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)
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/2∫T/2 p(t) e- jn2Π Ft dt
Therefore
The Fourier spectrum of xs(t) can begiven by
Interchanging order of integration and summation yields aliasing formula
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.
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