Fourier-Based Signal & System Design

The point of banning fft and conv is that using them would have made the assignment about calling a function correctly, not understanding what the function does. Task 1 and 2 implement the Fourier transform as a direct Riemann-sum approximation of the integral definition, summing discrete samples against complex exponentials by hand. Task 3 builds convolution the same way, an explicit O(N²) sliding-sum implementation rather than the O(N log N) trick conv uses internally.

The last task points the hand-rolled machinery at a genuine 9-second recording of my own voice, not a synthetic test signal. A spectral breakdown of the raw audio shows most of the energy sitting below 3 kHz with a broadband noise floor above it, so a 3.4 kHz low-pass filter, designed and applied with the exact same explicit Fourier and convolution code from Task 2, strips the noise while leaving speech intact. The filtered signal is then run back through Task 3's sampling and reconstruction pipeline at two rates: one comfortably above the filter's Nyquist limit, one deliberately below it. At 5 kHz the reconstruction error against the filtered signal comes out roughly 2,000× worse than at 8 kHz, the theory and the artifact matching up by ear, not just on a spectrum plot. It's the only project on this site with a literal playable output rather than a number or a log line.