Generate a square pulse of sampling frequency 150 hz the

Question

Problem 1 -

Expand the function f(x) = x² in a Fourier series valid on the interval - π ≤ x ≤ π. Plot both f and the partial of its Fourier series,

S_N(x) = _k=0∑^N a_k cos (kx) + b_k sin (kx)

For N = 1, 2, 5, 7. Observe hoe the graphs of the partial sums S_N(x) approximate the graph of f. Plot the same graph over the interval - 2π ≤ x ≤ 2π.

Problem 2 -

Generate a square pulse of sampling frequency 150 Hz. The time vector magnitude is 1 sec and the width of the rectangle is 0.2.

a) Plot the amplitude of the square pulse signal as a function of time.

b) Using fft command (fast fourier transform), find the frequency vector and plot the power spectrum of a square pulse (Frequency Vs Power).

Problem 3 -

Let y(t) = e^{-t^2/10} (sin(2t) + 2 cos(4t) + 0.4sintsin10t) be the function used to generate a signal S over the interval [0, 2π). The signal is sampled at 28 equally spaced nodes.

a) Find the fft of this signal and plot the signal and the filtered signal with FFT.

b) Use the functions: compress.m and fftcomp.m to plot the filtered signal with a compression of 75%.

Problem 4 -

Let a signal S be represented by the function x = sin(30πt) + sin(80πt) on the time interval [0, 99].

a) Compute the DFT (Discrete fourier transform) of the signal, the magnitude and phase of the transformed sequence.

b) Plot the magnitude and the phase using subplot.

Attachment:- Assignment File.rar