1. ABE425 Engineering Measurement Systems Fourier Transform, Sampling theorem, Convolution and Digital Filters Dr. Tony E. Grift
Dept. of Agricultural & Biological Engineering
University of Illinois

2. Agenda Sampling theorem (Shannon) Effect of sampling frequency
Aliasing
Nyquist/Shannon 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 on interval

4. Fourier series in complex form

5. Fourier series in complex form

6. Exponential Fourier Series on

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

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

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

10. Let’s analyze a signal and its Fourier Transform
Highest frequency
in input signal

11. Let’s sample the signal at 128 Samples/second
Do you still recognize
the signal?

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

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

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

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

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

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

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

19. 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)

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

21. Filtering a signal in Fourier domain
Time domain
Fourier domain
Filter
Do Fourier Transform (FT) on input, multiply by filter, do Inverse FT (IFT) OR:

22. Filtering a signal using convolution
Time domain
Fourier domain
Filter
Input
kernel
Convolution
Stay in time domain, convolve input with IFT(Filter)

23. Filtering a signal in Fourier domain OR using convolution
Time domain
Fourier domain
Filter
Convolution
Input
kernel
Do Fourier Transform (FT) on input, multiply by filter, do Inverse FT (IFT) OR:
Stay in time domain, convolve input with IFT(filter)

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

25. Convolution in time domain = Multiplication in Fourier domain

26. A product of two functions in the Fourier (and Laplace) domain is equivalent to the Convolution of their counterparts in the time domain. To filter a signal you can do two things 1) Do a Fourier Transform on your signal, multiply by your choice of filter in the Fourier Domain, and Transform back to the time domain. 2) Stay in the time domain and convolve your signal with the Inverse Fourier Transform of your filter. Q: Which method is easier to implement in hardware?

27. 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

28. From brick to sinc
Frequency
Time

29. The Fourier Transform of the Dirac delta function = 1
L’Hopital’s rule

30. 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 Nyquist (sometimes 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 destroy your measurements during digitization. Always get rid of noise before sampling.

31. Case study: Measure the power levels of a tub grinder in action

32. Tubgrinder experiments were conducted using 30 bales (Miscanthus and Switchgrass) and varying screen sizes
Crop
Screen size
MxG
Switch grass
6.35 mm (¼-inch)
9.53 mm (3/8-inch)
12.7 mm (1/2-inch)
25.4 mm (1-inch)
38.1 mm (1.5-inch)

33. Measurement of Torque and RPM yielded power, and integrated over time, grinding energy requirement
Accumulated mass

34. Grinding Miscanthus through a 1 inch screen costs about 0
Grinding Miscanthus through a 1 inch screen costs about 0.16 PIHV (including grinder!)

35. Low-pass convolution filtering of comminution data allowed determination of idle power
Idle power (5 kW)

36. Power data after low-pass filtering with 0.5 Hz cutoff
Max power (90 kW)

37. ABE425 Engineering Measurement Systems Fourier Transform, Sampling theorem, Convolution and Digital Filters The End
Dept. of Agricultural & Biological Engineering
University of Illinois