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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 jω ), x [ n ] = r n cos( ω n + φ ) ⇒ y [ n ] = fl fl H ( re jω ) fl fl r n cos ( ω n + φ + ∠ H ( re jω ) ) • 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 elementwise product of DFT’s 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 filter’s system function. 2 . If the input is the realvalued sinusoid x [ n ] = cos( ω n + φ ) = 1 2 · e jφ e jω n + 1 2 · e jφ e jω n , then, by linearity, the output is given by y [ n ] = 1 2 · H ( e jω ) e jφ e jω n + 1 2 · H ( e jω ) e jφ e jω n Since H ( e jω ) = H * ( e jω ), 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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This note was uploaded on 10/18/2011 for the course ENEE 241 taught by Professor Staff during the Spring '08 term at Maryland.
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