Discrete-Time Fourier Transform Pair 2.5 2000/08/09 2003/07/24 10:53:30 GMT-5 Don Johnson dhj@rice.edu Prashant Singh prash@ece.rice.edu Richard G. Baraniuk richb@rice.edu Mariyah Poonawala mariyah@rice.edu analog digital discrete-time Fourier Transform frequency Nyquist Computing discrete-time frequencies by the use of Fourier transforms. When we obtain the discrete-time signal via sampling an analog signal, the Nyquist frequency corresponds to the discrete-time frequency 1 2 . To show this, note that a sinusoid at the Nyquist frequency 1 2 T s has a sampled waveform that equals Sinusoid at Nyquist Frequency 1/2T 2 1 2 T s n T s n 1 n The exponential in the DTFT at frequency 1 2 equals 2 n 2 n 1 n , meaning that the correspondence between analog and discrete-time frequency is established: Analog, Discrete-Time Frequency Relationship f D f A T s where f D and f A represent discrete-time and analog frequency variables, respectively. The aliasing figure provides another way of deriving this result. As the duration of each pulse in the periodic sampling signal p T s t narrows, the amplitudes of the signal's spectral repetitions, which are governed by the Fourier series coefficients of p T s t , become increasingly equal. Examination of the periodic pulse signal reveals that as Δ decreases, the value of c 0 , the largest Fourier coefficient, decreases to zero: c 0 A Δ T . Thus, to maintain a mathematically viable Sampling Theorem, the amplitude A must increase as 1 Δ , becoming infinitely large as the pulse duration decreases. Practical systems use a small value of Δ , say 0.1 T s and use amplifiers to rescale the signal. Thus, the sampled signal's spectrum becomes periodic with period 1 T s . Thus, the Nyquist frequency 1 2 T s corresponds to the frequency 1 2 . The inverse discrete-time Fourier transform is easily derived from the following relationship: 1 2 1 2 f 2 f m f n 1 m n 0 m n Therefore, we find that f 1 2 1 2 S 2 f 2 f n f 1 2 1 2 m m s m 2 f m 2 f n m m s m f 1 2 1 2 2 f m n s n The Fourier transform pairs in discrete-time are Fourier Transform Pairs in Discrete Time S 2 f n n s n 2 f n Fourier Transform Pairs in Discrete Time s n f 1 2 1 2 S 2 f 2 f n