Chapter 4 sampling of continous-time signals 4. 1 periodic sampling .2 discrete-time processing of continuous-time signals 4.3 continuous-time processing of discrete-time signal 4. 4 digital processing of analog signals 5 changing the sampling rate using discrete-time processing
Chapter 4 sampling of continous-time signals 4.5 changing the sampling rate using discrete-time processing 4.1 periodic sampling 4.2 discrete-time processing of continuous-time signals 4.3 continuous-time processing of discrete-time signal 4.4 digital processing of analog signals
4.1 periodic sampling 1. ideal sample x[s]=x(o1-=X(I T:sample period fs-1/T sample rate s=2兀/T: sample rate x(1) x2(D) x s( 2T-T0 2T t 2T-T0 T 2T rIn 101234n 2-101234
4.1 periodic sampling 1.ideal sample x[n] x (t)| x (nT) = c t=nT = c T：sample period fs=1/T:sample rate Ωs=2π/T:sample rate
CD x2() xn=xc(nT) Figure 4.1 ideal continous-time-to-discrete-time(C/D)converter
Figure 4.1 ideal continous-time-to-discrete-time(C/D)converter
time normalization t→t/T=n C/ converter s(t onversion from impulse train x() to discrete-time x团n]=x(n7 sequence Figure 4.2(a)mathematic model for ideal C/D
Figure 4.2(a) mathematic model for ideal C/D =− = − n (t nT) time normalization t→t/T=n
AcUs F ture 43 gt BN 2) S(n) frequency spectrum change of ideal sample -29 29 30.Ω Xu C2-C2N≥2N A)=7∑x(a No aliasing C2-2N< aliasing 米1 (3-9x dX(e0)=Xs(1g)l=0/7 aliasing frequency 丌 O=QT XC((O-k2)/7
Figure 4.3 s −N N frequency spectrum change of ideal sample s −N N aliasing frequency No aliasing aliasing =− = − k s c s X j k T X j ( ( )) 1 ( ) = T =− = = − = k c s T j X j k T T X e X j ( ( 2 )/ ) 1 ( ) ( ) | / / 2 s 2
Period =2in time domain 2. 1and w=0. 1 are the same 05 .5 012345 trigonometric function property -y coa(s'Tan =co2(0 av)
cos(2.1n) = cos(0.1n) Period =2πin time domain： w=2.1πand w=0.1πare the same trigonometric function property
high frequency is changed into low frequency in time domain: w-l. l and w=0are the same 0.5 0 -0.5 10 trigonometric function property co?(r]s)=c02(0)
cos(1.1n) = cos(0.9n) high frequency is changed into low frequency in time domain： w=1.1π and w=0.9πare the same trigonometric function property
2. ideal reconstruction D/C Figure 4.10(b) ideal D/C converter
2.ideal reconstruction Figure 4.10(b) ideal D/C converter
ideal reconstruction in frequency domain Xeon) Xso) >2N 2 Hr(n) N<<(23-N) F1 gur 4 X()
Figure 4.4 ideal reconstruction in frequency domain = s / 2
EXAMPLE Figure 4.5 Take sinusoidal signal for example to understand aliasing from frequency domain X(j9) X、(j9) No aliasing 丌丌 X、(j) 40 ↑,↑ .-9 (3-)(23-)
EXAMPLE Figure 4 . 5 Take sinusoidal signal for example to understand aliasing from frequency domain s − 0