Fast Frequency and Response Measurements using FFTs

1. Fast Frequency and Response Measurements using FFTs

Alain Moriat,
Senior Architect
Fri. 12:45p
Pecan (9B)
www.natinst.com

2. Accurately Detect a Tone

What is the exact frequency and amplitude of
a tone embedded in a complex signal?
How fast can I perform these measurements?
How accurate are the results?
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3.

Presentation Overview
Why use the frequency domain?
FFT – a short introduction
Frequency interpolation
Improvements using windowing
Error evaluation
Amplitude/phase response measurements
Demos
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4. Clean Single Tone Measurement

2
20
d BV
Time signal
Vo lt
1
FFT Spectrum
0
0
-20
-1
-40
-2
0.0
-60
0.2
0.4
0.6
Clean sine tone
Easy to measure
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0.8 ms 1.0
kHz
0
5 10 15 20 25 30 35 40 45 50
Clean tone spectrum

5. Noisy Tone Measurement

2
20
d BV
Time signal
Vo lt
1
Our signal
FFT Spectrum
0
0
-20
-1
-40
-2
0.0
-60
0.2
0.4
0.6
0.8 ms 1.0
Noisy signal
Difficult to measure in
the time domain
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kHz
0
5 10 15 20 25 30 35 40 45 50
Noisy signal spectrum
Easier to measure

6. Fast Fourier Transform (FFT) Fundamentals (Ideal Case)

2
20
d BV
Time signal
Vo lt
1
0
0
-20
-1
-40
-2
0.0
FFT Spectrum
0.1
0.2
0.3
F sam p lin g = 100 kH z
T im e re s = 10 u s
0.4 ms 0.5
-60
0
5 10 15 20 25 30 35 40 45 50
kHz
Re c o rd size = 50 sam p le s
F re q . re s = 2 kH z
The tone frequency is an exact multiple of the
frequency resolution (“hits a bin”)
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7. FFT Fundamentals (Realistic Case)

2
20
d BV
Time signal
Vo lt
1
0
0
-20
-1
-40
-2
0.0
FFT Spectrum
0.1
0.2
0.3
F sam p lin g = 100 kH z
T im e re s = 10 u s
0.4 ms 0.5
-60
0
5 10 15 20 25 30 35 40 45 50
kHz
Re c o rd size = 50 sam p le s
F re q . re s = 2 kH z
The tone frequency is not a multiple of the
frequency resolution
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8. Input Frequency Hits Exactly a Bin

0
dB
Only one bin
is activated
-10
-20
-30
-40
-50
-60
-6
-5
-4
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-3
-2
-1
0
1
2
3
4
5 Bin6

9. Input Frequency is +0.01 Bin “off”

Input Frequency is +0.25 Bin “off”
0
dB
Real top
-10
Highest Bin
-20
Next Highest
Bin
-30
-40
-50
-60
-6
-5
-4
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-3
-2
-1
0
1
2
3
4
5 Bin6

10. Input Frequency is +0.25 Bin “off”

Input Frequency is +0.50 Bin “off”
0
dB
Highest
side-lobes
-10
-20
-30
-40
-50
-60
-6
-5
-4
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-3
-2
-1
0
1
2
3
4
5 Bin6

11. Input Frequency is +0.50 Bin “off”

Input Frequency is +0.75 Bin “off”
0
dB
The Side
lobe levels
decrease
-10
-20
-30
-40
-50
-60
-6
-5
-4
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-3
-2
-1
0
1
2
3
4
5 Bin6

12. Input Frequency is +0.75 Bin “off”

Input Frequency is +1.00 Bin “off”
0
dB
Only one
bin is
activated
-10
-20
-30
-40
-50
-60
-6
-5
-4
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-3
-2
-1
0
1
2
3
4
5 Bin6

13. Input Frequency is +1.00 Bin “off”

The Envelope Function
1.1
1.0
Real top
0.8
Highest Bin = a
0.6
Next highest
Bin = b
0.4
0.2
0.0
-0.2
-0.3
-4
-3
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-2
-1
0
1
2
3
Bin4

14. The Envelope Function

The Mathematics
Envelope function:
Sin( π bin)
Env
(π bin)
Bin offset:
b
Δbin
(a b)
Real amplitude:
(π Δbin)
Amp a
Sin( π Δbin)
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15. The Mathematics

Demo
Amplitude and frequency detection by
Sin(x) / x interpolation
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16. Demo

Aliasing of the Side-Lobes
Highest Bin =
Bin 4
0
dB
-10
-20
Aliased Bin =
“Negative Bin 4”
-30
-40
-50
-60
0
1
2
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3
4
5
6
7
8
9
Bin
10

17. Aliasing of the Side-Lobes

Weighted Measurement
Apply a Window to the signal
2
2
Vo lt
1
Vo lt
1
0
0
-1
-1
-2
0.0
-2
0.0
0.1
0.2
0.3
0.4 ms 0.5
Hanning window – one period of ( 1 - COS )
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0.1
0.2
0.3
0.4 ms 0.5

18. Weighted Measurement

Weighted Spectrum Measurement
Apply a Window to the Signal
Without Window
20
dBV
0
20With
dBV
0
-20
-20
-40
-40
-60
-60
0
5
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10
15
20kHz25
0
Hanning Window
5
10
15
20kHz25

19. Weighted Spectrum Measurement

Rectangular and Hanning Windows
0
dB
Side lobes
for Hanning
Window are
significantly
lower than
for
Rectangular
window
-10
-20
-30
-40
-50
-60
-6
-5
-4
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-3
-2
-1
0
1
2
3
4
5 Bin6

20. Rectangular and Hanning Windows

Input Frequency Exactly Hits a Bin
0
dB
Three bins
are
activated
-10
-20
-30
-40
-50
-60
-6
-5
-4
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-3
-2
-1
0
1
2
3
4
5 Bin6

21. Input Frequency Exactly Hits a Bin

Input Frequency is +0.25 Bin “off”
0
dB
More bins
are
activated
-10
-20
-30
-40
-50
-60
-6
-5
-4
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-3
-2
-1
0
1
2
3
4
5 Bin6

22. Input Frequency is +0.25 Bin “off”

Input Frequency is +0.50 Bin “off”
0
dB
Highest
side-lobes
-10
-20
-30
-40
-50
-60
-6
-5
-4
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-3
-2
-1
0
1
2
3
4
5 Bin6

23. Input Frequency is +0.50 Bin “off”

Input Frequency is +0.75 Bin “off”
0
dB
The Side
lobe levels
decrease
-10
-20
-30
-40
-50
-60
-6
-5
-4
www.natinst.com
-3
-2
-1
0
1
2
3
4
5 Bin6

24. Input Frequency is +0.75 Bin “off”

Input Frequency is +1.00 Bin “off”
0
dB
Only three
bins
activated
-10
-20
-30
-40
-50
-60
-6
-5
-4
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-3
-2
-1
0
1
2
3
4
5 Bin6

25. Input Frequency is +1.00 Bin “off”

The Mathematics for Hanning ...
Sin( π bin)
Envelope: Env
2
(π bin) (1 bin )
(a - 2b)
Bin Offset: Δbin
(a b)
(π Δbin)
Amplitude: Amp a
(1 Δbin 2 )
Sin( π Δbin)
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26. The Mathematics for Hanning ...

A LabVIEW Tool
Tone detector LabVIEW virtual instrument (VI)
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27. A LabVIEW Tool

Demo
Amplitude and frequency detection using a
Hanning Window (named after Von Hann)
Real world demo using:
The NI-5411 ARBitrary Waveform Generator
The NI-5911 FLEXible Resolution Oscilloscope
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28. Demo

Frequency Detection Resolution
1000.00
Freq error (ppm)
ppm
100.00
10.00
1.00
0.10
0.01
1
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10
Signal periods
100

29. Frequency Detection Resolution

Amplitude Detection Resolution
1000.00
Amplitude error (ppm)
ppm
100.00
10.00
1.00
0.10
0.01
1
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10
Signal periods
100

30. Amplitude Detection Resolution

Phase Detection Resolution
1000.00
Phase error (mdeg)
mdeg.
100.00
10.00
1.00
0.10
0.01
1
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10
Signal periods
100

31. Phase Detection Resolution

Conclusions
Traditional counters resolve 10 digits in one
second
FFT techniques can do this in much less than
100 ms
Another example of 10X for test
Similar improvements apply to amplitude and
phase
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32. Conclusions

(Notes Page Only)
Traditional Counters Resolve 10 digits in one
second
FFT Techniques can do this in much less than
100 ms
Another example of 10X for test
Similar improvements apply to Amplitude and
Phase
www.natinst.com