Π ϖ j j j e x e h e y example input 1 α α n u n x

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ϖ ϖ j j j e X e H e Y = Example: Input: 1 | | ], [ ] [ < = α α n u n x n , LTI system: 1 | | ], [ ] [ < = β β n u n h n Find the output.
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We have ϖ ϖ α ˆ ˆ 1 1 ) ( j j e e X - - = , ϖ ϖ β ˆ ˆ 1 1 ) ( j j e e H - - = The DTFT of the output is ϖ ϖ ϖ ϖ ϖ α β ˆ ˆ ˆ ˆ ˆ 1 1 1 1 ) ( ) ( ) ( j j j j j e e e X e H e Y - - - - = = Using the result of the previous example, the output in the time domain is ] [ ] [ ] [ )} ( { ] [ 1 1 ˆ 1 n u n u n u e Y F n y n n n n j β α β α β α β α β β α α ϖ - - = - = = + + - - - Please check the answer using the linear convolution ] [ ) ( ] [ 0 n u n y n k k n k = - = β α Frequency Response of Cascade System As we know that the impulse response of two systems with frequency responses H 1 ( ϖ ˆ j e ) and H 2 ( ϖ ˆ j e ) in cascade is the convolution of the two impulse responses, the frequency response of the cascade system is the product of their frequency responses. ) ( ) ( ) ( ˆ 2 ˆ 1 ˆ ϖ ϖ ϖ j j j e H e H e H = The result shows that the frequency response of the cascade system is the product of the frequency responses of individual systems. This relation does not depend on the order of the two systems in cascade.
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Recall that the impulse response of two LTI systems in cascade is the convolution of the impulse responses of the two systems, i.e. ] [ * ] [ ] [ 2 1 n h n h n h = . The convolution property of DTFT shows that the frequency response of h [ n ] is given by the product of ) ( ˆ 1 ϖ j e H and ) ( ˆ 2 ϖ j e H , i.e. ) ( ) ( ] [ ˆ 2 ˆ 1 0 ˆ ϖ ϖ ϖ j j n n j e H e H e n h = = - ) ( ) ( ] [ * ] [ ] [ ˆ 2 ˆ 1 2 1 ϖ ϖ j j e H e H n h n h n h = Convolution Multiplication Example: Consider two FIR filters with the coefficients as follows {2, 4, 6, 4, 2} and {1, -2, 2, -1}. Find the overall impulse response. Use the frequency response to find the overall impulse response. Frequency response of LTI1: 4 ˆ 3 ˆ 2 ˆ ˆ ˆ 1 2 4 6 4 2 ) ( ϖ ϖ ϖ ϖ ϖ j j j j j e e e e e H - - - - + + + + = Frequency response of LTI2: 3 ˆ 2 ˆ ˆ ˆ 2 2 2 1 ) ( ϖ ϖ ϖ ϖ j j j j e e e e H - - - - + - = The overall frequency response is ) ( ) ( ) ( ˆ 2 ˆ 1 ˆ ϖ ϖ ϖ j j j e H e H e H = ) 2 4 6 4 2 ( 4 ˆ 3 ˆ 2 ˆ ˆ ϖ ϖ ϖ ϖ j j j j e e e e - - - - + + + + = ) 2 2 1 ( 3 ˆ 2 ˆ ˆ ϖ ϖ ϖ j j j e e e - - - - + -
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7 ˆ 6 ˆ 5 ˆ 4 ˆ 3 ˆ 2 ˆ ˆ 2 0 2 2 2 2 0 2 ϖ ϖ ϖ ϖ ϖ ϖ ϖ j j j j j j j e e e e e e e - - - - - - - - + - + - + + = The overall impulse response is ] 7 [ 2 ] 5 [ 2 ] 4 [ 2 ] 3 [ 2 ] 2 [ 2 ] [ 2 ] [ - - - - - + - - - + = n n n n n n n h δ δ δ δ δ δ The overall impulse response using convolution of ] [ ] [ ] [ 2 1 n h n h n h = - + - + - + - + = ]) 4 [ 2 ] 3 [ 4 ] 2 [ 6 ] 1 [ 4 ] [ 2 ( n n n n n δ δ δ δ δ ]) 3 [ ] 2 [ 2 ] 1 [ 2 ] [ ( - - - + - - n n n n δ δ δ δ . + - + - + - + - + = ]) 4 [ 2 ] 3 [ 4 ] 2 [ 6 ] 1 [ 4 ] [ 2 ( n n n n n δ δ δ δ δ + - - - - - - - - - - ]) 5 [ 4 ] 4 [ 8 ] 3 [ 12 ] 2 [ 8 ] 1 [ 4 ( n n n n n δ δ δ δ δ + - + - + - + - + - ]) 6 [ 4 ] 5 [ 8 ] 4 [ 12 ] 3 [ 8 ] 2 [ 4 ( n n n n n δ δ δ δ δ - - - - - - - - - - ]) 7 [ 2 ] 6 [ 4 ] 5 [ 6 ] 4 [ 4 ] 3 [ 2 ( n n n n n δ δ δ δ δ ] 7 [ 2 ] 5 [ 2 ] 4 [ 2 ] 3 [ 2 ] 2 [ 2 ] [ 2 - - - - - + - - - + = n n n n n n δ δ δ δ δ δ The time domain convolution and frequency domain frequency response give the same result. Sinusoidal response of discrete-time systems (digital filters) Consider a discrete-time system with impulse response h [ n ]. The output is given by the linear convolution
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-∞ = - = k k n x k h n y ] [ ] [ ] [ Assume that the input ) ˆ ( ] [ φ ϖ + = n j Ae n x is a complex sinusoid (complex exponential signal) with the normalized radian frequency ϖ ˆ . The output
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