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1. Let Xc(t) = be a continuous-time periodic signal with period T0. The signal is band limited such that its Fourier series coefficients are (-4 < k < 4 )
a) Let T = /N be a sampling where :
i) Find minimum N that meets Nyquists criterion.
ii) Plot the sampled signal x[n] (real and imaginary parts ) for the corresponding N.
iii) Plot the DTFT of the sampled signal x[n]. Note that the 'fit' function in Matlab calculates the DFT, use it in order to plot the desired DTFT.
b) What is the minimal number of samples (as a function of N from the previous
question) required in order to compute all Fourier coefficients ak ? explain .
c) Now, we replace T with Tn = T0/2.25 :
i) plot the same figures as in (a) for the new Tn.
ii) is it possible to compute the Fourier coefficients ak in this case ?
if yes, what is the minimal number of required samples? And how to compute. if not, explain why.
- The Equations Didn T Appear As I Wrote Them In The Written Question Please Check The Image I Uploaded To See The Questi 1 (174.18 KiB) Viewed 32 times
1. Let x (t) Σakek be a continuous-time periodic signal with = k=-4 period To. The signal is band limited such that its Fourier series coef- ficients are {ak = 5+ k}% __4 and zero for |k| > 4. (a) Let T = To/N be a sampling, where N E N. i. Find minimum N that meets Nyquist's criterion. ii. Plot the sampled signal x[n] (real and imaginary parts) for the corresponding N. iii. Plot the DTFT of the sampled signal x[n]. Note that the 'fft' function in matlab calculates the DFT, use it in order to plot the desired DTFT. (b) What is the minimal number of samples (as a function of N from the previous
question) required in order to compute all Fourier coefficients ak? Explain. (c) Now, we replace T with TN = To/2.25 i. Plot the same figures as in (a) for the new TN. ii. Is it possible to compute the Fourier coefficients ak in this case? If yes, what is minimal number of required samples? And how to compute. If not, explain why. 1 Useful functions may be: 'plot' (for continues-time functions) ; 'stem' (for discrete time functions) and 'fft' (in order to compute different Fourier transforms numerically). All functions have a detailed docu- mentation inside Matlab.