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# lect7 - THE DISCRETE – TIME FOURIER TRANSFORM(DTFT The...

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THE DISCRETE – TIME FOURIER TRANSFORM (DTFT) The Discrete-Time Fourier Transform of a signal, ) ( n x , is simply its z-transform evaluated on the unit circle. Thus, given ) ( n x , the DTFT is given by -∞ = - = n n j j e n x e X w w ) ( ) ( . (7.1) Clearly, the DTFT is a periodic function of w with period p 2 . It then becomes clear that (7.1) is simply a fourier series expansion of ) ( w j e X with Fourier coefficients, ) ( n x . By the formula for calculating the coefficients of a Fourier Series expansion, we have the inversion formula for the DTFT - = p p w w w p d e e X n x n j j ) ( 2 1 ) ( . (7.2) The formulas (7.1) and (7.2) define the DTFT and its inversion formula, and this relationship is denoted in the usual way as ) ( ) ( w j e X n x . The physical meaning of the DTFT is evident in the inversion formula (7.2). It provides a representation of a discrete-time signal, ) ( n x , as a sum (integral) of pure complex sinusoids, n j e w . The function, ) ( w j e X , is essentially the complex amplitude of the sinusoid of frequency w . Thus, ) ( w j e X describes the

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frequency content of ) ( n x , and is often called the spectrum of ) ( n x . Notice that the discrete-time signal contains frequencies only between p - and p . To see why this makes sense, consider a pure complex sinusoid of frequency w , n j e w . Let us increase its frequency by the amount 2 π . We find n j n j e e w p w = + ) 2 ( . Thus, increasing the frequency by p 2 results in exactly the same signal. In other words, in discrete time with time base equal to unity any two complex sinusoids separated in frequency by p 2 are identical. On the other hand, it is not difficult to see that any contiguous range of frequencies of total width p 2 represent different signals. For this reason, in representing a given discrete-time signal, we are free to use any contiguous range of frequencies of total width p 2 . To illustrate this further, note that because the integrand in (7.2) has period p 2 , the range of integration can be changed to any contiguous range of frequencies of total width p 2 without changing the result; = = - range n j j n j j d e e X d e e X n x p w w p p w w w p w p 2 ) ( 2 1 ) ( 2 1 ) ( . Thus, we may consider ) ( n x to be composed of frequencies from p - to p , or from 0 to p 2 , etc.
Consider now that ) ( n x is the input to a system ) ( z H with output ) ( n y Since ) ( ) ( ) ( z X z H z Y = , if we substitute w j e z = we have ) ( ) ( ) ( w w w j j j e X e H e Y = , as illustrated below. The equation, ) ( ) ( ) ( w w w j j j e X e H e Y = , is the basis for the concept of filtering a discrete time signal by an LTI system, as shown in the following example. Example 7.1 ) ( ) ( ) ( z H z X z Y = ) ( z X ) ( w j e H ) ( n x ) ( n y ) ( w j e X ) ( w j e Y w 2 p 2 p - ) ( w j e X w 4 p 4 p - ) ( w j e H ) ( z H

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When the signal ) ( n x is filtered by ) ( w j e H , the output spectrum is that shown below. When dealing with discrete time signals, we should always keep in mind that their spectra are periodic functions with period p 2 .
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lect7 - THE DISCRETE – TIME FOURIER TRANSFORM(DTFT The...

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