1EZ44_0E

1EZ44_0E
Time Domain Measurements
using
Vector Network Analyzer
ZVR
Application Note 1EZ44_0E
Subject to change
19 May 1998, Olaf Ostwald
Products:
ZVR with option ZVR-B2
ZVC with option ZVR-B2
ZVRE with option ZVR-B2
ZVCE with option ZVR-B2
ZVRL with option ZVR-B2
CONTENTS
PAGE
2
TYPICAL APPLICATION
1
ABSTRACT
2
2
TYPICAL APPLICATION
2
3
THEORY
4
3.1
IMPULSE AND STEP RESPONSES
4
3.2
FINITE PULSE WIDTH
7
3.3
ALIASING
8
3.4
BANDPASS AND LOWPASS MODE
9
4
WINDOWS
10
5
GATES
12
6
MEASUREMENT EXAMPLES
13
6.1
STEPPED AIRLINE
13
6.2
FAST MEASUREMENTS
14
7
REFERENCES
15
7.1
ON TIME
15
Fig. 1 DUT
7.2
ON VECTOR NETWORK ANALYZER
FAMILY ZVR
15
8
ORDERING INFORMATION
15
9
ANNEX
16
9.1
TIME DOMAIN REPRESENTATIONS
16
Fig. 2 shows two main reflections (pulses with
highest amplitudes) caused by the two totalreflecting line ends. Other smaller pulses can
also be seen. They are caused by multireflections between the power splitter (12 dB mismatched at each output) and the line ends.
9.2
SPECTRA OF FILTER FUNCTIONS
16
1
DOMAIN TRANSFORM
As a typical example, fig. 2 shows results of a
reflection measurement in the time domain. The
DUT (Fig. 1) consists of a power splitter. A
short-circuited 270 mm long coaxial cable is
connected to one output of the power splitter,
and another coaxial line with an electrical length
of approx. 2750 mm is connected to the other
output. The end of this cable is left open-ended
in the first case (case Open).
ABSTRACT
Vector Network Analyzers of the ZVR family are
able to measure magnitude and phase of
complex S-parameters of a device under test
(DUT) in the frequency domain. By means of
the inverse Fourier transform the measurement
results can be transformed to the time domain.
Thus, the impulse or step response of the
DUT is obtained, which gives an especially clear
form of representation of its characteristics. For
instance, faults in cables can thus be directly
localized. Moreover, special time domain filters,
so-called gates, are used to suppress
unwanted signal components such as
multireflections. The measured data “gated“ in
the time domain are then transformed back to
the frequency domain and an S-parameter
representation without the unwanted signal
components is obtained as a function of
frequency. As usual, other complex or scalar
parameters such as impedance or attenuation
can then be calculated and displayed.
1EZ44_0E.DOC
Fig. 2
Example of a reflection measurement in
time domain (impulse response)
The measurement results in the frequency
domain are represented by the trace shown in
Fig. 3. In contrast to the time domain in which
the representation is simple and clear and the
different signal components can be easily
distinguished, the trace in the frequency domain
is obviously not easy to interpret.
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19 May 1998
Fig. 3
Measurement results in frequency
domain
Any interesting reflection can be selectively
removed by means of a gate, i.e. a filter in the
time domain. Such a reflection might for example be the second-highest pulse (MARKER
∇2 in vicinity of the center of Fig. 2) resulting
from the total reflection at the open end of the
longer cable. The impulse response filtered in
the time domain by a suitable gate (gate center
= 19 ns and gate span = 1.5 ns) is shown in Fig.
4. As can be seen, all other pulses but the
interesting one are suppressed by the gate.
Fig. 5
Measurement results in frequency
domain after time domain filtering
(gating)
Now, as an experiment, the open-ended cable
is terminated by a matched load (Fig. 1: case
Match). The diagram in Fig. 6 shows the really
impressive measurement results which are
obtained despite of the numerous reflections of
the test setup. Again, the gated frequency
response of the DUT is displayed. Corresponding to the low reflection of the matched load
used, the trace is in immediate vicinity of the
center point of the Smith chart. The scaling
value of the outer circle of the enlarged section
around the center of the Smith chart is -20 dB.
The measured values are less than -42 dB.
Fig. 4 Measurement results in time domain
with active gate
Fig. 6
If, in the next step, the gated impulse response
is transformed back to the frequency domain, a
frequency response (see Fig. 5) will be displayed, which is only representing the transfer
function (in reflection) of the power splitter and
the open ended cable.
1EZ44_0E.DOC
3
Measurement results in the frequency
domain with active time gate. DUT of
Fig. 1 with matched load at the end of
the long cable.
19 May 1998
3
THEORY
Each linear and time invariant network can
alternatively be represented in the time domain
by its impulse response h(t) or in the frequency domain by its transfer function H(f). The relation between the two forms of representation is
given by the Fourier Transform as follows:
∞
h ( t). e
H ( f)
j . 2 . π . f. t
dt
∞
H( f ) . e
h( t )
∞
j . 2 . π . f. t
( 2)
Fig. 7 Step response
This form of representation clearly shows the
variation of the impedance along the DUT. The
zero line in the middle of the diagram indicates
zero reflection, thus representing the reference
Z
Zo
Z
Zo
( 3)
The operation sequence for the desired
representation in the time domain is
[MODE] TIME DOMAIN: DEF TRANSF TYPE.
In the menu one can select between the
impulse response (IMPULSE) and the step
response (STEP).
3.1
df
The step response can be obtained by integration of the impulse response h(t). Fig. 7 shows
the step response for the DUT in Fig. 1.
1EZ44_0E.DOC
S 11
( 1)
Via Fourier transform, the impulse response is
transformed to the spectral representation of
the network in the frequency domain. The other
way round, the data measured in the frequency
domain by the network analyzer can be transformed to the time domain using Inverse Fourier Transform.
∞
impedance Zo. Positive values stand for higher
impedances than the reference impedance
(Z > Zo) and negative values for lower impedances (Z < Zo). In general, the relationship
between the measured reflection coefficient S11
and the impedance Z is as follows:
IMPULSE AND STEP RESPONSES
As already mentioned in the previous section
the impulse response and step response offer
different advantages for the representation of
measurement results in the time domain.
Mathematically, these two forms of representation are equivalent. They can be converted
into each other by differentiation or integration.
Historically, the step response offers technical
advantages since it is easier to generate a
single steep edge, i.e. a voltage step than two
voltage steps one after the other within a very
short time interval, i.e. a voltage impulse. This
advantage, however, is superseded by the
possibility to measure the transfer function of
the DUT first in the frequency domain and to
transform it then to the time domain in quasireal-time. The effort required for the numerical
calculation of the two forms of representation is
about the same if a sufficiently fast processor is
used.
It is recommend to use the step response if the
impedance characteristics of the DUT are of
interest. The impulse response, however,
should be made use of in the most of other
cases, especially for the determination of
discontinuities. A further advantage of the
impulse response is that in contrast to the step
response its magnitude can always be sensefully interpreted even if bandpass mode is used.
This will be further dealt with in section 3.4
Bandpass and Lowpass Mode.
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19 May 1998
The following diagrams (Figs 8 to 11) are examples of the impulse and step responses of
typical DUTs (starting with an open, via different
resistive loads till a short circuit). In each
diagram, the impulse response is displayed in
the upper part and the step response in the
lower part with the same scaling.
The lowpass mode (see section 3.4) was used
in all figures. For a correct representation of the
step response in the lowpass mode the DCvalue of the displayed S-parameter is of
importance. This value can be entered using the
following sequence: [MODE] TIME DOMAIN:
DEF TRANSF TYPE: LOWPASS DC
S-PARAM.
Fig. 8
Impulse and step response of an open
(LOWPASS DC S-PARAM = 1)
Fig. 10 Impulse and step response of a
25 Ω resistor
(LOWPASS DC S-PARAM = -1/3)
Fig. 9
Impulse and step response of a
75 Ω resistor
(LOWPASS DC S-PARAM = 0.2)
Fig. 11 Impulse and step response of a short
(LOWPASS DC S-PARAM = -1)
1EZ44_0E.DOC
5
19 May 1998
The impulse and step response characteristics
of complex impedances are highly interesting.
The measurement results shown in the following diagrams (Figs 12 to 17) were obtained with
single inductors and capacitors grounded at one
end (Figs 12 and 13), and with the series (Figs
14 and 15) or parallel connection (Figs 16 and
17) of an inductor or a capacitor respectively.
with a 50 Ω resistor.
For a qualitative interpretation of the measured responses it is useful to first illustrate the
step response: at the very beginning of a step
only the high-frequency behavior of the DUT
affects the step response due to the steepness
of the step. Thus at the first moments of a step
stimulus a capacitor reacts similar as a through
connection, whereas an inductor first appears
as an interruption.
Fig. 12 Impulse and step response of a 150 nH
inductor
(LOWPASS DC S-PARAM = -0.97)
Fig. 14 Series L
Impulse and step response of series
connection of a 150 nH inductor and
a 50 Ω resistor
(LOWPASS DC S-PARAM = 0)
Fig. 13
Fig. 15 Series C
Impulse and step response of series
connection of a 15 pF capacitor and
a 50 Ω resistor
(LOWPASS DC S-PARAM = 1)
Impulse and step response of a
15 pF capacitor
(LOWPASS DC S-PARAM = 1)
1EZ44_0E.DOC
6
19 May 1998
The signal components of the step response
occurring later in time correspond to lower and
lower frequency components down to DC. Thus
only the low-frequency behavior of the DUT has
an effect on the later part of the step response.
A capacitor now reacts like an interruption
whereas an inductor for low frequencies is
similar to a direct a through connection.
A typical example is the previous measurement
(Fig. 17 Parallel C: lower diagram). The characteristic of the measured step response can be
explained as follows: first the capacitor acts as
a short and is responsible for the negative edge
of the step response. (Negative signal components of the step response signify low-impedances.) With time, the capacitor charges up
and more and more acts as an open. The parallel connection consisting of capacitor and resistor is then equivalent to a single resistor. Since
the value of the resistor in the example is equal
to the reference impedance (R = Zo), no reflection will occur due to the matching. Thus the
step response again attains zero.
The characteristic of the impulse response
(Fig. 17: upper diagram) can be imagined by a
differentiation of the step response.
To correctly interpret the measurement results
displayed in the time domain by the network
analyzer note that due to
• finite span and
• frequency-discrete measurement
Fig. 16 Parallel L
Impulse and step response of parallel
connection of a 150 nH inductor and a
50 Ω resistor
(LOWPASS DC S-PARAM = -1)
of the network analyzer neither ideal (Dirac)
impulses nor ideal (rectangular) steps can be
represented.
3.2
FINITE PULSE WIDTH
The limited span of the network analyzer (e.g.
ZVR with fMAX = 4 GHz) widens the pulses in
the time domain. Mathematically, this behavior
can be explained as follows: first, an infinitely
wide frequency range is assumed. Now in fact,
via Fourier transform, infinitely narrow Dirac
pulses can be obtained. Because of the actual
finite frequency span, the frequency domain
data are multiplied by a rectangular weighting
function which takes the value 1 for the actual
frequency range (e.g. 9 kHz to 4 GHz) of the
network analyzer and which is otherwise zero.
This multiplication in the frequency domain
corresponds to a convolution of ideal Dirac
pulses with an si function in the time domain.
Fig. 17 Parallel C
Impulse and step response of parallel
connection of a 15 pF capacitor and a
50 Ω resistor
(LOWPASS DC S-PARAM = 0)
1EZ44_0E.DOC
si( x)
7
sin( x)
x
(4)
19 May 1998
3.3
The width ∆T of the si impulses is inversely
proportional to the span ∆F of the frequency
range:
∆T
2
∆F
(5)
For a span of, say, 4 GHz the pulse is widened
to approx. 500 ps as can be observed in the
following diagram (Fig. 18).
ALIASING
The data in frequency domain are not measured
continuously versus frequency but only at a
finite number of discrete frequency points.
This causes the time domain data after transformation to be repetitively replicated. This
phenomenon is called aliasing and can be
explained as follows:
The frequency discrete measurements can
assumed to be derived from an ideally continuous spectrum, which is multiplied by a comb
spectrum in the frequency domain. In the time
domain this corresponds to a convolution of the
time response with a periodic Dirac impulse
sequence. This results in the aliasing effect of a
frequently repetition of the original time response. The representation in the time domain
will thus become ambiguous. The time interval
∆t between the repetitions in the time domain is
called the ambiguity range. It can be calculated from the frequency step width ∆f in the
frequency domain as ∆t =1/∆f. This relationship
is illustrated in the following diagram (Fig. 19).
t=1/f
10000
Fig. 18 Widened si impulse due to finite span
(e.g. ∆F = 4 GHz).
The pulse width ∆T is approx. 500 ps.
Besides a widened pulse, Fig. 18 shows another characteristic of si pulses which are perceived as interference in practice: i.e. the occurrence of ringing (side lobes) to the left and right
of the (main) pulse. According to the si function
the highest (negative) side amplitudes to the left
and right of the main pulse are as follows:
π
sin 3.
2
3.
π
= 0.212
1000
100
t/ns
10
1
0,1
(6)
0,1
10
100
1000
10000
f/MHz
2
This corresponds to a side lobe suppression of
only 13.46 dB. The side lobes can be reduced
by suitable weighting methods in the frequency domain that are also called profiling
or windowing. For more details see section 4.
1
Fig. 19 Relationship between
frequency step width ∆f and ambiguity
range ∆t, or between
span ∆F and width ∆T of pulses
1EZ44_0E.DOC
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19 May 1998
For example, with the full frequency range of
ZVR of 4 GHz and 400 measurement points
used, the frequency step width is approximately
10 MHz. An aliasing signal such as illustrated in
Fig. 20 is thus obtained every ∆t = 100 ns.
By the way, the previous diagram of Fig. 19 can
also be used to convert for the relationship
between ∆F (SPAN) described above (Eq. 5)
and the resulting widening ∆T of the pulses in
the time domain.
Fig. 20
Example of aliasing. The ambiguity
range is ∆t = 100 ns
In Fig. 20, time domain signals can only be
clearly assigned within the ambiguity range of
∆t = 100 ns.
If a wider ambiguity range ∆t is required, the
frequency points have to be arranged in a more
dense frequency grid ∆f. For that, either the
SPAN ∆F can be reduced by which however the
time domain resolution is affected, or the
NUMBER OF POINTS can be increased which
reduces the measurement speed.
It should be noted that the analyzer is limited to
a maximum number of points of 2004 for all
four display channels together. The number of
2001 points is the limit per channel as it is for
common measurements in the frequency
domain. With the Fast Fourier Transform [1]
used instead of the more comfortable Chirp
transform [2], the number of points per channel
is reduced to the next lower power-of-two
number i.e. to a maximum value of 1024 points.
1EZ44_0E.DOC
For the lowpass mode described in the
following, the maximum possible number of
points is halved (for the negative frequencies)
and reduced by one (DC value).
3.4
BANDPASS AND LOWPASS MODE
Besides the default setting bandpass mode
(BANDPASS) the network analyzer also offers
the lowpass mode (LOWPASS). The bandpass
mode is generally recommended for scalar
applications. It allows an arbitrary number of
points and an arbitrary frequency range and it is
suitable to display the magnitude of the impulse
response. It does however not provide any
information about the measured values at zero
frequency, i.e. DC value, and the spectrum is
limited to positive frequencies only. As a
consequence, the impulse and step
responses are therefore complex and their
phases thus depend upon the distance between
the DUT and the reference plane. This has to
be considered when the real or imaginary part is
displayed in the time domain. It is therefore
generally not recommended to use the
bandpass mode when the sign of the measured
reflection coefficient is of interest. The lowpass
mode is required for this purpose.
In the lowpass mode (LOWPASS), the frequency grid is arranged such that an exact
extrapolation to zero frequency is possible. This
is based on the condition that the step width ∆f
between the frequency points in the frequency
domain is equal to the START frequency fSTART,
in other words:
fSTART = ∆f
(7)
A frequency grid meeting this requirement (7) is
called a harmonic grid since the frequency
value at each frequency point is an integer multiple of the START frequency. (Often, the
following more general definition is used:
START frequency fSTART should be an integer
multiple n of the step width ∆f, i.e. fSTART = n·∆f.
For the network analyzers of the ZVR family the
more stringent definition with n=1 is valid.)
The network analyzer is able to create a harmonic grid automatically by using the function
SET FREQS LOWPASS. For further informations about this topic please refer to the operating manual.
9
19 May 1998
Fig. 21 Lowpass grid
After generation of the required harmonic grid,
the network analyzer is able
1. to add an additional frequency point at zero
frequency (DC value) to the frequency grid
and
2. to mirror the data measured at positive
frequencies around zero frequency to the
negative frequencies in a conjugate complex
way.
In the lowpass mode the width of the frequency
domain is thus doubled (see Fig. 21) and in
contrast to the bandpass mode the resolution in
the time domain is improved by the factor of two
for the step response as well as for the impulse
response. Furthermore a real time response
(imaginary part = 0) is obtained. Compared to
Fig. 18 the pulse in Fig. 22 is clearly narrower.
The two diagrams have an identical scaling.
The figure shows the width of the si pulse
reduced by the factor of two which is now only
250 ps instead of 500 ps.
Fig. 22 Width of si pulse (∆F = 4 GHz) halved
in lowpass mode in comparison with
Fig. 18. The pulse width ∆T is now
approx. 250 ps.
1EZ44_0E.DOC
Besides the reduction of the pulse width the
pulse in Fig. 22 has a negative amplitude in
contrast to Fig. 18. Since the DUT was in both
cases a shorted line (S11 = -1), the measured
negative amplitude of Fig. 22 fully complies with
the expectations whereas Fig. 18 needs to be
afterwards explained: The reason for the
apparently incorrect amplitude of Fig. 18 has
already been given above. It is due to the
relationship between the bandpass mode and
its complex time response. For that the phase
of the time response becomes delaydependent, which corresponds to alternating
amplitudes of the real and imaginary parts. The
lowpass mode on the other hand provides a
real time response (imaginary part = 0) and
thus always the correct sign and amplitude of
the reflection coefficient of the shorted line end.
4
WINDOWS
As already described in section 3.2 the pulses in
the time domain are widened as a result of the
limited frequency range and ringing occurs (side
lobes). Especially the latter is disadvantageous
for time domain measurements since fraudulent
echos may be originated and the resolution as
well as measurement accuracy impaired.
Suitable windowing (also called: profiling) the
measured frequency domain data is considered
to be a remedy. Windowing is essentially an
attenuation of spectral components in the vicinity of the START and STOP frequencies. For
that, the analyzer offers different windows [3]
that are listed in the following table (TABLE 1).
Frequency domain
Window
NO
PROFILING
LOW FIRST
SIDELOBE
NORMAL
PROFILE
STEEP
FALLOFF
ARBITRARY
SIDELOBES
Time domain
Gate
STEEPEST
EDGES
STEEP
EDGES
NORMAL
GATE
MAXIMUM
FLATNESS
ARBITRARY
GATE SHAPE
Filter function
Rectangle
Hamming
Hann
Bohman
DolphChebichev
TABLE 1 Filter functions for available
windows and gates
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19 May 1998
Depending on the selected window (profile) the
shape of the time domain impulses (and time
domain steps) are influenced differently. This is
illustrated in the following figures. In the quad
channel quad split display below, Fig. 23 shows
the measured reflection of an OPEN standard
directly connected to PORT 1 of the network
analyzer. NO PROFILING was used in channel 1 (top left). In channel 2 (bottom left) a
profile (window) with a LOW FIRST SIDELOBE
was selected. The NORMAL PROFILE recommended for general applications was used in
channel 3 (top right), and the STEEP FALLOFF
(of side lobes) window in channel 4 (bottom
right).
Fig. 23
Quad channel quad split display of
impulses with different windows
1. NO PROFILING (see Fig. 23 top left or - if
colored printout is available - red trace in Fig.
24) indicates that no profiling (windowing)
was used. The rectangular frequency range
limitation is maintained. The impulses
remain si-shaped and have a relatively narrow main lobe but substantial side lobes.
The highest side lobe is approx. 13 dB below
the main lobe.
2. LOW FIRST SIDELOBE (see Fig. 23 bottom
left, or green trace in Fig. 24) uses the filter
function according to R. W. Hamming for
windowing in the frequency domain. This
window yields far smaller side lobes (approx.
43 dB) than those of the rectangular window
but the side lobes are hardly more
attenuated even far off the main lobe. The
first side lobe however is very strongly reduced. In the time domain this is indicated by
a widened main lobe and clearly reduced
ringing which however does not fully
disappear even far from the main pulse.
3. NORMAL PROFILE (see Fig. 23 top right, or
black trace in Fig. 24) makes use of the filter
function according to Julius von Hann. In the
frequency domain it is characterized by a
first side lobe of approx. 32 dB, i.e. slightly
higher than the Hamming filter. The other
side lobes however are attenuated more
strongly. This results in a visible first side
lobe in the time domain even with linear
scaling, but all other side lobes further away
are neglectable for most applications.
4. STEEP FALLOFF (see Fig. 23 bottom right,
or blue trace in Fig. 24) makes use of a Bohman filter which is characterized by the
steepest falloff of side lobe amplitudes of all
filters used. Thus, time domain pulses with
practically no side lobes are obtained. The
drawback is however that the pulse width is
doubled compared to the rectangular
window.
Fig. 24
Quad channel overlay display of
impulses with different windows
5. Especially high flexibility is offered by the
ARBITRARY SIDELOBES window. A filter
function according to Dolph-Chebichev is
used which is characterized by constant side
lobe amplitudes both in the frequency and
time domain. The user may select an arbitrary side lobe suppression between 10 dB
and 120 dB. As for the other windows the
same also applies here: high side lobe
suppression is combined with wide main
lobe and vice versa.
For examples see Annex.
1EZ44_0E.DOC
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19 May 1998
The effects of different windows on the shape of
the time domain impulses as illustrated in Figs
23 and 24 can also be observed on the display
of the network analyzer itself. To do this, for
example the reflection (S11) of PORT 1 of the
analyzer is measured in the time domain after
connecting an OPEN standard to PORT 1. The
key sequence
[MODE] TIME DOMAIN: DEF TRANSF TYPE
calls up the menu for defining the time domain
transform. By selecting one of the five possible
windows (from NO PROFILING to ARBITRARY
SIDELOBES) the different effects on the
measured impulse or step responses can
directly be observed on the display of the
analyzer. Moreover, the effect of different side
lobe suppressions can also be examined.
The Hann filter (NORMAL PROFILE) is
recommended for most of the general
applications. It is a suitable compromise
between high side lobe attenuation and
relatively small main lobe widening.
5
GATES
The TIME GATE is one of the most powerful
tools of the option Time Domain ZVR-B2. It
allows to filter out components, impedance
discontinuities or line faults that are spatially
separated, in the time domain. Due to the different distances to the reference plane of the
network analyzer the associated reflections
arrive at the test port at different times and can
thus be measured separately from each other in
the time domain. For transmission measurements direct transmissions can be distinguished in the time domain from indirect
transmissions (multipath), i.e. multireflected
transmissions (triple transit) or signal components with different propagation speeds. These
capabilities have already been mentioned in
section 2 Typical Application. The real benefit of
the gate function is the transformation of the
impulse response of the DUT filtered in the time
domain back to the frequency domain. The
interesting part of the DUT can thus be
displayed without unwanted discontinuities (see
Figs 2 to 6).
It is important to quickly perform all required
calculations so that measurement results can
be displayed in virtually real time even after two
Fourier transforms including time domain gating
(see also section 6.2).
The relationships between frequency domain
and time domain described in section 4 on
windowing also apply conversely between the
1EZ44_0E.DOC
time domain and frequency domain. During
filtering in the time domain using gates it is
therefore important to observe the shape of the
gates besides their position and span. A
rectangular gate in the time domain does not
yield optimum results in the frequency domain
as might have been expected. When the impulse response filtered in the time domain by a
rectangular gate is transformed back, a frequency domain response is obtained which may
quickly change versus frequency but whose
form is falsified. This behavior is analogous to
the si impulse in the time domain created by the
rectangular window in the frequency domain. It
has a relatively narrow main lobe with a fast
cutoff rate (rapid changes can be seen) but
significant side lobes (causing ripple in the
frequency domain).
1. The gate with the STEEPEST EDGES is
therefore recommended for use only if discontinuities that are very close to each other
have to be separated. The drawback here is
a falsified frequency domain response resulting from the high side lobe amplitudes (max.
13 dB) of a rectangular filter spectrum which
causes ripple on the trace.
2. STEEP EDGES is a filter (according to
Hamming) whose side lobe amplitudes are
far smaller (43 dB) but that hardly fall off
further with increasing order.
3. NORMAL GATE makes use of a filter according to Hann. It is a good compromise
between relatively low first side lobe (32 dB),
steep falloff of further side lobes, not too
wide main lobe and relatively flat passband
of main lobe. It is therefore recommended
for general applications.
4. MAXIMUM FLATNESS is a Bohman filter
with the flattest main lobe of all filters used
(nearly no passband ripple) but is also the
widest main lobe. The side lobes are very
low. It is suitable for use if the responses to
be separated are very far from each other.
5. ARBITRARY GATE SHAPE is a DolphChebichev filter giving the same amplitude
for all side lobes. The desired side lobe
suppression can be chosen by the user
between 10 dB and 120 dB below the main
lobe. It allows individual optimization of the
gate shape for especially critical measurement tasks.
The five types of time gates have already been
given in TABLE 1 together with the possible
windows in the frequency domain. For further
information about gate shapes and their spectra
see diagrams in the Annex.
12
19 May 1998
6
MEASUREMENT EXAMPLES
6.1
STEPPED AIRLINE
A stepped airline comprising a section with a
characteristic impedance of 25 Ω (thickened
center conductor) is an especially instructive
example that is often used to verify the
measurement accuracy of network analyzers.
Fig. 25 shows schematically a stepped airline,
which is then terminated at one end with a 50 Ω
matched load.
Fig. 25 Stepped airline
Impedance steps from first 50 Ω to 25 Ω and
then from 25 Ω to 50 Ω cause reflections that
can easily be seen in the time domain (Fig. 26).
The magnitude of the reflection coefficient at
the line discontinuities is 33.3% as calculated
via equation (3). This corresponds to a return
loss of 9.54 dB.
Fig. 26 Impulse response of a stepped airline
1EZ44_0E.DOC
Alternatively to the impulse response, the step
response can be displayed (Fig. 27). It gives a
clear representation of the DUT’s impedance
characteristic. The measured zero of the step
response can be seen in the left of the following
diagram. It corresponds to the characteristic
impedance of 50 Ω. The trace then falls to the
value -0.333 (MARKER ∇3) which exactly
corresponds to the reflection coefficient expected for 25 Ω. Then, the trace slightly increases
and 500 ps later (this interval corresponds to
the length of the 25 Ω section of 75 mm) goes
back again. A closer look at Fig. 27 reveals that
the step response does not exactly return to the
original value (zero) but only attains -0.04. This
is partly due to the loss of energy (approx. 11%)
of the transmitted step signal which is caused
by the first line discontinuity. A further loss of
energy is caused by the attenuation of the air
line. Because of these two energy losses the
amplitude of the step signal at the second
impedance discontinuity (at the end of the 25 Ω
section) is smaller than at the beginning and the
positive step in the amplitude is thus lower than
the negative step at the beginning.
Fig. 27
13
Step response of stepped airline
19 May 1998
The line attenuation can even be seen within
the 25 Ω section of the airline. The central
section of the trace of Fig. 27 has been
enlarged. This section corresponds to the 25 Ω
section of the airline and is shown in Fig. 28
with a higher resolution.
the transients of the generator and receiver of
the analyzer will be set.
Moreover, Fast Fourier Transform is to be
selected as transformation method ([MODE]
TIME DOMAIN: DEF TRANSF TYPE: FFT)
since it is faster than the Chirp transform. However, with FFT all measurement points in the
time domain are always evenly distributed over
the ambiguity range ∆t (= reciprocal step width
∆f in the frequency domain). This is why the
number of measurement points is reduced
when a time span narrower than the ambiguity
range is displayed.
On the other hand, the Chirp transform allows to
set any arbitrary time span and always displays
the full number of points irrespective of the
ambiguity range. It is therefore more convenient
to use but not as fast as FFT.
Fig. 28
Amplitude falloff of the step response
due to attenuation of the airline
The Dolph-Chebichev filter (ARBITRARY SIDELOBES) with a side lobe suppression of 70 dB
was selected as window in the frequency
domain. A smooth trace is thus obtained that
corresponds to the constant attenuation of the
airline. With an unsuitable window selected, an
interfering ripple of the trace would be
produced.
6.2
FAST MEASUREMENTS
For many practical applications it is extremely
important that the Fourier transform of the data
measured in the frequency domain to the time
domain, and the subsequent gating in the time
domain, as well as the retransformation back to
the frequency domain is performed fast enough
for the DUT to be aligned in real time. The network analyzers of the ZVR family offer various
ways for speed optimization.
It has to be noted that transformed data cannot,
of course, be displayed more quickly than the
data acquisition in the frequency domain is performed. Therefore, the IF BANDWIDTH has to
be sufficiently high for the SWEEP TIME to be
as small as desired. If required, the NUMBER
OF POINTS has to be reduced to, say 100
points. To meet highest speed demands, FAST
MODE in [MODE] menu should be switched on.
The IF bandwidth will then automatically be
switched to the maximum possible value of
26 kHz (FULL) and in addition reduced times for
1EZ44_0E.DOC
Olaf Ostwald, 1ES3
Rohde & Schwarz
19 May 1998
14
19 May 1998
7
REFERENCES
7.1
ON TIME
[1]
[2]
[3]
7.2
[4]
[5]
DOMAIN TRANSFORM
H. Marko: Methoden der Systemtheorie, SpringerVerlag, 1982.
L. R. Rabiner et al.: The Chirp z-transform Algorithm,
IEEE Trans. on Audio and Electroacoustics, vol. AU17, no.2, June 1969, pp. 86-92.
F. J. Harris: On the Use of Windows for Harmonic
Analysis with Discrete Fourier Transform, Proc. of
the IEEE, vol. 66, no.1, Jan. 1978, pp. 51-83.
ON VECTOR NETWORK ANALYZER
FAMILY ZVR
O. Ostwald: 3-Port Measurements with Vector
Network Analyzer ZVR, Appl. Note 1EZ26_1E.
H.-G. Krekels: Automatic Calibration of Vector
Network Analyzer ZVR, Appl. Note 1EZ30_1E.
[6]
O. Ostwald: 4-Port Measurements with Vector
Network Analyzer ZVR, Appl. Note 1EZ25_1E.
[7]
T. Bednorz: Measurement Uncertainties for
Vector Network Analysis, Appl. Note
1EZ29_1E.
[8]
P. Kraus: Measurements on FrequencyConverting DUTs using Vector Network
Analyzer ZVR, Appl. Note 1EZ32_1E.
[9]
J. Ganzert: Accessing Measurement Data and
Controlling the Vector Network Analyzer via
DDE, Appl. Note 1EZ33_1E.
[10] J. Ganzert: File Transfer between Analyzers
FSE or ZVR and PC using MS-DOS Interlink,
Appl. Note 1EZ34_1E.
[11] O. Ostwald: Group and Phase Delay Measurements with Vector Network Analyzer ZVR,
Appl. Note 1EZ35_1E.
[12] O. Ostwald: Multiport Measurements using
Vector Network Analyzer, Appl. Note
1EZ37_1E.
[13] O. Ostwald: Frequently Asked Questions about
Vector Network Analyzer ZVR, Appl. Note
1EZ38_3E.
8
ORDERING INFORMATION
Order designation
Type
Frequency
range
Order No.
Vector Network Analyzers (test sets included) *
3-channel, unidirectional,
50 Ω, passive
3-channel, bidirectional,
50 Ω, passive
3-channel, bidirectional,
50 Ω, active
4-channel, bidirectional,
50 Ω, passive
4-channel, bidirectional,
50 Ω, active
3-channel, bidirectional,
50 Ω, active
4-channel, bidirectional,
50 Ω, active
ZVRL
9 kHz to 4 GHz
1043.0009.41
ZVRE
9 kHz to 4 GHz
1043.0009.51
ZVRE
300 kHz to 4 GHz
1043.0009.52
ZVR
9 kHz to 4 GHz
1043.0009.61
ZVR
300 kHz to 4 GHz
1043.0009.62
ZVCE
20 kHz to 8 GHz
1106.9020.50
ZVC
20 kHz to 8 GHz
1106.9020.60
Alternative Test Sets *
75 Ω SWR Bridge for ZVRL (instead of 50 Ω) 1)
75 Ω, passive
ZVR-A71
9 kHz to 4 GHz
1043.7690.18
75 Ω SWR Bridge Pairs for ZVRE and ZVR (instead of 50 Ω) 1)
75 Ω, passive
75 Ω, active
ZVR-A75
ZVR-A76
9 kHz to 4 GHz
300 kHz to 4 GHz
1043.7755.28
1043.7755.29
AutoKal
Time Domain
Mixer Measurements 2)
Reference Channel Ports
Power Calibration 3)
3-Port Adapter
Virtual Embedding
Networks 4)
4-Port Adapter (2xSPDT)
4-Port Adapter (SP3T)
ZVR-B1
ZVR-B2
ZVR-B4
ZVR-B6
ZVR-B7
ZVR-B8
ZVR-K9
0 to 8 GHz
same as analyzer
same as analyzer
same as analyzer
same as analyzer
0 to 4 GHz
same as analyzer
1044.0625.02
1044.1009.02
1044.1215.02
1044.1415.02
1044.1544.02
1086.0000.02
1106.8830.02
ZVR-B14
ZVR-B14
0 to 4 GHz
0 to 4 GHz
1106.7510.02
1106.7510.03
Controller (German) 5)
Controller (English) 5)
Ethernet BNC for ZVR-B15
Ethernet AUI for ZVR-B15
IEC/IEEE-Bus Interface for
ZVR-B15
ZVR-B15
ZVR-B15
FSE-B16
FSE-B16
FSE-B17
-
1044.0290.02
1044.0290.03
1073.5973.02
1073.5973.03
1066.4017.02
Generator Step Attenuator
PORT 1
Generator Step Attenuator
PORT 2 6)
Receiver Step Attenuator
PORT 1
Receiver Step Attenuator
PORT 2
External Measurements,
7)
50 Ω
ZVR-B21
same as analyzer
1044.0025.11
ZVR-B22
same as analyzer
1044.0025.21
ZVR-B23
same as analyzer
1044.0025.12
ZVR-B24
same as analyzer
1044.0025.22
ZVR-B25
10 Hz to 4 GHz
(ZVR/E/L)
20 kHz to 8 GHz
(ZVC/E)
1044.0460.02
Options
1)
[14] A. Gleißner: Internal Data Transfer between
Windows 3.1 / Excel and Vector Network
Analyzer ZVR, Appl. Note 1EZ39_1E.
[15] A. Gleißner: Power Calibration of Vector
Network Analyzer ZVR, Appl. Note 1EZ41_2E
[16] O. Ostwald: Pulsed Measurements on GSM
Amplifier SMD ICs with Vector Analyzer ZVR,
Appl. Note 1EZ42_1E.
1EZ44_0E.DOC
To be ordered together with the analyzer.
Harmonics measurements included.
Power meter and sensor required.
4)
Only for ZVR or ZVC with ZVR-B15.
5)
DOS, Windows 3.11, keyboard and mouse included.
6)
For ZVR or ZVC only.
7)
Step attenuators required.
2)
3)
* Note:
Active test sets, in contrast to passive test sets, comprise internal bias
networks, e.g. to supply DUTs.
15
19 May 1998
9
ANNEX
Frequency
domain
Window
NO
PROFILING
LOW FIRST
SIDELOBE
NORMAL
PROFILE
STEEP
FALLOFF
ARBITRARY
SIDELOBES
9.1
Time domain
Gate
Filter function
STEEPEST
EDGES
STEEP
EDGES
NORMAL
GATE
MAXIMUM
FLATNESS
ARBITRARY
GATE SHAPE
Rectangle
Hamming
Hann
Bohman
DolphChebichev
TIME DOMAIN REPRESENTATIONS
Some examples of impluse responses are
given. They show the reflection of an OPEN
standard directly connected to PORT 1 of the
network analyzer. Different windows are used:
For the filter function according to Dolph-Chebichev any side lobe suppressions between 10 dB
and 120 dB can be set. Values 20 dB, 40 dB,
80 dB and 120 dB are given as an example in
the Annex.
9.2
SPECTRA OF FILTER FUNCTIONS
The spectra of the available filter functions are
shown. Two different scalings are used.
(Please ignore irrelevant abscissa labeling).
The diagrams in the left column show the
spectra in the vicinity of the main lobe. The right
column shows the characteristic of the spectra
far from the main lobe.
1EZ44_0E.DOC
16
19 May 1998
Time Domain Representations
Rectangular filter
Dolph-Chebichev filter (20 dB)
Hamming filter
Dolph- Chebichev filter (40 dB)
Hann filter
Dolph- Chebichev filter (80 dB)
Bohman filter
Dolph- Chebichev filter (120 dB)
Spectra of Filter Functions
Spectra in the vicinity of main lobe:
Spectra far away from the main lobe:
Rectangular filter
Rectangular filter
Hamming filter
Hamming filter
Hann filter
Hann filter
Bohman filter
Bohman filter
Spectra of Filter Functions:
Dolph-Chebichev filters with different side lobe suppressions
Spectra in the vicinity of main lobe:
Spectra far away from the main lobe:
Dolph-Chebichev (20 dB side lobe suppression)
Dolph-Chebichev (20 dB side lobe suppression)
Dolph-Chebichev (40 dB side lobe suppression)
Dolph-Chebichev (40 dB side lobe suppression)
Dolph-Chebichev (80 dB side lobe suppression)
Dolph-Chebichev (80 dB side lobe suppression)
Dolph-Chebichev (120 dB side lobe suppression)
Dolph-Chebichev (120 dB side lobe suppression)
Relation between step width and ambiguity range or between frequency span and pulse width.
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