Signals and Systems Fall 2003 Lecture #9 2 October 2003 The Convolution Property of the ctFt 2. Frequency Response and Lti Systems Revisited 3. Multiplication property and Parseval's relation The dt fourier transform
Signals and Systems Fall 2003 Lecture #9 2 October 2003 1. The Convolution Property of the CTFT 2. Frequency Response and LTI Systems Revisited 3. Multiplication Property and Parseval’s Relation 4. The DT Fourier Transform
The ct fourier transform Pair X lw) X u) (t)e Ju dt FT (Analysis Equation () X Gjw)e dw-Inverse Ft (Synthesis Equation Last lecture: some properties Today urther exploration
The CT Fourier Transform Pair Last lecture: some properties Today: further exploration (Synthesis Equation) (Analysis Equation)
Convolution property (t)=h(t)*(t)←→Y(ju)=H(j)X(j where h()→H(j) a consequence of the eigenfunction property Xiju dw coefficient a h(t) H(jw)aejwt H(u)- X(ju)dw H(w) 2 H(jw)X(w)eo da Synthesis equation Yu) y
Convolution Property A consequence of the eigenfunction property: Synthesis equation for y(t)
The frequency response revisited impulse response () y(t)=h(t)米x() Y(w)=hgw)X(w) frequency response The frequency response of a Ct Lti system is simply the Fourier transform of its impulse response Example#1: c(t) Recall 2丌6( Y(ju)=H(ju)X(j)=H(j)2m6(u-Wo)=2xH(jdo)6(u-o) v inverse FT y(t)=lwo)
The Frequency Response Revisited The frequency response of a CT LTI system is simply the Fourier transform of its impulse response Example #1: impulse response frequency response
Example #2: a differentiator da(t dt an ltI system Differentiation property: Y()=jw x(jw H 1)Amplifies high frequencies(enhances sharp edges) Larger at high 2)+丌/2 phase shift(j=c phase shift wo cos wot wo sin(wot+ dt cOS wl -wo sin wot wo cos(wot+
Example #2: A differentiator 1) Amplifies high frequencies (enhances sharp edges) Larger at high ωo phase shift Differentiation property:
Example #3: Impulse response of an Ideal Lowpass Filter HgO SIn w Questions Sinc Is this a causal system? No Define:sinc)ssin丌 2)What is h(0)? 6 h(0) Hlu du 2 3)What is the steady-state value of the step response, i.e. S(oo)? (t) h(t) h(t)dt=H(0)=1
Example #3: Impulse Response of an Ideal Lowpass Filter 2) What is h(0)? No. Questions: 1) Is this a causal system? 3) What is the steady-state value of the step response, i.e. s(∞)?
Example#4: Cascading filtering operations H1(j0) Hojo) HgO)=H,GO) H,gjo egH1(j0)=H2(j0) H(j0) Hgo= HGjo) has a sharper frequency
Example #4: Cascading filtering operations H(jω)
Example #5: sin 4t sin &t Y(w)=X(w 丌t (t) h(t) XGo) HojO) Ygo) 4π Example #6: T 米已 a+ 已 Gaussian x Gaussian=Gaussian= Gaussian s Gaussian= gaussian
Example #5: Gaussian × Gaussian = Gaussian ⇒ Gaussian ∗ Gaussian = Gaussian Example #6:
Example #2 from last lecture 1/a X(w) (t)e Ju dt a dt 0+/e-w… t 3 X(jo)=1/(a2+o) ∠Ⅹ(j0)=tan(oa) π/2 1/a 兀/4 1/ay2 a a a a /2
Example #2 from last lectu r e
Example #7: °·。 米 Y(j)=H(u)X(j)三 (1+j):(2+j a rational function of jw, ratio of polynomials of jw V Partial fraction expansion Y 1+w 2+ju v inverse FT
Example #7:
