j e cos 1 g else n n h 2 1 3 1 3 1 2 j j j e e e H j j e e H 3 cos 2 1 h cos n

J e cos 1 g else n n h 2 1 3 1 3 1 2 j j j e e e h j

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j e )] cos( 1 [ (g) else n n h 0 2 , 1 , 0 ) ( 3 1 3 1 ) ( 2 j j j e e e H j j e e H 3 cos 2 1 ) ( (h) ) ( ) cos( ) ( n u n r n h o n 33 3 1 j j j e e e
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j n o n j e n r e H 0 ) cos( ) ( 2 2 ) cos( 2 ) cos( 1 ) ( j j o j o j e r e r e r e H 1.12 THE INVERSE FOURIER TRANSFORM This is defined as: d e H n h n j ) ( 2 1 ) ( If ) ( j e H is expressed as a power series of complex exponents, then ) ( n h is the coefficient of n j e Example 3.2: Determine the inverse Fourier transform of the following (a) ) 2 cos 1 ( 3 1 ) ( j e H Applying Eulers’ identity, 3 3 3 1 ) ( j j j e e e H Note the coefficient of j e are 3 1 3 1 3 1 , , for 1 , 0 , 1 n respectively which equals ) ( n h (b) 4 4 3 3 2 2 1 1 1 ) ( j j j j j j e a e a e a ae ae e H 0 , ) ( n a n h n PROPERTIES OF THE FOURIER TRANSFORM As an exercise, prove the properties given below. 34
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(a) Linearity – Use the principle of superposition If ) ( ) ( ) ( 2 1 n h n h n x then ) ( ) ( ) ( ) ( 2 1 j j j e H z e H e X (b) Periodicity The Fourier transform is a periodic function of with period 2 ) ( ) ( ) 2 ( j k j e H e H (c) Magnitude and phase functions ) ( j e H is generally complex and may be written as ) ( ) ( ) ( j I j R j e jH e H e H denoting the real and imaginary responses ) ( ) ( ) ( ) ( ) ( 2 2 2 j I j R j j j e H e H e H e H e H ) ( / ) ( arctan ) ( arg j R j I j e H e H e H (d) Fourier transform of a delayed sequence If ) ( ) ( o n n x n y and if ) ( ) ( j e X n x F Then ) ( ) ( j n j e X e n y F o (e) Fourier transform of the convolution of two sequences  k n x n h k n x k h n y ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( j j j e H e X e Y (f) Fourier transform of the product of two sequences ) ( ) ( ) ( n x n h n y ) ( ) ( ) ( j j j e X e H e Y (periodic convolution) 35
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4.0 THE DISCRETE FOURIER TRANSFORM (DFT) The DFT, denoted by ) ( k H , is a sequence obtained by sampling the continuous time Fourier transform ) ( j e H at a number of finite points. That is: N k e H k H k j / 2 ) ( ) ( , 1 0 N k . The DFT is a sequence of length N . This is referred to as N point DFT. Consider a non periodic discrete sequence ) ( n h . Its Fourier transform ) ( j e H is continuous in and periodic. As a consequence of sampling at N k k / 2 the DFT is also periodic with period N We wish recover our original sequence ) ( n h from the sampled ) ( k H .
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