertainty
Measurement error due to low signal-to-noise ratio
141
142
148
156
5.11
Sweep time and update rate
159
6
Frequent Measurements and Enhanced Functionality
162
6.1
6.1.1
6.1.2
6.1.3
Phase noise measurements
Measurement procedure
Selection of resolution bandwidth
Dynamic range
162
162
165
167
6.2Measurements on pulsed signals
6.2.1 Fundamentals
6.2.2 Line and envelope spectrum
6.2.3 Resolution filters for pulse measurements
6.2.4 Analyzer parameters
6.2.5 Pulse weighting in spurious signal measurements
172
173
177
182
184
185
Table
of
Contents
6.2.5.1 Detectors, time constants
6.2.5.2 Measurement bandwidths
186
190
6.3Channel and adjacent-channel power measurement
6.3.1 Introduction
6.3.2Key parameters for adjacent-channel
power measurement
6.3.3Dynamic range in adjacent-channel
power measurements
6.3.4Methods for adjacent-channel power measurement
using a spectrum analyzer
6.3.4.1 Integrated bandwidth method
6.3.4.2Spectral power weighting with modulation filter
(IS-136, TETRA, WCDMA)
6.3.4.3 Channel power measurement in time domain
6.3.4.4 Spectral measurements on TDMA systems
190
190
198
200
201
References
204
The current spectrum analyzer models from Rohde & Schwarz
207
Block diagram of spectrum analyzer described in this book
220
193
194
195
195
Measurement Tips
Measurements in 75 W system
Measurement on signals with DC component
Maximum sensitivity
Identification of intermodulation products
Improvement of input matching
33
37
101
112
147
Fundamentals
of
S p e ct r u m A n a l y s i s
I n t r o d u ct i o n
1
Introduction
One of the most frequent measurement tasks in radiocommunications
is the examination of signals in the frequency domain. Spectrum analyzers required for this purpose are therefore among the most versatile and
widely used RF measuring instruments. Covering frequency ranges of
up to 40 GHz and beyond, they are used in practically all applications of
wireless and wired communication in development, production, installation and maintenance efforts. With the growth of mobile communications, parameters such as displayed average noise level, dynamic range
and frequency range, and other exacting requirements regarding functionality and measurement speed come to the fore. Moreover, spectrum
analyzers are also used for measurements in the time domain, such as
measuring the transmitter output power of time multiplex systems as a
function of time.
This book is intended to familiarize the uninitiated reader with the field
of spectrum analysis. To understand complex measuring instruments
it is useful to know the theoretical background of spectrum analysis.
Even for the experienced user of spectrum analyzers it may be helpful
to recall some background information in order to avoid measurement
errors that are likely to be made in practice.
In addition to dealing with the fundamentals, this book provides an
insight into typical applications such as phase noise and channel power
measurements.
For further discussions of this topic, refer also to Engelson [1-1] and
[1-2].
7
Signals
2
Signals
2.1
Signals displayed in time domain
A
jlm
In the time domain the amplitude of electrical signals is plotted versus
time – a display mode that is customary with oscilloscopes. To clearly
illustrate these waveforms, it is advantageous to use vector projection.
The relationship between the two display modes is shown in Fig. 2-1 by
way of a simple sinusoidal signal.
X0 t
Re
1
0.8
0.6
0.4
0.2
0
–0.2
–0.4
–0.6
–0.8
–1
0
0.5 T0
T0
1.5 T0
2 T0
t
Fig. 2-1 S
inusoidal signal displayed by projecting a complex rotating vector on
the imaginary axis
The amplitude plotted on the time axis corresponds to the vector projected on the imaginary axis (jIm). The angular frequency of the vector
is obtained as:
w 0 = 2 ⋅ p ⋅ f0 where w0
f0
(Equation 2-1)
angular frequency
signal frequency
A sinusoidal signal with x(t)= A · sin(2 · p · f0 · t ) can be described as
x(t)=A · Im{e j · 2p·f 0 ·t }.
8
Relationship Between Time
2.2
and
Frequency Domain
Relationship between time and frequency domain
Electrical signals may be examined in the time domain with the aid of
an oscilloscope and in the frequency domain with the aid of a spectrum
analyzer (see Fig. 2-2).
A
0
t
in
ma
e do
Tim
t
A
A
0
Frequ
ency d
omain
f
f
Fig. 2-2 Signals examined in time and frequency domain
The two display modes are related to each other by the Fourier transform (denoted F ), so each signal variable in the time domain has a characteristic frequency spectrum. The following applies:
+∞
{ }
X f ( f ) = F x (t ) = ∫ x (t ) ⋅ e- j 2 pft dt (Equation 2-2)
-∞
and
{
}
+∞
x (t ) = F -1 X f ( f ) = ∫ X f ( f ) ⋅ e j 2 pft dt (Equation 2-3)
-∞
where
F{x (t)} Fourier transform of x(t)
F –1 {X(f )} inverse Fourier transform of X(f )
x (t )
signal in time domain
Xf(f )
complex signal in frequency domain
To illustrate this relationship, only signals with periodic response in the
time domain will be examined first.
9
Signals
Periodic signals
According to the Fourier theorem, any signal that is periodic in the time
­domain can be derived from the sum of sine and cosine signals of different frequency and amplitude. Such a sum is referred to as a Fourier
­series. The following applies:
x (t ) =
A0
2
∞
∞
n=1
n=1
+ ∑ An ⋅ sin(n ⋅ w0 ⋅ t ) + ∑ B0 ⋅ cos(n ⋅ w0 ⋅ t ) (Equation 2-4)
The Fourier coefficients A0, An and Bn depend on the waveform of ­signal
x(t ) and can be calculated as follows:
2
A0 =
T0
An =
Bn =
2
T0
2
T0
T0
∫
x (t )dt T0
∫
x (t ) ⋅ sin(n ⋅ w0 ⋅ t )dt
(Equation 2-6)
0 T0
∫
x (t ) ⋅ cos(n ⋅ w0 ⋅ t ) dt (Equation 2-7)
0
where A0 2
(Equation 2-5)
0
x(t )
n
T0
w 0
DC component
signal in time domain
order of harmonic oscillation
period
angular frequency
Fig. 2-3b shows a rectangular signal approximated by a Fourier series.
The individual components are shown in Fig. 2-3a. The greater the number of these components, the closer the signal approaches the ideal rectangular pulse.
10
Relationship Between Time
and
a)
Frequency Domain
Harmonics
x(t)
n=1
n=5
n=3
n=7
0
b)
Sum of harmonics
t
x(t)
0
Fig. 2-3 Approximation of a
rectangular signal by
summation of various
sinusoidal oscillations
t
In the case of a sine or cosine signal a closed-form solution can be found
for Equation 2-2 so that the following relationships are obtained for the
complex spectrum display:
{ (
F sin 2 ⋅ p ⋅ f0 ⋅ t
)} = 1j ⋅ d ( f -f ) = -j ⋅ d ( f -f ) (Equation 2-8)
)} = d ( f -f ) (Equation 2-9)
0
0
and
{ (
F cos 2 ⋅ p ⋅ f0 ⋅ t
0
where d(f -f0) is a Dirac functiond(f -f0) = ∞ if f -f0 = 0, and f = f0
d(f -f0) = 0, otherwise
+∞
∫ d ( f - f ) df
0
-∞
11
= 1
Signals
It can be seen that the frequency spectrum both of the sine signal and
cosine signal is a Dirac function at f0 (see also Fig. 2-5a). The Fourier
transforms of sine and cosine signal are identical in magnitude, so that
the two signals exhibit an identical magnitude spectrum at the same
frequency f0 .
To calculate the frequency spectrum of a periodic signal whose time
characteristic is described by a Fourier series in accordance with Equation 2-4, each component of the series has to be transformed. Each of
these elements leads to a Dirac function, that is a discrete component
in the frequency domain. Periodic signals therefore always exhibit discrete spectra which are also referred to as line spectra. Accordingly, the
spectrum shown in Fig. 2-4 is obtained for the approximated rectangular signal of Fig. 2-3.
| X(f)|
f0
3f0
7f0 f
5f0
Fig. 2-4 Magnitude spectrum of
approximated rectangular signal shown in
Fig. 2-3
Fig. 2-5 shows some further examples of periodic signals in the time and
frequency domain.
Non-periodic signals
Signals with a non-periodic characteristic in the time domain cannot
be described by a Fourier series. Therefore the frequency spectrum of
such signals is not composed of discrete spectral components. Non-periodic signals exhibit a continuous frequency spectrum with a frequencydependent spectral density. The signal in the frequency domain is calculated by means of a Fourier transform (Equation 2-2).
Similar to the sine and cosine signals, a closed-form solution can be
found for Equation 2-2 for many signals. Tables with such transform
pairs can be found in [2-1].
For signals with random characteristics in the time domain, such as
noise or random bit sequences, a closed-form solution is rarely found.
12
Relationship Between Time
and
Frequency Domain
The frequency spectrum can in this case be determined more easily by a
numeric solution of Equation 2-2.
Fig. 2-6 shows some non-periodic signals in the time and frequency
domain.
a)
Time domain
Frequency domain
|A|
–
A
0
0
t
Sinusoidal signal
T0
f
1
f0 = ––
T0
b)
|A|
–
A
0
0
t
fT – fS
f
fT + fS
fT
Amplitude-modulated signal
c)
Âp
A
sin x
Envelope si(x ) = –––
x
|A|
–
U
·P
sin n · ––
Tp
U
Ân· fp = Âp · –– · 2 · –––––––––
U
Tp
n · –– · P
Tp
(
0
Tp
U
0
t
Periodic rectangular signal
?1
U
?2
U
)
?3
U
1
––
Tp
Fig. 2-5 Periodic signals in time and frequency domain (magnitude spectra)
13
f
Signals
a)
Time domain
Frequency domain
|A|
–
A
0
0
t
Band-limited noise
f
b)
|A|
–
A
1
sin x
Envelope si(x) = _____
x
0
TBit
t
1/TBit
Random bit sequence
2/TBit
3/TBit
f
c)
I
A
0
lg| –A|
t
Q
A
0
t
QPSK signal
fC
f
Fig. 2-6 Non-periodic signals in time and frequency domain
Depending on the measurement to be performed, examination may be
useful either in the time or in the frequency domain. Digital data transmission jitter measurements, for example, require an oscilloscope. For
determining the harmonic content, it is more useful to examine the signal in the frequency domain:
14
Relationship Between Time
and
Frequency Domain
The signal shown in Fig. 2-7 seems to be a purely sinusoidal signal with
a frequency of 20 MHz. Based on the above considerations one would
expect the frequency spectrum to consist of a single component at
20 MHz.
On examining the signal in the frequency domain with the aid of a
spectrum analyzer, however, it becomes evident that the fundamental
(1st order harmonic) is superimposed by several higher-order harmonics i.e.multiples of 20 MHz (Fig. 2-8). This information cannot be easily
obtained by examining the signal in the time domain. A practical quantitative assessment of the higher-order harmonics is not feasible. It is
much easier to examine the short-term stability of frequency and amplitude of a sinusoidal signal in the frequency domain compared to the
time domain (see also chapter 6.1 Phase noise measurement).
1
Fig. 2-7 Sinusoidal signal (f = 20 MHz) examined on oscilloscope
Ch1 500 mV
M 10.0 ns
15
CH1
–560 mV
Configuration
20
Ref 20 dBm
and
Control Elements
Att 50 dB
of a
S p e ct r u m A n a l y z e r
*RBW 300 kHz
*VBW
3 kHz
SWT 175 ms
1
10
1 AP
CLRWR 0
Marker 1 [T1 CNT]
14.61 dBm
20.000 MHz
Delta 2 [T1]
–45.25 dB
A
20.08800000 MHz
–10
PRN
–20
2
–30
–40
–50
–60
–70
–80
Center 39 MHz
6.2 MHz/
Span 62 MHz
Fig. 2-8 T he sinusoidal signal of Fig. 2-7 examined in the frequency domain with
the aid of a spectrum analyzer
16
F o u r i e r A n a l y z e r (FFT A n a l y z e r )
3Configuration and Control Elements of a
Spectrum Analyzer
Depending on the kind of measurement, different requirements are
placed on the maximum input frequency of a spectrum analyzer. In view
of the various possible configurations of spectrum analyzers, the input
frequency range can be subdivided as follows:
u AF range
up to approx. 1 MHz
u RF range
up to approx. 3 GHz
u microwave range
up to approx. 40 GHz
u millimeter-wave range
above 40 GHz
The AF range up to approx. 1 MHz covers low-frequency electronics as
well as acoustics and mechanics. In the RF range, wireless communication applications are mainly found, such as mobile communications and
sound and TV broadcasting, while frequency bands in the microwave
or millimeter-wave range are utilized to an increasing extent for broadband applications such as digital radio links.
Various analyzer concepts can be implemented to suit the frequency
range. The two main concepts are described in detail in the following
sections.
3.1
Fourier analyzer (FFT analyzer)
As explained in chapter 2, the frequency spectrum of a signal is clearly
defined by the signal’s time characteristic. Time and frequency domain
are linked to each other by means of the Fourier transform. Equation 2-2
can therefore be used to calculate the spectrum of a signal recorded in
the time domain. For an exact calculation of the frequency spectrum
of an input signal, an infinite period of observation would be required.
Another prerequisite of Equation 2-2 is that the signal amplitude should
be known at every point in time. The result of this calculation would be a
continuous spectrum, so the frequency resolution would be unlimited.
It is obvious that such exact calculations are not possible in practice. Given certain prerequisites, the spectrum can nevertheless be determined with sufficient accuracy.
17
Configuration
and
Control Elements
of a
S p e ct r u m A n a l y z e r
In practice, the Fourier transform is made with the aid of digital signal
processing, so the signal to be analyzed has to be sampled by an analog-digital converter and quantized in amplitude. By way of sampling
the continuous input signal is converted into a time-discrete signal and
the information about the time characteristic is lost. The bandwidth of
the input signal must therefore be limited or else the higher signal frequencies will cause aliasing effects due to sampling (see Fig. 3-1). According to Shannon’s law of sampling, the sampling frequency fS must be at
least twice as high as the bandwidth Bin of the input signal. The following applies:
fS ≥ 2 ⋅ B in where fS
Bin
TS
and
fS =
1 TS
(Equation 3-1)
sampling rate
signal bandwidth
sampling period
For sampling lowpass-filtered signals (referred to as lowpass signals)
the minimum sampling rate required is determined by the maximum
signal frequency fin,max . Equation 3-1 then becomes:
fS ≥ 2 ⋅ fin, max (Equation 3-2)
If fS = 2 · fin,max , it may not be possible to reconstruct the signal from the
sampled values due to unfavorable sampling conditions. Moreover, a
lowpass filter with infinite skirt selectivity would be required for band
limitation. Sampling rates that are much greater than 2 · fin, max are therefore used in practice.
A section of the signal is considered for the Fourier transform. That is,
only a limited number N of samples is used for calculation. This process
is called windowing. The input signal (see Fig. 3-2a) is multiplied with a
specific window function before or after sampling in the time domain. In
the example shown in Fig. 3-2, a rectangular window is used (Fig. 3-2b).
The result of multiplication is shown in Fig. 3-2c.
18
F o u r i e r A n a l y z e r (FFT A n a l y z e r )
a)
A
fS
––
2
A
Sampling with
sampling rate fS
fin
fS fS + fin
fS – fin
t
fin
b)
A
A
fS
fin,max < ––
2
3 fS f
2 fS
3 fS f
2 fS
3 fS f
fS
––
2
t
fin,max
2 fS
fin,max fS
c)
A
fin,max
A
fS
> ––
2
fA
fin,max > ––
2
Aliasing
fS
––
2
t
fin, max fS
Fig. 3-1 S
ampling a lowpass signal with sampling rate fS a), b) fin, max < fS/2, c) fin, max > fS/2, therefore ambiguity exists due to aliasing
The calculation of the signal spectrum from the samples of the signal
in the time domain is referred to as a discrete Fourier transform (DFT).
Equation 2-2 then becomes:
X (k ) =
N -1
∑ x (nT ) ⋅ e
n =0
S
-j 2 pk n /N
(Equation 3-3)
where kindex of discrete frequency bins,
where k = 0, 1, 2, …
n
index of samples
x(nTS) samples at the point n · TS , where n = 0, 1, 2, …
Nlength of DFT, i. e. total number of samples used for
calculation of Fourier transform
19
Configuration
and
Control Elements
of a
S p e ct r u m A n a l y z e r
The result of a discrete Fourier transform is again a discrete frequency
spectrum (see Fig. 3-2d). The calculated spectrum is made up of individual components at the frequency bins which are expressed as:
()
f k = k⋅
where
fS
N
f (k)
k
fS
N
= k⋅
1
N ⋅ TS
(Equation 3-4)
discrete frequency bin
index of discrete frequency bins, where k = 0, 1, 2 …
sampling frequency
length of DFT
It can be seen that the resolution (the minimum spacing required
between two spectral components of the input signal for the latter being
displayed at two different frequency bins f (k) and f (k + 1) depends on
the observation time N ·TS . The required observation time increases
with the desired resolution.
The spectrum of the signal is periodicized with the period fS through
sampling (see Fig. 3-1). Therefore, a component is shown at the frequency
bin f (k = 6) in the discrete frequency spectrum display in Fig. 3-2d. On
examining the frequency range from 0 to fS in Fig. 3-1a, it becomes evident that this is the component at fS-fin .
In the example shown in Fig. 3-2, an exact calculation of the signal spectrum was possible. There is a frequency bin in the discrete frequency
spectrum that exactly corresponds to the signal frequency. The following requirements have to be fulfilled:
u the signal must be periodic (period T0)
u the observation time N ·TS must be an integer multiple of the
period T0 of the signal.
These requirements are usually not fulfilled in practice so that the result
of the Fourier transform deviates from the expected result. This deviation is characterized by a wider signal spectrum and an amplitude error.
Both effects are described in the following.
20
F o u r i e r A n a l y z e r (FFT A n a l y z e r )
a)
1
Input signal x(t)
Samples
|–X(f)|
| –A|
A
0
–1
b)
1
TA 0
Te
0
t
fin = 1
Tin
|W(f)|
–
Window w(t )
f
| –A|
A
0
c)
1
N · TS
0
t
0
x (t) · w(t)
1 0 –––
1
– –––
N · TS N · TS
f
N=8
A
0
–1
0
d)
1
t
|X( f) · W (f)|k = 2
x (t ) · w(t ), continued periodically
k= 6
| –A|
A
0
–1
0
N · TLS
k = 0 k = 1 fe
frequency bins
t
fA
––
2
1 f
––––
N · TA
Fig. 3-2 D
FT with periodic input signal. Observation time is an integer multiple
of the period of the input signal
21
Configuration
a)
1
and
Control Elements
Samples
Input signal x(t)
of a
S p e ct r u m A n a l y z e r
|X(f
– )|
|A|
–
A
0
–1
b)
1
TS 0
0
t
Te
|W
– (f )|
Window w(t)
fin = 1
Tin
f
| –A|
A
0
N · TS
c)
1
0
t
0
f
1 0 –––
1
– –––
N · T S N · TS
x(t) · w(t)
N= 8
A
0
–1
0
d)
1
t
|X(f
– ) · –W(f )|
x(t) · w(t), continued periodically
|A|
–
A
0
N=8
–1
0
N · TS
k = 0 k = 1 fin
frequency bins
t
fS
––
2
f
1 f –f
–––
S
in
N · TS
Fig. 3-3 D
FT with periodic input signal. Observation time is not an integer
multiple of the period of the input signal
22
F o u r i e r A n a l y z e r (FFT A n a l y z e r )
The multiplication of input signal and window function in the time
domain corresponds to a convolution in the frequency domain (see [2-1]).
In the frequency domain the magnitude of the transfer function of the
rectangular window used in Fig. 3-2 follows a sine function:
W ( f ) = N ⋅ TS⋅ si (2pf ⋅ N ⋅ TS/2) = N ⋅ TS⋅
sin (2pf ⋅ N ⋅ TS/2)
2pf ⋅ N ⋅ TS/2
(Equation 3-5)
where W (f ) windowing function in frequency domain
N ·TS window width
In addition to the distinct secondary maxima, nulls are obtained at multiples of 1 / (N ·TS). Due to the convolution by means of the window
function the resulting signal spectrum is smeared, so it becomes distinctly wider. This is referred to as leakage effect.
If the input signal is periodic and the observation time N ·TS is an
integer multiple of the period, there is no leakage effect of the rectangular window since, with the exception of the signal frequency, nulls
always fall within the neighboring frequency bins (see Fig. 3-2d).
If these conditions are not satisfied, which is the normal case, there
is no frequency bin that corresponds to the signal frequency. This case
is shown in Fig. 3-3. The spectrum resulting from the DFT is distinctly
wider since the actual signal frequency lies between two frequency bins
and the nulls of the windowing function no longer fall within the neighboring frequency bins.
As shown in Fig. 3.3d, an amplitude error is also obtained in this case.
At constant observation time the magnitude of this amplitude error
depends on the signal frequency of the input signal (see Fig. 3-4). The
error is at its maximum if the signal frequency is exactly between two
frequency bins.
23
Configuration
and
Control Elements
of a
S p e ct r u m A n a l y z e r
max.
amplitude error
Fig. 3-4 Amplitude error
caused by rectangular
­windowing as a function
of signal frequency
fin
f(k)
Frequency bins
By increasing the observation time it is possible to reduce the absolute
widening of the spectrum through the higher resolution obtained, but
the maximum possible amplitude error remains unchanged. The two
effects can, however, be reduced by using optimized windowing instead
of the rectangular window. Such windowing functions exhibit lower secondary maxima in the frequency domain so that the leakage effect is
reduced as shown in Fig. 3-5. Further details of the windowing functions
can be found in [3-1] and [3-2].
To obtain the high level accuracy required for spectrum analysis a flattop window is usually used. The maximum level error of this windowing function is as small as 0.05 dB. A disadvantage is its relatively wide
main lobe which reduces the frequency resolution.
24
F o u r i e r A n a l y z e r (FFT A n a l y z e r )
Rectangular window
Hann window
Amplitude error
Leakage
f
f
Fig. 3-5 L eakage effect when using rectangular window or Hann window
(MatLab® simulation)
The number of computing operations required for the Fourier transform
can be reduced by using optimized algorithms. The most widely used
method is the fast Fourier transform (FFT). Spectrum analyzers operating on this principle are designated as FFT analyzers. The configuration
of such an analyzer is shown in Fig. 3-6.
Input
A
RAM
D
Lowpass
Memory
FFT
Display
Fig. 3-6 Configuration of FFT analyzer
To adhere to the sampling theorem, the bandwidth of the input signal
is limited by an analog lowpass filter (cutoff frequency fc = fin, max) ahead
of the A/D converter. After sampling the quantized values are saved in
a memory and then used for calculating the signal in the frequency
domain. Finally, the frequency spectrum is displayed.
Quantization of the samples causes the quantization noise which causes
a limitation of the dynamic range towards its lower end. The higher the
resolution (number of bits) of the A/D converter used, the lower the
quantization noise.
25
Configuration
and
Control Elements
of a
S p e ct r u m A n a l y z e r
Due to the limited bandwidth of the available high-resolution A/D converters, a compromise between dynamic range and maximum input frequency has to be found for FFT analyzers. At present, a wide dynamic
range of about 100 dB can be achieved with FFT analyzers only for lowfrequency applications up to 100 kHz. Higher bandwidths inevitably
lead to a smaller dynamic range.
In contrast to other analyzer concepts, the phase information is not
lost during the complex Fourier transform. FFT analyzers are therefore
able to determine the complex spectrum by magnitude and phase. If
they feature sufficiently high computing speed, they even allow realtime
analysis.
FFT analyzers are not suitable for the analysis of pulsed signals (see
Fig. 3-7). The result of the FFT depends on the selected section of the
time function. For correct analysis it is therefore necessary to know certain parameters of the analyzed signal, such as the triggering a specific
measurement.
A
N · TS = n · T0
Window
0
N · TS
T0
t
A
A
f
A
1
––
T0
f
Fig. 3-7 F FT of pulsed signals. The result depends on the time of the
­measurement
26
1
––
T0
f
A n a l y z e r s O p e r a t i n g A cc o r d i n g
to the
Heterodyne Principle
3.2Analyzers operating in accordance with the heterodyne
principle
Due to the limited bandwidth of the available A/D converters, FFT analyzers are only suitable for measurements on low-frequency signals. To
display the spectra of high-frequency signals up to the microwave or
millimeter-wave range, analyzers with frequency conversion are used.
In this case the spectrum of the input signal is not calculated from the
time characteristic, but determined directly by analysis in the frequency
domain. For such an analysis it is necessary to break down the input
spectrum into its individual components. A tunable bandpass filter as
shown in Fig. 3-8 could be used for this purpose.
Tunable
bandpass filter
Amplifier
Detector
Input
Display
y
x
Sawtooth
A
Tunable bandpass filter
Fig. 3-8 Block diagram of
spectrum analyzer
with tunable
bandpass filter
fin
The filter bandwidth corresponds to the resolution bandwidth (RBW) of
the analyzer. The smaller the resolution bandwidth, the higher the spectral resolution of the analyzer.
Narrowband filters tunable throughout the input frequency range
of modern spectrum analyzers are however technically hardly feasible. Moreover, tunable filters have a constant relative bandwidth with
27
Configuration
and
Control Elements
of a
S p e ct r u m A n a l y z e r
respect to the center frequency. The absolute bandwidth therefore
increases with increasing center frequency so that this concept is not
suitable for spectrum analysis.
Spectrum analyzers for high input frequency ranges therefore usually operate in accordance with the principle of a heterodyne receiver.
The block diagram of such a analyzer is shown in Fig. 3-9.
Mixer
IF amplifier
IF filter
Logarithmic
amplifier
Envelope
detector
Video filter
Input
Local oscillator
y
x
Sawtooth
Fig. 3-9 Block diagram of spectrum analyzer operating on heterodyne principle
The heterodyne receiver converts the input signal with the aid of a mixer
and a local oscillator (LO) to an intermediate frequency (IF). If the local
oscillator frequency is tunable (a requirement that is technically feasible), the complete input frequency range can be converted to a constant intermediate frequency by varying the LO frequency. The resolution of the analyzer is then given by a filter at the IF with fixed center
frequency.
In contrast to the concept described above, where the resolution filter as a dynamic component is swept over the spectrum of the input signal, the input signal is now swept past a fixed-tuned filter.
The converted signal is amplified before it is applied to the IF filter which determines the resolution bandwidth. This IF filter has a constant center frequency so that problems associated with tunable filters
can be avoided.
28
A n a l y z e r s O p e r a t i n g A cc o r d i n g
to the
Heterodyne Principle
To allow signals in a wide level range to be simultaneously displayed
on the screen, the IF signal is compressed using of a logarithmic amplifier and the envelope determined. The resulting signal is referred to as
the video signal. This signal can be averaged with the aid of an adjustable lowpass filter called a video filter. The signal is thus freed from
noise and smoothed for display. The video signal is applied to the vertical deflection of a cathode-ray tube. Since it is to be displayed as a function of frequency, a sawtooth signal is used for the horizontal deflection
of the electron beam as well as for tuning the local oscillator. Both the
IF and the LO frequency are known. The input signal can thus be clearly
assigned to the displayed spectrum.
A
IF filter
Input signal
converted to IF
fIF
A
f
IF filter
Input signal
converted to IF
Fig. 3-10 Signal “swept past”
resolution filter in
heterodyne receiver
fIF
29
f
Configuration
and
Control Elements
of a
S p e ct r u m A n a l y z e r
In modern spectrum analyzers practically all processes are controlled by
one or several microprocessors, giving a large variety of new functions
which otherwise would not be feasible. One application in this respect is
the remote control of the spectrum analyzer via interfaces such as the
IEEE bus.
Modern analyzers use fast digital signal processing where the input
signal is sampled at a suitable point with the aid of an A/D converter and
further processed by a digital signal processor. With the rapid advances
made in digital signal processing, sampling modules are moved further
ahead in the signal path. Previously, the video signal was sampled after
the analog envelope detector and video filter, whereas with modern spectrum analyzers the signal is often digitized at the last low IF. The envelope of the IF signal is then determined from the samples.
Likewise, the first LO is no longer tuned with the aid of an analog
sawtooth signal as with previous heterodyne receivers. Instead, the LO
is locked to a reference frequency via a phase-locked loop (PLL) and
tuned by varying the division factors. The benefit of the PLL technique
is a considerably higher frequency accuracy than achievable with analog tuning.
An LC display can be used instead of the cathode-ray tube, which
leads to more compact designs.
3.3
Main setting parameters
Spectrum analyzers usually provide the following elementary setting
parameters (see Fig. 3-11):
Frequency display range
The frequency range to be displayed can be set by the start and stop frequency (that is the minimum and maximum frequency to be displayed),
or by the center frequency and the span centered about the center frequency. The latter setting mode is shown in Fig. 3-11. Modern spectrum
analyzers feature both setting modes.
Level display range
This range is set with the aid of the maximum level to be displayed (the
reference level), and the span. In the example shown in Fig. 3-11, a reference level of 0 dBm and a span of 100 dB is set. As will be described later,
the attenuation of an input RF attenuator also depends on this setting.
30
M a i n S e tt i n g P a r a m e t e r s
Frequency resolution
For analyzers operating on the heterodyne principle, the frequency resolution is set via the bandwidth of the IF filter. The frequency resolution
is therefore referred to as the resolution bandwidth (RBW).
Sweep time (only for analyzers operating on the heterodyne principle)
The time required to record the whole frequency spectrum that is of
interest is described as sweep time.
Some of these parameters are dependent on each other. Very small resolution bandwidths, for instance, call for a correspondingly long sweep
time. The precise relationships are described in detail in chapter 4.6.
Fig. 3-11 Graphic display of recorded spectrum
31
P r a ct i c a l R e a l i z a t i o n
of an
Analyzer Operating
on the
­H e t e r o d y n e P r i n c i p l e
4Practical Realization of an Analyzer Operating on the
­Heterodyne Principle
This chapter provides a detailed description of the individual components of an analyzer operating on the heterodyne principle as well as
of the practical implementation of a modern spectrum analyzer for the
frequency range 9 kHz to 3 GHz/7 GHz. A detailed block diagram can be
found on the fold-out page at the end of the book. The individual blocks
are numbered and combined in functional units.
4.1
RF input section (frontend)
Like most measuring instruments used in modern telecommunications,
spectrum analyzers usually feature an RF input impedance of 50 W. To
enable measurements in 75 W systems such as cable television (CATV),
some analyzers are alternatively provided with a 75 W input impedance.
With the aid of impedance transformers, analyzers with 50 W input may
also be used (see measurement tip: Measurements in 75 W system).
A quality criterion of the spectrum analyzer is the input VSWR, which is
highly influenced by the frontend components, such as the attenuator,
input filter and first mixer. These components form the RF input section
whose functionality and realization will be examined in detail in the following.
A step attenuator (2)* is provided at the input of the spectrum analyzer for the measurement of high-level signals. Using this attenuator,
the signal level at the input of the first mixer can be set.
The RF attenuation of this attenuator is normally adjustable in 10 dB
steps. For measurement applications calling for a wide dynamic range,
attenuators with finer step adjustment of 5 dB or 1 dB are used in some
analyzers (see chapter 5.5: Dynamic range).
*
The colored code numbers in parentheses refer to the block diagram at the end of the
book.
32
RF I n p u t S e ct i o n (F r o n t e n d )
Measurements in 75 W system
In sound and TV broadcasting, an impedance of 75 W is more
common than the widely used 50 W. To carry out measurements
in such systems with the aid of spectrum analyzers that usually
feature an input impedance of 50 W, appropriate matching pads
are required. Otherwise, measurement errors would occur due to
mismatch between the device under test and spectrum analyzer.
The simplest way of transforming 50 W to 75 W is by means
of a 25 W series resistor. While the latter renders for low insertion loss (approx. 1.8 dB), only the 75 W input is matched, however,
the output that is connected to the RF input of the spectrum analyzer is mismatched (see Fig. 4-1a). Since the input impedance of
the spectrum analyzer deviates from the ideal 50 W value, measurement errors due to multiple reflection may occur especially
with mismatched DUTs.
Therefore it is recommendable to use matching pads that
are matched at both ends (e. g. Π or L pads). The insertion loss
through the attenuator may be higher in this case.
a)
1007
757
Source
Spectrum
analyzer
257
Zin = 50 7
Zout = 75 7
b)
507
757
Fig. 4-1 Source
Input matching
to 75 W using
external
matching pads Zout = 75 7
Spectrum
analyzer
Matching
pad
Zin = 50 7
33
P r a ct i c a l R e a l i z a t i o n
of an
Analyzer Operating
on the
­H e t e r o d y n e P r i n c i p l e
The heterodyne receiver converts the input signal with the aid of a mixer
(4) and a local oscillator (5) to an intermediate frequency (IF). This type
of frequency conversion can generally be expressed as:
m ⋅ fLO ± n ⋅ fin = fIF where
m, n
fLO
fin
fIF
(Equation 4-1)
1, 2, …
frequency of local oscillator
frequency of input signal to be converted
intermediate frequency
If the fundamentals of the input and LO signal are considered (m, n = 1),
Equation 4-1 is simplified to:
fLO ± fin = fIF (Equation 4-2)
or solved for fin
fin = fLO ± fIF (Equation 4-3)
With a continuously tunable local oscillator, a further input frequency
range can be implemented at constant frequency. For specific LO and
intermediate frequencies, Equation 4-3 shows that there are always two
receive frequencies for which the criterion set by Equation 4-2 is fulfilled (see Fig. 4-2). This means that in addition to the wanted receive
frequency there are also image frequencies. To ensure unambiguity of
this concept, input signals at such unwanted image frequencies have to
be rejected with the aid of suitable filters ahead of the RF input of the
mixer.
34
RF I n p u t S e ct i o n (F r o n t e n d )
Conversion
A
Input filter
Image frequency
reponse
$ f = fIF
fin, u
fIF
fLO
fin, o
f
Fig. 4-2 Ambiguity of heterodyne principle
Conversion
A
Overlap of
input and image
frequency range
LO frequency range
Image frequency
range
Input frequency
range
fIF
fin, min
fLO,min
fim,min
fe,max
fLO, max fim,max
f
Fig. 4-3 Input and image frequency ranges (overlapping)
Fig. 4-3 illustrates the input and image frequency ranges for a tunable
receiver with low first IF. If the input frequency range is greater than
2 · fIF , the two ranges are overlapping, so an input filter must be implemented as a tunable bandpass for image frequency rejection without
affecting the wanted input signal.
To cover the frequency range from 9 kHz to 3 GHz, which is typical of
modern spectrum analyzers, this filter concept would be extremely complex because of the wide tuning range (several decades). Much less complex is the principle of a high first IF (see Fig. 4-4).
35
P r a ct i c a l R e a l i z a t i o n
A
of an
Analyzer Operating
on the
­H e t e r o d y n e P r i n c i p l e
Conversion
Input filter
Input frequency range
fIF = fLO – fin
LO frequency
range
fIF
Image frequency
range
fIF = fim – fLO
f
Fig. 4-4 Principle of high intermediate frequency
In this configuration, image frequency range lies above the input frequency range. Since the two frequency ranges do not overlap, the image
frequency can be rejected by a fixed-tuned lowpass filter. The following
relationships hold for the conversion of the input signal:
fIF = fLO - fin (Equation 4-4)
and for the image frequency response:
fIF = fim - fLO (Equation 4-5)
Frontend for frequencies up to 3 GHz
The analyzer described here uses the principle of high intermediate frequency to cover the frequency range from 9 kHz to 3 GHz. The input
attenuator (2) is therefore followed by a lowpass filter (3) for rejection of
the image frequencies. Due to the limited isolation between RF and IF
port as well as between LO and RF port of the first mixer, this lowpass
filter also serves for minimizing the IF feedthrough and LO ­reradiation
at the RF input.
In our example the first IF is 3476.4 MHz. For converting the input frequency range from 9 kHz to 3 GHz to an upper frequency of 3476.4 MHz,
the LO signal (5) must be tunable in the frequency range from
3476.40 MHz to 6476.4 MHz. According to Equation 4-5, an image frequency range from 6952.809 MHz to 9952.8 MHz is then obtained.
36
RF I n p u t S e ct i o n (F r o n t e n d )
Measurement on signals with DC component
Many spectrum analyzers, in particular those featuring a very
low input frequency at their lower end (such as 20 Hz), are DCcoupled, so there are no coupling capacitors in the signal path
between RF input and first mixer.
A DC voltage may not be applied to the input of a mixer
because it usually damages the mixer diodes. For measurements
of signals with DC components, an external coupling capacitor
(DC block) is used with DC-coupled spectrum analyzers. It should
be noted that the input signal is attenuated by the insertion loss
of this DC block. This insertion loss has to be taken into account
in absolute level measurements.
Some spectrum analyzers have an integrated coupling capacitor to prevent damage to the first mixer. The lower end of the frequency range is thus raised. AC-coupled analyzers therefore have
a higher input frequency at the lower end, such as 9 kHz.
Due to the wide tuning range and low phase noise far from the carrier
(see chapter 5.3: Phase noise) a YIG oscillator is often used as local oscillator. This technology uses a magnetic field for tuning the frequency of
a resonator.
Some spectrum analyzers use voltage-controlled oscillators (VCO)
as local oscillators. Although such oscillators feature a smaller tuning
range than the YIG oscillators, they can be tuned much faster than YIG
oscillators.
To increase the frequency accuracy of the recorded spectrum, the LO signal is synthesized. That is, the local oscillator is locked to a reference signal (26) via a phase-locked loop (6). In contrast to analog spectrum analyzers, the LO frequency is not tuned continuously, but in many small
steps. The step size depends on the resolution bandwidth. Small resolution bandwidths call for small tuning steps. Otherwise, the input signal may not be fully recorded or level errors could occur. To illustrate
this effect, a filter tuned in steps throughout the input frequency range
is shown in Fig. 4-5. To avoid such errors, a step size that is much lower
than the resolution bandwidth (such as 0.1 · BN) is selected in p
­ ractice.
37
praCTiCal realizaTion
a)
of an
analyzer operaTing
on The
heTerodyne prinCiple
Input signal
A
fin
Tuning step >> resolution bandwidth
A
Displayed spectrum
fin
b)
Input signal
A
fin
Tuning step >> resolution bandwidth
A
Displayed spectrum
fin
Fig. 4-5 Effects of too large tuning steps a) input signal is completely lost b) level error in display of input signal
The reference signal is usually generated by a temperature-controlled
crystal oscillator (TCXO). To increase the frequency accuracy and longterm stability (see also chapter 5.9: Frequency accuracy), an oven-controlled crystal oscillator (OCXO) is optionally available for most spectrum analyzers. For synchronization with other measuring instruments,
the reference signal (usually 10 MHz) is made available at an output
connector (28). The spectrum analyzer may also be synchronized to an
externally applied reference signal (27). If only one connector is available for coupling a reference signal in or out, the function of such connector usually depends on a setting internal to the spectrum analyzer.
38
RF I n p u t S e ct i o n (F r o n t e n d )
As shown in Fig. 3-9, the first conversion is followed by IF signal processing and detection of the IF signal. With such a high IF, narrowband
IF filters can hardly be implemented, which means that the IF signal in
the concept described here has to be converted to a lower IF (such as
20.4 MHz in our example).
2nd conversion
A
Image rejection
filter
2nd IF
Image
1st IF
f
2nd LO
Fig. 4-6 Conversion of high 1st IF to low 2nd IF
With direct conversion to 20.4 MHz, the image frequency would only
be offset 2 · 20.4 MHz = 40.8 MHz from the signal to be converted at
3476.4 MHz (Fig. 4-6). Rejection of this image frequency is important
since the limited isolation between the RF and IF port of the mixers
signals may be passed to the first IF without conversion. This effect
is referred to as IF feedthrough (see chapter 5.6: Immunity to interference). If the frequency of the input signal corresponds to the image frequency of the second conversion, this effect is shown in the image frequency response of the second IF. Under certain conditions, input signals may also be converted to the image frequency of the second conversion. Since the conversion loss of mixers is usually much smaller than
the isolation between RF and IF port of the mixers, this kind of image
frequency response is far more critical.
Due to the high signal frequency, an extremely complex filter with high
skirt selectivity would be required for image rejection at a low IF of
20.4 MHz. It is therefore advisable to convert the input signal from the
first IF to a medium IF such as 404.4 MHz as in our example. A fixed
LO signal (10) of 3072 MHz is required for this purpose since the image
frequency for this conversion is at 2667.6 MHz. Image rejection is then
39
P r a ct i c a l R e a l i z a t i o n
of an
Analyzer Operating
on the
­H e t e r o d y n e P r i n c i p l e
simple to realize with the aid of a suitable bandpass filter (8). The bandwidth of this bandpass filter must be sufficiently large so that the signal will not be impaired even for maximum resolution bandwidths. To
reduce the total noise figure of the analyzer, the input signal is amplified
(7) prior to the second conversion.
The input signal converted to the second IF is amplified again, filtered
by an image rejection bandpass filter for the third conversion and converted to the low IF of 20.4 MHz with the aid of a mixer. The signal thus
obtained can be subjected to IF signal processing.
Frontend for frequencies above 3 GHz
The principle of a high first IF calls for a high LO frequency range
(fLO, max = fin, max + f1st IF). In addition to a broadband RF input, the first
mixer must also feature an extremely broadband LO input and IF output – requirements that are increasingly difficult to satisfy if the upper
input frequency limit is raised. Therefore this concept is only suitable for
input frequency ranges up to 7 GHz.
To cover the microwave range, other concepts have to be implemented
by taking the following criteria into consideration:
uThe frequency range from 3 GHz to 40 GHz extends over more
than a decade, whereas 9 kHz to 3 GHz corresponds to approx. 5.5
decades.
uIn the microwave range, filters tunable in a wide range and with
narrow relative bandwidth can be implemented with the aid of
YIG technology [4-1]. Tuning ranges from 3 GHz to 50 GHz are fully
­realizable.
Direct conversion of the input signal to a low IF calls for a tracking
bandpass filter for image rejection. In contrast to the frequency range
up to 3 GHz, such preselection can be implemented for the range above
3 GHz due to the previously mentioned criteria. Accordingly, the local
oscillator need only be tunable in a frequency range that corresponds to
the input frequency range.
In our example the frequency range of the spectrum analyzer is thus
enhanced from 3 GHz to 7 GHz. After the attenuator, the input signal
is split by a diplexer (19) into the frequency ranges 9 kHz to 3 GHz and
3 GHz to 7 GHz and applied to corresponding RF frontends.
40
RF I n p u t S e ct i o n (F r o n t e n d )
In the high-frequency input section, the signal passes a tracking YIG filter (20) to the mixer. The center frequency of the bandpass filter corresponds to the input signal frequency to be converted to the IF. Direct
conversion to a low IF (20.4 MHz, in our example) is difficult with this
concept due to the bandwidth of the YIG filter. It is therefore best to convert the signal first to a medium IF (404.4 MHz) as was performed with
the low-frequency input section.
In our example, a LO frequency range from 2595.6 MHz to
6595.6 MHz would be required for converting the input signal as upper
sideband, (that is for fIF = fin- fLO). For the conversion as lower sideband (fIF = fLO- fin), the local oscillator would have to be tunable from
3404.4 MHz to 7404.4 MHz.
If one combines the two conversions by switching between the upper
and lower sideband at the center of the input frequency band, this concept can be implemented even with a limited LO frequency range of
3404.4 MHz to 6595.6 MHz (see Fig. 4-7).
A
Tracking preselection
Input signal converted
as lower sideband
LO frequency range
Input frequency range
fIF
f
Input signal converted
as upper sideband
A
LO frequency range
Input frequency range
fIF
fin, max
fin,min
Input frequency range
= Tuning range of bandpass filter
Fig. 4-7 Conversion to a low IF; image rejection by tracking preselection
41
f
P r a ct i c a l R e a l i z a t i o n
of an
Analyzer Operating
on the
­H e t e r o d y n e P r i n c i p l e
The signal converted to an IF of 404.4 MHz is amplified (23) and coupled into the IF signal path of the low-frequency input section through
a switch (13).
Upper and lower frequency limits of this implementation are determined
by the technological constraints of the YIG filter. A maximum frequency
of about 50 GHz is feasible.
In our example, the upper limit of 7 GHz is determined by the tuning
range of the local oscillator. There are again various possibilities for
converting input signals above 7 GHz with the specified LO frequency
range:
Fundamental mixing
The input signal is converted by means of the fundamental of the LO
signal. For covering a higher frequency range with the specified LO frequency range it is necessary to double, for instance, the LO signal frequency by means of a multiplier before the mixer.
Harmonic mixing
The input signal is converted by a means of a harmonic of the LO signal
produced in the mixer due to the mixer’s nonlinearities.
Fundamental mixing is preferred to obtain minimal conversion loss,
thereby maintaining a low noise figure for the spectrum analyzer. The
superior characteristics attained in this way, however, require complex
processing of the LO signal. In addition to multipliers (22), filters are
required for rejecting subharmonics after multiplying. The amplifiers
required for a sufficiently high LO level must be highly broadband since
they must be designed for a frequency range that roughly corresponds
to the input frequency range of the high-frequency input section.
Conversion by means of harmonic mixing is easier to implement but
implies a higher conversion loss. A LO signal in a comparatively low frequency range is required which has to be applied at a high level to the
mixer. Due to the nonlinearities of the mixer and the high LO level, harmonics of higher order with sufficient level are used for the conversion.
Depending on the order m of the LO harmonic, the conversion loss of the
mixer compared to that in fundamental mixing mode is increased by:
42
RF I n p u t S e ct i o n (F r o n t e n d )
DaM = 20 dB ⋅ lg m
(Equation 4-6)
where DaMincrease of conversion loss compared to that in
­fundamental mixing mode
m
order of LO harmonic used for conversion
The two concepts are employed in practice depending on the price class
of the analyzer. A combination of the two methods is possible. For example, a conversion using the harmonic of the LO signal doubled by a multiplier would strike a compromise between complexity and sensitivity at
an acceptable expense.
External mixers
For measurements in the millimeter-wave range (above 40 GHz), the frequency range of the spectrum analyzer can be enhanced by using external harmonic mixers. These mixers also operate on the principle of harmonic mixing, so that a LO signal in a frequency range that is low compared to the input signal frequency range is required.
The input signal is converted to a low IF by means of a LO harmonic
and an IF input inserted at a suitable point into the IF signal path of the
low-frequency input section of the analyzer.
In the millimeter-wave range, waveguides are normally used for
conducted signal transmission. Therefore, external mixers available
for enhancing the frequency range of spectrum analyzers are usually
waveguides. These mixers do not normally have a preselection filter and
therefore do not provide for image rejection. Unwanted mixture products have to be identified with the aid of suitable algorithms. Further
details about frequency range extension with the aid of external harmonic mixers can be found in [4-2].
43
P r a ct i c a l R e a l i z a t i o n
4.2
of an
Analyzer Operating
on the
­H e t e r o d y n e P r i n c i p l e
IF signal processing
lg (Hv ( f )) / dB
IF signal processing is performed at the last intermediate frequency
(20.4 MHz in our example).
Here the signal is amplified again and the resolution bandwidth
defined by the IF filter.
The gain at this last IF can be adjusted in defined steps (0.1 dB steps
in our example), so the maximum signal level can be kept constant in
the subsequent signal processing regardless of the attenuator setting
and mixer level. With high attenuator settings, the IF gain has to be
increased so that the dynamic range of the subsequent envelope detector and A/D converter will be fully utilized (see chapter 4.6: Parameter
dependencies).
The IF filter is used to define that section of the IF-converted input
signal that is to be displayed at a certain point on the frequency axis.
Due to the high skirt selectivity and resulting selectivity characteristics,
a rectangular filter would be desirable. The transient response, however,
of such rectangular filters is unsuitable for spectrum analysis. Since
such a filter has a long transient time, the input signal spectrum could
be converted to the IF only by varying the LO frequency very slowly
to avoid level errors from occurring. Short measurement times can be
achieved through the use of Gaussian filters optimized for transients.
The transfer function of such a filter is shown in Fig. 4-8.
0
–3
–6
–60
f0
f
Fig. 4-8 Voltage transfer function
of Gaussian filter
In contrast to rectangular filters featuring an abrupt transition from
passband to stopband, the bandwidth of Gaussian filters must be specified for filters with limited skirt selectivity. In spectrum analysis it is
common practice to specify the 3 dB bandwidth (the frequency spacing
44
IF S i g n a l P r o c e s s i n g
HV(f )
H 2V(f )
between two points of the transfer function at which the insertion loss
of the filter has increased by 3 dB relative to the center frequency).
HV, 0
H 2V, 0
Voltage
transfer
function
Pulse
bandwidth
BI
0.5
f0
Power
transfer
function
Noise
bandwidth
BN
0.5
f
f0
f
Fig. 4-9 Voltage and power transfer function of Gaussian filter
For many measurements on noise or noise-like signals (e. g. digitally modulated signals) the measured levels have to be referenced to the measurement bandwidth, in our example the resolution bandwidth. To this
end the equivalent noise bandwidth BN of the IF filter must be known
which can be calculated from the transfer function as follows:
BN =
1
H V2, 0
+∞
()
⋅ ∫ H V2 f ⋅ df (Equation 4-7)
0
where BN
noise bandwidth
HV(f ) voltage transfer function
HV, 0value of voltage transfer function at center of band
(at f0)
This can best be illustrated by looking at the power transfer function
(see Fig. 4-9). The noise bandwidth corresponds to the width of a rectangle with the same area as the area of the transfer function HV2 (f ). The
effects of the noise bandwidth of the IF filter are dealt with in detail in
chapter 5.1 Inherent noise.
45
P r a ct i c a l R e a l i z a t i o n
of an
Analyzer Operating
on the
­H e t e r o d y n e P r i n c i p l e
For measurements on correlated signals, as can typically be found in the
field of radar, the pulse bandwidth is also of interest. In contrast to the
noise bandwidth, the pulse bandwidth is calculated by integration of the
voltage transfer function. The following applies:
BI =
1
H V, 0
+∞
()
⋅∫ H V f ⋅ df (Equation 4-8)
0
where BI
pulse bandwidth
HV( f ) voltage transfer function
HV,0value of voltage transfer function at center of band
(at f0)
The pulse bandwidth of Gaussian or Gaussian-like filters corresponds
approximately to the 6 dB bandwidth. In the field of interference measurements, where spectral measurements on pulses are frequently carried out, 6 dB bandwidths are exclusively specified. Further details of
measurements on pulsed signals can be found in chapter 6.2.
Chapter 6 concentrates on pulse and phase noise measurements.
For these and other measurement applications the exact relationships
between 3 dB, 6 dB, noise and pulse bandwidth are of particular interest. Table 4-1 provides conversion factors for various filters that are
described in detail further below.
Initial value is
3 dB bandwidth
4 filter circuits
(analog)
5 filter circuits
(analog)
Gaussian filter
(digital)
1.480 · B3 dB
1.464 · B3 dB
1.415 · B3 dB
Noise bandwidth (BN)
1.129 · B3 dB
1.114 · B3 dB
1.065 · B3 dB
Pulse bandwidth (BI)
1.806 · B3 dB
1.727 · B3 dB
1.506 · B3 dB
Initial value is
6 dB bandwidth
3 dB bandwidth (B3 dB)
0.676 · B6 dB
0.683 · B6 dB
0.707 · B6 dB
Noise bandwidth (BN)
0.763 · B6 dB
0.761 · B6 dB
0.753 · B6 dB
Pulse bandwidth (BI)
1.220 · B6 dB
1.179 · B6 dB
1.065 · B6 dB
6 dB bandwidth
(B6 dB)
Table 4-1 R
elationship between 3 dB / 6 dB bandwidths and noise and pulse
bandwidths
46
IF S i g n a l P r o c e s s i n g
If one uses an analyzer operating on the heterodyne principle to record
a purely sinusoidal signal, one would expect a single spectral line in
accordance with the Fourier theorem even when a small frequency
span about the signal frequency is taken. In fact, the display shown in
Fig. 4-10 is obtained.
0
Ref 0 dBm
*RBW 10 kHz
*VBW 30 Hz
SWT 680 ms
Att 30 dB
T
T2
1 AP –10
CLRWR
–20
–30
–40
Marker 1 [T1]
–5.16 dBm
1.00000000 GHz
ndB [T1] 3.00 dB
BW 9.80000000 kHz
Temp 1 [T1 ndB]
–81.62 dBm
999.95000000 MHz
Temp 2 [T2 ndB]
–8.22 dBm
1.00000500 GHz
A
PRN
–50
–60
–70
–80
–90
–100
Center 1 GHz
10 kHz/
Span 100 kHz
Fig. 4-10 IF filter imaged by a sinusoidal input signal
The display shows the image of the IF filter. During the sweep, the input
signal converted to the IF is “swept past” the IF filter and multiplied with
the transfer function of the filter.
A schematic diagram of this process is shown in Fig. 4-11. For reasons of
simplification the filter is “swept past” a fixed-tuned signal, both kinds
of representations being equivalent.
47
praCTiCal realizaTion
A
of an
Input
signal
analyzer operaTing
on The
heTerodyne prinCiple
IF filter
f
A
Image of
resolution
bandwidth
f
Fig. 4-11 IF filter imaged by an input signal “swept past” the filter (schematic representation of imaging process)
As pointed out before, the spectral resolution of the analyzer is mainly
determined by the resolution bandwidth, that is, the bandwidth of the
IF filter. The IF bandwidth (3 dB bandwidth) corresponds to the minimum frequency offset required between two signals of equal level to
make the signals distinguishable by a dip of about 3 dB in the display
when using a sample or peak detector (see chapter 4.4.). This case is
shown in Fig. 4-12a. The red trace was recorded with a resolution bandwidth of 30 kHz. By reducing the resolution bandwidth, the two signals
are clearly distinguishable (Fig. 4-12a, blue trace).
48
IF S i g n a l P r o c e s s i n g
If two neighboring signals have distinctly different levels, the weaker
signal will not be shown in the displayed spectrum at a too high resolution bandwidth setting (see Fig. 4-12b, red trace). By reducing the resolution bandwidth, the weak signal can be displayed.
In such cases, the skirt selectivity of the IF filter is also important
and is referred to as the selectivity of a filter. The skirt selectivity is specified in form of the shape factor which is calculated as follows:
SF60/3 =
B60 dB
B3 dB
(Equation 4-9)
where B3 dB 3 dB bandwidth
B60 dB 60 dB bandwidth
For 6 dB bandwidths, as is customary in EMC measurements, the shape
factor is derived from the ratio of the 60 dB bandwidth to the 6 dB bandwidth.
The effects of the skirt selectivity can clearly be seen in Fig. 4-13. One
Kilohertz IF filters with different shape factors were used for the two
traces. In the blue trace (SF = 4.6), the weaker signal can still be recognized by the dip, but a separation of the two signals is not possible in the
red trace (SF = 9.5) where the weaker signal does not appear at all.
49
P r a ct i c a l R e a l i z a t i o n
of an
Ref –10 dBm
Analyzer Operating
Att 20 dB
–10
on the
­H e t e r o d y n e P r i n c i p l e
*RBW 3 kHz
*VBW 3 kHz
SWT 45 ms
*
A
–20
1AP –30
CLRWR
–40
PRN
–50
–60
–70
–80
–90
–100
–110
Center 100.015 MHz
Ref –10 dBm
20 kHz/
Att 20 dB
Span 200 kHz
*RBW 3 kHz
*VBW 1 kHz
SWT 135 ms
–10
*
–20
A
–30
–40
PRN
–50
–60
–70
–80
–90
–100
–110
Center 100 MHz
20 kHz/
Span 200 kHz
Fig. 4-12 S
pectrum of input signal consisting of two sinusoidal carriers with
same and with different level, recorded with different resolution
bandwidths (blue traces RBW = 3 kHz, red traces RBW = 30 kHz)
50
IF S i g n a l P r o c e s s i n g
Ref Lvl
–10 dBm
RBW 1 kHz
VBW 200 kHz
SWT 300 ms
RF Att 20 dB
Unit
dBm
–10
*
–20
A
–30
SF = 9.5
–40
SF = 4.6
–50
1SA
2AP
–60
–70
–80
–90
–100
–110
Center 100 MHz
2 kHz/
Span 20 kHz
Fig. 4-13 T wo neighboring sinusoidal signals with different levels recorded with
a resolution bandwidth of 1 kHz and a shape factor of 9.5 and 4.6
If the weaker signal is to be distinguished by a filter with a lower skirt
selectivity, the resolution bandwidth has to be reduced. Due to the longer transient time of narrowband IF filters, the minimum sweep time
must be increased. For certain measurement applications, shorter sweep
times are therefore feasible with filters of high skirt selectivity.
As mentioned earlier, the highest resolution is attained with narrowband IF filters. These filters, however, always have a longer transient
time than broadband filters, so contemporary spectrum analyzers provide a large number of resolution bandwidths to allow resolution and
measurement speed to be adapted to specific applications. The setting
range is usually large (from 10 Hz to 10 MHz). The individual filters are
implemented in different ways. There are three different types of ­filters:
u analog filters
u digital filters
u FFT
51
P r a ct i c a l R e a l i z a t i o n
of an
Analyzer Operating
on the
­H e t e r o d y n e P r i n c i p l e
Analog IF filters
Analog filters are used to realize very large resolution bandwidths. In
the spectrum analyzer described in our example, these are bandwidths
from 100 kHz to 10 MHz. Ideal Gaussian filters cannot be implemented
using analog filters. A very good approximation, however, is possible
at least within the 20 dB bandwidth so that the transient response is
almost identical to that of a Gaussian filter. The selectivity characteristics depend on the number of filter circuits. Spectrum analyzers typically
have four filter circuits, but models with five filter circuits can be found,
too. Shape factors of about 14 and 10 can thus be attained, whereas an
ideal Gaussian filter exhibits a shape factor of 4.6.
The spectrum analyzer described in our example uses IF filters that
are made up of four individual circuits. Filtering is distributed so that
two filter circuits each (29 and 31) are arranged before and after the IF
amplifier (30). This configuration offers the following benefits:
u The filter circuits ahead of the IF amplifier provide for rejection of
mixture products outside the passband of the IF filter. Intermodulation products that may be caused by such signals in the last IF
amplifier without prefiltering can thus be avoided (see chapter 5.2:
Nonlinearities).
u The filter circuits after the IF amplifier are used to reduce the noise
bandwidth. If they were arranged ahead of the IF amplifier, the
total noise power in the subsequent envelope detection would be
distinctly higher due to the broadband noise of the IF amplifier.
Digital IF filters
Narrow bandwidths can best be implemented with the aid of digital signal processing. In contrast to analog filters, ideal Gaussian filters can
be realized. Much better selectivity can be achieved using digital filters
instead of analog filters at an acceptable circuit cost. Analog filters consisting of five individual circuits, for instance, have a shape factor of
about 10, whereas a digitally implemented ideal Gaussian filter exhibits
a shape factor of 4.6. Moreover, digital filters feature temperature stability, are free of aging effects and do not require adjustment. Therefore
they feature a higher accuracy regarding bandwidth.
The transient response of digital filters is defined and known. Using
suitable correction factors, digital filters allow shorter sweep times
than analog filters of the same bandwidth (see chapter 4.6: Parameter
­dependencies).
52
IF S i g n a l P r o c e s s i n g
In contrast to that shown in the block diagram, the IF signal after the
IF amplifier must first be sampled by an A/D converter. To comply with
the sampling theorem, the bandwidth of the IF signal must be limited
by analog prefilters prior to sampling. This band limiting takes place
before the IF amplifier so that intermodulation products can be avoided,
as was the case for analog filters. The bandwidth of the prefilter is variable, so depending on the set digital resolution bandwidth, the smallest
possible bandwidth can be selected. The digital IF filter provides for limiting the noise bandwidth prior to envelope detection.
The digital IF filter can be implemented by configurations as described
in [3-1] or [3-2]. In our example, the resolution bandwidths from 10 Hz to
30 kHz of the spectrum analyzer are realized by digital filters.
FFT
Very narrow IF bandwidths lead to long transient times which considerably reduce the permissible sweep speed. With very high resolution it
is therefore advisable to calculate the spectrum from the time characteristic – similar to the FFT analyzer described in chapter 3.1. Since very
high frequency signals (up to several GHz) cannot directly be sampled
by an A/D converter, the frequency range of interest is converted to the
IF as a block, using a fixed-tuned LO signal, and the bandpass signal is
sampled in the time domain (see Fig. 4-14). To ensure unambiguity, an
analog prefilter is required in this case.
For an IF signal with the center frequency fIF and a bandwidth B,
one would expect a minimum sampling rate of 2 · (fIF + 0.5 ·B) in accordance with the sampling theorem (Equation 3-1). If the relative bandwidth, however, is small (B/fIF« 1), then undersampling is permissible to
a certain extent. That is, the sampling frequency may be lower than that
resulting from the sampling theorem for baseband signals. To ensure
unambiguity, adherance to the sampling theorem for bandpass signals
must be maintained. The permissible sampling frequencies are determined by:
2 ⋅ fIF + B
k +1
where
fS
fIF
B
k
≤ fS ≤
2 ⋅ fIF - B
k
(Equation 4-10)
sampling frequency
intermediate frequency
bandwidth of IF signal
1, 2, …
53
P r a ct i c a l R e a l i z a t i o n
of an
Analyzer Operating
on the
­H e t e r o d y n e P r i n c i p l e
The spectrum can be determined from the sampled values with the aid
of the Fourier transform.
Conversion
A
Analog
bandpass
filter
fIF
fLO
f
Span
A
D
RAM
FFT
Display
Fig. 4-14 Spectrum analysis using FFT
The maximum span that can be analyzed at a specific resolution by
means of an FFT is limited by the sampling rate of the A/D converter
and by the memory available for saving the sampled values. Large spans
must therefore be subdivided into individual segments which are then
converted to the IF in blocks and sampled.
While analog or digital filter sweep times increase directly proportional
to the span, the observation time required for FFT depends on the
desired frequency resolution as described in chapter 3.1. To comply with
sampling principles, more samples have to be recorded for the FFT with
increasing span so that the computing time for the FFT also increases.
At sufficiently high computing speed of digital signal processing, distinctly shorter measurement times than that of conventional filters can
be attained with FFT, especially with high span/BN ratios (see chapter
4.6 Parameter dependencies).
The far-off selectivity of FFT filters is limited by the leakage effect,
depending on the windowing function used. The Hann window described
in chapter 3.1 is not suitable for spectrum analysis because of the ampli54
Determination
of
Video Voltage
and
Video Filters
tude loss and the resulting level error. A flat-top window is therefore
often used to allow the leakage effect to be reduced so that a negligible
amplitude error may be maintained. This is at the expense of an observation time that is by a factor of 3.8 longer than that of a rectangular window. The flat-top window causes a wider representation of the windowing function in the frequency domain (corresponding to the convolution
with a Dirac function in the frequency domain). When the flat-top window is implemented, a shape factor of about 2.6 can be attained, which
means that selectivity is clearly better than when analog or digital IF filters are used.
FFT filters are unsuitable for the analysis of pulsed signals (see chapter 3.1). Therefore it is important for spectrum analyzers to be provided
with both FFT and conventional filters.
4.3
Determination of video voltage and video filters
Information about the level of the input signal is contained in the level
of the IF signal, such as amplitude-modulated signals in the envelope
of the IF signal. With the use of analog and digital IF filters, the envelope of the IF signal is detected after filtering the last intermediate frequency (see Fig. 4-15).
Envelope
AIF
AVideo
Envelope
detection
0
0
t
1
fIF
Fig. 4-15 Detection of IF signal envelope
55
t
P r a ct i c a l R e a l i z a t i o n
of an
Analyzer Operating
on the
­H e t e r o d y n e P r i n c i p l e
VVideo
This functional configuration is similar to analog envelope detector circuitry used to demodulate AM signals (see Fig. 4-16). The IF signal is
detected and the high-frequency signal component eliminated by a lowpass filter and the video voltage is available at the output of this circuit.
VIF
^
t
BIF
VIF
VVideo
R
C
VVideo
0
t
Video filter
^
VIF
0
VIF
V Video
0
fIF
f
Video filter
fc = R · C
0
fg
fIF
2 fIF f
BVideo
Fig. 4-16 Detection of IF signal envelope by means of envelope detector
For digital bandwidths, the IF signal itself is sampled, i. e. the envelope is
determined from the samples after the digital IF filter. If one looks at the
IF signal represented by a complex rotating vector (cf. chapter 2.1), the
envelope corresponds to the length of the vector rotating at an angular
velocity of wIF (see Fig. 4-17). The envelope can be determined by forming
the magnitude using the Cordic algorithm [4-3].
56
j lm
Determination
of
Video Voltage
and
Video Filters
Samples
A
Vid
e
o
XIF
Re
Fig. 4-17 IF signal with sinusoidal input
signal, represented by complex
rotating vector
Due to envelope detection, the phase information of the input signal
gets lost, so that only the magnitude can be indicated in the display. This
is one of the primary differences between the envelope detector and the
FFT analyzer as described in chapter 3.1.
The dynamic range of the envelope detector determines the dynamic
range of a spectrum analyzer. Modern analyzers feature a dynamic range
of about 100 dB. It has no sense to simultaneously display so much different values in a linear scale. The level is usually displayed in a logarithmic scale on the spectrum analyzer. The IF signal can therefore be amplified with the aid of a log amplifier (32) ahead of the envelope detector
(33), thereby increasing the dynamic range of the display.
The resulting video voltage depends on the input signal and the selected
resolution bandwidth. Fig. 4-18 shows some examples. The spectrum
analyzer is tuned to a fixed frequency in these examples, so the displayed span is 0 Hz (zero span).
57
P r a ct i c a l R e a l i z a t i o n
of an
Analyzer Operating
on the
­H e t e r o d y n e P r i n c i p l e
a)
Ain
0
t
AVideo
AIF
0
t
1
––
fe
fin
f
f
fIF
AIF
0
t
0
t
0
|AIF|
fin
f
0
fm
BIF
f
fIF
0
f
AIF
AVideo
fm
Ain
0
0
0
t
t
1
––
fm
t
1
––
fIF
1
__
fin
|AIF|
fin
fm
f
0
BIF
| AVideo|
| Ain|
0
t
| AVideo|
| Ain|
c)
f
1
__
fIF
1
__
fe
0
0
AVideo
Ain
0
BIF
| AVideo|
|AIF|
b)
t
1
---fIF
| Ain|
0
0
f
fIF
0
fm
f
fm
Fig. 4-18 V
ideo signal (orange traces) and IF signal after IF filter (blue traces)
for various input signals (green traces) and resolution bandwidths
a) sinusoidal signal b) AM signal, resolution bandwidth smaller than
twice the modulation bandwidth c) AM signal, resolution bandwidth
greater than twice the modulation bandwidth
58
Determination
of
Video Voltage
and
Video Filters
d)
0
t
| Ain|
0
0
|AIF|
f
AVideo
AIF
0
t
BIF
fIF
0
t
| AVideo|
Ain
f
0
f
Fig. 4-18 (continued) Video signal (orange traces) and IF signal after IF filter
(blue traces) for various input signals (green traces) and resolution
bandwidths d) noise
The envelope detector is followed by the video filter (35) which defines
the video bandwidth (BV). The video filter is a first order lowpass configuration used to free the video signal from noise, and to smooth the trace
that is subsequently displayed so that the display is stabilized. In the
analyzer described, the video filter is implemented digitally. Therefore,
the video signal is sampled at the output of the envelope detector with
the aid of an A/D converter (34) and its amplitude is quantized.
Similar for the resolution bandwidth, the video bandwidth also limits the maximum permissible sweep speed. The minimum sweep time
required increases with decreasing video bandwidth (chapter 4.6.1).
The examples in Fig. 4-18 show that the video bandwidth has to be
set as a function of the resolution bandwidth and the specific measurement application. The detector used also has be taken into account in
the video bandwidth setting (chapter 4.5). The subsequent considerations do not hold true for RMS detectors (chapter 4.4 D
­ etectors).
For measurements on sinusoidal signals with sufficiently high signal-to-noise ratio a video bandwidth that is equal to the resolution bandwidth is usually selected. With a low S/N ratio the display can however
be stabilized by reducing the video bandwidth. Signals with weak level
are thus shown more distinctly in the spectrum (Fig. 4-19) and the measured level values are stabilized and reproducible. In the case of a sinusoidal signal the displayed level is not influenced by a reduction of the
video bandwidth. This becomes quite clear when looking at the video
voltage resulting from the sinusoidal input signal in Fig. 4-18a. The video
59
P r a ct i c a l R e a l i z a t i o n
of an
Analyzer Operating
on the
­H e t e r o d y n e P r i n c i p l e
signal is a pure DC voltage, so the video filter has no effect on the overall level of the video signal.
Ref –40 dBm
–40
–50
–60
–70
1AP
–80
CLRWR
–90
–100
–110
–120
–130
A
PRN
EXT
Center 100 MHz
Ref –40 dBm
–40
–50
–60
–70
1AP –80
CLRWR
–90
–100
–110
–120
–130
Att 10 dB
RBW 300 kHz
VBW 1 MHz
SWT 2.5 ms
1 MHz/
Att 10 dB
Span 10 MHz
RBW 300 kHz
*VBW 300 Hz
SWT 280 ms
B
Center 100 MHz
1 MHz/
Span 10 MHz
Fig. 4-19 S
inusoidal signal with low S/N ratio shown for large (top) and small
(bottom half of screen) video bandwidth
To obtain stable and reproducible results of noise measurements, a narrow video bandwidth should be selected. The noise bandwidth is thus
reduced and high noise peaks are averaged. As described in greater
detail in chapter 4.4, the displayed average noise level will be 2.5 dB
below the signal´s RMS value.
Averaging should be avoided when making measurements on pulsed
signals. Pulses have a high peak and a low average value (depending on
mark-to-space ratio). In order to avoid too low display levels, the video
bandwidth should be selected much greater than the resolution bandwidth (Fig. 4-20). This is further discussed in chapter 6.2.
60
D e t e ct o r s
Ref –20 dBm
–20
–30
–40
1 AP
–50
CLRWR
–60
–70
–80
–90
–100
–110
–120
Att 10 dB
Marker 1 [T1]
–38.30 dBm
1.00000000 GHz
1
A
PRN
EXT
Center 1 GHz
Ref –40 dBm
–20
–30
–40
1 AP
–50
CLRWR
–60
–70
–80
–90
–100
–110
–120
*RBW 1 MHz
*VBW 10 MHz
SWT 2.5 ms
20 MHz/
*RBW 1 MHz
*VBW 100 kHz
SWT 5 ms
Att 10 dB
Span 200 MHz
Marker 1 [T1]
–43.50 dBm
1.00000000 GHz
1
Center 1 GHz
20 MHz/
B
Span 200 MHz
Fig. 4-20 P
ulsed signal recorded with large and small video bandwidth (top and
bottom half of screen); note amplitude loss with small video bandwidth
(see marker)
4.4
Detectors
Modern spectrum analyzers use LC displays instead of cathode ray
tubes for the display of the recorded spectra. Accordingly, the resolution
of both the level and the frequency display is limited.
The limited resolution of the level display range can be remedied by
using marker functions (see chapter 4.5: Trace processing). Results can
then be determined with considerably high resolution.
Particularily when large spans are displayed, one pixel contains the
spectral information of a relatively large subrange. As explained in chapter 4.1, the tuning steps of the 1st local oscillator depend on the resolution bandwidth so that several measured values, referred to as samples
or as bins, fall on one pixel. Which of the samples will be represented
by the pixel depends on the selected weighting which is ­determined by
61
P r a ct i c a l R e a l i z a t i o n
of an
Analyzer Operating
on the
­H e t e r o d y n e P r i n c i p l e
the detector. Most of the spectrum analyzers feature min peak, max
peak, auto peak and sample detectors. The principles of the detectors is
shown in Fig. 4-21.
N=5
Video voltage
Samples
f
Max Peak
A
Max Peak
RMS
AV
RMS
AV
Auto Peak
Sample
Sample
Auto Peak
Min Peak
Min Peak
Pixel n
Pixel n + 1
f
Displayed sample
Figs 4-21 Selection of sample to be displayed as a function of detector used
62
D e t e ct o r s
Logarithmic
amplifier
lin
IF signal
log
Max Peak
Envelope
detector
A/D converter
Sample
A
D
Display
Video
filter
Fig 4-22 Analog realization of detectors
Min Peak
These detectors can be implemented by analog circuits as shown in
Fig. 4-22. In this figure, the weighted video signal is sampled at the output of the detector. In the spectrum analyzer described, the detectors
(36 to 39) are implemented digitally, so that the video signal is sampled
ahead of the detectors (in this case even ahead of the video filter). In
addition to the above detectors, average and RMS detectors may also be
realized. Quasi-peak detectors for interference measurements are implemented in this way.
Max peak detector
The max peak detector displays the maximum value. From the samples
allocated to a pixel the one with the highest level is selected and displayed. Even if wide spans are displayed with very narrow resolution
bandwidth (span/RBW >> number of pixels on frequency axis), no
input ­signals are lost. Therefore this type of detector is particularly useful for EMC measurements.
Min peak detector
The min peak detector selects from the samples allocated to a pixel the
one with the minimum value for display.
Auto peak detector
The auto peak detector provides for simultaneous display of maximum
and minimum value. The two values are measured and their levels displayed, connected by a vertical line (see Fig. 4-21).
Sample detector
The sample detector samples the IF envelope for each pixel of the trace
to be displayed only once. That is, it selects only one value from the
samples allocated to a pixel as shown in Fig. 4-21 to be displayed. If the
span to be displayed is much greater than the resolution bandwidth
63
P r a ct i c a l R e a l i z a t i o n
of an
Analyzer Operating
on the
­H e t e r o d y n e P r i n c i p l e
(span/RBW >> number of pixels on frequency axis), input signals are
no longer reliably detected. The same unreliability applies when too
large tuning steps of the local oscillator are chosen (see Fig. 4-5). In this
case, signals may not be displayed at the correct level or may be completely lost.
RMS detector
The RMS (root mean square) detector calculates the power for each pixel
of the displayed trace from the samples allocated to a pixel. The result
corresponds to the signal power within the span represented by the
pixel. For the RMS calculation, the samples of the envelope are required
on a linear level scale. The following applies:
VRMS =
1 N 2
⋅ ∑v N i=1 i
(Equation 4-11)
where VRMS RMS value of voltage
Nnumber of samples allocated to the pixel concerned
vi
samples of envelope
The reference resistance R can be used to calculate the power:
P=
2
VRMS
R
(Equation 4-12)
AV detector
The AV (average) detector calculates the linear average for each pixel of
the displayed trace from the samples allocated to a pixel. For this calculation the samples of the envelope are required on a linear level scale.
The following applies:
VAV =
1 N
⋅ ∑v N i=1 i
(Equation 4-13)
where VAV
average voltage
Nnumber of samples allocated to the pixel concerned
vi
samples of envelope
Like with the RMS detector, the reference resistance R can be used to
calculate the power (Equation 4-12).
64
D e t e ct o r s
Quasi peak detector
This is a peak detector for interference measurement applications with
defined charge and discharge times. These times are laid down by CISPR
16-1 [4-4] for instruments measuring spurious emission. A detailed
description of this type of detector can be found in chapter 6.2.5.1.
With a constant sampling rate of the A/D converter, the number of samples allocated to a certain pixel increases at longer sweep times. The
effect on the displayed trace depends on the type of the input signal and
the selected detector. They are described in the following section.
Effects of detectors on the display of different types of input signals
Depending on the type of input signal, the different detectors partly provide different measurement results. Assuming that the spectrum analyzer is tuned to the frequency of the input signal (span = 0 Hz), the
envelope of the IF signal and thus the video voltage of a sinusoidal input
signal with sufficiently high signal-to-noise ratio are constant. Therefore,
the level of the displayed signal is independent of the selected detector
since all samples exhibit the same level and since the derived average
value (AV detector) and RMS value (RMS detector) correspond to the
level of the individual samples.
This is different however with random signals such as noise or noiselike signals in which the instantaneous power varies with time. Maximum and minimum instantaneous value as well as average and RMS
value of the IF signal envelope are different in this case.
The power of a random signal is calculated as follows:
 +T


2

1
1
P = ⋅ lim  ⋅∫ v 2 t dt  
R T →∞ T T

 
2
()
(Equation 4-14)
or for a certain limited observation time T
t+
T
2
1 1
P = ⋅ ⋅ ∫ v 2 t dt R T T
t-
()
(Equation 4-15)
2
65
P r a ct i c a l R e a l i z a t i o n
of an
Analyzer Operating
on the
­H e t e r o d y n e P r i n c i p l e
In the specified observation time T, a peak value can also be found for
the instantaneous power. The relationship between the peak value and
power can be expressed by the crest factor as follows:
P 
CF = 10 dB ⋅ lg  S  P 
(Equation 4-16)
where CF
crest factor
PSpeak value of instantaneous power
in observation time T
P
power
With noise, any voltage values may occur theoretically, so the crest factor
would be arbitrarily high. Since the probability for very high or very low
voltage values is low, a crest factor of about 12 dB is usually obtained in
practice for Gaussian noise observed over a sufficiently long period.
Digitally modulated signals often exhibit a spectrum similar to noise.
However, the crest factor usually differs from that for Gaussian noise.
Fig. 4-23 shows the peak and RMS values of Gaussian noise and of a
IS-95 CDMA signal (forward channel).
66
D e t e ct o r s
a) Crest factor 12 dB
Ref –50 dBm
RBW 3 MHz
VBW 10 MHz
SWT 100 s
Att 10 dB
–50
A
SGL
–55
1 RM*
VIEW
–60
2 PK*
VIEW –65
PRN
–70
EXT
–75
–80
–85
–90
–95
–100
Center 2.2 GHz
10 s/
b) Crest factor 13.8 dB
Ref –10 dBm
RBW 3 MHz
VBW 10 MHz
SWT 100 s
Att 20 dB
–10
*
A
SGL
–15
1 RM*
CLRWR
–20
2 PK*
CLRWR
–25
PRN
–30
EXT
–35
–40
–45
–50
–55
–60
Center 2.2 GHz
10 s/
Fig. 4-23 P
eak (red traces) and RMS values (blue traces) of Gaussian noise (a)
and of a IS-95 CDMA signal (b), recorded with max peak and RMS
detectors
67
P r a ct i c a l R e a l i z a t i o n
of an
Analyzer Operating
on the
­H e t e r o d y n e P r i n c i p l e
The effects of the selected detector and of the sweep time on the results
of measurements on stochastic signals are described in the following.
Max peak detector
When using the max peak detector, stochastic signals are overweighted
so that the maximum level is displayed. With increasing sweep time the
dwell time in a frequency range allocated to a pixel is also increased. In
the case of Gaussian noise the probability that higher instantaneous
values will occur also rises. This means that the levels of the displayed
­pixels also become higher (see Fig. 4-24a).
With a small ratio between span and resolution bandwidth, the noise
displayed for short sweep times is equal to that displayed with a sample
detector, since only one sample is recorded per pixel.
Min peak detector
When using the min peak detector, stochastic signals are underweighted
so that the minimum level is displayed. The noise displayed on the spectrum analyzer is strongly suppressed. In the case of Gaussian noise the
probability that lower instantaneous values will occur increases with
increasing sweep time. This means that the levels of the displayed ­pixels
also become lower (see Fig. 4-24a).
If measurements are carried out on sinusoidal signals with low signal-to-noise ratio, the minimum of the noise superimposed on the signal will also be displayed so that the level measurements yield too low
values.
With a small ratio between span and resolution bandwidth, the noise
displayed for short sweep times is equal to that displayed with a sample
detector, since only one sample is recorded per pixel.
Auto peak detector
When using the auto peak detector, the results of the max peak and min
peak detectors are displayed simultaneously, the two values being connected by a line. With increasing sweep time the displayed noise band
becomes distinctly wider.
With a small ratio between span and resolution bandwidth, the noise
displayed for short sweep times is equal to that displayed with a sample
detector, since only one sample is recorded per pixel.
68
D e t e ct o r s
a)
Ref –50 dBm
Att 10 dB
*RBW 1 MHz
VBW 3 MHz
*SWT 10 s
–50
*
A
SGL
–60
1 RM*
CLRWR
–70
2 PK*
VIEW
–80
PRN
–90
–100
–110
–120
–130
–140
–150
Center 1.5 GHz
10 MHz/
Span 100 MHz
b)
Ref –50 dBm
Att 10 dB
*RBW 1 MHz
VBW 3 MHz
*SWT 10 s
–50
*
A
SGL
–60
1 MI*
CLRWR
–70
2 MI*
VIEW
–80
PRN
–90
–100
–110
–120
–130
–140
–150 Center 1.5 GHz
10 MHz/
Span 100 MHz
Fig. 4-24 D
isplayed noise varying as a function of sweep time, with max peak
detector (a) and min peak detector (b), sweep time 2.5 ms (blue trace)
and 10 s (red trace)
69
P r a ct i c a l R e a l i z a t i o n
of an
Analyzer Operating
on the
­H e t e r o d y n e P r i n c i p l e
Sample detector
As shown in Fig. 4-21, the sample detector always displays a sample
recorded at a defined point in time. Due to the distribution of the instantaneous values, the trace displayed in the case of Gaussian noise therefore varies about the average value of the IF signal envelope resulting
from noise. This average value is 1.05 dB below the RMS value. If noise
is averaged over a narrow video bandwidth (VBW < RBW) using the logarithmic level scale, the displayed average value is an additional 1.45 dB
too low. The displayed noise is then 2.5 dB below the RMS value.
In contrast to the other detectors the sweep time has no effect on the
displayed trace since the number of the recorded samples is independent of the sweep time.
RMS detector
The RMS detector allows measurement of the actual power of an input
signal irrespective of its temporal characteristic. When using a sample
or max peak detector, the relationship between RMS value and peak
value must be precisely known for determining the power of signals
with random instantaneous value. This knowledge is not required when
using an RMS detector.
The RMS value displayed by a specific pixel is calculated from all
­samples pertaining to this pixel. By increasing the sweep time, the number of samples available for the calculation is increased, thus allowing smoothing of the displayed trace. Smoothing by reducing the video
bandwidth or by averaging over several traces (see chapter 4.5) is neither ­permissible nor necessary with the RMS detector. The measurement results would be falsified, since the displayed values would be too
low (max. 2.51 dB). To avoid any falsification of results, the video bandwidth should be at least three times the resolution bandwidth when
using the RMS detector.
AV detector
The AV detector determines the average value from the samples using
the linear level scale. The actual average value is thus obtained irrespective of the type of input signal. Averaging of logarithmic samples (log
average) would yield results that were too low since higher signal ­levels
are subject to greater compression by logarithmation. By increasing the
sweep time, several samples are available for calculating the average
value that is displayed by a specific pixel. The displayed trace can thus
be smoothed.
70
D e t e ct o r s
A narrow video bandwidth causes averaging of the video signal. If samples of the linear level scale are applied to the input of the video filter, the
linear average of the samples is formed when reducing the video bandwidth. This corresponds to the function of the AV detector so that smoothing by means of narrow video bandwidths is permissible in this case.
The same holds true for the analyzer described here, since samples
with linear level scale are applied to the input of the video filter when
the AV detector is used (see block diagram).
If the video bandwidth is reduced, the displayed noise converges for max
peak, min peak, auto peak and sample detectors since the samples are
averaged by the video filter before they are weighted by the detector. If
a linear envelope detector is used to determine the IF signal envelope,
samples with linear scale are averaged by the video filter. The resulting display corresponds to the true average value and hence to the displayed noise when using an AV detector. If the IF signal is log-amplified
before the video voltage is formed, the resulting averaged samples are
lower than the true average value. In the case of Gaussian noise the difference is 1.45 dB (see Fig. 4-25a). Since the linear average of the video
voltage resulting from Gaussian noise is already 1.05 dB below the RMS
value, the samples obtained are all 2.5 dB lower than those obtained
with the RMS detector (see Fig. 4-25a). Due to this known relationship
an RMS detector is not required to determine the Gaussian noise power.
The power can be calculated from the samples collected by the sample
detector, taking into account a correction factor of 2.5 dB.
This relationship does not apply to other random signals whose
instantaneous values are not in line with the Gaussian distribution (for
example, digitally modulated signals, see Fig. 4-25b). If the crest factor
is unknown, the power of such signals can only be determined using an
RMS detector.
Averaging over several measurements
As described in the following chapter, modern analyzers feature the possibility of averaging traces over several measurements (trace average).
This method of averaging partly leads to results different from those
when using narrowband video filters.
Depending on whether the recorded trace is displayed on a linear or
logarithmic level scale, linear or logarithmic samples are used for averaging. Whether the trace is falsified by averaging depends on the display mode.
71
P r a ct i c a l R e a l i z a t i o n
of an
Analyzer Operating
on the
­H e t e r o d y n e P r i n c i p l e
a)
Ref –87 dBm
RBW 300 kHz
VBW 3 MHz
*SWT 5 s
Att 10 dB
–87
*
A
SGL
1 RM*
–88
VIEW
–89
2 AP*
VIEW
–90
RMS
3 AV*
–91
CLRWR
Average (lin)
1.05 dB
–92
2.50 dB
1.45 dB
PRN
EXT
Auto Peak (log)
–93
–94
–95
–96
–97
Center 100 MHz
1 MHz/
Span 10 MHz
b)
Ref –24 dBm
Att 10 dB
RBW 3 MHz
VBW 10 MHz
SWT 1 s
–24
*
A
SGL
1 RM*
VIEW –25
–26
2 AP*
VIEW
–27
RMS
3 AV*
–28
CLRWR
Average (lin)
–29
PRN
EXT
>2.50 dB
–30
–31
Auto Peak (log)
–32
–33
–34
Center 2.2 GHz
100 ms/
Fig. 4-25 M
easurement of Gaussian noise (a) and IS-95 CDMA signal (b) using
RMS and AV detectors (green and red traces) as well as auto peak
detector with averaging over narrow video bandwidth (blue trace)
72
Trace Processing
In the case of averaging over several measurements, the displayed
noise levels do not converge for max peak, min peak and sample detectors. The average is derived from the maximum and minimum values,
whereas with the use of the video filter, the samples are averaged prior
to weighting and therefore converge.
The sample detector yields the average noise level. With logarithmic
level display, the displayed average value is 1.45 dB too low, as already
explained above. With linear level display and large video bandwidth
(VBW ≥ 10 · RBW) the true average is obtained, as with the AV detector.
When using the auto peak detector, averaging over several traces is
not recommended since the maximum and minimum value is displayed.
When the trace average function is activated, automatic switchover to
sample detector is often made.
For the RMS detector, trace averaging is permitted neither in the linear nor in the logarithmic level mode.
4.5
Trace processing
As was explained in chapter 4.4, linear samples are required for AV and
RMS detectors. For displaying the traces on a logarithmic level scale
when these detectors are used, the detectors are followed by a log amplifier (40) which may be optionally activated.
In modern spectrum analyzers, the measurement results are digitized
before they are displayed. This allows many different methods of trace
evaluation (41).
Measured data memory
Several traces can be stored in modern analyzers and simultaneously
displayed. This function is particularly useful for comparative measurements.
Trace average
With the aid of this function a displayed trace can be smoothed by averaging over several measurements (sweeps). The user can enter the number of sweeps to be averaged.
Depending on the input signal and the detector used, this way of
averaging may lead to other results than averaging by reducing the
video bandwidth.
73
P r a ct i c a l R e a l i z a t i o n
of an
Analyzer Operating
on the
­H e t e r o d y n e P r i n c i p l e
Marker functions
Marker functions are particularly useful for the evaluation of recorded
traces. They allow frequency and level to be displayed at any point of the
trace. The limited display accuracy due to the constrained screen resolution can thus be remedied. In addition to functions which set the marker
automatically to a signal with maximum level, level differences between
signals can also be directly displayed using the delta marker feature.
Modern spectrum analyzers feature enhanced marker functions
allowing, for instance, direct noise or phase noise measurements, without manual setting of bandwidth or correction factors (see Fig. 4-26).
The precise frequency of a displayed signal can also be determined with
the aid of a marker and a count function (signal count). In many cases
the spectrum analyzer can thus replace a frequency counter.
Tolerance masks (limit lines)
Limit values to be adhered to by the device under test can easily be
checked with the aid of tolerance masks. To simplify use in production,
recorded traces are automatically checked for violation of the specified
limit values and the result is output in form of a “pass” or “fail” message
(see Fig. 4-27).
Channel power measurement
In the case of digitally modulated signals, power often has to be measured within one channel or within a specific frequency range. Channel power is calculated from the recorded trace, with special functions
being provided for this purpose by modern spectrum analyzers. Adjacent-channel power measurement with the aid of a spectrum analyzer is
described in detail in chapter 6.3.
74
Trace Processing
Fig. 4-26 Marker functions for easy phase noise measurement of an input signal
Fig. 4-27 Evaluation of traces with the aid of limit lines
75
P r a ct i c a l R e a l i z a t i o n
4.6
of an
Analyzer Operating
on the
­H e t e r o d y n e P r i n c i p l e
Parameter dependencies
Some of the analyzer settings are interdependent. To avoid measurement
errors, these parameters are coupled to one another in normal operating mode of modern spectrum analyzers. That is, upon varying one setting all other dependent parameters will be adapted automatically. The
parameters can, however, also be set individually by the user. In such a
case it is especially important to know relationships and effects of various settings.
4.6.1
Sweep time, span, resolution and video bandwidths
Through the use of analog or digital IF filters, the maximum permissible sweep speed is limited by the transient time of the IF filter and video
filter. The transient time has no effect if the video bandwidth is larger
than the resolution bandwidth. In this case, the required transient time
increases inversely with the square of the resolution bandwidth, so with
a decrease of the resolution bandwidth by the factor n the required minimum sweep time becomes n2 longer. The following applies:
TSweep = k ⋅
Df
2
B IF
(Equation 4-17)
where TSweepminimum sweep time required (with specified span
and resolution bandwidth)
BIF
resolution bandwidth
Df
span
k
proportionality factor
The proportionality factor k depends on the type of filter and the permissible transient response error. For analog filters made up of four
or five individual circuits, the proportionality factor k is 2.5 (maximum
transient response error approx. 0.15 dB). With digitally implemented
Gaussian filters, the transient response is known and exactly reproducible. Compared to analog filters, higher sweep speeds without amplitude
loss can be obtained through appropriate correction factors independent of the type of input signal. A k factor of 1 can thus be attained. Fig.
4-28 shows the required sweep time for a span of 1 MHz as a function of
the resolution bandwidth.
76
Parameter Dependencies
10 7
Min. sweep time / s
10 6
k=1
k = 2.5
FFT filter (real)
FFT filter (theoretical)
10 5
10 4
10 3
10 2
10 1
1
10
–1
10 – 2
10 – 3
10 – 4
10 – 5
10 – 6
1 Hz
10 Hz
100 Hz
1 kHz
10 kHz
100 kHz
1 MHz
Resolution bandwith
Fig. 4-28 T heoretically required sweep time as a function of resolution bandwidth at a span of 1 MHz. Example of sweep times that can be attained
with FFT filters in a modern spectrum analyzer
If the video bandwidth is smaller than the resolution bandwidth, the
required minimum sweep time is influenced by the transient time of
the video filter. Similar to the IF filter, the transient time of the video filter increases with decreasing bandwidth. The video filter is usually a 1st
order lowpass, or a simple RC section if implemented in analog form.
Therefore there is a linear relationship between video bandwidth and
sweep time. Reducing the video bandwidth by a factor n results in an n
times longer sweep time.
Upon failure to attain the minimum sweep time, the IF filter or video
filter cannot reach steady state, causing an amplitude loss and distorted
signal display (frequency offset). A sinusoidal signal, for instance, would
be displayed neither at the correct level nor correct frequency (see
Fig. 4-29). Moreover, the effective resolution would be degraded due to
the widened signal display.
77
P r a ct i c a l R e a l i z a t i o n
of an
Analyzer Operating
on the
RBW 3 kHz
VBW 3 kHz
SWT 5 ms
UNCAL
Ref Lvl –20 dBm
­H e t e r o d y n e P r i n c i p l e
RF Att
Unit
10 dB
dBm
–20
*
A
–30
–40
–50
1AVG
2VIEW
–60
1SA
2SA
–70
–80
–90
–100
–110
–120
Center 1 GHz
20 kHz/
Span 200 kHz
Fig. 4-29 A
mplitude loss if minimum sweep time required is not attained (blue
trace)
To avoid measurement errors due to short sweep times, resolution bandwidth, video bandwidth, sweep time and span are coupled in normal
operating mode of modern spectrum analyzers.
Resolution bandwidth is automatically adapted to the selected span.
Long sweep times due to narrow resolution bandwidths at large spans
or poor resolution due to high resolution bandwidths at small spans are
thus avoided. Handling of a spectrum analyzer becomes much easier.
The coupling ratio between span and resolution bandwidth can often
be set by the user.
Partial coupling of the parameters is also possible. With manual setting of the resolution and video bandwidths, the sweep time can, for
instance, be adapted automatically.
When using manual settings, if the minimum sweep time is not adhered
to, a warning is usually displayed (UNCAL in Fig. 4-29 upper left corner).
With FFT filters, the transient time is replaced by the observation
time required for a specific resolution (Equation 3-4). In contrast to the
sweep time with analog or digital filters, the observation time is independent of the span, so even if the span were increased, the observation
time would not be increased for constant resolution. The observation
78
Parameter Dependencies
time as a function of the resolution (yellow trace) shown in Fig. 4-28 is
therefore independent of the span.
In practice, larger spans are made up of several subranges. At a specific resolution, the resulting observation time is required for each subrange. The total observation time is directly proportional to the number
of subranges. The attainable measurement time therefore is distinctly
longer than the theoretically expected one. Fig. 4-28 shows sweep times
that can be attained with a modern spectrum analyzer using FFT filters.
It is clearly shown that high span-to-resolution bandwidth ratios allow
greatly reduced sweep times with FFT filters, especially when using very
narrow resolution bandwidths.
In modern spectrum analyzers, the video bandwidth can be coupled to
the resolution bandwidth. When varying the IF bandwidth, the video
bandwidth is automatically adapted. The coupling ratio (the ratio
between re­solution and video bandwidth) depends on the application
mode and therefore has to be set by the user (see chapter 4.3). In addition to the user-defined entry of a numeric value, the following options
are often available:
u Sine
BN/BV = 0.3 to 1
u Pulse
BN/BV = 0.1
u Noise
BN/BV = 10
In the default setting, the video bandwidth is usually selected so that
maximum averaging is achieved without increasing the required sweep
time with the video filter. With a proportionality factor k = 2.5 (Equation 4-17), the video bandwidth must be at least equal to the resolution bandwidth (BN/BV = 1). If the IF filter is implemented digitally, a
proportionality factor k = 1 can be attained through appropriate compensation as described above, and the minimum sweep time required
can be reduced by a factor of 2.5. To ensure steady state of the video
filter despite the reduced sweep time, the video bandwidth selected
should be about three times greater than the resolution bandwidth
(BN/BV = 0.3).
79
P r a ct i c a l R e a l i z a t i o n
of an
Analyzer Operating
on the
­H e t e r o d y n e P r i n c i p l e
4.6.2 Reference level and RF attenuation
Spectrum analyzers allow measurements in a very wide level range that
is limited by the inherent noise and the maximum permissible input
level (see chapter 5.1 and chapter 5.4). With modern analyzers this level
range may extend from -147 dBm to +30 dBm (with a resolution bandwidth of 10 Hz), thus covering almost 180 dB. It is not possible however
to reach the two range limits at a time since they require different settings and the dynamic range of log amplifiers, envelope detectors and
A/D converters is much smaller anyway. Within the total level range only
a certain window can be used which must be adapted by the user to the
specific measurement application by selecting the reference level (maximum signal level to be displayed). The RF attenuation aRF and the IF
gain g IF are to be adjusted as a function of the ­reference level.
To avoid overdriving or even damaging of the first mixer and sub­
sequent processing stages, the high-level input signals must be attenuated by the analyzer’s attenuator (see Fig. 4-30). The attenuation required
for a specific reference level depends on the dynamic range of the first
mixer and subsequent stages. The level at the input of the first mixer (i. e.
the mixer level) should be distinctly below the 1 dB compression point.
Due to nonlinearities, products are generated in the spectrum analyzer
whose levels increase over-proportionally with increasing mixer level. If
the mixer level is too high, these products may cause interference in the
displayed spectrum so that the socalled intermodulation-free range will
be reduced.
80
Parameter Dependencies
a)
L
Strong input signal
Dynamic range limit
of logarithmic amplifier /
A/D converter
aRF
IF gain
Dynamic range
limit
(reference level)
RF attenuation
Max.
input level
gIF
Dynamic range
Mixer level
Input
1st mixer
Logarithmic amplifier
Envelope detector
A/D converter
b)
L
Max.
input level
Weak input signal
Dynamic range limit of logarithmic amplifier /
A/D converter
Input
1st mixer
IF gain
aRF
range
Dynamic
Dynamic range
limit
(reference level)
RF attenuation
Mixer level
gIF
Logarithmic amplifier
Envelope detector
A/D converter
Fig. 4-30 A
daptation of RF attenuation and IF gain to maximum signal level
level to be displayed (max. signal level = reference level)
81
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