sampling interval in the frequency domain is An,it corresponds to replication of signals in time domain at every 2/An.In order to recover x(t)from i(t)by time windowing,x(t)should be time-limited toTo,and sampling interval should be small enough so that △0<0 2π We have basically the same result in the discrete-time domain. 5 X(e) x(m] 个 The sampling interval Aw should satisfy the following condition: △w<M If we denote the number of frequency samples from 0 to 2m as N,it is required that 2π N= △w >M. Under this condition,x[n]can be perfectly recovered from the samples of the DTFT: 4W=六∑X肉w,n=0.1w-1 w-1 k=0 The Discrete Fourier Series Let n]be a periodic signal with period N(We will use to denote periodic signals).Consider representing this signal by a Fourier series corresponding to a linear combination of harmonically related complex exponentials,where ek n]=eikn 3sampling interval in the frequency domain is ΔΩ, it corresponds to replication of signals in time domain at every 2π/ΔΩ. In order to recover x(t) from x˜(t) by time windowing, x(t) should be time-limited to T0, and sampling interval should be small enough so that 2π ΔΩ < . T0 We have basically the same result in the discrete-time domain. The sampling interval Δω should satisfy the following condition: 2π Δω < . M If we denote the number of frequency samples from 0 to 2π as N, it is required that 2π N = > M. Δω Under this condition, x[n] can be perfectly recovered from the samples of the DTFT: 1 N−1 x[n] = � X[k]W−nk, n = 0, 1, . . . , N − 1. N N k=0 The Discrete Fourier Series Let x˜[n] be a periodic signal with period N (We will use ˜ to denote periodic signals). Consider representing this signal by a Fourier series corresponding to a linear combination of harmonically 2πk related complex exponentials ejωn, where ω = N . ek[n] = e 3 2π j N kn