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 (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 . 2M 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: