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1. Write a MATLAB or Octave program that, given a sequence, f[k], of samples in the time domain, calculates a filtered output y[k], given by the formula y[k]=-a_2[k-1]-a_1y[k-2]-a_0y[k-3]+b_3f[k]+b_2f[k-1]+b_1f[k-2]+b_0f[k-3]. The values y are outputs, k is an index into a sequence, (so if "now"=k, one sample in the past is k-1), there are coefficient vectors a and b, which can be input parameters to the function. Make up a sequence f. Plot the input f and that output, y.
2. What do we know about the coefficient vectors a, b, if we know the filter is a finite impulse response filter?
3. Use a symmetric sequence for the coefficient sequence b, and zero for the values of the coefficient sequence a. Calculate the FFT of the sequence y, call it Y. Plot the phase of Y vs. frequency.(See https://www.embedded.com/design/real-time-and-performance/4008837/DSP-Tricks-An-odd-way-to-build-a-simplified-FIR-filter-structure for a description of symmetric coefficients.)
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