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Texas Instruments Using Windowing With SNRBoost 3G Technology Application notes
Application Report
SLAA445 – August 2010
Using Windowing With SNRBoost3G Technology
Sourabh Gupta and Vinod Paliakara
.................................................................... High-Speed Converters
ABSTRACT
Coherency is a well-known requirement when using FFT techniques to examine the spectrum of the
output of an analog-to-digital converter (ADC). SNRBoost3G technology results in loss in coherency, and
this can be seen as an unstable noise floor in the spectrum. Windowing of the ADC output is a well-known
solution to restore coherency and stable spectrum.
However, windowing also modifies the amplitude of the fundamental signal and the SNR values. This
masks the real improvement in SNR due to the SNRBoost3G technology.
This application report explains the scaling caused by different window functions that can be used to
recover the actual SNR of the original signal (prior to windowing).
Contents
Introduction ..................................................................................................................
SNRBoost3G Technology Causes Loss in Coherency ..................................................................
Using Windowing ............................................................................................................
3.1
Example .............................................................................................................
Appendix A
Theory ...............................................................................................................
1
2
3
1
2
3
4
5
List of Figures
3G
On................................................................................ 2
1
Spectrum of Signal With SNRBoost
2
Full-Band SNR Comparison With SNRBoost3G Disabled and Enabled .............................................. 2
3
Time Domain Data With SNRBoost3G Disabled and Enabled (With No Input Signal).............................. 3
4
Unstable Spectrum With SNRBoost3G Enabled ......................................................................... 3
5
Stable Spectrum With SNRBoost3G Enabled and After Windowing .................................................. 4
6
Spectrum of ADC Output Sequence
7
Spectrum of Window Function ............................................................................................ 6
8
Spectrum of Signal After Windowing ..................................................................................... 6
.....................................................................................
5
List of Tables
1
2
1
........................................................................................................
Window Functions ..........................................................................................................
Effect of Windowing
4
7
Introduction
FFT techniques are commonly used to examine the spectrum of the output of an analog-to-digital
converter (ADC). One of the requirements when using FFT is to ensure coherency between the analog
input signal and the sampling clock of the ADC.
When using an ADC with SNRBoost3G technology, loss in coherency is observed. This can be seen as an
unstable noise floor in the spectrum. Windowing of the ADC output is a well-known solution to restore
coherency and stable spectrum. However, windowing also modifies the amplitude of the fundamental
signal and the SNR values. This masks the real improvement in SNR due to SNRBoost3G technology.
This application note explains the scaling caused by different window functions that can be used to
recover the actual SNR of the original signal (prior to windowing).
SLAA445 – August 2010
Using Windowing With SNRBoost3G Technology
Copyright © 2010, Texas Instruments Incorporated
1
SNRBoost3G Technology Causes Loss in Coherency
www.ti.com
SNRBoost3G Technology Causes Loss in Coherency
2
SNRBoost3G technology works by improving the noise within a select band of frequencies. At the same
time, the noise degrades outside the selected frequency band (Figure 1).
0
fIN = 40 MHz
fIN = 185 MHz
Fund Ampl = −2 dBFS
IN-BAND SNR = 72.4 dBFS
(16.25 MHz to 76.25 MHz)
−10
−20
−30
Amplitude (dB)
−40
−50
−60
−70
−80
−90
−100
−110
−120
0
10
20
30
40
50
60
70
80
90
Frequency (MHz)
Figure 1. Spectrum of Signal With SNRBoost3G On
Note that although the noise in the selected band improves with SNRBoost3G enabled, the noise over the
entire band degrades (compared to SNRBoost3G disabled). For example, the full-band SNR is 66.91 dBFS
with SNRBoost3G disabled and 32.73 dBFS with SNRBoost3G enabled (Figure 2).
0
0
fIN = 40 MHz
fIN = 185 MHz
Fund Ampl = −2 dBFS
FULL-BAND SNR = 66.9 dBFS
(0 to Nyquist)
−10
−20
−30
−20
−30
−40
Amplitude (dB)
Amplitude (dB)
−40
−50
−60
−70
−50
−60
−70
−80
−80
−90
−90
−100
−100
−110
−110
−120
fIN = 40 MHz
fIN = 185 MHz
Fund Ampl = −1.97 dBFS
FULL-BAND SNR = 32.7 dBFS
(0 to Nyquist)
−10
0
10
20
30
40
50
60
70
80
90
−120
0
10
Frequency (MHz)
20
30
40
50
60
70
80
90
Frequency (MHz)
Figure 2. Full-Band SNR Comparison With SNRBoost3G Disabled and Enabled
In the time domain, this SNR degradation shows up as an increase in the range of code variations. For
example, the output code range is 1 LSB with SNRBoost3G disabled and 113 LSBs with SNRBoost3G
enabled (Figure 3).
2
Using Windowing With SNRBoost3G Technology
Copyright © 2010, Texas Instruments Incorporated
SLAA445 – August 2010
Using Windowing
1140
1140
1120
1120
1100
1100
1080
1080
Output Code (LSB)
Output Code (LSB)
www.ti.com
1060
1040
1020
1060
1040
1020
1000
1000
980
980
960
960
940
0
4000
940
8000 12000 16000 20000 24000 28000 32000
0
4000
8000 12000 16000 20000 24000 28000 32000
Sample Number
Sample Number
Figure 3. Time Domain Data With SNRBoost3G Disabled and Enabled (With No Input Signal)
The increased noise results in loss of coherency. The FFT spectrum of this data has a noise floor that is
not uniformly flat and changes from one capture to the next.
0
0
FFT for Run 1
FFT for Run 2
Fin = 40 MHz
Fs = 185 MHz
Fund Ampl = −2 dBFS
IN−BAND SNR = 72.4 dBFS
(16.25 MHz to 76.25 MHz)
−10
−20
−30
−20
−30
−40
Amplitude (dB)
Amplitude (dB)
−40
−50
−60
−70
−50
−60
−70
−80
−80
−90
−90
−100
−100
−110
−110
−120
Fin = 40 MHz
Fs = 185 MHz
Fund Ampl = −2 dBFS
IN−BAND SNR = 73.1 dBFS
(16.25 MHz to 76.25 MHz)
−10
0
10
20
30
40
50
60
70
80
90
−120
0
10
Frequency (MHz)
20
30
40
50
60
70
80
90
Frequency (MHz)
Figure 4. Unstable Spectrum With SNRBoost3G Enabled
3
Using Windowing
Windowing can be used to overcome the loss in coherency. Any of the common window functions
(Hanning, Hamming, Blackman-Harris, etc.) can be applied to the ADC output data. The resulting signal is
now coherent with a stable noise floor in the FFT spectrum.
SLAA445 – August 2010
Using Windowing With SNRBoost3G Technology
Copyright © 2010, Texas Instruments Incorporated
3
Using Windowing
www.ti.com
0
0
Fin = 40 MHz
Fs = 185 MHz
Fund Ampl = −7.37 dBFS
IN−BAND SNR = 78.1 dBFS
(16.25 MHz to 76.25 MHz)
FFT for Run1
−10
−20
−30
−20
−30
−40
Amplitude (dB)
Amplitude (dB)
−40
−50
−60
−70
−50
−60
−70
−80
−80
−90
−90
−100
−100
−110
−110
−120
Fin = 40 MHz
Fs = 185 MHz
Fund Ampl = −7.37 dBFS
IN−BAND SNR = 78.1 dBFS
(16.25 MHz to 76.25 MHz)
FFT for Run 2
−10
0
10
20
30
40
50
60
70
80
−120
90
0
10
20
Frequency (MHz)
30
40
50
60
70
80
90
Frequency (MHz)
Figure 5. Stable Spectrum With SNRBoost3G Enabled and After Windowing
The windowing has one side effect: the amplitude of all the frequency bins in the spectrum are reported
smaller than their actual values. For a narrow-band signal, the fundamental signal and the rest of the
spectrum are modified by different amounts as shown in Table 1.
Table 1. Effect of Windowing
Window Function
Delta SNR (dB)
Delta Signal Amplitude (dB)
Hamming
4.01
5.35
Hanning
4.26
6.02
Blackman Harris
5.88
8.90
Use Table 1 to determine the SNR and signal amplitude of the original signal without windowing as
follows:
SNR (pre-windowing) = SNR (post-windowing) – Delta SNR
Signal Amplitude (pre-windowing) = Signal Amplitude (post-windowing) + Delta Signal Amplitude where
SNR and Signal Amplitude are reported in dBFS values.
In this way, the correct SNR of the signal with SNRBoost3G enabled can be determined.
3.1
Example
With the preceding method, the improvement in SNR using SNRBoost3G technology in the ADS58C48 can
be determined.
Consider a case where the sample clock is 184.32 MSPS and the input signal bandwidth is 60 MHz
centered at 46 MHz. With SNRBoost3G disabled, the in-band SNR reported is 68.6 dBFS. After the
SNRBoost3G technology is turned on, a Hamming window is applied to the ADC output data. The spectrum
of the windowed signal reports an in-band SNR of 78.1 dBFS.
Using Table 1, find the in-band SNR (pre windowing) with SNRBoost3G ON:
= 78.1 dBFS – 4.01 dB
= 74.09 dBFS
This means that the SNRBoost3G technology resulted in an in-band SNR improvement:
= SNR (pre-windowing) with SNRBoost3G on – SNR with SNRBoost3G off
= 74.01 dBFS – 68.6 dBFS
= 5.49 dB within the band of 60 MHz.
4
Using Windowing With SNRBoost3G Technology
Copyright © 2010, Texas Instruments Incorporated
SLAA445 – August 2010
www.ti.com
Appendix A Theory
The aim of this section is to determine the change in spectral power after windowing. To find this, a single
tone input signal is used that is coherent and has known power. The input signal is then multiplied by the
window function. The difference in the power of the tone between the input signal and the windowed
signal gives the desired result.
Consider two N-point sequences V(n) and W(n), that represent the ADC output data and the window
function sequence, respectively.
æ 2p
ö
V(n) = A cos ç
k n+ q÷
è N
ø
(1)
æ 2p
ö
W(n) = m1+m2 cos ç
w n÷
N
1
è
ø
(2)
For simplicity of analysis, the ADC output sequence is assumed to be a single tone with all other tones
having zero power. Because the spectrum from N/2 to N–1 is a mirror image of the spectrum from 0 to
N/2–1, the spectrum is represented only in the range 0 to N/2–1, after adjusting the power of each tone.
V (n) = A cos(
2p
k n +q )
N
Total Signal Power represented in
0 to N/2
Amplitude
k
A
A/ 2
A/ 2
0
Amplitude
N/2
FFT bin number
N-k
N
0
k
N/2
FFT bin number
Figure 6. Spectrum of ADC Output Sequence
Similarly, the spectrum of the window function is shown in Figure 7.
SLAA445 – August 2010
Using Windowing With SNRBoost3G Technology
Copyright © 2010, Texas Instruments Incorporated
5
Appendix A
www.ti.com
2p
W (n) = m + m cos(
w n)
1
2
N -1
Amplitude m
1
m2
0
N/2
FFT bin number
Figure 7. Spectrum of Window Function
Windowing involves multiplication of the ADC output sequence by the window function.
So,
Vo (n) = V(n) W(n)
æ 2p
ö
æ 2p
ö
kn + q ÷ (m1 + m 2 cos ç
wn ÷
» A cos ç
è N
ø
è N
ø
m A
m A
æ 2p
ö
æ 2p
ö
æ 2p
ö
= m1 Acos ç
kn + q ÷ + 2 cos ç
(k - w)n + q ÷ + 2
cos ç
(k + w)n + q ÷
N
2
N
2
N
è
ø
è
ø
è
ø
(3)
The spectrum of the signal after windowing is shown in Figure 8.
vo (n) = W (n) v(n)
m1 A
Amplitude
0
m2 A
m2 A
2
2
k-w k k+w
N/2
FFT bin number
Figure 8. Spectrum of Signal After Windowing
Note that each tone with bin number 'k' in the input signal spreads into three tones.
• One main tone (at bin k) having amplitude m1A and
• Two side tones (at bins k ± w) having amplitudes m2A / 2.
The change in signal amplitude of the main tone is due to windowing = 20 log (m1)
Note that it is assumed that the input signal has zero amplitude noise components. With non-zero noise
components, the input signal can be represented as:
6
Using Windowing With SNRBoost3G Technology
Copyright © 2010, Texas Instruments Incorporated
SLAA445 – August 2010
Appendix A
www.ti.com
N-1
å
æ 2p
ö
V(n) = Acos ç
kn + q ÷ +
è N
ø
r=0
æ 2p
ö
Br cos ç
r n+ fr ÷
è N
ø
(4)
Similar to the signal tone, each noise component is also spread into three tones having
m B
amplitudes m1Br and 2 r
2
(5)
The total power of all noise components after windowing is
2
1
=m
N -1
å
r =0
Br 2 +
æ 2 m 2
= çm + 2
1
2
è
m22 N -1
4
å
r =0
Br 2 +
m 22 N -1
4
å
r =0
Br 2
ö N -1 2
÷ å Br
ø r=0
(6)
N-1
å
Because
Br2
r =0
represents the noise power of the ADC output sequence, Equation 6 can be re-written
as:
æ
m 2ö
Noise power after windowing = ç m12 + 2 ÷ ´ ADC output noise power .
ç
2 ÷
è
ø
(7)
æ
m 2
So, the change in the SNR after windowing = 10 logç m12 + 2
ç
2
è
ö
÷
÷
ø
(8)
Table 2 lists the changes in signal amplitude and SNR after windowing for some common window
functions.
Table 2. Window Functions
Window Function
m1
m2
w
Delta SNR (dB)
Delta Signal Amplitude (dB)
Hamming
0.54
–0.46
1
4.01
5.35
Hanning
0.5
–0.5
1
4.26
6.02
This is a para to keep from creating a blank page
SLAA445 – August 2010
Using Windowing With SNRBoost3G Technology
Copyright © 2010, Texas Instruments Incorporated
7
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