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Unformatted text preview: Discrete–time Fourier Series and Fourier Transforms We now start considering discrete–time signals. A discrete–time signal is a function (real or complex valued) whose argument runs over the integers, rather than over the real line. We shall use square brackets, as in x [ n ], for discrete–time signals and round parentheses, as in x ( t ), for continuous–time signals. This is the notation used in EECE 359 and EECE 369. Discrete–time signals arise in two ways. Firstly, the signal could really be representing a discrete sequence of values. For example, x [ n ] could be the n th digit in a string of binary digits being transmitted along some data bus in a computer. Or it could be the maximum temperature for day number n . Secondly, a discrete–time signal could arise from sampling a continuous–time signal at a discrete sequence of times. Periodic Signals Just as in the continuous–time case, discrete–time signals may or may not be periodic. We start by considering the periodic case. Imagine an application in which we have to measure some function x ( t ), that is periodic of period 2 ‘ , and compute its Fourier coefficients from the measurements. We can think of x ( t ) as the amplitude of some periodic signal at time t . Because we can only make finitely many measurements, we cannot determine x ( t ) for all values of t . Suppose that we measure x ( t ) at N equally spaced values of t “covering” the full period 0 ≤ t < 2 ‘ . Say at t = 0 , 2 ‘ N , 2 2 ‘ N , ··· , ( N- 1) 2 ‘ N . Because we do not know x ( t ) for all t we cannot compute the complex (1) Fourier coefficient c k = 1 2 ‘ Z 2 ‘ x ( t ) e- ik π ‘ t dt (1) exactly. But we can get a Riemann sum approximation to it using only t ’s for which x ( t ) is known. All we need to do is divide the domain of integration up into N intervals each of length 2 ‘ N . For t in the t y y = x ( t ) e- ik π ‘ t 2 ‘ 2 ‘ N 2 2 ‘ N 3 2 ‘ N interval n 2 ‘ N ≤ t < ( n + 1) 2 ‘ N , we approximate the integrand x ( t ) e- ik π ‘ t by its value at t = n 2 ‘ N , which is x ( n 2 ‘ N ) e- ik π ‘ n 2 ‘ N = x ( n 2 ‘ N ) e- 2 πi kn N . So we approximate the integral over n 2 ‘ N ≤ t < ( n + 1) 2 ‘ N by the area of a rectangle of height x ( n 2 ‘ N ) e- 2 πi kn N and width 2 ‘ N . This gives c k ≈ c ( N ) k = 1 2 ‘ N- 1 X n =0 x ( n 2 ‘ N ) e- 2 πi kn N 2 ‘ N = 1 N N- 1 X n =0 x ( n 2 ‘ N ) e- 2 πi kn N (1) We are using complex Fourier series rather than sin / cos Fourier series only because the computations are cleaner. It is perfectly possible to use sin / cos Fourier series instead. Alternatively, the sin / cos Fourier series coefficients can be easily computed from the complex ones as we did in the notes “Fourier Series”....
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This note was uploaded on 12/20/2010 for the course E E 338 taught by Professor Vicky during the Spring '10 term at University of Alberta.
- Spring '10