# Anti-aliasing Prefilter (6B) Young Won Lim 6/8/12

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1 ni-aliasing Prefiler (6B)

2 Copyrigh (c) Young W. Lim. Permission is graned o copy, disribue and/or modify his documen under he erms of he GNU Free Documenaion License, Version. or any laer version published by he Free Sofware Foundaion; wih no Invarian Secions, no Fron-Cover exs, and no Back-Cover exs. copy of he license is included in he secion eniled "GNU Free Documenaion License". Please send correcions (or suggesions) o his documen was produced by using OpenOffice and Ocave.

3 Sampler Ideal Sampling x() Pracical Sampling τ x() = x(n ) δ( n ) x() x(n ) p( n ) CF CF X ( f ) = x() e j π f d? 6B Prefiler 3

4 Zero Order Hold (ZOH) / rec( ) / τ x ZOH ( ) = n= / n x[n] rec( ) 6B Prefiler 4

5 Square Wave CFS () Coninuous ime Fourier Series C k = x( ) e j k d x() = n= C k e + j k C k = + / x / ( ) e j k d = π Fundamenal Frequency C k = / + / x () e j k d + / j k = / e [ d = + / e j k j k ] / C k sin(k /) k / = e j k / e + j k / = sin (k /) j k k / k = / C k + Period = 6B Prefiler 5

6 Square Wave CFS () C k = sin(k / ) k / = sin( k /) k / = π Fundamenal Frequency sin(k /) = sin(k //) = sin(k π/ ) = C k C k = k = ±, ±4, ±6, = ±, ±4, ±6, sin (k /) k / k a k + Period = = / π 6B Prefiler 6

7 CF and CFS Coninuous ime Fourier ransform periodic Coninuous ime Signal X j = x e j d x = X j e j d + Coninuous ime Fourier Series Periodic Coninuous ime Signal C k = x( ) e j k d x() = n= C k e + j k Period = + + Period = 6B Prefiler 7

8 CF CFS periodic Coninuous ime Signal Coninuous ime Fourier ransform x( ) + s, x () x( ) Periodic Coninuous ime Signal Coninuous ime Fourier Series = π Period = + x () = n= x( n ) + 6B Prefiler 8

9 CF and CFS as, = = / C k sin(k /4) k / = 4 = / 4 C k sin(k /8) k / = 8 = /8 C k sin(k /6) k / B Prefiler 9

10 = = Sampling () Ideal Sampling Pracical Sampling τ X X τ τ 6B Prefiler

11 = = Sampling () Ideal Sampling Frequency Domain CF X * CF = π CF 6B Prefiler

12 = = Sampling (3) Pracical Sampling Frequency Domain τ CF X * τ CF = π a k sin (k /) k / k τ CF 6B Prefiler

13 Convoluion wih Impulse rain X (w) G() = X () F () = F (w) X ( w)d w F () X ( w) flip F () X ( w) shif F () F (w) 6B Prefiler 3

14 Convoluion wih Sinc Impulse rain X (w) G() = X () F () = F (w) X ( w)d w F () X ( w) flip F () X ( w) shif F () F (w) 6B Prefiler 4

15 CF of Sampled Signal x() = x (n ) δ( n ) CF π π X ( f ) = + π x() e j π f d + π = x(n ) δ( n ) e j π f d = x(n ) e j π f n d x() = x (n ) δ( n ) CF X ( f ) = x(n ) e j π f n 6B Prefiler 5

16 Periodiciy in Frequency f s π π + π + π f s = π f s = π = π f = x() = x(n ) δ( n ) CF X ( f ) = x(n ) e j π f n e jπ( f + f s) n = e j π( f ) n f s = Period = Sampling Frequency f s X ( f ) = X ( f + f s ) 6B Prefiler 6

17 ime Sequence x() x() x() x() Conver o ime Sequence x[n] Ideal Sampling p( ) = δ( n ) Sampling Period p( ) x[n] ime Sequence 6B Prefiler 7

18 DF of a ime Sequence x() x() = x (n ) δ( n ) CF π X ( f ) = π + π + π x(n ) e j π f n x[n] = x(n ) π π + π + π x[n], Sampling Period DF X ( f ) = x [n] e j π f n Here, X(f) does no denoe he CF of x() 6B Prefiler 8

19 Discree ime Fourier ransform () x[n] = x(n ) π π + π + π x[n], Sampling Period DF X ( f ) = x[n] e j π f n Normalized ngular Frequency π f = π f / = π f f s = 4π π +π +4π = DF x[n], Sampling Period X (e j ) = x[n] e j n 6B Prefiler 9

20 Discree ime Fourier ransform () π π + π + π f s = π f s = π = f s f s + f s + f s f X ( f ) = x[n] e j π f n + + f / f s Normalized ngular Frequency π f = π f / = π f = f s 4π π +π +4π = π f / f s X (e j ) = x[n] e j n = π f 4π f s π f s +π f s +4π f s Nyquis Inerval 6B Prefiler

21 Discree ime Fourier ransform (3) π π + π + π f s = π f s = π = X ( f ) bsolue Frequency X ( f ) = x[n] e j π f n f s f s + f s + f s f Normalized ngular Frequency π f = π f / = π f = f s X (e j ) Normalized ngular Frequency uni circle emphasize he periodic naure X (e j ) = x[n] e j n 4π π +π +4π = π f / f s 6B Prefiler

22 Fourier Series Inerpreaion π π + π + π x() = x (n ) δ( n ) CF X ( f ) = x(n ) e j π f n f s / x(n ) = f s + f s / X ( f ) e + j π f n df CFS X ( f ) = x(n ) e j π f n +π = π = π f / f s X () e + j n d π d f f s = d π Fourier Series Coefficiens x(n ) X ( f ) Coninuous Periodic Funcion View as a Fourier Series Expansion 6B Prefiler

23 Numerical pproximaion X ( f ) = lim X ( f ) x() CF X ( f ) = x() e + j π f d f x(n ) e j π f n x() X ( f ) X ( f ) x() = CF π π x (n ) δ( n ) X ( f ) = + π + π x(n ) e j π f n 6B Prefiler 3

24 = Specrum Replicaion () Ideal Sampling x() x() = x(n ) δ( n ) X s() = δ( n ) s() = e + j π m f s m = x() = x( ) e + j π m f s x() Shif Propery X ( f ) = X ( f m f s ) 6B Prefiler 4

25 = Specrum Replicaion () S ( f ) = δ( f m f s ) m = Frequency Domain X ( f ) Convoluion in Frequency X ( f ) = X ( f ) S ( f ) = X ( f f ' )S ( f ' ) d f ' S ( f ) * = m = X ( f f ' )δ( f ' m f s ) d f ' π π + π + π X ( f ) = X ( f m f s ) X ( f ) π π + π + π 6B Prefiler 5

26 X i ( f ) X ( f ) X ( f ) nalog signal Ideal Prefiler H(f) Bandlimied signal Ideal Sampler - DC Sampled signal f s f s + f s f s 4 f s 3 4 f s f s 6B Prefiler 6

27 6B Prefiler 7

28 6B Prefiler 8

29 References [] hp://en.wikipedia.org/ [] J.H. McClellan, e al., Signal Processing Firs, Pearson Prenice Hall, 3 [3] graphical inerpreaion of he DF and FF, by Seve Mann [4] R. G. Lyons, Undersanding Digial Signal Processing, 997 [5] VR: Enhancing DC resoluion by oversampling [6] S.J. Orfanidis, Inroducion o Signal Processing