IEEE TRANSACTIONS ON ACOUSTICS, SPEECH, AND SIGNAL PROCESSING, VQL. ASSP-28, NO* I, FEBRUARY 1980 107 Equations (8), (9), (1 O), and. ( 12) are the desired design equations for digital spectrum analysis using the lo-sinh win- dow. To design a window for prescribed values of R and Af requires simply the computation of from (IO), the subse- quent computation of N using (9), and finally, the computa- Fig 1. System for decorrelating quantization errors in a uniform tion of w(n) using the subroutine [3] for lo [ 1. With these quantizer using pseudorandom noise. equations It is no more difficult to use this window fox spec- trum analysis than to use one of the multitude of other win- dows that are commonly used [ 61. The Io-sinh window, how- nique for seducing the effect of quantization noise in PCM ever, is much more flexible than almost all the other windows; speech coding. As described in Section 11, the procedure con- and, using (1 2) and the other equations given here, it is con- sists of using dither noise to ensure that the quantization er- venient to explore the tradeoffs between record length, spectral rors can be modeled as additive signal-independent noise, and then reducing this noise through the use of any of a variety resolution, and leakage in digital~ spectrum analysis. of noise reduction systems. The procedure is illustrated in REFERENCES Section 111. D. Slepian, H. 0.Pollak, and H. J, Landau, “Prolate spheroidal 11, PROCEDURE FOR QUANTIZATION NOISE REDUCTION wave functions, Fourier analysis, and uncertainty-X,” Bell @st. Tech. J., vol. 40, pp. 43-63, Jan. 1961. For linear quantization, in which the step sizes are fixed and J* F. Kaiser, “Digital Filters,” System A nal’sis by Di@td Com- equal, if the input signal fluctuates sufficiently and if the puter, E‘. F, Kuo and J. F. Kaiser, Eds. New York: Wiley, 1966, quantization step size is small enough, the quantization error pp. 218-285. can be modeled.as additive uniformly- distributed white noise - 3 ‘Wonrecursive digital filter design using the Io-sinh window that is statistically uncorrelated with the signal [ 31. ‘When the function,” in Proc. 1974 IEEE Inf. Symp. Circuits Syst., Apr. step size becomes sufficiently large, the quantization error be- 1974, pp. 20-23. comes signal dependent. For such cases, however, it is well A. V. Oppenheim and R. W. Schafer, Digital sigPzal Processing. known [ 4 1 -[ 61 that through the use of “dither” noise, as il- Englewood Cliffs, NJ: Prentice-Hall, 1975, pp. 242-247, lustrated in Fig. l, the quantization error can be replaced by Hamming, Digital Filters. Englewood Cliffs, Prentice- R, W. NJ: white noise which is uniformly distributed and statistically Hall, 1977, pp. 149-159. in- F. J. Harris, “On the use of windows for harmonic andysis with dependent of the signal, In this system, z(n) is a pseudo- the discrete Fourier transform,” hoc, IEEE, vola 66, pp. 51-83, random uniformly-distributed white-noise sequence, with the Jan. 1978. probability density function I A A P,(,>(z) = for - - <z<- n 2 -2 0 otherwise (1) Reduction of Quantization Noise In PCM Speech Coding where A is the quantizer step size. A similar technique [ 43 has been used in image processing to remove the contouring effect JAE S. LTM AND ALAN V. OPPENHEIM evident in uniform image quantization. For nonuniform fixed quantization with dither noise the quantizer can be represented conceptually in the form of Absrract-A new technique to reduce the effect of quantization noise ~;i~*2 where FI.1 is a specified nonlinearity, ln the imple- in PCM speech coding is proposed. The procedure consists of using mentation of a nonuniformquantizer (in the form of F~~.2) dither noise to ensure that the quantization errors can be modeled as the effects are additivewhite noise to the additive signal-independent noise, and then .-reducing this noise through nonlinearityF-1 .I the use of a noise reduction system. The procedure is illustrated with In this correspon&e we propose a system to reduce the examples. effects of quantization noise for general nonuniform quantiza- tion systems of the form of Fig. 2. The system exploits the I. INTRODUCTION fact that, in the system of Fig. 2, the quantization error is at an intermediate stage as additive white signal-independenl Wide-band speech coding systems typically rely on general noise, Recently, a number of procedures have been proposed waveform coding techniques [ 1] , [ 23, such as instantaneous and developed [7] for the enhancement speech degraded quantization (PCM), for which the step size is fixed, any of a of by variety of forms of adaptive quantization, and differential additive uncorrelated background noise, Our procedure for reducing the effects quantization then corresponds to schemes, such as delta modulation, adaptive delta modulation, of in- continuously variable slope delta modulation, etc. Instanta- serting a noise reduction system into the system of Fig. 2, neous quantization with fixed step sizes has the advantage that as indicated in Fig. 3. The sequences x^(n) and ;(n) in the the quantizer and coder are particularly straightforward, al- figure represent an estimate of x(n) and sin). A system similar to Fig. 3 was considered by Lim [ with encouraging results though as the step size increases, the quantization effects be- 83 reducing the quantization noise the context of im- come severe. In this correspondence we propose a new tech- in in PCM age coding. , Manuscript received May 31, 1979; revised September 6, 1979- This 111. EXAMPLES work was sponsored by the Department of the Air Farce. The views and conclusions contained in this document are those of the con- We illustrate the characteristics of the system of Fig. 3 tractor and should not be interpreted as necessarily representing the through an example in which uniform quantization is assumed official policies, either expressed or implied, of the United -States so that the nonlinearity F[*] is eliminated. The noise’reduc- Government. tion system used is the revised linearized maximum a poste- The authors are with Lincoln Laboratory, Massachusetts Institute of riuri estimation (RLMAP) speech enhancement system [ 91, Technology, Lexington, MA 02173. [ 101, In this system speech is modeled on a short-time basis 0096-3518/80/0200-0107$00.75O 1980 IEEE 108 XEEE TRANSACTIONS ON ACOUSTICS, SPEECH, AND SIGNAL PRQCESSIMG, VOL. ASSP-28, NO. 3, FEBRUARY 1980 Pig. 2. System for decorrelating quantization enors in a nonuniform fixed-level quantizer using pseudorandom noise. Fig. 3. A system for the reduction of quantization noise in PCM speech coding with a nonuniform fixed-level qua-ntizer, TIME TIME TIME (a Fig. 4. (a) A segment of voiced speech. (b) Speech waveform of Fig, 4(a) coded by a PCM system with a 2-bit uniform quantizer. (c) Result of using pseudorandom noise for the speech waveform of Fig. 4(a) in a PCM system with a 2-bit uniform quantizer. (d) Result of the application of a noise seduction system to the waveform sf Fig. 4(c). as the response of an all-pole system excited by white Gaussian of quantization is quite visible in the staircase shape of the noise. With this model sf speech, the all pole parameters are waveform. In Fig. 4(c) is shown the result of adding and then estimated by maximum a posteriori (MAP) estimation proce- subtracting dither noise in a PCM system with a 2-bit uniform cedure from the noisy speech. This involves solving sets of quantizer. In Fig. 4(d) is shown the result obtained by apply- linear equations in an iterative manner. The estimated all-pole ing the RLMAP speech enhancement system to the wavehrm parameters are then used to design a filter that filters the in Fig. 4(c). noisy speech to reduce additive noise. Further details on the Figs. 5 and 6 illustrate the procedure as carried out on the RLMAP speech enhancement system- can be found in [SI, sentence “line up at the screen door” spoken by a male E 101 speaker. In Fig. 5(a) is S~QW~the spectrogram of the original In Fig 4(a) is shown a segment of noise-free voiced speech. sentence. Fig. 5(b), (c), and (dl correspond to spectrograms In Fig. 4(b) is S~QWIIthe speech waveform of Fig. 4(a) coded of the speech in Fig. 5(a) based on PCM coding, PCM coding by a PCM system with a 2-bit uniform quantizer. The effect with addition and subtraction of dither noise, and PCM coding IEEE TRANSACTIONS ON ACOUSTICS, SPEECH, AND .SIGNAL PROCESSING, VOL. ASSP-28, NO. 1, FEBRUARY 1980 109 t f T t t - TIME TIME 0) Fig. 6. (a) Spectrogram of speech in Fig. S(a) coded by a PCM system with a 4-bit uniform quantizer. (b) Spectrogram of speech in Pig. 5(a) obtained by using pseudorandom noise in a PCM system with a 4-bit uniform quantizer. (c) Spectrogram of speech obtained by applying a noise reduction system to speech in Fig. 6(b). based on a specific noise reduction system support the basic % TIME hypothesis and are very encouraging. While the basis for the procedure is associated with fixed nonuniform quantization, (dl it is speculated that it can be generalized to other quantization Fig. 5. (a) Spectrogram of an English sentence “line up at the screen schemes. such as differential auantization. Furthermore. it is with addition and subtraction of dither noise followed by the REFERENCES RLMAP speech enhancement system, respectively, The PCM [ 11 L. R, Rabiner and R. W, Schafer, Dlgital Processing of Speech coding system used in Fig. 5 is again a 2-bit uniform quantizer.