Chapter 4:Sampling of Continuous-Time Signals 4.0 Introduction 4.1 Periodic Sampling 4.2 Frequency-Domain Representation of Sampling 4.3 Reconstruction of a Bandlimited Signal from its Samples 4.4 Discrete-Time Processing of Continuous- Time signals 4.5 Continuous-time Processing of Discrete- Time Signal 2
2 Chapter 4: Sampling of Continuous-Time Signals 4.0 Introduction 4.1 Periodic Sampling 4.2 Frequency-Domain Representation of Sampling 4.3 Reconstruction of a Bandlimited Signal from its Samples 4.4 Discrete-Time Processing of ContinuousTime signals 4.5 Continuous-time Processing of DiscreteTime Signal
4.0Introduction Discrete-Time Processing of Continuous-Time signals >Continuous-time signal processing can be implemented through a process of sampling, discrete-time processing,and the subsequent reconstruction of a continuous-time signal. C/D Discrete-time D/C x(n] system y[n] T 下：sampling period xn,1/:sampling frequency (rate)in Hz -oK<n<oo 2,=2π/T,(rad/s) sampling frequency(rate)in radians/s 3
and the subsequent reconstruction of a continuous-time signal. 3 4.0 Introduction ➢Continuous-time signal processing can be implemented through a process of sampling, ( ), f=1/T: sampling frequency (rate) in Hz x n x nT c n = − =s 2 , / T rad s ( ) T: sampling period discrete-time processing, Discrete-Time Processing of Continuous-Time signals sampling frequency(rate) in radians/s
C/D converter Unit 00 4.1 Periodic impulse s(0=∑6t-nT) traim Sampling 1=-o0 单位冲激串 Conversion from impulse train x(0 to discrete-time x(n]=xc(nT) Continuous- sequence time signal impulse train sampling T=T1 x( x,(4)=x(☑）∑δ(t-nT) (冲激串采样) n= 4T-27 2T4工 T =∑x.(nT)ò(t-nT） x(n] n=-00 sampling period x[n]=x(nT) Sampling sequence -2-101234 (采样序列) 4
4 ( ) n t nT =− = − 4.1 Periodic Sampling Continuoustime signal Unit impulse train ( ) ( ) c n x nT t nT =− = − ( ) ( ) ( ) s c n x t t nT x t =− = − [ ] ( ) c x n = x nT impulse train sampling T: t sampling period n 单位冲激串 Sampling sequence (采样序列) (冲激串采样)
4.2 Frequency-Domain Representation of Sampling 单位冲激串的傅立叶变换：S2)-2平∑δ2-k2) T:sample period;fs=1/T:sample rate;s-2/T:sample rate s()为单位冲激串函数，可展开傅立叶级数 sG∑ot-n0=之aenr=72e2w s(t) tycoidi- Is() ejkS2stF→2π6(S2-k2s) 2π +S(j2) sU0)2平∑s0-k2) 2π 2π T T 5
5 T：sample period; fs=1/T:sample rate;Ωs=2π/T:sample rate ( ) ( ) n s t t nT =− = − ( ) ( ) 2 s k S j k T =− = − ( ) ( ) 2 s k S j k T =− = − k k a e jk ts =− = s(t)为单位冲激串函数，可展开傅立叶级数 1 k s jk t T e =− = 2 ( ) F s jk ts e k → − -T 1 t 0 T s t( ) … … / 2 / 2 ( ) 1 1 s T jk t k T a e dt T T t + − − = = 0 S j ( ) … … 2 T 2 T 2 T − 单位冲激串的傅立叶变换： 4.2 Frequency-Domain Representation of Sampling s t( )
4.2 Frequency-Domain Representation of Sampling s0)∑δt-nsU22牙∑2-k2) x(④=x.0s0=x0∑6t-nI)-∑x6-n) (冲激串采样) xn X.-sx.U(-0do 利用调制性质：时域相乘， =2元2妥立0-2)xUQ-00 频域卷积，来求X,(2) =号260-k0)X0Q-6do=7ΣX(-kD》 X(2)的周期延拓 then represent X(ei)of x[n] xe(t) x(t in terms of x,(j). -4-3-2-101234 -2T-T 0 T 2T
( ) ( ) ( ) 1 * 2 X j X j S j s c = c c ( ) ( ) ( ) ( ) n n x t t nT x nT t nT =− =− = − = − 6 4.2 Frequency-Domain Representation of Sampling x n[ ] ( ) ( ) n s t t nT =− = − x t x t s t s c ( )= ( ) ( ) ( ) ( ) FT 2 s k S j k T =− ⎯→ = − ( ( )) 1 c k X j k s T =− = − then represent of x[n] in terms of . X j s ( ) ( ) ( ) 1 ( ) 2 2 s c k k j d T X =− − = − − ( ) ( ) 1 ( ) s c k X j T k d − =− = − − ( ) ( ) 1 ( ) 2 S j X j c d − = − ( ) j X e 利用调制性质:时域相乘, 频域卷积,来求 ( ) X j s X j c ( ) 的周期延拓 (冲激串采样)