Section 8 Convolution and Deconvolution

Transcription

1 APPLICATIONS IN SIGNAL PROCESSING Secio 8 Covoluio ad Decovoluio This docume illusraes several echiques for carryig ou covoluio ad decovoluio i Mahcad. There are several operaors available for hese fucios: he summaio operaor ad Mahcad's discree Fas Fourier Trasform fucios df ad idf. You provide: f(), ad g(), he wo sigals o covolve, T, he ime ierval over which he fucios are ozero, N, he umber of sample pois. A quick approximaio o he covoluio ad decovoluio of wo coiuous fucios is performed by samplig, ad applyig a FFT. Aleraely, you ca eer: wo files coaiig discree ime pulse rai ampliudes. The covoluio is performed direcly wih a summaio o he wo fiie series, ad plos of all fucios ad heir covoluios are geeraed. Refereces William McC. Sieber, Circuis, Sigals, ad Sysems, The MIT Press (Cambridge, 986). Backgroud Covoluio Whe a sigal passes hrough a filer, he oupu is he covoluio of he ipu fucio wih he impulse respose of he filer. Covoluios ca be edious if o impossible o perform o complicaed sigals. Foruaely, he covoluio heorem saes ha covoluio i he ime domai is equivale o muliplicaio i he frequecy domai. To compue he covoluio of wo sigals, fid heir frequecy compoes by Fourier rasformaio, muliply he compoes ogeher ad fid he iverse rasform of his produc. The coiuous-ime Fourier iegrals for a operiodic fucio ad is rasform are give by: FFTs x()= X(f) e 2 π f df X (f) = x () e 2 π f Discree Fourier rasforms (DFTs) are summaios raher ha iegrals. The various symmeries ad periodiciies of he expoeial erms i hese summaios ca be exploied o make he calculaio faser, ad such a calculaio is ermed a fas Fourier rasform (FFT). For a quick approximaio o a coiuous-ime Fourier rasform, we ca sample coiuous fucios f() ad h() o creae he pulse rais x() ad h(), ad he apply FFTs. d

2 Mahcad Implemeaio This docume shows wo ways of carryig ou covoluios by Fourier rasformig: usig he buil-i FFT operaors, ad usig he summaio operaor. The docume also illusraes recovery of he origial sigal by decovoluio. Approximae Covoluio Usig FFTs This example shows a quick approximaio o he covoluio of wo coiuous fucios by samplig hem, muliplyig heir FFTs ad he akig he iverse rasform o arrive a he covolved pulse rai. The fial coiuous ime soluio is he recovered by ierpolaio. The rouie assumes ha he fucios are oly o-zero over a fiie ime T, ad i reurs a fucio h givig heir covoluio over he ierval [, 2T]. Firs, defie wo coiuous fucios i ime f ad g, ad eer for T he loges ime for which eiher fucio is ozero, assumig he sigals sar a ime. Also, choose a value for N, he umber of sample pois. Two sigals: f() ( ) ( ) si ( 4 π ) H () Loges ime for which eiher sigal is ozero: Number of sample pois: N 28 T f ( ) Nex, defie a rage variable for sampled ime ad creae a vecor from he rage variable: N 2 T Usig he ime samples, cosruc vecors x ad h wih Mahcad's vecorize operaor, so ha hey coai he sampled fucios: x f () h H() The covoluio of he sampled f ad g is give by: y 2 T idf df(x) df(h) N where he square roo of N is a ormalizaio facor. The lierp fucio ierpolaes liearly bewee he elemes of he discree covoluio, providig a approximaely coiuous covoluio of f wih g. g() lierp(, Re (y), ),. T 2 T

3 f () H() g() T Fig. 8. Plos of f(), H() ad he approximae coiuous ime covoluio Approximae Decovoluio Usig FFTs You ca carry ou decovoluio efficiely by iverig his procedure: divide he FFT of he oupu sequece by he FFT of oe of he ipus. Noe ha "divide" here meas divide each erm of oe sequece by he correspodig erm of he oher. I geeral, some erms i he deomiaor sequece will be zero, bu you ca usually avoid a divisio by zero error ad sill obai a accurae decovoluio by addig a small safey facor o each erm of he deomiaor. ε 6 (a small safey facor) The decovoluio is give by x' N 2 T Re idf df(y) df(h) + ε x' f () Fig. 8.2 Decovolved sigal compared wih origial sigal

4 The plo shows ha he sequece give by he decovoluio maches he origial sampled ipu fucio f. Noice ha he ierpolaio is o really ecessary wih his may pois, sice he graphig fucio will coec hem smoohly ayway. Discree Covoluio of Two Fiie Sequeces This example uses Mahcad's sum ad mod operaors o fid he covoluio of wo fiie sequeces. The arrays S ad S2 are he wo sequeces o be covolved. S S2 [ 2222] T T The legh of he covoluio will be oe less ha he sum of he leghs of he wo sequeces. This meas ha each sequece mus have a legh of K. M legh(s) N legh(s2) K M+ N m M N Pad each sequece wih zeros o he correc legh i a ew vecor: X S H S2 H M + X + i N m M+ 2 M+ 2 The covoluio is give by C i X H i mod ( i+ K, K)

5 S m m S C Fig. 8.3 Ipu pulse rais ad heir covoluio You ca also use he FFT echiques of he firs wo examples for fiie sequeces. You will eed o pad he sequeces o a legh equal he sum of heir origial leghs. Follow his procedure i he las example for a model.