f_ 00 X(w) exp(7w77T 0)&7. : , - oo < 0 < oo (1.4) where each replica is shifted horizontally by an integer multiple of the angular sampling frequency wo = Proof: Using the inverse Fourier transform Eq. 1.1.2 at a sampling time t = nTo, we have exp(7w77T 0)&7. This integral can be decomposed into an infinite sum, of 27r/To length each, to give -(2k+1)7/To exp(3wnTo)dw. Using variable substitution w = - tc, this becomes

doi:10.1201/9781482276237-8

Chapter

ABSTRACT

The sampling theorem, Eq. 1.4, provides a practical way of computing the discretetime Fourier transform of a sampled signal, Xdt (9), directly from the Fourier transform of the original continuous-time signal, X(w). This involves three steps: 1) linear scaling of the frequency axis according to B = wTo; 2) summation of an infinite number of frequency-shifted (continuous-time) Fourier transforms; and 3) scaling of the amplitude axis by 1/To.