Spectrum of a down-sampled signal provided the signal x(n) whose spectrum is X(ω) or X(ejω) we need to search the spectrum of y(n), the down-sampled version of x(n), shown by y(n) ↔ Y(ω).
Suppose the periodic train of impulses, p(n), with period M
The discrete Fourier series shows of p(n) is
The Fourier coefficients are provided by
Thus the DFS for p(n) is
Describe the signal x'(n)
The sequence x¢(n) consists of values of x(n) whenever n = 0, ±M, ±2M, ..., and zeros in between that points.
Describe the down-sampled version y(n)
y(n) = x'(Mn) = x(Mn) p(Mn) = x(Mn)
The signal y(n) has of values of x(Mn) at n = 0, ±1, ±2, ..., but no zeros in between.
With y(n) = x'(Mn) = x(Mn) our objective is to search the spectrum Y(ω).Take in mind that X(ω) periodic in ω since x(n) is a discrete-time sequence; and the similar is true of Y(ω). Now the z-transform of y(n) is
Set Mn = k: then n = k/M and the summation limits n = {- ∞ to ∞} become k = {- ∞ to ∞}. Thus
Here x'(n) = 0 except when n is a multiple of M. Substituting x(n) p(n) for x'(n) in the above equation,
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