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1 Appndix: Nots on signal procssing Capturing th Spctrum: Transform analysis: Th discrt Fourir transform A digital spch signal such as th on shown in Fig. 1 is a squnc of numbrs. Fig. 1: Transform analysis dcomposs this squnc of numbrs into a wightd sum of othr (componnt) tim sris. Th componnt tim sris must b prcisly dfind. Diffrnt transform analyss ar basd on diffrnt dfinitions of componnt tim sris. Th most popular transform usd is th Fourir Transform. In a Fourir transform, th componnt tim sris ar complx xponntials. Th transform analysis dtrmins th wights of th componnt tim sris that compris th givn signal bing analyzd. Th complx xponntial Th complx xponntial is a complx sum of two sinusoids. jq = cosq + j sinq Th ral part is a cosin function. Th imaginary part is a sin function. A complx xponntial tim sris is a complx sum of two tim sris jwt = cos(wt) + j sin(wt) Two complx xponntials of diffrnt frquncis ar orthogonal to ach othr. i... jt jt dt 0 if Figur 2 shows a st of orthogonal Fig. 2: complx xponntial tim sris of th sam frquncy.

2 A signal su uch as th on n in Fig. 1 is xprssd as a sum of svral such com mplx xponntial tim srris, of diffrn nt frqunci s. Th numbr of such tim m sris (and d thrfor th h numbr off frquncis into which th signal is analyzd) is dcidd by th allgorithm usd d to obtain th h transform. Fig. 3 showss thr stss of complx xponntial t tim sris. Fig. 3: t Fourir trransform Th discr Fourir transform of a discrt signal is oftn calld Discrt Fourir Transsform, or DFTT. In Fig. 3, th cofficin nts (or wightts) A, B, and C, C for xampl, would b obtaind o by a DFT. Th disccrt Fourir transform m dcomposss th signal in nto th sum of o a finit num mbr of comp plx xponnttials. In fact, it i dcomposs a signal in nto xactly ass many xpon nntials as thr ar sampls in th sign nal bing ot b dcomp posd into a sum of a finit numbr of complx c analyzd. An apriodicc signal canno xponntials. Or into a sum of any countabl c stt of priodic signals. s Th discrt d Fourir transform t signal biing analyzd is xactly on priod of an n infinitly lon ng signal. In actually assums that th Fourir spctrum of th infinitly longg priodic siggnal, of which th analyzd d rality, it computs th o priod. data ar on Considr th t signal in Fig. F 4. Fig. 4:

3 Th discrt Fourir transform of th abov signal actually computs th Fourir spctrum of th priodic signal shown in Fig. 5. Not that th spctrum xtnds from infinity to +infinity. Th priod of this signal is 31 sampls in this xampl. Fig. 5: Th k th point of a Fourir transform is computd as: X[k] x[n] is th n th point in th analyzd data squnc. X[k] is th valu of th k th point in its Fourir spctrum. is th total numbr of points in th squnc. Not that th (+k) th Fourir cofficint is idntical to th k th Fourir cofficint X[ k] 1 n 0 1 n0 x[ n] x[ n] j 2kn 1 n0 j 2n x[ n] j2kn j 2 k n 1 n0 x[ n] 1 n0 j 2kn x[ n ] j2n j 2knk X[k] Discrt Fourir transform cofficints ar gnrally complx. jq has a ral part cosq and an imaginary part sinq jq = cosq + j sinq As a rsult, vry X[k] has th form X[k] = X r al[k] + jx imagina ry[k] A magnitud spctrum rprsnts only th magnitud of th Fourir cofficints X magnitud [ k] = sqrt(x ral [k] 2 + X imag [k] 2 ) A powr spctrum is th squar of th magnitud spctrum X powr [k] = X ral [k] 2 + X im mag[k] 2 For spch rcognition, w usually us th magnitud or powr spctra A discrt Fourir transform of an point squnc will only comput uniqu frquncy componnts, i.. th DFT of an point squnc will hav points. Th point DFT rprsnts frquncis in th continuous tim signal that was digitizd to obtain th digital signal. Th 0 th point in th DFT rprsnts 0Hz, or th DC componnt of th signal. Th ( 1) th point in th DFT rprsnts (

4 1)/ tims th sampling frquncy. All DFT points ar uniformly spacd on th frquncy axis btwn 0 and th sampling frquncy. Fig. 6 shows a 50 point sgmnt of a dcaying sin wav sampld at 8000 Hz. Th corrsponding 50 point magnitud DFT is shown in Fig. 6. Th 51 st point (shown in rd) is idntical to th 1 st point. Fig. 6: Th Fast Fourir Transform (FFT) is simply a fast algorithm to comput th DFT. It utilizs symmtry in th DFT computation to rduc th total numbr of arithmtic oprations gratly. Th tim domain signal can b rcovrd from its DFT as: x[ n] 1 1 k 0 X j 2kn [ k] Windowing Th DFT of on priod of th sinusoid shown in th Fig. 6 computs th Fourir sris of th ntir sinusoid from infinity to +infinity. Fig. 6: a sinusoid; on priod of th sinusoid; (c) DFT of

5 (c) Th DFT of any squnc computs th Fourir sris for an infinit rptition of that squnc. Th DFT of a partial sgmnt of a sinusoid (Fig. 7) computs th Fourir sris of an inifinit rptition of that sgmnt, and not of th ntir sinusoid. This will not giv us th DFT of th sinusoid itslf! Fig. 7: Partial sgmnt of a sinusoid; corrsponding infinit priodic signal; (c) DFT of ; (d) DFT of th corrct sinusoid (c)

6 (d) Th diffrnc btwn Fig. 7 (c) and Fig. 7 (d) occurs du to two rasons: Th transform cannot know what th signal actually looks lik outsid th obsrvd window. Rathr, it infrs what happns outsid th obsrvd window from what happns insid. As a rsult, a signal such as Fig. 8 cannot b infrrd. Fig. 8: Th transform cannot infr th signal outsid th sn window as such. It infrs th signal shown in 7 instad. Th implicit rptition of th obsrvd signal introducs larg discontinuitis at th points of rptition. Ths ar shown ncircld in grn in Fig. 8. This distorts vn our masurmnt of what happns at th boundaris of what has bn rliably obsrvd. Th actual signal (whatvr it is) is unlikly to hav such discontinuitis. Fig. 8: discontinuitis at th points of rplication in th signal infrrd by th transform Whil w can nvr know what th signal looks lik outsid th window, w can try to minimiz th discontinuitis at th boundaris. W do this by multiplying th signal with a window function, as in Fig. 9. W call this procdur windowing. W rfr to th rsulting signal as a windowd signal. Fig. 9: windowing; chang in th cntral rgions of th slctd sgmnt du to windowing; (c) infrrd windowd signal

7 (c) Windowing attmpts to kp th windowd signal similar to th original in th cntral rgions, as shown in Fig. 9, and rduc or liminat th discontinuitis in th implicit priodic signal, as in Fig. 9(c). Th DFT of th windowd signal shown in Fig. 10 is shown in Fig. 10. It dos not hav any artfacts introducd by discontinuitis in th signal. Oftn it is also a mor faithful rproduction of th DFT of th complt signal whos sgmnt w hav analyzd. Fig. 10: a windowdd signal; magnitud spctrum of th wndowd signal in

8 Fig. 11 summarizs th advantags of windowing in trms of th changs achivd in th signal spctrum: Fig. 11: agnitud spctrum of original sgmnt; agnitud spctrum of windowd signal; (c) agnitud spctrum of complt sin wav (c) As w s in Fig. 9, Windowing is not a prfct solution. Th original (unwindowd) sgmnt is idntical to th original (complt) signal within th sgmnt. Th windowd sgmnt is oftn not idntical to th complt signal anywhr. Svral windowing functions hav bn proposd that strik diffrnt tradoffs btwn th fidlity in th cntral rgions and th smoothing at th boundaris. Fig. 9 uss a Hamming window. This is on of a class of windows calld cosin windows. Som cosin windows ar: (In th following, window lngth is, Indx bgins at 0) Hamming: w[n] = cos(2pn/) Hanning: w[n] = cos(2pn/)

9 Blackman: cos(2pn/) cos(4pn/) Gomtric windows ar anothr catgory of common windows. Som of ths ar shown in Fig. 12. Fig. 12: Gomtric windows: Rctangular (boxcar); Triangular (Bartltt); (c) Trapzoid Zro Padding W can pad zros to th nd of a signal to mak it a dsird lngth. This is usful if th FFT (or any othr algorithm w us) rquirs signals of a spcifid lngth. (on xampl is a radix 2 FFT computation algorithm : it rquirs signals of lngth 2 n, whr n is a natural numbr). Th consqunc of zro padding is to chang th priodic signal whos Fourir spctrum is bing computd by th DFT. Zro padding is shown in Fig. 13, which shows a zro paddsam as th DFT of th unpaddd signal, with additional spctral sampls insrtd in btwn. It dos not containn any additional information ovr th original DFT. It also signal and its DFT. Th DFT of th zro paddd signal in Fig. 13 is ssntially th dos not contain lss information. Fig. 13: an xampl of a zro paddd signal; th signal; its DFT

10 Fig. 14 futhr illustrats th consquncs of zro padding. Fig. 14: Th lft panls show th signals, and th right panls show th magnitud spctra. Th ffcts of windowing ar not th sam as th ffcts of zro padding. Th DFT of th zro paddd signal is ssntially th sam as th DFT of th unpadddd signal, with additional spctral sampls insrtd in btwn. It dos not contain any additional information ovr th original DFT. It also dos not contain lss information. Fig. 15 illustrats th spcial cas of zro padding windowd signals. Wil windowing rsults in signals that appar to b lss discontinuous at th dgs, th rgularization of th signal is only illusory. W also do not introduc any nw information into th signal by mrly padding it with zros.

11 Fig. 15: zro paddd signal signal as prcivd by th transform (c) signal magnitud spctrum of th (c) Othr xampls of magnitud spctra ar shown in Fig. 16. Fig. 16: Lft panls show th signalss and th right panls show th corrsponding magnitud spctra.

12 Numbr of points in a DFT Fig. 17 shows 128 sampls from a spch signal sampld at Hz. Fig. 17 and 17(c) show th Th first 65 points of a 128 point DFT, and th first 513 points of a 1024 point DFT rspctivly. Th magnitud spctrum is ar mor dtaild in 17(c). Fig. 17:

13 (c)

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