Parseval's Theorem 2.4 2000/08/09 2005/05/09 20:14:46 GMT-5 Don Johnson dhj@rice.edu Don Johnson dhj@rice.edu Shawn Stewart mrshawn@alumni.rice.edu Parseval's Theorem More about Parseval's Theorem. The properties of the discrete-time Fourier transform mirror those of the analog Fourier transform. The DTFT properties table shows similarities and differences. One important common property is Parseval's Theorem. n s n 2 f 1 2 1 2 S 2 f 2 To show this important property, we simply substitute the Fourier transform expression into the frequency-domain expression for power. f 1 2 1 2 S 2 f 2 f 1 2 1 2 n n s n 2 f n m m s n 2 f m n , m n , m s n s m f 1 2 1 2 2 f m n Using the orthogonality relation, the integral equals δ m n , where δ n is the unit sample. Thus, the double sum collapses into a single sum because nonzero values occur only when n m , giving Parseval's Theorem as a result. We term n n s n 2 the energy in the discrete-time signal s n in spite of the fact that discrete-time signals don't consume (or produce for that matter) energy. This terminology is a carry-over from the analog world. Suppose we obtained our discrete-time signal from values of the product s t p T s t , where the duration of the component pulses in p T s t is Δ . How is the discrete-time signal energy related to the total energy contained in s t ? Assume the signal is bandlimited and that the sampling rate was chosen appropriate to the Sampling Theorem's conditions. If the sampling frequency exceeds the Nyquist frequency, the spectrum of the samples equals the analog spectrum, but over the normalized analog frequency f T . Thus, the energy in the sampled signal equals the original signal's energy multiplied by T.