ps6_fall09

# ps6_fall09 - Department of Electrical and Computer...

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Department of Electrical and Computer Engineering Digital Speech Processing Homework No. 6 Problem 1 The complex cepstrum, ˆ x [ n ], of a sequence x [ n ] is the inverse Fourier transform of the complex log spectrum ˆ X ( e ) = log | X ( e ) | + j arg[ X ( e )] Show that the cepstrum c [ n ], deﬁned as the inverse Fourier transform of the log magnitude, is the even part of ˆ x [ n ]; i.e., show that c [ n ] = ˆ x [ n ] + ˆ x [ - n ] 2 Problem 2 A linear time-invariant system has the transfer function: H ( z ) = 8 1 - 4 z - 1 1 - 1 6 z - 1 (a) Find the complex cepstral coeﬃcients, ˆ h [ n ], for all n . (b) Sketch ˆ h [ n ] versus n for the range - 10 n 10. (c) Solve for the (real) cepstrum coeﬃcients, c [ n ], for all n . Problem 3 Consider a ﬁnite length minimum phase sequence x [ n ] with complex cepstrum ˆ x [ n ], and a sequence y [ n ] = α n x [ n ] with complex cepstrum ˆ y [ n ]. (a) If 0

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ps6_fall09 - Department of Electrical and Computer...

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