Method and apparatus for reducing correlated errors in subband coding systems with quantizers

A method and apparatus for reducing correlated errors in subband coding systems with quantizers is disclosed. A subband coding system comprises a plurality of subband analysis filters to divide the frequency spectrum of the input signal into subbands, individual subband quantizers for coding each subband by a preselected number of quantization levels, corresponding subband decoders and subband synthesis filters. The transfer function of each of the subband synthesis filters is advantageously determined based on the transfer functions of the subband analysis filters as well as on the characteristics of the quantizer used to code the corresponding subband. Specifically, the synthesis filter transfer functions may be based on a perfect reconstruction filter bank or a quadrature mirror filter bank, as well as on the gain factors of a gain plus additive noise linear model for the Lloyd-Max quantizers used to code the corresponding subbands. That portion of the error between the input signal and the replica signal as reconstructed by the system which is correlated to the input signal may be advantageously reduced or eliminated, irrespective of that portion of the error which is uncorrelated to the input signal. Thus, the total error in a final signal may be advantageously reduced by the subsequent application of prior art techniques for the reduction of random, uncorrelated noise.

FIELD OF THE INVENTION 
The present invention relates generally to digital waveform coding systems 
and more specifically to subband coding systems incorporating quantizers. 
BACKGROUND OF THE INVENTION 
In a system for the communication and/or storage of signals representing, 
for example, speech, image or video information, the signals to be 
transmitted or stored are often coded or compressed to reduce the amount 
of data required to represent them. One technique useful for achieving 
such signal compression while maintaining signal quality of the 
subsequently decoded signal is subband coding. In subband coding the 
frequency spectrum of the signal to be coded is divided into a plurality 
of subbands by a bank of bandpass filters (the analysis filter bank). Each 
subband is, in effect, translated to zero frequency by modulation 
techniques, and then sampled (or resampled) at its Nyquist rate (twice the 
width of the band). Each individual subband signal is then digitally 
encoded, typically by a quantizer with a preselected number of 
quantization levels. 
On reconstruction, the encoded subband signals are decoded and translated 
back to their original locations in the spectrum. These reconstructed 
subband signals are then combined using synthesis filters to give a close 
replica of the original signal. With this technique, each subband can be 
encoded according to criteria (including perceptual criteria) that are 
specific to that band. In particular, the number of bits per sample 
(dependent on the number of quantizer levels) in each band can be 
individually allocated, thereby separately controlling the reconstruction 
error variance in each band. In this way, the inherent tradeoff between 
bits per sample and reconstructed signal quality can be optimized based on 
the characteristics of the type of input signal. For example, in the case 
of speech signals, a relatively larger number of bits per sample are 
typically used in the low to medium frequency bands, where pitch and 
formant structure are advantageously preserved for faithful reproduction 
of the signal. 
The principals of subband coding techniques are described generally in N. 
S. Jayant and P. Noll, Digital Coding of Waveforms: Principles and 
Applications to Speech and Video, ch. 11, Prentice-Hall, Englewood Cliffs 
N.J., 1984, and in Subband Image Coding (J. W. Woods, ed.), ch. 2, Kluwer 
Academic Publishers, Boston Mass., 1991, each of which is hereby 
incorporated by reference. In addition, aspects of the use of subband 
coding techniques to reduce bitrates for digital speech communication are 
described in detail in U.S. Pat. No. 4,048,443, issued on Sep. 13, 1977, 
to R. E. Crochiere et al., and assigned to the assignee of the present 
invention. The Crochiere patent is also hereby incorporated by reference. 
The application of subband coding techniques to still images and to video 
signals are described, e.g., in Subband Image Coding. 
Much of the work on subband coding techniques has been directed to 
reconstructing the original signal from individual subband signals. 
Specifically, such work has emphasized recreating as closely as possible 
the original (unquantized) input signal, while ignoring any loss of 
information due to coding (quantization) effects. It is well known that 
the bandpass filters used in the analysis filter bank can never have 
perfectly sharp cut-offs (as do "brick wall" filters). Thus, one effect of 
dividing the original signal into subbands and subsequently recombining 
the subbands is to produce errors relating from signals from other 
subbands. The effect of these errors should be reduced or eliminated by 
the synthesis filters used to process the subband signals. 
Early efforts in this direction addressed the aliasing effects which occur 
when overlapping subbands are sampled at a frequency less than twice the 
entire width of the band (including all of the corresponding analysis 
filter's roll-off). One result of these efforts was the Quadrature Mirror 
Filter (QMF) technique, described, e.g., in Jayant and Noll, ch. 11. Using 
QMF techniques, aliasing effects resulting from the reconstruction of the 
original signal from its overlapping subbands may be entirely eliminated 
by synthesis filters having transfer functions based on the transfer 
functions of the analysis filters. In other efforts, so-called perfect 
reconstruction filter sets have been developed in which the original 
(unquantized) input signal can be replicated perfectly in the absence of 
coding errors based on appropriate synthesis filters. Again, this is 
accomplished by using synthesis filters having transfer functions based on 
the transfer functions of the analysis filters. 
Given this state of the art, a typical approach used by designers of 
subband coding systems is to use a perfect reconstruction (or 
alias-cancellation QMF) filter bank and then to select subband quantizers. 
However, the selection of the quantizers has not been an integral part of 
the design of the filter banks. As used in this discussion, overall 
(total) reconstruction error is determined as the difference between the 
input signal applied to the analysis filter bank and the resultant output 
(replica) signal produced by combining the outputs of the synthesis 
filters. Since the quantization error cannot be eliminated by the design 
of the filters (a quantization by its nature results in a loss of 
information), prior subband coding systems have sought to achieve minimum 
overall reconstruction error using perfect reconstruction filter banks and 
separately optimized (minimal loss) quantizers. Moreover, such optimal, 
minimum error quantizers, so-called Lloyd-Max quantizers, are well known 
in the art. 
SUMMARY OF THE INVENTION 
The limitations of the prior art techniques are overcome and a technical 
advance is made in accordance with the present invention based, in part, 
on a recognition that it is advantageous in subband coders with quantizers 
to reduce that part of the reconstruction error which is correlated with 
the input signal, even at the expense of increasing uncorrelated (random) 
error. It has been found that if most or all of the correlated error is 
eliminated it becomes possible to ultimately achieve a reconstructed 
signal with less total error than the aforementioned prior approach. This 
is so because well-known random noise removal techniques may be 
advantageously applied to the output of the subband coding system to 
reduce uncorrelated error. Thus, even though a subband coding system in 
accordance with one aspect of the present invention may produce a 
resultant reconstructed signal with more total error than a prior system, 
it will nonetheless be preferred because its error is exclusively or 
primarily uncorrelated with the input signal. Typical error from optimal 
quantization methods (such as Lloyd-Max quantization)includes both 
correlated and uncorrelated error. Therefore, an illustrative embodiment 
of the present invention advantageously incorporates characteristics of 
the quantizers in the design of synthesis filters. Thus, in accordance 
with an illustrative embodiment of the present invention, a technique is 
provided for reducing correlated errors in subband coding systems with 
quantizers. In accordance with one aspect of this embodiment, a subband 
synthesis filter for processing signals in one subband has a transfer 
function based on both the transfer function of the subband analysis 
filter bank and on the characteristics of the corresponding quantizer used 
to code that subband. 
In accordance with another aspect of an illustrative embodiment of the 
present invention, a plurality of subband synthesis filters each has a 
transfer function based on both the transfer function of the subband 
analysis filter bank and on the characteristics of the quantizer used to 
code the corresponding subband. The outputs of these subband synthesis 
filters are then combined to produce a replica signal representative of 
the input signal. 
In accordance with a further aspect of an illustrative embodiment, subband 
synthesis filters for each subband have transfer functions which reduce 
error in the replica signal that is correlated with the input signal. 
In accordance with another aspect of an illustrative embodiment, each 
subband synthesis filter has a transfer function G given by 
G=(1/.alpha.)T. Here, T is the transfer function of a perfect 
reconstruction filter section for the corresponding subband based on the 
subband analysis filter bank, and .alpha. is a gain factor relating to the 
gain plus additive noise model of the Lloyd-Max quantizer for the 
corresponding subband. 
In accordance with yet another aspect of an illustrative embodiment, each 
subband synthesis filter has a transfer function G given by 
G=(1/.alpha.)T. Here, T is the transfer function of a quadrature mirror 
filter section for the corresponding subband based on the subband analysis 
filter bank, and .alpha. is a gain factor relating to the gain plus 
additive noise model of the Lloyd-Max quantizer for the corresponding 
subband. 
In accordance with a further aspect of an illustrative embodiment, a method 
and apparatus for decoding a subband coded input signal filters each coded 
subband signal with a corresponding subband synthesis filter. Each subband 
synthesis filter has a transfer function based on the transfer function of 
the subband analysis filter bank and on the characteristics of the 
corresponding quantizer used to code that subband. 
In accordance with yet a further aspect of an illustrative embodiment, a 
method and apparatus for decoding a subband coded input signal reduces the 
error in the resultant decoded signal which is correlated with the 
original input signal. 
In accordance with another aspect of an illustrative embodiment, a method 
and apparatus for decoding a subband coded input signal reduces the total 
error in the resultant decoded signal. First, the error which is 
correlated with the original input signal is reduced. Then, a noise 
removal technique is applied to reduce the uncorrelated error as well.

DETAILED DESCRIPTION 
Introduction 
For clarity of explanation, the illustrative embodiment of the present 
invention is presented as comprising individual functional blocks. The 
functions represented by these blocks may be provided through the use of 
either shared or dedicated hardware, including, but not limited to, 
hardware processors capable of executing software. Illustrative 
embodiments may comprise digital signal processor (DSP) hardware, such as 
the AT&T DSP16 or DSP32C, and software performing the operations discussed 
below. Very large scale integration (VLSI) hardware embodiments of the 
present invention, as well as hybrid DSP/VLSI embodiments, may also be 
advantageous in some circumstances. 
FIG. 1 shows a prior art subband coding system, familiar to those of 
ordinary skill in the art, illustrated for image or speech coding. The 
system comprises encoder 15, decoder 23 and appropriate input and output 
devices. In the case of speech, microphone 12 may be used to input speech 
signal X to encoder 15, and speaker 34 may be used to project 
reconstructed (output) speech signal Y provided by decoder 23. In the case 
of still images, scanner 13 may be used to input image signal X to encoder 
15, and printer 35 may be used to print out reconstructed (output) image 
signal Y provided by decoder 23. In the case of video, camera 14 may be 
used to input video signal X to encoder 15, and display 36 may be used to 
display reconstructed (output) image signal Y provided by decoder 23. Not 
shown explicitly are analog to digital converters and corresponding 
digital to analog converters to convert from and to signals for input or 
output devices which supply or require analog signals. Such converters 
will be used in appropriate cases as will be clear to those skilled in the 
art. In each case, encoder 15 comprises analysis filters 16-1 to 16-n, 
downsamplers 18-1 to 18-n, subband coders 20-1 to 20-n and multiplexer 22. 
Decoder 23 comprises demultiplexer 24, subband decoders 26-1 to 26-n, 
upsamplers 28-1 to 28-n, synthesis filters 30-1 to 30-n and combiner 32. 
Specifically, encoder 15 codes input signal X to produce a coded input 
signal W for storage or transmission. Analysis filters 16-1 to 16-n and 
corresponding downsamplers 18-1 to 18-n make up the analysis filter bank 
of the subband coding system. This filter bank splits the input signal 
into n subband channels, translating each to zero frequency by a 
modulation process. Moreover, each subband is downsampled by a factor of 
n. Downsampling by a factor m comprises the process of selecting every 
m'th sample. Although it is possible to perform subband coding with 
downsampling. by a factor m&lt;n, where n is the number of channels, it is 
most efficient and most common that m=n. This is known as a critically 
sampled filter bank. 
Next, subband coders 20-1 to 20-n individually code each subband signal by 
quantization functions Q.sub.1 to Q.sub.n respectively. Each of these 
quantization functions quantizes the sampled value of the subband signal 
into one of a predetermined number of discrete quantization levels. Note 
that each coder may advantageously quantize its corresponding subband 
signal into a different number of levels, as described above. In addition 
to the quantization, coders 20-1 to 20-n transform the quantized values 
into an encoded representation. The final step performed by encoder 15 is 
the combination of each coded subband signal into a single coded input 
signal W by multiplexer 22. This coded input signal may then be stored for 
later retrieval and decoding or transmitted across a communication channel 
for decoding at a receiving end. Alternatively, the individual coded 
subband signals may be stored or transmitted separately. In this latter 
case, neither multiplexer 22 of encoder 15 nor demultiplexer 24 of decoder 
23 is included, and coded input signal W represents a plurality of 
separate signals. 
Additional coding or other processing for transmission or storage may be 
used in particular applications. Thus, for example, when encoder 15 and 
decoder 23 are used in a telecommunications context, additional switching 
and channel coding may be used to connect a subscriber at a first location 
with another subscriber at a second location via any of a variety of 
communications channels. 
Decoder 23 decodes the coded input signal W to produce replica signal Y, 
which is representative of input signal X. Demultiplexer 24 separates the 
combined coded input signal back into individual coded subband signals, 
which are, in turn decoded by subband decoders 26-1 to 26-n. Specifically, 
each subband decoder performs the inverse function of the corresponding 
subband coder in encoder 15. For example, subband decoder 26-1 performs 
function Q.sub.1.sup.-1 and subband decoder 26-n performs function 
Q.sub.n.sup.-1, where Q.sub.1.sup.-1 extracts the quantized values from 
the encoded representation. Of course, the information lost by the 
quantization process in encoder 15 cannot be restored, since the functions 
Q.sub.1 to Q.sub.n are many-to-one functions. The decoded subband signals 
are then upsampled by a factor of n by upsamplers 28-1 to 26-n 
(corresponding to the downsampling performed by downsamplers 18-1 to 18-n 
in encoder 15). 
Next, the decoded subband signals pass through synthesis filters 30-1 to 
30-n (the synthesis filter bank). Each subband signal is modulated back to 
its original spectral location before finally being combined by combiner 
32 to produce replica signal Y. Moreover, it is the task of synthesis 
filters 30-1 to 30-n to address the reconstruction problems described 
above. In particular, an appropriate choice of a combination of an 
analysis filter bank and a synthesis filter bank can eliminate aliasing 
errors (e.g., a QMF design) or guarantee perfect reconstruction of the 
original signal (in the absence of coding errors). 
QMF and Perfect Reconstruction Filter Banks 
Without loss of generality, the subband coding system of FIG. 1 can employ 
a value of n=2. The output of the system in the absence of quantization 
(i.e., assume coding functions Q.sub.1 to Q.sub.n are identity functions) 
is 
EQU Y(z)=(1/2)[G.sub.1 (z)H.sub.1 (z)+G.sub.2 (z)H.sub.2 (z)]X(z)+(1/2)[G.sub.1 
(z)H.sub.1 (-z)+G.sub.2 (z)H.sub.2 (-z)]X(-z) (1) 
The component X(-z) is the aliased version of the signal; alias 
cancellation systems are designed to remove this part of the signal. For 
example, the quadrature mirror filter (QMF) technique uses the following 
choice of filters: 
EQU H.sub.1 (z)=G.sub.1 (z)=H(z), H.sub.2 (z)=-G.sub.2 (z)=H(-z).(2) 
As is known in the prior art, once the filters are chosen as in (2), it is 
not possible to obtain perfect reconstruction of the signal, i.e., 
Y(z)=X(z) (except for trivial two-tap filters in the FIR, or 
finite-input-response, case). Note, however, that by numerically 
approximating perfect reconstruction, filters of extremely high quality 
can be designed. 
To achieve both alias cancellation and perfect reconstruction, it is clear 
from equation (1) that the filters must satisfy the following two 
equations: 
EQU G.sub.1 (z)H.sub.1 (z)+G.sub.2 (z)H.sub.2 (z)=2, (3) 
and 
EQU G.sub.1 (z)H.sub.1 (-z)+G.sub.2 (z)H.sub.2 (-z)=0. (4) 
Note, however, that such filters would achieve perfect reconstruction only 
in the absence of quantization. Thus, it is an object of the present 
invention to incorporate the quantization process in the design of the 
filter banks. 
Gain Plus Additive Noise Model for Lloyd-Max Quantization 
A well known optimal quantizer is the Lloyd-Max quantizer, described, e.g., 
in Jayant and Noll, ch. 4, which is hereby incorporated by reference. For 
Lloyd-Max quantizers, it can be shown that 
EQU .sigma..sub.y.sup.2 =.sigma..sub.x.sup.2 -.sigma..sub.q.sup.2,(5) 
where .sigma..sub.q.sup.2, .sigma..sub.x.sup.2 and .sigma..sub.y.sup.2 are 
the variances of the quantization error, the quantizer input signal and 
the quantizer output signal, respectively. This quantizer is optimal in 
that it yields the minimum mean-squared error .sigma..sub.q.sup.2. It is 
also well known in the art that the Lloyd-Max quantizer can be modeled by 
a "gain plus additive noise" linear model. That is, its input/output 
relationship may be given by 
EQU y=.alpha.x+r, (6) 
where x and y are random variables representing the input and output of the 
quantizer, respectively, r is a random variable representing the additive 
noise, and .alpha. is the gain factor (.alpha..ltoreq.1). 
The gain plus additive noise model for the Lloyd-Max quantizer is shown 
diagrammatically in FIG. 2 as an instantiation of one subband coder 20 
from the system of FIG. 1. In particular, amplifier 38 applies a gain 
.alpha., where .alpha..ltoreq.1, to quantizer input (random) variable x. 
Then adder 40 adds in random variable r, the noise term, to produce 
quantizer output (random) variable y. 
FIG. 3 shows the subband coding system of FIG. 1 with an illustration of 
the gain plus additive noise model for the Lloyd-Max quantizers 
incorporated therein. In particular, coder 20-1 comprises amplifier 38-1, 
which applies gain .alpha..sub.1, .alpha..sub.1 .ltoreq.1, and adder 40-1, 
which adds in random noise signal R.sub.1. Similarly, coder 20-n is shown 
comprising amplifier 38-n, which applies gain .alpha..sub.n, .alpha..sub.n 
.ltoreq.1, and adder 40-n, which adds in random noise signal R.sub.n. 
As is well known in the art, the gain factor .alpha. may be specifically 
chosen so that the additive noise component will be uncorrelated to the 
input signal. In fact, this is an important feature of the quantizer and 
associated model. In particular, for zero mean, unity-variance input 
signals, it is advantageous to choose 
EQU .alpha.=1-(.sigma..sub.q.sup.2 /.sigma..sub.x.sup.2). (7) 
Note that for input signals which are not zero mean, unity-variance 
signals, it can be readily determined that it is only necessary to 
normalize the quantizer input and output (random) variables x and y. In 
this manner, .alpha. is derived based on the mean and standard deviation 
of the input signal. In particular, choose 
EQU .alpha.=1-(E(q.sup.2)/E(x.sup.2)), (8) 
where E(q.sup.2) and E(x.sup.2) are the second moments of q and x, 
respectively. The gain plus additive noise model for the Lloyd-Max 
quantizer is described, e.g., in Jayant and Noll, ch. 4. 
Component Error Analysis 
It is well known in the art to separate the error of a subband coding 
system into components in order to investigate the influence and severity 
of each. For example, an analysis of the error components in a subband 
coding system using QMF's is described in P. H. Westerink, J. Biemond and 
D. E. Boekee, "Scaler Quantization Error Analysis for Image Subband Coding 
Using QMF's," IEEE Trans. Signal Processing, vol. 40, pp. 421-428, 
February 1992, which is hereby incorporated by reference. 
Without loss of generality, the subband coding system of FIG. 3 may employ 
n=2 in which coders 20-1 and 20-2 are Lloyd-Max quantizers with 
quantization functions Q.sub.1 and Q.sub.2, respectively. Furthermore, the 
gain plus additive noise linear models for Q.sub.1 and Q.sub.2 may be 
assumed to yield gains of .alpha..sub.1 and .alpha..sub.2, respectively, 
and additive noise components of R.sub.1 (z) and R.sub.2 (z), 
respectively, as shown. Then, the output of the subband coding system 
(including quantization)is 
##EQU1## 
Therefore, the total error (difference between the input signal X and the 
output signal Y) is 
##EQU2## 
This error can be decomposed into three constituent parts. The signal error 
E.sub.S (z) is the term with X(z), namely 
EQU E.sub.S (z)=(1/2)[.alpha..sub.1 G.sub.1 (z)H.sub.1 (z)+.alpha..sub.2 
G.sub.2 (z)H.sub.2 (z)-2]X(z), (11) 
the aliasing error E.sub.A (z) is the term with X(-z), namely 
EQU E.sub.A (z)=(1/2)[.alpha..sub.1 G.sub.1 (z)H.sub.1 (-z)+.alpha..sub.2 
G.sub.2 (z)H.sub.2 (-z)]X(-z), (12) 
and the random error E.sub.R (z)is 
EQU E.sub.R (z)=G.sub.1 (z)R.sub.1 (z.sup.2)+G.sub.2 (z)R.sub.2 (z.sup.2).(13) 
Thus, the total error E(z) can be represented as 
EQU E(z)=E.sub.S (z)+E.sub.A (z)+E.sub.R (z). (14) 
Changing Synthesis According to Quantization 
According to an illustrative embodiment of the present invention, the 
synthesis filters to be used in a subband coding system are advantageously 
based not only on the analysis filters used, but also on the 
characteristics of the quantizers. Consider, for example, the subband 
coding system of FIG. 3 with n=2 as analyzed above. Analysis filters 16-1 
and 16-2 are chosen to have respective transfer functions H.sub.1 and 
H.sub.2. Synthesis filters 30-1 and 30-2, with respective transfer 
functions T.sub.1 and T.sub.2, are chosen to form, in combination with 
analysis filters 16-1 and 16-2, a perfect reconstruction system in the 
absence of quantization. It is well known in the art that such transfer 
functions T.sub.1 and T.sub.2 can always be determined given transfer 
functions H.sub.1 and H.sub.2. As described above, prior art subband 
coding design techniques would commonly choose synthesis filters having 
transfer functions T.sub.1 and T.sub.2, and then choose optimal quantizers 
with the desired number of quantization levels for each subband. 
According to an illustrative embodiment of the present invention, however, 
different synthesis filters will be used. In particular, the synthesis 
filters in accordance with this embodiment of the present invention have 
transfer functions which are based not only on T.sub.1 and T.sub.2, 
respectively, but based on the corresponding quantizers as well. Note 
first that the transfer functions H.sub.1 and H.sub.2 together with the 
transfer functions T.sub.1 and T.sub.2 must satisfy the conditions 
specified in equations (3) and (4) above, since they form a perfect 
reconstruction filter set. Specifically, then, 
EQU T.sub.1 (z)H.sub.1 (z)+T.sub.2 (z)H.sub.2 (z)=2, (15) 
and 
EQU T.sub.1 (z)H.sub.1 (-z)+T.sub.2 (z)H.sub.2 (-z)=0. (16) 
FIG. 4 shows the subband coding system of FIG. 3 modified according to an 
illustrative embodiment of the present invention. In particular, synthesis 
filters 42-1 and 42-2 are chosen to have transfer functions G.sub.1 and 
G.sub.2, respectively, so that 
EQU G.sub.1 (z)=(1/.alpha..sub.1)T.sub.1 (z), (17) 
and 
EQU G.sub.2 (z)=(1/.alpha..sub.2)T.sub.2 (z), (18) 
where .alpha..sub.1 and .alpha..sub.2 are, once again, the respective gains 
of the gain plus additive noise linear model for quantization functions 
Q.sub.1 and Q.sub.2, as implemented by coders 20-1 and 20-2, respectively. 
Substituting equations (17) and (18) into equations (11), (12) and (13), 
and then applying equations (15) and (16), gives signal error E.sub.S (z) 
as 
EQU E.sub.S (z)=(1/2)[T.sub.1 (z)H.sub.1 (z)+T.sub.2 (z)H.sub.2 
(z)-2]X(z)=0,(19) 
aliasing error E.sub.A (z) as 
EQU E.sub.A (z)=(1/2)[T.sub.1 (z)H.sub.1 (-z)+T.sub.2 (z)H.sub.2 
(-z)]X(-z)=0,(20) 
and random error E.sub.R (z) as 
EQU E.sub.R (z)=(1/.alpha..sub.1)T.sub.1 (z)R.sub.1 
(z.sup.2)+(1/.alpha..sub.2)T.sub.2 (z)R.sub.2 (z.sup.2). (21) 
Therefore, by choosing synthesis filters having transfer functions 
according to equations (17) and (18), all signal-dependant error has been 
eliminated. Only signal-independent error E.sub.R (z) remains. Therefore, 
known (random) noise removal techniques may be advantageously applied in 
order to reduce E.sub.R (z). Note, however, that the random error 
component of the total error has, in fact, been increased as compared to 
prior art techniques. This is the result of dividing the terms in equation 
(21) by .alpha..sub.1, where .alpha..sub.1 .ltoreq.1, as compared with the 
terms of equation (13). Moreover, note that the total error, namely, 
E.sub.S (z)+E.sub.A (z)+E.sub.R (z), may or may not be reduced from that 
of prior art techniques, depending on the relative magnitude of 
.alpha..sub.1 and .alpha..sub.2 as compared with that of E.sub.S (z) and 
E.sub.A (z)in equations (11) and (12). Nonetheless, the total error 
remaining after noise removal will advantageously be reduced by the 
application of a standard random noise removal technique. One such 
technique is typically applied in an image processing context as 
described, for example, in P. Chan and J. Lim, "One-Dimensional Processing 
for Adaptive Image Restoration," IEEE Trans. Acoust., Speech, and Signal 
Processing, vol. ASSP-33, pp. 117-125, February 1985, which is hereby 
incorporated by reference. 
Referring to FIG. 4, random noise reducer 44 accepts replica signal Y 
(representative of input signal X), processes it according to a prior art 
noise reduction technique such as the cited Chan and Lim process to 
produce improved replica signal Z. The signal Z is then provided to output 
devices such as speaker 34 (in the case of speech), printer 35 (in the 
case of still images) or display 36 (in the case of video). 
Generalizing to the n subband case, synthesis filters 42-1 to 42-n are 
chosen to have transfer functions G.sub.1 to G.sub.n, respectively, such 
that 
EQU G.sub.i (z)=(1/.alpha..sub.i)T.sub.i (z), (22) 
for i=1 to n. Again, T.sub.1 to T.sub.n are the transfer functions for 
respective perfect reconstruction synthesis filters 30-1 to 30-n for the 
system of FIG. 3. .alpha..sub.1 to .alpha..sub.n are the respective gain 
factors for the Lloyd-Max quantization functions Q.sub.1 to Q.sub.n, as 
implemented by coders 20-1 to 20-n, respectively. With all 
signal-dependent error eliminated, the only remaining (random) error is 
EQU E.sub.R (z)=(1/.alpha..sub.1)T.sub.1 (z)R.sub.1 (z.sup.2)+. . 
.+(1/.alpha..sub.n)T.sub.n (z)R.sub.n (z.sup.2). (23) 
FIG. 5 shows an illustrative embodiment of a digital signal processor 
implementation of the subband coding system of the present invention. In 
particular, the illustrated system comprises digital signal processor 
(DSP) 52, bidirectional transmission line 50, input data store 54, output 
data store 56, filter coefficient store 58, quantizer parameter store 60 
and encoded signal store 62. In appropriate cases, the various storage 
elements 54, 56, 58, 60 and 62 will be combined in one or a few data 
storage elements under the control of well known memory management 
apparatus or programs internal or external to DSP 52. The illustrated 
system may be used to encode an input signal X supplied to DSP 52 by input 
data store 54. The encoded signal may then be stored in encoded signal 
store 62 for later retrieval or may be communicated to another system via 
bidirectional transmission line 50. The system may also be used to produce 
a replica signal Y from an encoded input signal, which may be retrieved 
from encoded signal store 62 or received from another system through 
bidirectional transmission line 50. The replica signal is stored by DSP 52 
in output data store 56. The analysis filter transfer functions and the 
synthesis filter transfer functions are implemented as described above in 
DSP 52 based on the filter coefficients provided by filter coefficient 
store 58. The quantization functions are implemented as described above 
based on quantization parameters provided by quantizer parameter store 60. 
That is, each of the sampling, filtering, quantizing, and translating 
operations are individually well known in the art. Similarly, the 
derivation of quantizing parameters, transfer function formulation and 
other described signal processing functionality are individually well 
known in the art. The present disclosure teaches how these techniques can 
be modified to produce the advantageous results of the present invention. 
Multidimensional Case Generalization 
The preceding description of an illustrative embodiment of the present 
invention processes unidimensional input signals such as speech. The 
invention is not so limited. That is, the technique is readily applied to 
multidimensional input signals such as still images (two dimensions) or 
video signals (three dimensions). Specifically, given an m-dimensional 
input signal, one chooses the synthesis filters to have transfer functions 
G.sub.i (z), such that 
EQU G.sub.i (z)=(1/.alpha..sub.i)T.sub.i (z), (24) 
where z is the m-dimensional z-transform vector, to eliminate all 
signal-dependent errors. Furthermore, the remaining, random, error will be 
EQU E.sub.R (z)=(1/.alpha..sub.1)T.sub.1 (z)R.sub.1 (z.sup.D)+ . . 
.+(1/.alpha..sub.n)T.sub.n (z)R.sub.n (z.sup.D), (25) 
where D is the sampling matrix representing the sampling lattice, and 
z.sup.D denotes multidimensional upsampling. Multidimensional subband 
coding techniques involved in these operations are well known in the art, 
as described, e.g., in J. Kovacevic and M. Vetterli, "Non-separable 
multidimensional perfect reconstruction filter banks and wavelet bases for 
R.sup.n," IEEE Trans. Inform. Th., special issue on Wavelet Transforms and 
Multiresolution Signal Analysis, vol. 38, pp. 533-555, March 1992, which 
is hereby incorporated by reference. 
Use With QMF Filter Banks 
The techniques used in the preceding illustrative embodiments of the 
present invention can also be advantageously applied to systems using QMF 
filter banks, rather than perfect reconstruction filter banks. In 
particular, again consider the subband coding system of FIG. 3 with n=2. 
Now, however, it is to be assumed that analysis filters 16-1 and 16-2 and 
synthesis filters 30-1 and 30-2 have transfer functions such that the 
filter banks in combination form a quadrature mirror filter (QMF) system, 
rather than a perfect reconstruction system. Specifically, according to 
equation (2), let analysis filters 16-1 and 16-2 have transfer functions 
H(z) and H(-z), respectively, and let transfer functions T.sub.1 and 
T.sub.2 be H(z) and -H(-z), respectively. Then, according to another 
illustrative embodiment of the present invention, let synthesis filters 
42-1 and 42-2 of the improved subband coding system of FIG. 4 have 
transfer functions G.sub.1 and G.sub.2, respectively, so that 
EQU G.sub.1 (z)=(1/.alpha..sub.1)T.sub.1 (z)=(1/.alpha..sub.1)H(z),(26) 
and 
EQU G.sub.2 (z)=(1/.alpha..sub.2)T.sub.2 (z)=-(1/.alpha..sub.2)H(-z),(27) 
where .alpha..sub.1 and .alpha..sub.2 are, once again, the respective gains 
of the gain plus additive noise linear model for quantization functions 
Q.sub.1 and Q.sub.2, as implemented by coders 20-1 and 20-2, respectively. 
In this case, letting H.sub.1 (z)=H(z), H.sub.2 (z)=H(-z), and substituting 
equations (26) and (27) into equation (9), the resulting output of the 
system becomes 
EQU Y(z)=(1/2)[H.sup.2 (z)-H.sup.2 (-z)]X(z)+(1/.alpha..sub.1)H(z)R.sub.1 
(z.sup.2)-(1/.alpha..sub.2)H(-z)R.sub.2 (z.sup.2). (28) 
Note that the aliasing term including X(-z) from equation (9) goes to zero 
in equation (28). The error between the output of the system and the input 
signal now becomes 
EQU E(z)=(1/2)[H.sup.2 (z)-H.sup.2 (-z)-2]X(z)+(1/.alpha..sub.1)H(z)R.sub.1 
(z.sup.2)-(1/.alpha..sub.2)H(-z)R.sub.2 (z.sup.2). (29) 
The error consists of only two terms. The first term including X(z) 
represents the QMF error (lack of perfect reconstruction error), and the 
second term is random error uncorrelated with the input signal. However, 
it is known in the art that the QMF error is typically almost negligible. 
Therefore, the technique of this illustrative embodiment of the present 
invention, even when based on QMF filter banks rather than perfect 
reconstruction filter banks, results in a subband coding system with 
almost entirely uncorrelated error. Thus, the aforementioned random noise 
reduction techniques may also be advantageously applied to a QMF-based 
system, resulting in an improvement over prior art subband coding systems. 
Although a number of specific embodiments of this invention have been shown 
and described herein, it is to be understood that these embodiments are 
merely illustrative of the many possible specific arrangements which can 
be devised to represent application of the principles of the present 
invention. Numerous and varied other arrangements can be devised in 
accordance with these principles by those of ordinary skill in the art 
without departing from the spirit and scope of the invention. 
Those skilled in the art will recognize that the techniques described may 
be applied to coding and decoding of signals of various kinds in a wide 
range of applications. As an illustration, the above-described techniques 
will advantageously find application in telecommunications applications in 
combination with well known transmission, switching and terminal 
equipment. While audio, image and video signals have been illustratively 
discussed above, it should be understood that processing of other signals 
from applications such as medical technology and seismology will, in 
appropriate cases be enhanced using the techniques of the present 
invention. Likewise, though perfect reconstruction and QMF filter systems 
have illustratively been described above, other particular systems and 
applications using them will benefit from the application of the present 
techniques. Further, though Lloyd-Max quantizers have illustratively been 
employed to illustrate the use of systems including quantization in 
combination with subband filters, it should be understood that the 
teachings of the present invention contemplate the use of other particular 
quantizers having characteristics which can be advantageously reflected in 
decoder filters of the overall system. 
While the above description has proceeded in terms of a particular linear 
transformation technique, i.e., subband coding, it should be understood 
that other linear transformation techniques may be used in an equivalent 
manner. As an example of such linear transformation, the well known 
Discrete Cosine Transform coding (DCT) technique may be used in a manner 
consistent with the above teachings. Thus, signals processed by a DCT 
transform analysis followed by a quantization may be transformed back to 
the original form using an inverse DCT synthesis process which is modified 
in accordance with the above teachings. 
It should also be understood that the term speech in connection with inputs 
to the above-described coders/decoders (codecs) is merely representative 
of the broader class of audio signals. Thus the term should be understood 
to include music and other high quality audio signals, as well as speech 
signals. The term audio, likewise, is, of course, understood to include 
speech.