ECE 6560 Multirate Signal Processing Decimation and Interpolation

Transcription

1 ECE 656 ultiate Signal Pocessing Deciation and Intepolation D. Badle J. Bazuin Westen icigan Univesit College of Engineeing and Applied Sciences Depatent of Electical and Copute Engineeing 93 W. icigan Ave. Kalaazoo I,

2 Refeences Cociee, R.E.; Rabine,.R.;, "Intepolation and deciation of digital signals A tutoial eview," Poceedings of te IEEE, vol.69, no.3, pp. 3-33, ac 98. ttp://ieeeploe.ieee.og/stap/stap.jsp?anube= R.E. Cociee and.r. Rabine, "ultiate Digital Signal Pocessing," Pentice-Hall, Inc., Englewood Cliffs, N.J., 983. ECE 656 2

3 Filte-Deciato Ipleentation () Deiving a coputational stuctue ECE p p n p w n p p p w p w

4 Filte-Deciato Ipleentation (2) ECE : fo Ipleentation: () Geneate polpase eleents (2) Su te polpase eleents

5 Filte-Deciato Ipleentation (3) Using a causal filte of lengt N= Ipleentation: () Geneate polpase eleents (2) Su te polpase eleents fo : Coefficient and Data Sets fo : fo : fo : ECE 656 5

6 Polpase Filte-Deciato ECE 656 6

7 ati Ipleentation ECE * ow Su Newest saple Oldest saple ocations based on pevious page : fo : fo : fo ows coluns

8 atlab Ipleentation Eaple atlab Code Eecution fo () given and (-(labda*-):-) Xati = flipud(flipl(esape(,,labda); Hati= esape(,,labda); Yo = su(xati.*hati,2); Y = su(yo); Updating Xati Xati(:,2:labda) = Xati(:,:labda-); Xati(:,) = flipud((:) T ); ECE 656 8

9 Filte and Deciate Input at 2 Hz, wit an output data ate onl at 4 Hz. Copute te 36 tap filte once eve 5 input data ccles, but te data locations ust be aintained. Wat if we too te 36 tap filte an patitioned it into 5 sapled filtes? i i : 5 : end Eac filte would get one new input eve 5 cloc ccles Tese is a polpase ipleentation ECE 656 9

10 ATAB Polpase Filte Deciato Cap5_2. labda = lengt()/poltaps; = poltaps; Pfilte = esape(,,labda); aa = zeos(,labda); sift = [zeos(labda-,) ee(labda-) ; zeos(,labda)]; fo ii = :nublocs aa = aa * sift; Tinde = +((ii-)*:(ii*-)); aa(:,) = flipud(testsig(tinde)); vect(:,ii) = su(((aa)).* Pfilte,2); end out = su(vect).; ECE 656

11 Te Z-Tansfo of w, and w w n w n,, l n,, 2 otewise l n n wn ep j2, n Te discete Fouie seies of an ipulse tain wit peiod Y z z Y Y z w z w z w ep j2 z l z l Note: w (n) is not deciated ECE 656 Fo: R.E. Cociee and.r. Rabine, "ultiate Digital Signal Pocessing," Pentice-Hall, Inc., Englewood Cliffs, N.J., 983. Capte 2, p

12 ECE Studing te Z-Tansfo (2) l z l j w z Y 2 ep 2 ep l z l j w z Y 2 ep l z l j W z Y Since z X z H z W l z l j X z l j H z Y 2 ep 2 ep Fo: R.E. Cociee and.r. Rabine, "ultiate Digital Signal Pocessing," Pentice-Hall, Inc., Englewood Cliffs, N.J., 983. Capte 2, p

13 ECE Z-Tansfo in tes of te Fequenc 2 ep T f wee w jw z w j l j X w j l j H e Y l jw ep 2 ep ep 2 ep l w X l w j H e Y l jw 2 ep 2 ep w j w j jw jw jw e X e H e X e H e Y 2 2 Note: Te Nquist zone ae included/aliased based on te filte esponse. Fo a low pass filte signal o filte jw jw jw jw e X e X e H e Y Fo: R.E. Cociee and.r. Rabine, "ultiate Digital Signal Pocessing," Pentice-Hall, Inc., Englewood Cliffs, N.J., 983. Capte 2, p

14 Bandpass Filte Wat would appen if a bandpass filtes wee used instead of a low pass filte? Y j w 2 jw jw jw j w 2 e H e X e H e X e It sould wo te sae wa. ECE 656 Fo: R.E. Cociee and.r. Rabine, "ultiate Digital Signal Pocessing," Pentice-Hall, Inc., 4 Englewood Cliffs, N.J., 983. Capte 2, p

15 Nquist Zones of a Polpase Filte Cap6_. and Cap6_2. agnitude and pase plots of non-deciated polpase filte eleents. Wen a cople filte/cople signal iing is pefoed, te pase plots ae onl equal in te Nquist band of te pefectl centeed cople iing signal. All otes sould povide pase sus tat cancel signal in te Nquist band. ECE 656 5

16 Cociee and Rabine Filte Deciation Suaized () n n wn n w n w Assue a causal FIR filte of lengt :,,, fo fo fo ECE 656 Fo: R.E. Cociee and.r. Rabine, "ultiate Digital Signal Pocessing," Pentice-Hall, Inc., 6 Englewood Cliffs, N.J., 983. Capte 2, p

17 Filte Deciation Suaized (2) et Te suation of -tap filtes. Te coputation is pefoed once eve input saples. ECE 656 Fo: R.E. Cociee and.r. Rabine, "ultiate Digital Signal Pocessing," Pentice-Hall, Inc., 7 Englewood Cliffs, N.J., 983. Capte 2, p

18 Vecto Intepetation as coluns of ows, wit coluns nubeed left-to-igt in tie as inceases and botto-to-top as inceases. as coluns of ows, wit coluns nubeed left-to-igt as inceases and top-to-botto as inceases. ( & ust convolve) To pefo point-wise ultiplication, if is stoed left-to-igt and top-tobotto, flipl and ten flipud and it will line up coectl fo point-wise ultiplication and suation. (Tis can be done to o to convolve). Tis appeas as a ati convolution of te saple vectos. ECE 656 Fo: R.E. Cociee and.r. Rabine, "ultiate Digital Signal Pocessing," Pentice-Hall, Inc., 8 Englewood Cliffs, N.J., 983. Capte 2, p

19 Polpase Ipleentation () Dela ine Based Bloc delas n 2 coefficients applied sequentiall Suation ECE 656 9

20 Polpase Ipleentation (2) Coutato Base n 2 Su ECE 656 2

21 Wat about iing Pio to Filte Deciation n n wn n W n n n ep j2 n N n nep j2 ep j2 n et N n ep j2 n n N ECE N n n N Note equivalence to a cople filte

22 ECE iing Continued et n j 2 ep j 2 ep 2 ep j Te cople weigting and suation of -tap filtes. Te coputation is pefoed once eve input saples. Note, fo one fequenc, use an IFFT fo all fequencies in.

23 Polpase Ipleentation n n wn n W W n W 2 W 2 Su ep j2 A Single Fequenc Digital Cannel Output. ECE W

24 ECE Polpase Ipleentation IFFT 2 n 2 A ulticannel Digital Cannelize o Analsis Ipleentation 2 ep j

25 Intepolate-Filte () Deiving a coputational stuctue w n, n n,,, 2 otewise n p wn p p n p p n p p n s p ECE

26 Intepolate-Filte (2) ECE p p n p n p p s n n n p s p s p s p s Note tat te o suation onl eists fo p = aing eist as signal outputs fo te suation s s

27 Intepolate-Filte (3) Using a causal filte of lengt Ipleentation: () Geneate polpase eleents (2) Output is individual eleents s s fo : s : s s Coefficient sets fo : s s fo :,,, fo fo fo ECE

28 Polpase Intepolato-Filte s s ECE

29 ati Ipleentation ECE * s Rows in (sift egistes) Newest saple Oldest saple s s Oldest saple Newest saple

30 ATAB Polpase Intepolate Filte Cap5_3. ivecto = zeos(labda,); sift2 = [zeos(,labda); ee(labda-) zeos(labda-,) ] ; fo ii = :nublocs ivecto = sift2 * ivecto; Tinde = +((ii-)*:(ii*-)); ivecto(,) = out(ii); ati(:,ii) = Pfilte * ivecto; end out2 = *esape((ati),nu_saples,); ECE 656 3

31 Vecto Intepetation n n s t s t as ows of lengt coluns, wit ows nubeed top-to-botto in tie as inceases (negative tie). as coluns of ows, wit coluns nubeed left-to-igt as inceases and top-to-botto as t inceases. To pefo vecto-ati ultiplication, if is stoed left-to-igt and top-to-botto it will line up coectl fo vecto ultiplication wit wic is inveted in tie. (Note: if is flipud, ten flipl, flipud and flipud te output vecto). Etact te output fo te colun based esult top-to-botto. Tis appeas as a ati ultiplication of te saple vectos. ECE 656 Fo: R.E. Cociee and.r. Rabine, "ultiate Digital Signal Pocessing," Pentice-Hall, Inc., 3 Englewood Cliffs, N.J., 983. Capte 2, p

32 Intepolation-Filte Polpase Ipleentation Coutato Base 2 n ECE

33 Wat about iing (Up-convesion) Afte Te Intepolation Filte wn n n n N n ep j2 s t ep j2 et s t N s N et n n s t s t tep j2 s N ECE N t N s t s t ep j2 ep j2 t s t n W Cople filte equivalence

34 iing Polpase Ipleentation t N BPF s t t s ep j2 s t s t placed in Nquist egion. Filte becoes a cople bandpass filte. ECE W W W 2 W 2 n

35 ECE ultiple Input Signal Ipleentation Eac is ied to a diffeent bin/band and sued wit all te ote pio to polpase intepolation. 2 ep N N t j s t t s 2 ep N t N t j s t s 2 n 2

36 Cascaded Eleents FFT intepolation-filteing Nquist ate TD saples to FD fequenc Cople TD sbols to FD output (siila to OFD sbol geneation) Filte-deciation wit FFT FD fequencies to naowband TD at te Nquist ate! FD sbols to Cople TD output (siila to OFD sbol eception) ECE

37 FD Geneation and Pocessing Foing Wideband and Refoing Naowband Filtes naowe tan te Nquist egions ae used fo geneating FD wavefos. A guad band between adjacent fequencies is tpicall used n n 2 FFT 2 2 IFFT 2 Sntesis OFD Sbol Basis Analsis ECE

38 Quadatue io Filte Pocessing Obseving Naowband and Refoing Wideband Significant filte estiction ae equied if te output is equied to appoiate te input! Quadatue io Filte Definition and Requieents n n 2 IFFT 2 2 FFT 2 Analsis Sntesis ECE

39 QF Applications Fequenc doain filteing o equalization Tie-Spectal Analsis wit econstuction Abitail tae signals apat and ten econstuct te Patial-Band Sntesis to one o oe abita bandwidts (univesal base station eceive) Patial-Band Analsis wit fequenc doain suation and fullband sntesis (univesal base station tansitte) Applications: cellula telepone base stations, satellite ela stations, etc. ECE