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wave
Plotting and Computing Fourier Transform for Time Series

Plotting Multiple Time Series

1. Plot for and sec on the same graph with domain and .2. The analytical formula for the Fourier transform of isCompute the discrete Fourier transform (DFT) for both sampled time series, and compare them to the analytical for both 's on the same graph.Hints:As numpy fft assumes signal starts from time 0, you can use the shift property of Fourier transform to first shift the to startfrom zero, and after fftshift(fft()) operations, multiply the spectrum by complex exponential sinusoid function.You need to sample the theoretical curve with w_axis = 2*pi*f_axis, or else rewrite it as if you'd rathersample it with f_axisAs a guide (so you can be confident of your fft utilization for the remainder of the lab), we expect that the amplitudes (use numpy.abs(...)) of the discrete FT and the continuous FT essentially match. The phase won’t necessarily match.3. Comment on the effect of filtering a general input time function by (i.e. convolution of with ), and explain thedifference in filtered output after applying Gaussian functions with or secs.4. Comment on how this is related to the time-frequency uncertainty principle (a signal cannot be infinitesimally sharp both in time andfrequency).

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