Elec4621 Lab3 - T1 2021
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Elec4621 Lab3 - T1 2021
March 2, 2021
This lab is essentially a practice Matlab exercise to illustrate the concepts of filtering and linear time-invariant systems.
1. A fifth order all-zero filter has the following transfer function,
where
(a) Find the impulse response of the filter.
Hint: There are a number of ways to do this: One way is to build a vector, v, containing all 5 zeroes, and then use the Matlab function, ‘poly’, to find the coefficients of the polyno- mial with these roots – the coefficients of the polynomial are the filter taps. Another way is to proceed from the second order sections of the transfer function and use polynomial multiplication to build 5-th order numerator. Then you will have the filter taps.
(b) Find and plot the magnitude and phase responses of the filter.
Hint: You should remember that the frequency response is ob- tained by substituting z = ejw . Therefore you can think of this as the evaluation of the transfer function on a fine grid of ω . Thus, you can construct a vector w of angular frequencies and evaluate the transfer function for this vector, then plot the amplitude and phase of the result.
(c) Apply the filter to the sunspot data from the previous lab. What do you observe?
(d) Suppose that we want to suppress the high frequency noise of the sunspot data to better see the 11 year and longer term cycles. Experiment with the zeros of the filter above to achieve this.
2. Consider the signal x(t) = A cos(ωt) where A = 1 and ω = 2πf with f = 110 Hz. In the rest of this part, and unless otherwise specified, you are to plot the signals in the time for 0 < t < 30 ms and in the frequency domain for _1000 < f < 1000 Hz.
(a) What is the minimum sampling frequency to avoid aliasing.
(b) Suppose we sample the signal at fs = 300 Hz. Plot the continuous and sampled signals.
(c) Plot the Fourier spectra of the continuous and sampled signals.
(d) Now suppose that we want to recover the continuous time signal. Give the specification of a suitable reconstruction filter to achieve this. Design a practical FIR filter to do the same.
(e) Apply the filter to the sampled signal and plot and compare the original and reconstructed signals. Also plot and compare their spectra.
(f) Now suppose that we want the reconstructed signal to have a frequency of 70 Hz. Suggest a way to achieve this via a suit- able choice of the sampling frequency and reconstruction filter. Implement this system and plot and compare the original and reconstructed signals both in the time and frequency domains.
(g) Now suppose that we want the reconstructed signal to have a frequency of 170 Hz. Repeat the previous part by suggesting a suitable choice of the sampling and reconstruction filter to achieve this.
2023-02-28