Interpolation and Pulse Shaping

Transcription

1 EE345S Real-Time Digial Signal Proceing Lab Spring 2006 Inerpolaion and Pule Shaping Prof. Brian L. Evan Dep. of Elecrical and Compuer Engineering The Univeriy of Texa a Auin Lecure 7

2 Dicree-o-Coninuou Converion Inerpolae a mooh coninuou-ime funcion hrough a equence of ample ( connec he do ) If f 0 < ½ f, hen ( 2π f T φ ) y[ n] = A co n 0 + would be convered ino ( 2π f φ ) y( ) = A co n Oherwie, aliaing ha occurred, and he converer would reconruc a coine wave whoe frequency i equal o he aliaed poiive frequency ha i le han ½ f 7-2

3 Dicree-o-Coninuou Converion General form of inerpolaion i um of weighed pule ~ y ( ) = y[ n] p( T n) n= Sequence y[n] convered ino coninuou-ime ignal ha i an approximaion of y() Pule funcion p() could be recangular, riangular, parabolic, inc, runcaed inc, raied coine, ec. Pule overlap in ime domain when pule duraion i greaer han or equal o ampling period T Pule generally have uni ampliude and/or uni area Above formula i dicree-ime convoluion for each value of 7-3

4 Inerpolaion From Table Uing mahemaical able of numeric value of funcion o compue a value of he funcion Compue f(.5) from able Zero-order hold: ake value o be f() o make f(.5) =.0 ( airep ) Linear inerpolaion: average value of neare wo neighbor o ge f(.5) = 2.5 Curve fiing: fi he hree poin in able o funcion x 2 o compue f(.5) = x f(x) x

5 Recangular Pule Zero-order hold Eay o implemen in hardware or ofware p ) = rec T = 0 if The Fourier ranform i T < T 2 2 oherwie ( in( π f T ) in P( f ) = T inc( π f T ) = T where inc( x) = π f T ( x) x In ime domain, no overlap beween p() and adjacen pule p( - T ) and p( + T ) In frequency domain, inc ha infinie wo-ided exen; hence, he pecrum i no bandlimied p() -½ T ½ T 7-5

6 Sinc Funcion 3π 2π inc(x) x π 0 π 2π 3π ( x) in inc( x) = x How o compue inc(0)? A x 0, numeraor and denominaor are boh going o 0. How o handle i? Even funcion (ymmeric a origin) Zero croing a x = ± π, ± 2π, ± 3π,... Ampliude decreae proporionally o /x 7-6

7 Triangular Pule Linear inerpolaion I i relaively eay o implemen in hardware or ofware, alhough no a eay a zero-order hold p ) = T = T 0 ( Overlap beween p() and i adjacen pule p( - T ) and p( + T ) bu wih no oher Fourier ranform i if T < T oherwie P ( f ) = T inc ( f ) 2 T p() -T T How o compue hi? Hin: The riangular pule i equal o / T ime he convoluion of recangular pule wih ielf In frequency domain, inc 2 ha infinie wo-ided exen; hence, he pecrum i no bandlimied 7-7

8 Sinc Pule Ideal bandlimied inerpolaion p( ) π = inc = T π in T π T rec In ime domain, infinie overlap beween oher pule Fourier ranform ha exen f [-W, W], where P(f) i ideal lowpa frequency repone wih bandwidh W In frequency domain, inc pule i bandlimied Inerpolae wih infinie exen pule in ime? Truncae inc pule by muliplying i by recangular pule Caue mearing in frequency domain (muliplicaion in ime P( domain i convoluion in frequency domain) f ) = T f T W = 2 T 7-8

9 Raied Coine Pule: Time Domain Pule haping ued in communicaion yem co( 2π α W ) p( ) = inc T ideal lowpa filer impule repone 2 6α W Aenuaion by / 2 for large o reduce ail W i bandwidh of an ideal lowpa repone α [0, ] rolloff facor Zero croing a = ± T, ± 2 T, 2 2 See handou G in reader on raied coine pule 7-9

10 Raied Coine Pule Specra Pule haping ued in communicaion yem Bandwidh: ( + α) W = 2 W f f raniion begin from ideal lowpa repone o zero if 0 f < f 2W π ( f W ) P( f ) = in if f f < 2W f 4W 2W 2 f 0 oherwie W = 2 T α = f W 7-0

11 When α = inc( 4W ) Full Coine Rolloff π f + co P f ) = 4W 2W p( ) = 2 2 6W 0 A = ± ½ T = ± / (4 W), p() = ½, o ha he pule widh meaure a half of he maximum ampliude i equal o T Addiional zero croing a = ± 3/2 T, ± 5/2 T, Advanage in communicaion yem? Eaier for receiver o exrac iming ignal for ynchronizaion Drawback in communicaion yem? Tranmied bandwidh double over inc pule Bandwidh generally carce in communicaion yem ( if 0 f < 2W oherwie 7 -

12 DSP Fir Demonraion Web ie: hp://uer.ece.gaech.edu/~dpfir Sampling and aliaing demonraion (Chaper 4) Sample inuoid y() o form y[n] Reconruc inuoid uing recangular, riangular, or runcaed inc pule p() by ~ y ( ) = y[ n] p( T n) n= Which pule give he be reconrucion? Sinc pule i runcaed o be four ampling perioid long. Why i he inc pule runcaed? Wha happen a he ampling rae i increaed? 7-2