信号与系统（英文版）_7. Sampling

7 Sampling 7. Sampling 7.1 Representation of a Continuous-time Signal by its Samples: The Sampling Theorem 7. 1.1 Impulse-train Sampling (1) Sampling p(t) ()=x(t)p() 3[X(j)*P(m) x() where ∑ d(t-nT) n=-0

7 Sampling 7.1 Representation of a Continuous-time Signal by its Samples: The Sampling Theorem 7. Sampling      = = [ ( )* ( )] 2 1 ( ) ( ) ( ) ( )    X j X j P j x t x t p t p p 7.1.1 Impulse-train Sampling x(t) p(t) xp (t)  + =− = = − n T where p(t)  (t)  (t nT) (1) Sampling

7 Sampling Time domain: x2()=x(1)61()=∑x(n)(-nT) 1=-00 0 p(t) T (t)

7 Sampling Time domain:  + =− =  = − n p T x (t) x(t)  (t) x(nT) (t nT)

7 Sampling Frequency domain x(1)<>X(jO) plt) K AS a Periodic signa. p()P(jo)=∑2mb(0-kO,)=∑0,(0-ko,) k=-00 ()4X2(10)=∑X(m-k)=∑X(o-kO,) 丌

7 Sampling     + =− + =− + =− + =− ⎯→ = − = − ⎯→ = − = −  ⎯→ = ⎯→ k s k s s p F p k s s k k s F k F S F X k T x t X j X k p t P j a k k Periodic signal T p t a x t X j ( ) 1 ( ) 2 ( ) ( ) ( ) ( ) 2 ( ) ( ) ( ) 1 ( ) ( ) ( ) . .                  Frequency domain:

7 Sampling XGo) PGw) 2 2 XpO) PoJo)

7 Sampling

7 Sampling (2) (Shannon) Sampling theorem Let x(t be a band-limited signal with X(o=0 for lo>OM. Then x(t) is uniquely determined by its samples X(nT),n=0,±1,2,…,f Os>2 OM, where @s=2T/T 20M is called Nyquist Rate Minimum distortionless sampling frequency

7 Sampling (2) (Shannon) Sampling theorem Let x(t) be a band-limited signal with X(j )=0 for ||> M . Then x(t) is uniquely determined by its samples x(nT),n=0,1,2,…, if s>2 M, where s =2/T . 2M is called Nyquist Rate. ( Minimum distortionless sampling frequency )

7 Sampling (3)Recovery System for sampling and reconstruction ∞ p(t)=∑8(t-nT) X() H(jo) X,()

7 Sampling (3) Recovery System for sampling and reconstruction:

7 Samplin p n X 2 Hojo <ω。<(o X (jo) DM (M

7 Sampling

7 Sampling 7. 1. 2 Sampling with a Zero-order Hold (1 Sampling system construction p() ( () X(t) X0(

7 Sampling 7.1.2 Sampling with a Zero-order Hold (1) Sampling system construction:

7 Sampling p()

7 Sampling

7 Sampling 2)Signal Recovery HGw) () M→)./0 o() h+() () H I Hr(jo) ≮H(u)

7 Sampling (2) Signal Recovery