Signals and systems Fall 2003 Lecture #11 9 October 2003 DTFT Properties and Examples 2. Duality in FS ft 3. Magnitude/Phase of Transforms and frequency responses
Signals and Systems Fall 2003 Lecture #11 9 October 2003 1. DTFT Properties and Examples 2. Duality in FS & FT 3. Magnitude/Phase of Transforms and Frequency Responses
Convolution Property example X(eu) e 小 y n xiNle ae Jw ratio of polynomials in A B ≠a:Y( eJw) PFE ae A, B- determined by partial fraction expansion yn= Aaun+ boeun Y 1-ae-yw dX(eJu ym]=(m+1)a2[m] d
Convolution Property Example
DT LTI SyStem Described by lccde's M ∑an-对=∑bxm一k k=0 From time-shifting property:xln-k←→c--X( ∑akc-1yY()=∑ bke JrX( k=0 M ke k=o ake jkw Rational function of ejo H(eJu) use PFe to get hn
DT LTI System Described by LCCDE’s — Rational function of e-j ω, use PFE to get h[n]
Example: First-order recursive system gIn-agIn-1=r with the condition of initial rest s causal ce )Y(e)=X(e°) Y H()= ae ja hn=aln
Example: First-order recursive system with the condition of initial rest ⇔ causal
DTFT Multiplication property yn]=x1{m2·x2m 2丌 X1()②X2(e1) Periodic convolution Derivation ∑x1m n=-0o ∑ Xi(eje )eien dea2lnle (X1(e)∑a2nle de 2 X2(e(u-6) X1e)X2(elu-ro)de 2
DTFT Multiplication Property