Show that the time-dependent Fourier transform, as defined by Eq. (10.18), has the following properties:

(a) Linearity:

If x[n] = ax₁[n] + bx₂[n], ? ? ? ??then? ? ? X[n, ?) = ?X₁[n, ?) + bX₂[n, ?).

(b) Shifting: If y[n] = x[n ? n₀], then Y[n, ?) = X[n ? n₀, ?).

(c) Modulation: If y[n] = e^j?0n?x[n], then Y[n, ?) = e^j?0n X[n, ? ? ?₀).

(d) Conjugate Symmetry: If x[n] is real, then X[n, ?) = X*[n, ??).

Transcribed Image Text:

X[n. À) = x(n + m]w[m]e jàm. E