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Unformatted text preview: ENEE 241 02 * READING ASSIGNMENT 24 Tue 05/12 Lecture Topics: convolution in the z-domain; frequency-selective filters Textbook References: sections 4.5, 4.6.5 Key Points: If X ( z ) def = X n =- x [ n ] z- n , then the input x [ ], impulse response h [ ] and output y [ ] of a linear time-invariant system are related by Y ( z ) = H ( z ) X ( z ) Thus convolution in the time domain amounts to multiplication in the (so-called) z-domain. Ideal filters have amplitude responses which are piecewise constant functions of . These responses are unattainable in practice. Ideal filters can be approximated by FIR filters. A longer coefficient vector affords a better approximation to an ideal amplitude response, at the expense of a longer delay between input and output. Theory and Examples: 1 . The z-transform of a sequence x = x [ ] is the power series defined by X ( z ) = X n =- x [ n ] z- n Convergence may be an issue if x [ ] has infnite duration. Notably, exponential sequences such as x [ n ] = z n (for all n ) do not have a z-transform except in the trivial case z = 0. 2 . The z-transform of a finite-duration sequence is always well defined; it was introduced earlier as the system function of a FIR filter: H ( z ) = M X n =0 b n z- n = X n =- h [ n ] z- n We have also seen that the system function H ( z ) of the cascade of two FIR filters (labeled 1 and 2) satisfies H ( z ) = H 1 ( z ) H 2 ( z ) This result was obtained by applying a two-sided exponential input to the cascade....
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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.
- Spring '08