Implementation of the Digital Phase Vocoder Using the Fast Fourier Transform

Transcription

1 EEE TRANSACTONS ON ACOUSTCS, SPEECH, AND SGNAL PROCESSNG, VOL. ASSP-24, NO. 3, JUNE signals with low frequencies. The characteristics will be odified due to the changing perforance of delta odulation. However, for this application, the perforance of delta odu- [] lation is relatively insensitive as long as the quantized noise is approxiately independent on a easured signal. Especially [2 for signals with very low-frequency coponents, the ethod oitting the of integrator available. be will [3 ACKNOWLEDGMENT 4 The author wishes to thank Prof. J. Kawata and Prof. T. [5 Kubota for their helpful coents and suggestions. The author also wishes to thank his any associates at the Depart- 6 ent of Electronics, Tokyo Electrical Engineering College, Tokyo, Japan. REFERENCES Y. shii, A real-tie signal processing syste for correlation and spectru analysis, J. SZCE (Japan), vol. 8, pp , Nov B. LuBow, Correlation entering new field with real-tie signal analysis, Electronics, vol. 3, pp. 75-8, Oct J. E. Abate, Linear and adoptive delta odulation, Proc. ZEEE, vol. 55, pp , Mar M. R. Winkler, High inforation delta odulation, in EEEZnt. Conv. Rec., pt. 8,963, pp A. Peled and B. Liu, A new approach to the realization of nonrecursive digital filters, ZEEE Trans. Audio Electroacoust., vol. AU-2, pp , Dec N. S. Jayant, On the power spectru of the staircase function in linear delta odulation, EEE Trans. Acoust., Speech, Signal F rocessing, vol. ASSP-23, pp , Apr pleentation of the Digital Phase Vocoder Using the Fast Fourier Transfor Abstract-This paper discusses a digital forulation of the phase vocoder, an analysis-synthesis syste providing a paraetric representation of a speech wavefor by its short-tie Fourier transfor. Such a syste is of interest both for data-rate reduction and for anipulating basic speech paraeters. The syste is designed to be an identity syste in the absence of any paraeter odifications. Coputational efficiency is achieved by eploying the fast Fourier transfor (FFT) algorith to perfor the bulk of the coputation in both the analysis and synthesis procedures, thereby aking the forulation attractive for ipleentation on a inicoputer. T. NTRODUCTON HE REPRESENTATON of a speech signal by its shorttie Fourier transfor is of interest both as a eans for data-rate reduction in counications and as a technique for anipulating the basic speech paraeters. Systes based on this representation are often referred to as phase vocoders since the paraeters obtained have traditionally been the agnitude and phase (or phase-derivative) of the shorttie Fourier transfor [. One difficulty in ipleenting such systes in digital for has been the rapid increase in the aount of coputation required as the nuber of frequency bands is ade large. Schafer and Rabiner [2] have shown how to greatly reduce Manuscript received May 0, 975; revised Deceber 9, 975. This work was supported by the Advanced Research Projects Agency, onitored by the ONR under Contract N C-095. The author is with the Departent of Electrical Engineering and Coputer Science, Research Laboratory of Electronics, Massachusetts nstitute of Technology, Cabridge, MA the aount of coputation required for the analysis procedure by forulating the syste such that ost of the coputation is perfored by the fast Fourier transfor (FFT) algorith. However, the coputation required for the direct ipleentation of the synthesis procedure is at least as great as that required for the direct analysis, and it has, therefore, reained a proble. n this paper, we present an analysis-synthesis syste based on the discrete short-tie Fourier transfor. This syste will be shown to be, atheatically, an identity syste if no paraeter odifications are introduced. The analysis procedure is a refineent of that proposed by Schafer and Rabiner in which the coplex ultiplies used to deodulate the channel signals are now eliinated. The synthesis procedure is new and is significantly ore efficient than the direct procedure [2]. The coputational savings is effected by reducing the nuber of interpolations required for each output value fro N (where N is the nuber of frequency bands in the representation) to and by perforing the reaining coputations using the FFT algorith (a savings of approxiately log, N versus N operations per output value).. FORMULATON Let x(n) represent saples of a speech wavefor. The crete short-tie Fourier transfor of x(n) is defined by dis- 00 X&) = x(r)h(n - r) w;;yk () y= -03 for k = 0,,.. *,N -, where W, = exp [j(2a/n)] and h(n)

2 244 TRANSACTONS EEE ON ACOUSTCS, SPEECH, AND PROCESSNG, SGNAL JUNE 976 is an appropriately chosen window. x&) ay be interpreted as N saples of a tie-varying spectru with k the index associated with frequency and n the index associated with tie. According to (), x&) is obtained at each tie saple n by weighting the sequence x(r) by the window h(n - v) and Fourier transforing the resulting sequence. n the next section it will be shown how to obtain x&) at a particular n by coputing a single discrete Fourier transfor (DFT) of a finite-duration sequence of length N. By properly choosing k(n), it can be guaranteed that the original sequence x(n) is exactly recoverable fro its shorttie transfor defined by (). Furtherore, x(n) is given in this case by Fig.. Digital fiiter-bank analog for discrete short-tie Fourier analysis. Fig. 2. Representation of the kth fiiter-bank channel in ters of the prototype low-pass fiiter k(n). Although the necessary and sufficient conditions for x(n) to be given by (2)~ can be derived directly fro (l), it is infora- invariant and thus copletely characterized by its unit-saple tive to interpret () and (2) in ters of a bank of digital band- response. Let &(n) represent the overall unit-saple response pass filters with contiguous passbands. Consider a set of N relating the output y(n) of the filter bank to the input x(n). coplex bandpass filters {hk(n)} with passbands equally Then spaced about the unit circle and with unit-saple responses &(n) = 2 hk(n) hk(n)=fh(n)wgk, k=0,;**,, (3) k=o where h(n) is a prototype low-pass filter with real unit-saple response. f these filters are cobined to for the structure shown in Fig., then the output of the kth filter, denoted by given yk(n), convolution is the by 0 yk(n) = x(r)kk(n - Y) [. k=o k=o = jp(n) WGk = h(n) - WEk y= - = h(n) [; SN] = 5 x(v) [; h(n - r) Wit: -M = y= - Nn) wn))n2 where 6((n))~ = for all n 0 od N and is zero otherwise. Thus, &) unit-saple is the siply response pro- h(n) the of totype low-pass filter sapled every N saples, specifically = - wgk x(r)h(n - r) wjp N y=- where Xk(n) is just the discrete short-tie Fourier transfor Now if Y(n) is to be to x(n), then '(') ust of x(n) given by (l). F~~~ and (4) a single channel ofthe itself be aunit saple. Therefore, necessary and sufficient confilter bank is seen to be equivalent to the structure shown in ditions fory(n) =x(n) for ' as Fig. 2. The output of the filter bank y(n) is given by the su of the ) h(0) =. N channelsyk(n), i.e., 'This result also follows directly fro () by ultiplying () by (/N) WGk and suing over k for 0 < k < N - to obtain -dn) = yk(n) - -rk nk k=o - xk(n) wgk =j'f X(r)h(n -) WN wn k=o k=o y= -to t is, therefore, clear that if x(n) is to be recovered fro xk(n) by eans of (2), then h(n) ust be chosen in such a anner that the output y(n) is identical to the input x@). The filter-bank syste depicted in Fig. is linear and shift = x(n +qn)h(-qn) *=-w = x(n) iff (5).

3 MPLEMENTATON PORTNOFF: OF THE DGTAL PHASE VOCODER 245 2) h(n) = 0 for Z = +N, *2N, +3N, * * *. (5) for every Rth value of n, where R dn. The sequences Xk(n) can then be reconstructed by interpolation as part of the synthesis procedure. f the sapling period R is chosen equal to N, which corresponds to the lowest sapling rate allowed by the sapling These conditions are equivalent to the stateent in the frequency doain that although each hk(n) is not necessarily an ideal bandpass filter, the su of their N frequency responses is unity for all frequencies. Observe that the conditions (5) are precisely those constraints on the unit-saple response of a digital interpolating filter [3]. Moreover, if these conditions are not satisfied, then &) will no longer be a unit saple, but a weighted sequence of unit saples with spacing N; hence y(n) will not be identical to x(n) and the resulting distortion will be perceived as reverberation in the output signal. The ost straightforward approach to designing the prototype low-pass filter h(n) is by windowing [4]. Specifically, the unit-saple response sin (nn/n) hideadn) = nn/n of an ideal low-pass filter with cutoff frequencies fic = k(n/n) is ultiplied by a sooth, finite-duration window (e.g., Haing [5], Kaiser [6], Dolph-Chebyshev [7]) to obtain h(n). The precise specifications of h(n) are deterined by the length and shape of the window; any h(n) designed in this anner will satisfy conditions (5). Alternatively, one ight eploy one of the recently proposed techniques for designing optiu (iniax) equiripple finite ipulse response (FR) interpolating filters [3],[8]. However, for a large nuber of frequency saples, the long length required for h(n) tends to ake these filters prohibi-.tively expensive to design. Furtherore, the additional aount of coputation incurred by using a suboptiu h(n) designed by windowing is probably sall copared with the total aount of coputation in the overall syste. The short-tie Fourier transfor provides a paraetric representation of the sequence x(n) in ters of the paraeters X,&). f X&) is coputed for k = 0,,.. -, N - and for all n, then N coplex paraeters are required for each saple of x(n). f x(n) is real, then this represents an increase in coplexity by a factor of 2N. There are, however, properties of the discrete short-tie Fourier transfor that can be exploited to reduce the nuber of paraeters required to represent x(n) to an average of approxiately one per saple ofx(n). First, if Xk(n) is viewed for a particular value of n as N equally spaced saples of a Fourier transfor, then, since x(n) is assued to b.e real, X&) is conjugate syetric in k; that is, theore, then the total nuber of real paraeters in the shorttie Fourier representation of x(n) is exactly equal to the duration (total nuber of saples) of Although it is theoretically possible to reconstruct the sequences X&) if they are sapled every R =N saples, in practice it is necessary to saple at a soewhat higher rate, because neither the low-pass filter nor the interpolator can be ipleented ideally. A procedure that is particularly well suited to designing interpolating filters for reconstructing the channel sequences is the algorith proposed by Oetken et al. [9] for designing optial FR digital interpolating filters. This procedure is attractive because it is a siple and efficient procedure for designing filters of very high order. Furtherore, the design algorith exploits the fact that the data to be interpolated can be oversapled to iprove the perforance of the fdter.. MPLEMENTATON OF THE ANALYS SYSTEM USNG THE FFT ALGORTHM f the nuber of frequency bands N is chosen to be a highly coposite nuber (usually an integral power of 2) then the FFT algorith can be eployed to copute efficiently the short-tie Fourier transfor X&) defined by (). Observe that () does not have the for of a DFT and, therefore, can- not be coputed directly with the FFT algorith. The liits on the suation are given as infinite, but in practice are finite ind deterined by the length of h(n). By recognizing X&) as saples, equally spaced in frequency, of the (continuous-valued) Fourier transfor of x(r)h(r - n), X&) can be expressed as the DFT of ann-point sequence obtained by tie-doain aliasing of x(r)h(n - r). Substituting s = r - n into () gives X,(n) = x(n s)h(-s) w,-(n+s)k - = X&V-k))&G where ((n))~ denotes the least residue of n odulo N. Thus, if N is even, X&) is copletely specified by the values of Xk(n) for k = 0,, *.,N/2, and only N real paraeters are required (n.b., X&) is real for k =O and k=n/2). The second property of X,&) that allows a further reduction in the nuber of paraeters required to represent x(n) is apparent when X,&) is viewed for a particular value of k as a sequence in n. Fro Fig. 2 it can be seen that because it is the output of a low-pass filter with unit-saple response h(n), each such sequence is approxiately band-liited to the frequency range -n/n < i2 <n/n. Thus, it follows fro the sapling theore that it is only necessary to copute X&) or 2When this representation is used as a vocoder, data-rate reduction is achieved by quantizing the paraeters xk(n) [2].

4 246 TRANSACTONS EEE ACOUSTCS, SPEECH, X,(n) = W j p Z(n) WjiF, =o where (6) Z,(n)= x(n++n+)h(-n-). (7) = - E4 The expression Zk(n> = = o Z(n)WGk is recognized as the DFT of the N-point (in ) sequence? (n) for fixed n and can, therefore, be coputed directly with the FFT algorith once Z (n) has been fored. n addition to the coputational savings gained by coputing the short-tie Fourier transfor using the FFT, further savings ay be gained by avoiding the coplex ultiplications by WGnk in (6). Observing that x&) is given by -nk - xk(n) = WN where Tk(n) is the DFT of x,(n), we can exploit the property of the DFT that a circular shift in one doain corresponds to ultiplication by a coplex exponential in the other doain. Thus, by circularly shifting?,(n) prior to coputing its DFT, the ultiplications by Wink are avoided. Specifically, (6) can be rewritten as - x,t(n> = c x(( -n))ngz> wsk =o or xk(n) = x(n) Wi (8) =o where x (n) =?(( -n))n(n). Based on the preceding analysis, the procedure for coputing the discrete short-tie Fourier transfor coefficients x&) at a particular value of n is the following. Referring to Fig. 3, the input data sequence considered as a function of the duy index r is ultiplied by the window h(n - r) (in practice h(n) is often zero phase, in which case h(n - r) = h(r - n)). t is assued that h(n) is of finite duration and, in fact, chosen to have length equal to an even ultiple of N, plus one. The resulting weighted sequence is partitioned into sections each of length N such that ~ (r)l,=~ is the zeroth saple of one of the sections. The resulting N-point subsequences denoted by x$@) for 0 < <N - are then added together to for x,(n) = cxg(n), = 0,,..., N-. ;i,(n) is circularly shifted (in ) by n saples to obtain x(n? =Z((-n))N(nj, and its DFT is coputed by eans of the FFT algorith to give the desired X,&), i.e., >-n AND PROCESSNG, SGNAL JUNE 976 nf (b) Fig. 3. (a) Typical unit-saple response for prototype low-pass filter h(n). (b) h(n) shifted and superiposed on input sequence x(y). X&) = x(n) Wik k = 0,,.. *,N-. =o V. MPLEMENTATON OF THE SYNTHESSYSTEM USNG THE FFT ALGORTHM t has been shown that for any h(n) satisfying conditions (5) the sequence x(n) can be recovered fro its discrete shorttie Faurier transfor by the relation k=o According to Fig. 2,. this operation ay be interpreted as odulating each of the N signals X&) to the center frequencies ak = 2rk/N and suing the resulting signals. t was argued in Section that it is only necessary to copute X&) for every Rth value of n where R <N. Hence, the paraeters to the synthesizer will be assued to be the saples X&?) and not X,&). Clearly, each of the N signals Xk(rR) could be interpolated to get xk(n), which could then be used in (2) to copute x(n) directly [2]. Unfortunately, since x&) depends on n, (2) does not have the for of an (inverse) DFT and is coputationally intractable for large values of N. A synthesis procedure will now be forulated which, for a highly coposite nuber N, perits x(n) to be coputed fro the saples xk(rk?) using the FFT algorith. n addition to the coputational savings afforded by eploying the FFT, the nuber of coputations required to perfor the : R interpolation is reduced by the factor N. Let the input paraeters to the synthesizer be denoted by Sk(r), where Sk(r)=Xk(rR) forallrandk=o,l;.-,n-l. bt f(n) represent the unit-saple response of a : R FR interpolating filter with length 2QR +. The interpolated signals x&) are, therefore, given by r

5 PORTNOFF: MPLEMENTATON OF THE DGTAL PHASE VOCODER 24 where the liits on the su, deterined by the length of f(n), are L+(n) = + Q C o o o o o ~ o o o o o ~ o o and where [A4 eans the largest integer contained in M. Substituting x&) given by (9) into the synthesis equation (2) gives 0 n 0 (b-)r 0 R (r,+l)r Fig. 4. Net on which x&) is defined. Shaded points represent values associated with Sk(r) = xk(rr). Since the liits on both sus are finite, the order ofsuation can be interchanged to give 0 where 0 n 0 (ro-llr OR (ro+llr Fig. 5. Net on which x,(n) is defined. Shaded points represent values associated with s,(r) = x,(rr). Values along path = n od N are x(n) =x&). Thus, for fixed values of Y, s&) is the inverse DFT of sk(r) and can, therefore, be coputed by the FFT algorith. t is iportant to observe that s,(y) is periodic in n with period N. Since the FFT only coputes values of sn(r) for one period (n = 0,, *.., N - l), it is necessary to interpret the subscript h in ( ) as reduced odulo N. The synthesis procedure iplied by ( 0) and ( ) can be interpreted as follows. Consider the two-diensional net shown in Fig. 4. The points on the net represent the discrete set of points on which x&) is defined. Thk horizontal direction represents tie and the vertical frequency. The points corresponding to the values of X&) available to the synthesizer, i.e., every Rth colun, are indicated by shading. nverting (8) gives x,@) as the inverse DFT of x&) for each n, i.e., Furtherore, x, (pi) is defined on the net shown in Fig. 5. Because Sk(r)=Xk(rR), it follows that s,(y) =x,(rr) and, therefore, s,(y) is defined on the shaded points in Fig. 5. By coparing (2) with (2), it can be seen that the values of x(n) are given by the values of x,(n)l, En odn, which correspond to the points in, Fig. 5 on the helical path rn n od N. The operation defined by (O) is, therefore, interpreted as interpolating s,(r) to obtain the unknown values of x,(n), but only those values of x, (n) on the path En od N that are the values of x(n). The ipleentation of the synthesis procedure is, therefore, :,. *! *o,. *,,......;; t : :... p...t... p...: 3.. d... J (ro-l)r ror (ro+2)r (r,+l)r,. O f (c) Fig. 6. (a) Typical unit-saple response for :R FR digital interpolating filter. (b) Mask to extract values required for interpolation using f(n). (c) Net associated with x,(n). 0 indicates points representing s,(r) =x,(rr). 0 indicates points representingx(n) =x&). as follows. First, the values of s,(y) are obtained by inverse transforing Sk(r) using the FFT ( ). The values of x(n) are then obtained by interpolating s,(r) according to (0). Notice that for each value of x(n), 2Q values of s,(y) are required. n fact, for R consecutive values of x(n),~ these values are obtained fro the sae 2Q coluns. Thus, it is natural to copute x(n) in records of length R. For each output value, iagine a ask that extracts 2Q values of s,(y), as shown in Fig. 6. These values are then processed according to (0) to copute x(n). Successive output values are obtained by shift-

6 248 EEE TRANSACTONS ON ACOUSTCS, SPEECH, AND SGNAL PROCESSNG, VOL. ASSP-24, NO. 3, JUNE 976 ing the ask one saple at a tie along the path p and repeating the process. V. CONCLUSONS n od N We have discussed a new ipleentation of the digital phase vocoder, a syste that provides a paraetric representation of a sequence in ters of its discrete short-tie Fourier transfor. f no paraeter odifications are introduced, the syste has been shown to be, atheatically, an identity syste. The bulk of the coputation in both the analysis ad synthesis procedures is perfored by the FFT, thereby aking the syste attractive for ipleentation on a inicoputer (especially if a high-speed FFT processor is available). The syste described has been ipleented on a PDP-9 coputer using block floating-point arithetic. The syste is being used to odify certain paraefers of speech signals and currently allows as any as 52 frequency channels. When operated as an identity syste, the synthesized output differs in no perceptual or easurable way fro the input speech. ACKNOWLEDGMENT The author wishes to thank his advisor, Prof. A. V. Oppenhei, who carefully read and coented on this anuscript during its preparation. REFERENCES J. L. Flanagan and R.M. Golden, Phase vocoder, Bell Syst. Tech. J.,vo~. 45,~~ ,Nov R. W. Schafer and L. R. Rabiner, Design and siulation of a speech analysis-synthesis syste based on short-tie Fourier analysis, EEE Trans. Audio Electroacoust. (Special ssue on 972 Conference on Speech Counication and Processing), vol. AU-2,pp , June , A digital signal processing approach to interpolation, Proc. EEE, vol. 6,pp , June 973. R. W. Schafer, L. R. Rabiner, and 0. Herrann, FR digital filter banks for speech analysis, Bell Syst. Tech. J., vol. 54, pp ,Mar A. V. Oppenhei and R. W. Schafer, Digital Signal Processing. Englewood Cliffs, NJ: Prentice-Hall, 975. J. F. Kaiser, Nonrecursive digital filter design using the,-sinh window function, in Proc. 974 EEE nt. Syp. Circuits and Systes, San Francisco, CA, Apr. 974, pp H. D. Hels, Nonrecursive digital filters: Design ethods for achieving specifications on frequency, response, EEE Trans. Audio Electroacoust. (Special ssue on Digital Filters: The Proise of LS to Signal Processing), vol. AU-6, pp , Sept H. S. Hersey and R. M. Mersereau, An algorith to perfor iniax approxiation in the absence of the Haar condition, M..T. Res. Lab. Electron., Cabridge, MA, Quarterly Progress Rep. 4, pp , July 5,974. G. Oetken, T. W. Parks, and H. W. Schussler, New results in the design of digital interpolators, EEE Trans. Acoust., Speech, Signal Processing (Special ssue on 974 Arden House Workshop on Digiral Signal Processing), vol. ASSP-23, pp , June 975. Liit Cycles in the Cobinatorial ple.entation of Digital Filters TRAN-THONG, MEMBER, EEE, AND BEDE LU, FELLOW, EEE Abstct-The existence of liit cycles in cobinatorial filters using two s copleent truncation arithetic is investigated in this paper. Exact results for liit cycles of period one and two are presented. Soe results for longer period liit cycles are obtained using an effective value linear odel. Bounds on these liit cycles are also derived. The accessability of the liit cycles is briefly discussed.. NTRODUCTON OMBNATORAL FLTERS appeared recently in the literature [l], [2] as an alternative ethod for iple- C enting digital filters. These filters do not eploy hardware ultipliers. nstead, the coputation is carried out Manuscript received March 24, 975; revised Septeber 3,975 and Deceber 20, 975. This work was supported by the Air Force Office of Scientific Research, USAF, under Grant AF-AFOSR T.-Thi%g was with the Departent of Electrical Engineering, Princeton University, Princeton, NJ He is now with the Western Geophysical Copany, Houston, TX Liu is with the Departent of Electrical Engineering, Princeton University, Princeton, NJ with read only eory (ROM) and an accuulator. Consequently, they offer considerable saving in hardware and power consuption with the potential for increased operating speed. This paper is concerned with the stability of cobinatorial filters under zero input condition. The proble is different fro ost of the past work on liit cycles [3]-[5] in that the cobinatorial filter can be odeled as a digital filter with only one quantizer in each section instead of the usual one quantizer with each ultiplier. The stability of an idealized filter structure with one quantizer using either sign-agnitude truncation or rounding arithetic has been reported recently [6], [7], and the results are applicable to cobinatorial filters using these two types of arithetic. However, as a result of the eliination of ultipliers in these filters, an ipleentation with two s copleent is easier [l], [2]. n two s copleent arithetic the variance of the roundoff noise in rounding and in truncation are the sae. However, in the latter case, there is a dc offset which is easily coputed and can be reoved in the final conversion to an analog signal.