ECE 431 Digital Signal Processing Lecture Notes

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ECE 431Digital Signal ProcessingLecture NotesProf. Dan CobbContents1 Introduction22 Review of the DT Fourier De nition and Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Periodic Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 Time and Frequency Domain Analysis . . . . . . . . . . . . . . . . . . . . . . . . . Aliasing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Nyquist Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Anti-Aliasing Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Downsampling.

Since X ej and Y ej both have period 2ˇ;the linear (i.e. ordinary) convolution blows up (except in trivial cases): Z1 1 X 1 ej( 2 X 2 ej d = X1 i=1 Z ˇ(i+1) 2ˇi X 1 ej( X 2 ej d = X1 i=1

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