R21 - ENEE 241 02 READING ASSIGNMENT 21 Thu 04/30 Lecture...

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Unformatted text preview: ENEE 241 02* READING ASSIGNMENT 21 Thu 04/30 Lecture Topics: introduction to linear time-invariant filters; response to FIR filters to sinusoidal and exponential inputs; frequency response and system function Textbook References: sections 4.2.2, 4.3, 4.4.1, 4.4.3 Key Points: The input-output relationship y [ n ] = b x [ n ] + b 1 x [ n- 1] + + b M x [ n- M ] , n Z , describes a linear time-invariant system known as a finite impulse response (FIR) filter. If the input x [ ] to a FIR filter is the complex exponential sequence x [ n ] = z n (where z C ), 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. In particular, when x [ n ] = e jn , the output is given y [ n ] = H ( e j ) e jn , where H ( e j ) = b + b 1 e- j + + b M e- jM is the filters frequency response. If all coefficients b k are real-valued, then the amplitude response | H ( e j ) | is symmetric (even) about = 0 and , while the phase response H ( e j ) is antisymmetric (odd) about the same frequencies. The frequency response of an FIR filter with coefficient vector b = b 0: M can be obtained for N M + 1 uniformly spaced frequencies in [0 , 2 ) by zero-padding b to length N and computing a DFT. Theory and Examples: 1 . In developing the DFT, we considered discrete-time signals which are either vectors (i.e., consisted of finitely many samples) or periodic extensions thereof. We now turn our attention to general discrete-time signals, namely sequences such as x = x [ ] = { x [ n ] , n Z } If x is a linear combination of (not necessarily periodic) sinusoids, then it also has a spectrum . Its spectrum consists of the coefficients of these sinusoids given (or plotted) as a function of frequency....
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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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R21 - ENEE 241 02 READING ASSIGNMENT 21 Thu 04/30 Lecture...

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