Fourier Transform, Sampling theorem, Convolution and Digital Filters

1. Fourier Transform, Sampling theorem, Convolution and Digital Filters ABE425Engineering Measurement Systems

2. Agenda • Sampling theorem (Shannon) • Effect of sampling frequency • Aliasing • Nyquist/Shannon sampling theorem • Convolution in time domain = Multiplication in frequency domain • Signal reconstruction • Noise effects • How do you choose the reconstruction convolution kernel?

3. Classical Fourier series

4. Fourier series in complex form

5. A pulse train is a series of Dirac pulses at constant intervals (Dirac comb) Written as a complex Fourier Series Since the Fourier Transform of a Dirac pulse is 1:

6. The sampler (digitizer output is now the product of the input signal and the pulse train The Fourier transform of a time shift is: Therefore:

7. Now we can write the Fourier Transform of the sampler output as follows: The Spectrum of the original signal repeats itself around the sampling frequency

8. Let’s look at a signal and its Fourier Transform Highest frequency in input signal

9. Let’s sample the signal at 128 S/s Do you still recognize the signal?

10. You do not have the original signal! You only have samples. How do you reconstruct the original?

11. Can we still reconstruct the original @ 128 S/s? Brick function

12. Let’s sample the signal at 32 S/s ! Do you still recognize the signal?

13. Can we still reconstruct the original @ 32 S/s? Brick function

14. Now let’s sample the signal at 16 S/s ! Do you still recognize the signal?

15. Can we still reconstruct the original @ 16 S/s? Brick function

16. Noise can significantly distort a reconstructed signal! Solution, pre-filter your signal!

17. Nyquist/Shannon sampling theorem • When you sample a signal, make sure you use a sample rate at least twice as high as the maximum frequency in your input signal. • Problem: Do you know the highest frequency in your input singals? Usually not. • One option is to simply limit the highest frequency to what you are interested in by using a pre sampling filter (called anti-aliasing filter)

18. Terminology Time (s) Frequency (rad/s) Fourier Transform Sampling Brick function Reconstruction Convolution w/ F-1(brick) Aliasing Inverse Fourier Transform

19. Convolution example: Moving average filter • 1st iteration • 2nd iteration • General • Continuous case

20. Convolution in time domain = Multiplication in Fourier domain

21. If you multiply by a brick in the Fourier domain, what do you convolve with in the time domain? • You need the inverse Fourier Transform a brick! • The inverse FT of a brick function is a Sinc function

22. From brick to sinc Frequency Time

23. The FT of the Dirac delta = 1

24. Things to remember • Sampling causes higher frequencies in the Fourier domain (spectrum) • The original frequency spectrum repeats itself around the sample frequency • To reconstruct the signal from samples you need to low pass filter • You can only low pass filter if the sampling frequency is at least twice (in reality 5 or 10 times) as high as the highest frequency in your input signal. This is the Nyquist (Shannon) criterion. • If you do not adhere to the Nyquist criterion, you will NOT be able to reconstruct the original signal. You will introduce aliases, which are artificial frequencies caused by the sampling process. • Q? How do you know the highest frequency in your input signal? In general you don’t. Therefore you set it yourself by filtering before sampling (pre-sampling or anti-aliasing filter) • Noise can really destroy your measurements during digitization. Always get rid of noise before sampling.

25. The End