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Unformatted text preview: ENEE 241 02 * READING ASSIGNMENT 22 Tue 05/05 Lecture Topics: response of FIR filters to exponential inputs (continued); response of FIR filters to periodic inputs; cascaded filters Textbook References: sections 4.4.2, 4.4.4, 4.4.5 Key Points: For an FIR filter with frequency response H ( e j ), x [ n ] = r n cos( n + ) y [ n ] = fl fl H ( re j ) fl fl r n cos ( n + + H ( re j ) ) The response of an FIR filter to a periodic input can be computed by a circular convolution in the time domain, or, equivalently, by taking an element-wise product of DFTs in the frequency domain. Either technique can be used to generate an integer number of periods of the output signal. If two filters with system functions H 1 ( z ) and H 2 ( z ) are connected in series (cascade), the resulting filter has system function H ( z ) = H 1 ( z ) H 2 ( z ) , regardless of the order of the connection. Theory and Examples: 1 . We have seen that if x [ n ] = z n is the input to an FIR filter with coefficients b ,...,b M , then the output y [ ] is given by y [ n ] = H ( z ) z n , where H ( z ) = b + b 1 z- 1 + + b M z- M is the filters system function. 2 . If the input is the real-valued sinusoid x [ n ] = cos( n + ) = 1 2 e j e j n + 1 2 e- j e- j n , then, by linearity, the output is given by y [ n ] = 1 2 H ( e j ) e j e j n + 1 2 H ( e- j ) e- j e- j n Since H ( e- j ) = H * ( e j ), the expression above equals the sum of two complex conjugate terms, which is the same as twice the real part of either term: y [ n ] = < e n H...
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