Chapter 7 Sampling
1 Chapter 7 Sampling
Chapter 7 Sampling Continuous sampling Discrete Code Digital Signal Signal 1g Signal Continuous D/A Signal DSP x2() x() T 0 T 2T t x1(n)=x2(n7)=x(mr) x1()≠x2(t)≠x()
2 Chapter 7 Sampling Continuous Signal Discrete Signal sampling Code Digital Signal DSP Continuous D/A Signal -T 0 T 2T t x (t) 1 x (t) 2 x (t) 3 ( ) ( ) ( ) 1 2 3 x nT x nT x nT = = ( ) ( ) ( ) 1 2 3 x t x t x t
Chapter 7 Sampling 57.1 The Sampling Theorem 57. 1.1 Impulse-Train Sampling () p()=∑(-n7) x(7)x(2T n=-00 3T-2T-T0T 2T 3T 4T
3 Chapter 7 Sampling §7.1 The Sampling Theorem §7.1.1 Impulse-Train Sampling -3T -2T -T 0 T 2T 3T 4T t x(t) x(0) x(T) x(2T) x(t) x (t) p p(t) (t nT ) n = − + =−
Chapter 7 Sampling Sampling Theorem: Let x() be a band-limited signal with X(o)=0, o>OM then x(t) is uniquely determined by its samples x(nT)n=0,+1 if 2兀 0s>2OM where p()=∑6(-m7) HGo) P Hojo OM<oc<os=om 4
4 Chapter 7 Sampling Sampling Theorem: Let be a band-limited signal with then is uniquely determined by its samples if where ( ) M x(t) X j = 0 , x(t) x(nT),n = 0,1, s 2 M T s 2 = x (t) p x(t) x(t) p(t) (t nT ) n = − + =− H(j) M c s − M 0 T −c H(j) c
Chapter 7 Sampling he reconstruction of the signal nT HlO n=-00 P x Hojo 0 Ox(nT)sa on(t-nT) 5
5 Chapter 7 Sampling The reconstruction of the signal x (t) p x t r ( ) x(t) p(t) (t nT ) n = − + =− H(j) M c s − M 0 T −c H(j) c ( ) ( ) a H ( ) n x t x nT S t nT + =− = −
Chapter 7 Sampling §712 Natural Sampling P()= ∑ P(-n7) Hlo 1/4 n=-00 P Hla 0 OM<OC<Os=OM +0 x()={x()∑P(=nm)*6() 6
6 §7.1.2 Natural Sampling Chapter 7 Sampling x (t) p x t r ( ) x(t) ( ) ( ) n p t P t nT + =− = − H(j) M c s − M 0 0 1/ a −c H(j) c r ( ) ( ) ( ) ( ) n x t x t P t nT h t + =− = −
Chapter 7 Sampling 57. 1.3 Sampling with a Zero-Order Hold x( Zero-Order xo(O Hold x() x(7)x(27) -3T-2T-T0T 2T 3T. 4T 7
7 Chapter 7 Sampling §7.1.3 Sampling with a Zero-Order Hold x(t) Zero-Order Hold x(t) x (t) 0 x(0) x(T) x(2T) -3T -2T -T 0 T 2T 3T 4T t x t( ) ( ) p x t
Chapter 7 Sampling +∞ p(=2 (=m h2(o) P 0 Zero-Order hold x()=x2()1()=∑x(n)1(-nm) n=-00 8
8 Chapter 7 Sampling x (t) 0 x (t) p x(t) p(t) (t nT ) n = − + =− 0 1 T t h (t) 0 Zero-Order Hold x (t) x (t) h (t) x(nT )h (t nT ) n = p = − + =− 0 0 0
Chapter 7 Sampling Reconstruction SIt-nT Filter n=-0 P x xX h1(O) ( Zero-Order Hold p()=∑8(-m7) Xr x LPF Impulse-Train Sampling O<O<O.-0
9 Chapter 7 Sampling h (t) r x (t) r x (t) x(t) r = h (t) 0 x (t) 0 M c s − M LPF x (t) p x(t) x(t) p(t) (t nT ) n = − + =− Impulse-Train Sampling x (t) p x(t) p(t) (t nT ) n = − + =− Zero-Order Hold Reconstruction Filter
Chapter 7 Sampling 97.3 The Effect of Undersampling: Aliasing 欠采样 混叠 @s <20M x,()≠x() x,()=∑x(m7) Tsina(t-nr) r(t-nr) Let 0= 2 —T x()=∑x(n)Sn( t-nT x,(nr)=x(nr)
10 Chapter 7 Sampling §7.3 The Effect of Undersampling: Aliasing 欠采样 混叠 2 s M ( ) ( ) r x t x t ( ) ( ) ( ) ( ) sin c r n T t nT x t x nT t nT + =− − = − 2 s c T Let = = ( ) ( ) ( ) r a n x t x nT S t nT T + =− = − ( ) ( ) r x nT x nT =
