Pulse-train sampling Assignment Help

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Pulse-train sampling For the pulse-train sampling of signal x(t) by rectangular pulse-train p(t) resulting in sampled signal xs(t), we have

xs(t) = x(t) p(t)

660_Pulse-train sampling1.png

The Fourier series of p(t) can be given by p(t)= ∑ C e jn2Π Ft  

here Fs = 1/T and Fourier coefficients, are n = - 

Cn = 1/T -T/2T/2 p(t) e- jn2Π Ft dt

Therefore

1306_Pulse-train sampling2.png

The Fourier spectrum of xs(t) can begiven by

490_Pulse-train sampling3.png

 

Interchanging order of integration and summation yields aliasing formula

2329_Pulse-train sampling4.png

349_Pulse-train sampling.png

The sampled signal spectrum Xs(F) is sketched as follows. For the convenience of illustration we have supposed the base band spectrum, X(F), to be real valued; the maximum value of |X(F)| is taken to be 1. Xs(F) comprises of replicas of X(F), scaled by Fourier coefficients Cn and repeated at intervals of Fs. Particularly, the replica at the origin is simply X(F) scaled by C0.

Note that the magnitudes, |Cn|, have symmetry. In this particular case, as FM Fs-FM, there is no overlap among replicas in the graph of Xs(F). As a result of the original signal x(t) can be recovered by passing xs(t) through the low pass filter with the bandwidth B which satisfies the condition FM B Fs-FM, and a gain of 1/C0.

867_Pulse-train sampling5.png

 

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