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R24 - ENEE 241 02 READING ASSIGNMENT 24 Tue 05/12 Lecture...

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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 ω . 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 0 (for all n ) do not have a z -transform except in the trivial case z 0 = 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. 1
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δ H 1 h * h (2) (1) h (1) H 2 3 . By applying an input δ = δ [ · ] to the same cascade, we see (figure above) that the impulse response of the cascade is given by h = h (1) * h (2)
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