Frequency Response of an LSI System 1.2 2004/07/07 14:36:22 GMT-5 2004/07/16 09:22:56.297 GMT-5 Richard G. Baraniuk richb@rice.edu Robert Ahlfinger ahlfing@rice.edu Richard G. Baraniuk richb@rice.edu LSI ELEC 301 An introduction to the frequency response of an LSI system.

H : circulent matrix h : H 's 0th column H : DFT of h ; H F H h contain equivalent info! Effect of LSI system on an input x is easy to describe in the Fourier domain (frequency domain)...

Time Domain

Frequency Domain

ie: Y k H k X k ; 0 k N 1 ie: pointwise multiplication

2 point smoother

Compute frequency response H if LSI system... H k 1 N n N 1 0 h n 2 N k n 1 N n N 1 0 1 2 2 N k n 1 N 1 2 N k 1 N 2 N k 2 2 N k 2 2 N k 2 2 N N k N k H k 2 N N k

where 0 is low frequency, is high frequency, 2 is low frequency, and w k 2 n k Therefore it is a lowpass filter

ie: smoothing lowpass filtering

ie: h is a "square box". The Frequency response is (using answer from test 1): H k 1 N M N k N k N M 1 k Where M N k N k is the "Dirichlet Kernel." H 0 ? 0 0 L'Hopitâl's rule to the rescue... k 0 H k k 0 k numerator k 0 k denominator 1 N k 0 M N M N k k 0 N N k 0 1 N M M N

Edge Detector

H k 1 N n 0 N 1 h n 2 N k n 1 N 1 2 N k n 1 N 2 N k 2 2 N k 2 2 N k 2 1 N 2 N k N k

High Pass Filter

Circular Shift

H k 1 N n 0 N 1 h n 2 N k n 1 N 2 N k m H k 1 N , ∡ H k 2 N k m

Alll Pass Filter

That is, delay "linear phase shift." (slope m 2 N