EE123 Digital Signal Processing

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1 EE23 Digital Signal Processing Lecture 7B Sampling

2 What is this Phenomena?

3 Sampling of Continuous ime Signals (Ch.4) Sampling: Conversion from C. (not quantized) into D. (usually quantized) Reconstruction D. (quantized) to C. Why? Digital storage (audio, images, videos) Digital communications (fiber optics, cellular...) DSP (compression, correction, restoration) Digital synthesis (speech, graphics)

4 Sampling of C.. Signals ypical System: ADC A/D x c (t) Analog Anti-Aliasing Filter sampler t = n x[n] =x c (n ) Quantizer Discrete stuff (DSP, storage...) y[n] DAC D/A Reconstruction y c (t)

5 Ideal Sampling Model x c (t) C/D x[n] =x c (n ) x[n] n define impulsive sampling: x c (t) t Discrete and Continuous x s (t) x s (t) =+ x c (0) (t)+x c ( ) (t )+ x s (t) =x c X Continuous x c (t) t (t n ) n=

6 Ideal Sampling Model x s (t) =x c X n= (t n ) Not physical: used for modeling & derivations x[n] $ x s (t) $ x c (t) How is x[n] related to xs(t) in freq. domain?

7 Frequency Domain Analysis How is x[n] related to xs(t) in the Freq. Domain? x s (t) :C. X s (j ) = X n x c (n )e j n x[n] :D. X(e j! )= X n x[n]e j!n! = X(e j! )=X s (j ) =!/ X s (j ) =X(e j! )!= x s (t) =x c X n= (t n )

8 Frequency Domain Analysis How is xs(t) related to xc(t)? x s (t) =x c (t) X (t n ) n {z },s(t)

$s c n = Y:cU) Z...!f{f-n) {z _9.$

9 Frequency Domain Analysis How is xs(t) related to xc(t)? X 7 _ -I'O<? x (t) = x (t) X,(t If) n lh f{. ) dom,h. s c n = Y:cU) Z...!f{f-n) {z _9. s }

10 Frequency Domain Analysis How is xs(t) related to xc(t)? x s (t) =x c (t) X (t n ) n {z },s(t) s(t) $ S(j ) S(j ) = 2 X k= ( 2 k) = s 2 s(t) S(j ) 2

11 Frequency Domain Analysis X s (j ) = 2 X c(j ) S(j ) = X X c (j( k s )) s = 2 k= Xs is replication of Xc!

12 Frequency Domain Analysis s(t) 2 S(j ) 2 So, if : X c (j ) and s > 2 X s (j ) s /2 s! = X(e j ) s 2 =

13 Aliasing s(t) 2 S(j ) 2 So, if : X c (j ) and s < 2 X s (j ) s /2 s! = X(e j! ) s 2 =

14 Aliasing Q: What is the difference in acquisition between the two images?

15 Δk=/FOV FOV

16 IV tube

17 Reconstruction of Bandlimited Signals Nyquist Sampling hm: suppose xc(t) is bandlimited X c (j ) =0 8 if s > 2, then x c (t) can be uniquely determined from its samples x[n] =x c (n ) Bandlimitedness is the key to uniqueness x[n] x c (t) multiple signals go through the samples, but only one is bandlimited! n t

18 Reconstruction in Frequency Domain x[n] Convert to impulse train X s (j ) s /2 x s (t) H r (j ) x r (t) H r (j ) s /2 s /2 X r (j )

19 Reconstruction in ime Domain - h r (t) = 2 2 = 2 = t Z s /2 s /2 jt sj t e j t d s /2 s /2 e j s 2 t e j s 2 t 2j = t sin( s 2 t)= t sin( t) = sinc( t )

20 Reconstruction in ime Domain! X x r (t) =x s (t) h r (t) = x[n] (t n ) n = X x[n]h r (t n ) n h r (t) he sum of sincs gives xr(t) Unique signal bandlimited by s......

21 Aliasing If > s /2, x r (t) an aliased version of x c (t) Xs (j ) if apple s /2 X r (j ) = 0 otherwise

22 Anti-Aliasing ADC A/D x c (t) Analog Anti-Aliasing Filter HLP(jΩ) sampler t = n x[n] =x c (n ) Quantizer X c (j ) and s < 2 X s (j ) s /2 s X c (j )H LP (j ) and s < 2 X s (j ) s /2 s /2 s

23 Non Ideal Anti-Aliasing X c (j )H LP (j ) interference s /2 X(e j )

Signals & Systems. Chapter 7: Sampling. Adapted from: Lecture notes from MIT, Binghamton University, and Purdue. Dr. Hamid R.

Signals & Systems. Chapter 7: Sampling. Adapted from: Lecture notes from MIT, Binghamton University, and Purdue. Dr. Hamid R. Signals & Systems Chapter 7: Sampling Adapted from: Lecture notes from MIT, Binghamton University, and Purdue Dr. Hamid R. Rabiee Fall 2013 Outline 1. The Concept and Representation of Periodic Sampling