《信号与系统 Signals and Systems》课程教学资料（英文版）lecture11 Convolution Property example

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

Calculating periodic convolutions Suppose we integrate from-丌to丌: Y(eJu 2丌 1(e1)X2(la-=)de 2丌 where (c0),0≤丌 0 otherwise

Calculating Periodic Convolutions

sin(m/4)\2 Example: yn sin(rn/ 4 1|·T 7n Y(c)=X1(e)②X2(e) X(e e X(e/(o-B)i 0H2兀 Y(e jy

Example:

Duality in Fourier analysis Fourier Transform is highly symmetric CTFT: Both time and frequency are continuous and in general aperiodic Xw)et Same except for these differences X lw) Suppose f and g are two functions related by Te Let t=t and r=w: 1(t)=g(t)→X1(u)=f() Letr=- w and r=t:x2(t)=f(t)←→X2(j)=2xg(-)

Duality in Fourier Analysis Fourier Transform is highly symmetric CTFT: Both time and frequency are continuous and in general aperiodic Same except for these differences Suppose f(•) and g (•) are two functions related by Then

Example of ctft duality Square pulse in either time or frequency domain X1(j TUT1 TU/T1 X2(t) X2gjo W

Example of CTFT duality Square pulse in either time or frequency domain

DTES Discrete &e periodic in time Periodic discrete in frequency kwon k an+N k= k ∑n∪n=ak+N Duality in DTFS Suppose f. and gl are two functions related by fm=∑9rl groom N r= =∑ f merlo en Let m=n and r=-k 1]=fn]←→ak=N2 Let r=n and m= k 2n]=gmn]←→ak=fk

DTFS Duality in DTFS Then