DTFT Examples 2.9 2000/08/09 2004/08/09 11:22:06.768 GMT-5 Don Johnson dhj@rice.edu Prashant Singh prash@ece.rice.edu John Austin Cottrell jac3@rice.edu Mariyah Poonawala mariyah@rice.edu Don Johnson dhj@rice.edu Richard G. Baraniuk richb@rice.edu Shawn Stewart mrshawn@alumni.rice.edu analog digital discrete-time Fourier Transform Parseval's Theorem example Nyquist frequency DSP digital signal processing examples discrete-time Fourier transform How to compute discrete-time Fourier transforms for decaying sequences. Let's compute the discrete-time Fourier transform of the exponentially decaying sequence s n a n u n , where u n is the unit-step sequence. Simply plugging the signal's expression into the Fourier transform formula, Fourier Transform Formula S 2 f n a n u n 2 f n n 0 a 2 f n This sum is a special case of the geometric series. Geometric Series α α 1 n 0 α n 1 1 α Thus, as long as a 1 , we have our Fourier transform. S 2 f 1 1 a 2 f Using Euler's relation, we can express the magnitude and phase of this spectrum. S 2 f 1 1 a 2 f 2 a 2 2 f 2 S 2 f a 2 f 1 a 2 f No matter what value of a we choose, the above formulae clearly demonstrate the periodic nature of the spectra of discrete-time signals. shows indeed that the spectrum is a periodic function. We need only consider the spectrum between 1 2 and 1 2 to unambiguously define it. When a 0 , we have a lowpass spectrum — the spectrum diminishes as frequency increases from 0 to 1 2 — with increasing a leading to a greater low frequency content; for a 0 , we have a highpass spectrum ().

The spectrum of the exponential signal ( a 0.5 ) is shown over the frequency range -2 2 , clearly demonstrating the periodicity of all discrete-time spectra. The angle has units of degrees.

The spectra of several exponential signals are shown. What is the apparent relationship between the spectra for a 0.5 and a -0.5 ?

Analogous to the analog pulse signal, let's find the spectrum of the length- N pulse sequence. s n 1 0 n N 1 0 The Fourier transform of this sequence has the form of a truncated geometric series. S 2 f n 0 N 1 2 f n For the so-called finite geometric series, we know that Finite Geometric Series n n 0 N n 0 1 α n α n 0 1 α N 1 α for all values of α . Derive this formula for the finite geometric series sum. The "trick" is to consider the difference between the series'; sum and the sum of the series multiplied by α . α n n 0 N n 0 1 α n n n 0 N n 0 1 α n α N n 0 α n 0 which, after manipulation, yields the geometric sum formula. Applying this result yields (.) S 2 f 1 2 f N 1 2 f f N 1 f N f The ratio of sine functions has the generic form of N x x , which is known as the discrete-time sinc function, dsinc x . Thus, our transform can be concisely expressed as S 2 f f N 1 dsinc f . The discrete-time pulse's spectrum contains many ripples, the number of which increase with N , the pulse's duration.

The spectrum of a length-ten pulse is shown. Can you explain the rather complicated appearance of the phase?