Signals and Systems Fall 2003 Lecture #14 23 October 2003 Review/Examples of Sampling/Aliasing 2. DT Processing of CT Signal
Signals and Systems Fall 2003 Lecture #14 23 October 2003 1. Review/Examples of Sampling/Aliasing 2. DT Processing of CT Signals
Sampling review p()=∑8(tnT) HOjO) x(t T x) If X(u)=0, w>wM and ws => 2wM then, assuming we choose wm <Wc< Ws-WM r(t=a( Demo: Effect of aliasing on music
Sampling Review Demo: Effect of aliasing on music
Strobe demo Rotating disc X(t) p(t) T+△ Strobe A>0, strobed image moves forward, but at a slower pace A=0, strobed image still A<0, strobed image moves backward Applications of the strobe effect(aliasing can be useful sometimes) - E.g., Sampling oscilloscope
Strobe Demo ∆ > 0, strobed image moves forward, but at a slower pace ∆ = 0, strobed image still ∆ < 0, strobed image moves backward. Applications of the strobe effect (aliasing can be useful sometimes): — E.g., Sampling oscilloscope
DT Processing of Band-Limited CT signals Xan=Xc(nT ya[n]=yc(nT) C/D Discrete-Time D/C xc(t) y() conversion System conversion H(e Why do this? Inexpensive, versatile, and higher noise margin How do we analyze this system? We will need to do it in the frequency domain in both ct and dt In order to avoid confusion about notations specify CT frequency variable Q2--DT frequency variable(Q2=OT) Step 1: Find the relation between xe (t) and x[n], or X gjo) and Xd(ei2
DT Processing of Band-Limited CT Signals Why do this? — Inexpensive, versatile, and higher noise margin. How do we analyze this system? — We will need to do it in the frequency domain in both CT and DT — In order to avoid confusion about notations, specify ω — CT frequency variable Ω — DT frequency variable (Ω = ωΤ) Step 1: Find the relation between xc(t) and xd[n], or Xc(jω) and Xd(ejΩ)
Time-Domain Interpretation of C/D Conversion C/D conversion Conversion of p(t) impulse tra to discrete-time xn]=×c(nT) Note: Not fulll sequence analog/digital (A/D) conversion not quantizing Xp(t) p() he xn values T=T1 T=2T 0 T 2T 2T t
Time-Domain Interpretation of C/D Conversion Note: Not full analog/digital (A/D) conversion – not quantizing the x[n] values
Frequency-Domain Interpretation of C/D Conversion ∑6(t ∑xl(nm)6t-nm) 儿F ∑X(i(-k)=∑aa T k CT (periodic with period Ws=2TT )=∑ Ealnle-jis2r ∑xl(n) m二-0 DT periodic with period2丌 l Compare Eqs.(1)& (2 )and note Q2=wT Xd(e=xp CT DT
Frequency-Domain Interpretation of C/D Conversion Note: ωs ⇔ 2 π CT DT
Illustration of C/D Conversion in the frequency-Domain CcJo) (jo) Xc(jo) TET T =2T 2 ∧∧……∧八… 2 2TU X(e-) Ⅹa(e-) ●●● ●●● ●●● T Q=OT 2T Q=OT
Illustration of C/D Conversion in the Frequency-Domain X (e ) jΩ d X (e ) jΩ d Ω = ω T1 Ω = ω T2
D/C Conversion yan]>y(t) Reverse of the process of c/D conversion D/C conversion Conversion of discrete-time yp(t) T yc(t) sequence to impulse train Again, Q=WT Yplw)=Yd(ea)- reverses frequency scaling Yuju) TYa(e1),同<当- bandlimited 0. otherwise
D/C Conversion y d [ n] → y c ( t ) Reverse of the process of C/D conversion
Now the whole picture Hojo) xp(t)Conversion of ixa[n] ya[n]: Conversion of yp(t) 0-+( impulse train Ha(e/o) sequence to y() to sequence mpule train DT processing C/D D/C Overall system is time-varying if sampling theorem is not satisfied It is lti if the sampling theorem is satisfied, i.e. for bandlimited inputs x(t), with 2 When the input x(t) is band-limited (Xo)=0 ata>@m and the sampling theorem is satisfied(@s>2am),then Ycjd)=He(ju)Xc(ju)←→3e(t)=h(+)*c(t)LTI hanged
Now the whole picture • Overall system is time-varying if sampling theorem is not satisfied • It is LTI if the sampling theorem is satisfied, i.e. for bandlimited inputs xc(t), with • When the input xc(t) is band-limited (X(jω) = 0 at |ω| > ωΜ) and the sampling theorem is satisfied (ωs > 2ωM), then ω M < ωs 2 DT omege needs to changed
Frequency-Domain lustration of DT Processing of cT signals Ha(en),xa(e/) DT filter xa(e) Sampling Xp(o) Hp(o), Xp(io) DT freq→ CT freq 入∧ 0 Interpolate CT freq→ DT freq xa(el Hc(jo), Xc (jo) (LPF) equivalent CT filter T=2; c
Frequency-Domain Illustration of DT Processing of CT Signals Sampling DT filter Interpolate (LPF) ⇓ equivalent CT filter CT freq → DT freq DT freq → CT freq