Fourier Series & Transforms

The problem, from first principles

02-taylor approximates a function near one point by adding up powers of x - a. Fourier analysis approximates a function over a whole stretch by adding up waves instead: a repeating signal, however jagged, can be written as a sum of sines and cosines whose frequencies are whole multiples of a base frequency. The transform that finds those waves, and the inverse that rebuilds the signal from them, are among the most used algorithms in computing. This question builds both from their definitions, for a continuous periodic function (the series) and for a list of samples (the discrete transform).

From theory to code

Implement fourier_coefficients(f, period, num_terms, num_samples) to find the cosine and sine weights of a periodic function, fourier_series_eval(a0, a, b, period, t) to rebuild the function from them, then dft(x) and inverse_dft(spectrum) for the discrete transform of a sample list, and dominant_frequency(x, sample_rate) to read the strongest frequency out of a signal. The signatures and docstrings are already in the editor.

Constraints

• f takes a NumPy array of times and returns an array of the same shape. It repeats every period.
• fourier_coefficients estimates the integrals by averaging f at num_samples evenly spaced times over one period, starting at t = 0. It returns (a0, a, b) where a and b are arrays of length num_terms holding the weights for harmonics 1 through num_terms. The series is a0 / 2 + sum(a[k-1] * cos(2*pi*k*t/period) + b[k-1] * sin(2*pi*k*t/period)).
• fourier_series_eval accepts t as a float or an array and returns the same shape.
• dft and inverse_dft use the definition directly with a matrix of complex exponentials. They must not call np.fft. dft takes a 1D array of length N and returns a complex array of length N. inverse_dft returns a complex array that is the exact inverse.
• dominant_frequency returns the frequency in Hz of the largest-magnitude bin among frequencies from 1 bin up to the Nyquist bin, ignoring the zero-frequency (average) bin. sample_rate is in samples per second.

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