Anti-Image Postfilter (7B) Young Won Lim 7/27/12
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1 Ani-Image Posfiler (7B)
2 Copyrigh (c) 2012 Young W. Lim. Permission is graned o copy, disribue and/or modify his documen under he erms of he GNU Free Documenaion License, Version 1.2 or any laer version published by he Free Sofware Foundaion; wih no Invarian Secions, no Fron-Cover Texs, and no Back-Cover Texs. A copy of he license is included in he secion eniled "GNU Free Documenaion License". Please send correcions (or suggesions) o youngwlim@homail.com. This documen was produced by using OpenOffice and Ocave.
3 Sampler Ideal Sampling x() Ideal Reconsrucor ŷ() ŷ() 7B Posfiler 3
4 = = CTFT of Reconsrucors (1) = ±, ±2, ±3, h( ) = 0 1 h( ) 1 f s 2 X h() = sin(π /) = sin(π f s ) π / π f s CTFT H ( f ) 0 = 2π 1 * CTFT f s f s /2 /2 H ( f ) =, f f s /2 0, oherwise B Posfiler 4
5 Sampling (2) Effec of sampling f, f ± f s, f ± 2 f s, f ± 3 f s, Replace he original frequency f Wih he replicaed se of Ideal reconsrucor Exracs from a sampled All he frequency componens Tha lie wihin Nyquis inerval Removes all frequencies ouside ha inerval Lowpass filer Cuoff frequency [ f s 2, 2 ] 7B Posfiler 5
6 Sampling (2) Guard Band δ = f s 2 f max 7B Posfiler 6
7 saircase reconsrucor Ani-image Lowpass Posfiler ŷ() = y(nt ) Digial DAC y a () Analog Posfiler y POST () Analog ŷ() = y (nt ) y a () y POST () 7B Posfiler 7
8 ideal reconsrucor Ani-image Lowpass Posfiler ŷ() = y(nt ) Digial DAC y a () Analog Posfiler y POST () Analog A A f s f s /2 /2 f s f s /2 /2 7B Posfiler 8
9 saircase reconsrucor Ani-image Lowpass Posfiler ŷ() = y(nt ) Digial DAC y a () Posfiler y POST () Analog A A A f s f s /2 /2 f s f s /2 /2 f s f s /2 /2 surviving replica surviving replica Non-fla: Parially aenuaed 7B Posfiler 9
10 CTFT of Reconsrucors (1) = ±, ±2, ±3, h( ) = 0 h( ) 1 1 f s 2 h() = sin(π /) = sin(π f s ) π / π f s CTFT CTFT H ( f ) f s f s /2 /2 H ( f ) =, f f s /2 0, oherwise 7B Posfiler 10
11 * * = = Reconsruc via Convoluion Ideal Reconsrucor Pracical Reconsrucor Sampling frequency T = f T s = y a ( ) y a ( ) 7B Posfiler 11
12 Reconsrucors in Frequency Domain Ideal Reconsrucor ŷ() Pracical Reconsrucor ŷ() CTFT CTFT A A f s f s /2 /2 f s f s /2 /2 7B Posfiler 12
13 A f s f s /2 /2 Surviving specral replicas can be removed by an addiional lowpass filer Ani-image Posfiler A f s f s /2 /2 surviving replica surviving replica Non-fla: Parially aenuaed Cuoff Frequency f max f s 2 Freq domain (reconsrucor + posfiler) o remove he specral replicas as much as possible Time domain Effec of rounding off he corners of saircase oupu making smooher Two sage (Saircase Reconsrucor + Posfiler) simpliciy of implemenaion of reconsrucor : DAC generaing an analog oupu ha remains consan during T Emulae he ideal reconsrucor 7B Posfiler 13
14 Analog Reconsrucor x() T T ŷ() = + n = y(nt ) δ( nt ) Y a ( f ) = H ( f )Ŷ ( f ) + y a () = h( ') ŷ( ' ) d ' Ŷ a ( f ) = 1 T + m = Y ( f m f s ) y a () = + n = y(nt )h( nt ) 7B Posfiler 14
15 Impulse Response of Ideal Reconsrucor x() T Ŷ ( f ) = 1 T Y ( f ) f s 2 f 2 y() = + n = y(nt )h( nt ) h( ) = sin(π /T ) π /T = sin (π f s ) π f s 7B Posfiler 15
16 X i ( f ) X ( f ) X ( f ) Analog Ideal Prefiler H(f) Bandlimied Ideal Sampler - ADC Sampled 0 1 f 2 s 0 f s 0 1 f 2 s 2 4 f s 3 4 f s f s 7B Posfiler 16
17 7B Posfiler 17
18 7B Posfiler 18
19 References [1] hp://en.wikipedia.org/ [2] J.H. McClellan, e al., Signal Processing Firs, Pearson Prenice Hall, 2003 [3] A graphical inerpreaion of he DFT and FFT, by Seve Mann [4] R. G. Lyons, Undersanding Digial Signal Processing, 1997 [5] AVR121: Enhancing ADC resoluion by oversampling [6] S.J. Orfanidis, Inroducion o Signal Processing
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