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

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?

Classical Fourier series on interval

Fourier series in complex form

Fourier series in complex form

Exponential Fourier Series on

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:

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:

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

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

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

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?

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

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

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

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

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

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

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)

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

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:

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

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)

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

Convolution in time domain = Multiplication in Fourier domain

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?

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

From brick to sinc Frequency Time

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

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.

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

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)

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

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

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

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

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