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

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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 0 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 0 + b 1 z - 1 + · · · + b M z - M is the filter’s system function. In particular, when x [ n ] = e jωn , the output is given y [ n ] = H ( e ) e jωn , where H ( e ) = b 0 + b 1 e - + · · · + b M e - jωM is the filter’s frequency response. If all coefficients b k are real-valued, then the amplitude response | H ( e ) | is symmetric (even) about ω = 0 and π , while the phase response H ( e ) 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 π ) 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. 2 . Linear filters can be used to alter the spectra of sequences in an immediate, “real time”, fashion. A linear filter H acts as a (linear) transformation of an input sequence x to an output sequence y . (Since the input and output spaces are infinite-dimensional, this linear
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