# lect2 - Department of Computer Science and Electrical...

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Signal and Image Processing I Brian Lovell Department of Computer Science and Electrical Engineering 1 12/03/2003 Sines and Cosines From the phasor diagram, the difference between sines and cosines is just the point of origin ft Ff ft T Ffe fte f jf T t () ( ) () ( ) ( ) −⇔ ⇔− 2 2 0 0 π π Delay Theorem Time Delay Phase Shift Similarly Modulation Theorem ptrain1

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Signal and Image Processing I Brian Lovell Department of Computer Science and Electrical Engineering 2 12/03/2003 Phase Modulation Produce sinusoid with arbitrary phase cos sin + cos2 π ft φ cos( ) 2 π φ Often used to produce a stable FM modulator sin 2 π
Signal and Image Processing I Brian Lovell Department of Computer Science and Electrical Engineering 3 12/03/2003 How to produce sine from cosine? Need all-pass wideband 90 degree phase shifter. Hilbert transformer 90 degree phase shift Xt () \$ \$ X t d t = =∗ −∞ +∞ τ τ τ 1 filter impulse response

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Signal and Image Processing I Brian Lovell Department of Computer Science and Electrical Engineering 4 12/03/2003 Hilbert Transformer Frequency response: pure imaginary Re Xt () Im ) sgn( ) ( f j f H = f \$ Hilbert Transformer jXt \$ + Analytic signal
Signal and Image Processing I Brian Lovell Department of Computer Science and Electrical Engineering 5 12/03/2003 Even and Odd Functions Even Odd Not Odd T h(-t)=h(t) h(-t)=-h(t) g(-t)=g(t) g(-t)=-g(t) h(t) h(t) g(t) g(t) t t Not Even t t T

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Signal and Image Processing I Brian Lovell Department of Computer Science and Electrical Engineering 6 12/03/2003 Finding Even and Odd Parts h(t) t h(-t) t Ev(t) h(t)+h(-t) 22 t Od(t) h(t)-h(-t) t
Signal and Image Processing I Brian Lovell Department of Computer Science and Electrical Engineering 7 12/03/2003 Even and Odd Functions All functions can be split into even and odd parts in the following manner. Ev x H x H x () ( ) = + 2 Od x H x H x = 2 Note: for discrete sequences, this decomposition is a 2 point DFT

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