A trellis-based optimal parameter value selection for audio coding

Abstract

This paper considers the problem of selecting a set of parameter values from a given parameter space, in order to perform rate-distortion optimization in the context of audio compression. Due to interdependencies between parameters, separate optimization of parameter values is inherently suboptimal, yet a straightforward brute-force joint search involves prohibitive computational complexity. This work proposes a new method for joint rate-distortion optimization, while accounting for interparameter dependencies. The optimal solution is achieved, at significantly reduced complexity as compared to a brute-force search, by employing a Viterbi search over a trellis. Two objective distortion metrics are specifically considered: the average, and the maximum noise-to-mask ratio. Subjective (AB/MOS) and objective (average/maximum noise-to-mask ratio) tests demonstrate considerable gains at low bit rates of 16 kbps per channel for a 44.1-kHz sampled audio signal using the proposed approach.

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IEEE TRANSACTIONS ON AUDIO, SPEECH, AND LANGUAGE PROCESSING, VOL. 14, NO. 2, MARCH 2006 623
A Trellis-Based Optimal Parameter Value
Selection for Audio Coding
Ashish Aggarwal, Member, IEEE, Shankar L. Regunathan, Member, IEEE, and Kenneth Rose, Fellow, IEEE
Abstract—This paper considers the problem of selecting a set
of parameter values from a given parameter space, in order to
perform rate-distortion optimization in the context of audio com-
pression. Due to interdependencies between parameters, separate
optimization of parameter values is inherently suboptimal, yet
a straightforward brute-force joint search involves prohibitive
computational complexity. This work proposes a new method
for joint rate-distortion optimization, while accounting for in-
terparameter dependencies. The optimal solution is achieved, at
significantly reduced complexity as compared to a brute-force
search, by employing a Viterbi search over a trellis. Two objective
distortion metrics are specifically considered: the average, and
the maximum noise-to-mask ratio. Subjective (AB/MOS) and ob-
jective (average/maximum noise-to-mask ratio) tests demonstrate
considerable gains at low bit rates of 16 kbps per channel for a
44.1-kHz sampled audio signal using the proposed approach.
Index Terms—Advanced audio coder (AAC), audio coding,
bit allocation, dynamic programming, parameter selection, side-
information, trellis, Viterbi.
I. INTRODUCTION
A
mission of music over the Internet. Such applications benefit
substantiallyfrom improvedcompressionperformance.Current
audio coders such as MPEG Advanced Audio Coder (AAC)
[1], [2], AC3 [3], PAC [4], ATRAC [5], and G.722.1 [6] rely
heavily on the removal of perceptually irrelevant information
[7]–[10] from the source signal. For a thorough description of
current audio coding techniques, see [11]. Perceptually irrel-
evant information is exploited via calculation of the masking
threshold—the threshold below which a signal (or noise) is
rendered inaudible—which, in turn, involves time-adaptive
spectral shaping of the quantization noise. Shaping of the
quantization noise is a rate-distortion optimization performed
at the encoder. Noise shaping is typically achieved by varying
the granularity of the quantizer employed in the different
UDIO COMPRESSION is central to many multimedia
applications such as digital audio broadcasting and trans-
Manuscript received January 12, 2003; revised November 24, 2004.
This work was supported in part by the NSF under Grants MIP-9707764,
EIA-9986057, and EIA-0080134, the University of California MICRO
Program, Dolby Laboratories, Inc., Lucent Technologies, Inc., Mindspeed
Technologies, and Qualcomm, Inc. The associate editor coordinating the
review of this manuscript and approving it for publication was Dr. Ravi P.
Ramchandran.
A. Aggarwal is with Harman Consumer Group, Northridge, CA 93129 USA
(e-mail: aaggarwa@harman.com).
S. L. Regunathan is with Microsoft Corp., Redmond, WA 98052 USA
(e-mail: shrane@microsoft.com).
K. Rose is with the Department of Electrical and Computer Engineering,
University of California, Santa Barbara, CA 93106-9560 USA (e-mail: rose@
ece.ucsb.edu).
Digital Object Identifier 10.1109/TSA.2005.855833
frequency bands (or critical bands [7] that emulate the human
auditory system’s grouping of adjacent frequency bands). The
choice of quantizer granularity is one of the many parameters
whose values are chosen dynamically by the encoder in order to
perform rate-distortion optimization. We refer to the complete
set of such parameters as the “encoding parameters.” Selection
of encoding parameter values is central to the rate-distortion
optimization performed by the encoder.
Consider AAC for example. It performs spectral decompo-
sition of a frame of the audio signal, groups the spectral coef-
ficients into bands, and quantizes the coefficients using scalar
quantizers. Adaptive noise shaping is achieved by allowing
per-band scaling of the generic scalar quantizer by an appro-
priate scale factor (SF). Since the SF is shared by the entire
band, each band is commonly referred to as a scale factor band
(SFB). The quantized coefficient indices are entropy coded
using a possibly different Huffman codebook (HCB) for each
SFB. The choice of the HCB is made from a set of predesigned
codebooks. The SF and HCB values chosen per SFB form the
set of parameters which, together with the quantized coeffi-
cient indices, convey to the decoder all the information needed
to reconstruct the coefficients for the frame. These parameters
constitute the encoding parameters, whose values are deter-
mined by the encoder for every frame of the audio signal.
It is conceivable to obtain the optimal parameter values in a
rate-distortion sense using a straightforward brute-force search.
However, such an optimal scheme involves prohibitive com-
putational complexity due to the large size of the parameter
space. AAC allows for as many as 60 distinct SF values and
12 predesigned HCBs. For a frame of 44.1-kHz sampled audio
consisting of 49 SFBs the cardinality of the parameter space
reaches
—clearly putting brute-force search beyond
computational reach.
A suboptimal choice of parameter values can significantly
degrade the encoder’s compression performance. At rela-
tively high encoding rates, there exist multiple solutions
for which the quantization noise completely falls below the
masking threshold. In this case, a suboptimal choice in the rate-
distortion sense may not cause considerable subjective perfor-
mance degradation. However, when the signal is quantized at
low rates (for example, 16–48 kbps/channel for a 44.1-kHz
sampled signal) it is impossible to maintain all the quantization
noise below the masking threshold, and it is critical to care-
fully optimize the parameter values. Hence, computationally
efficient search for the optimal encoding parameter values
is an interesting and important problem in audio coding. It
is known to play a crucial role in other signal compression
applications as well [12]–[14]. In this paper we focus on the
1558-7916/$20.00 © 2006 IEEE

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624IEEE TRANSACTIONS ON AUDIO, SPEECH, AND LANGUAGE PROCESSING, VOL. 14, NO. 2, MARCH 2006
problem of optimally selecting encoding parameter values for
audio compression. The term “optimality” is employed in the
rate-distortion sense, i.e., the optimal selection is one which
minimizes the distortion measure for the prescribed total rate.
We outline the solution for two objective metrics: the average
and the maximum noise-to-mask ratios (NMR) [15]–[18]. Note
that this paper does not directly address the widely recognized
problem of finding an objective metric that adequately reflects
the subjective quality of reconstructed audio signals.
Selection of values for the encoding parameters is closely
related to the problem of bit allocation [14], [19] whose early
approaches employ high-resolution quantization theory to
arrive at a simple solution that is implementable by the popular
water-filling algorithm [20]–[22]. The algorithm attempts to
maintain a constant distortion (say, NMR) across the coeffi-
cients (or critical bands) and forms the basis for selection of
parameter values in various audio coding algorithms, such as
the so-called two-loop search (TLS) [23]. For a comprehensive
review of approaches to bit allocation and parameter value
selection, see [12], [13], [24]–[27]. Conventional water-filling
based approaches suffer from two major drawbacks. First,
the coefficients are not statistically independent, however,
conventional methods do not accurately account for these inter-
coefficient dependencies that exist in the spectral representation
[13]; and second, as we show later, their solution fails to distin-
guish between the objective measures considered in this paper.
Consequently, the choice of encoding parameter values may
be significantly suboptimal for either metric and the resulting
compression performance penalty may be considerable at low
bit rates. The importance of improved low bit rate performance
is further highlighted in the case of bit rate scalable (also,
embedded or layered) compression [28], [29] where multiple
low bit rate encoding modules are employed.
We propose a search algorithm which explicitly optimizes
for the interparameter dependencies that exist in the spectral
representation. To combat the prohibitive computational com-
plexityofthestraightforward brute-force solution, werecast the
problem as a search through a trellis, and employ dynamic pro-
gramming [30] to obtain the optimal solution at a drastically
reduced search complexity. The search is outlined for the two
objective metrics, which are both based on the NMR, namely,
ANMR and MNMR. The proposed trellis-based search is com-
pared with the water-filling approach of TLS described in [23]
as competing search modules in AAC (see Section V for fur-
ther details). Note that TLS is the best publicly disclosed search
method for AAC. Simulation results demonstrate substantial
improvement in the encoder’s low bit rate performance. For ex-
ample, on a standard critical test database from EBU-SQAM
[31],[32]comprisingof44.1-kHzsampled(mono)audiosignal,
the proposed search method operating at bit rates in the range
of 16–32 kbps, requires half the bit rate to achieve the same
objective (ANMR/MNMR) and subjective (AB/MOS) quality
as TLS. When implemented within a four-layer scalable coder
where each layer employs 16-kbps AAC encoding modules,
the proposed scheme achieved performance close to that of a
56 kbps nonscalable AAC coder. Furthermore, as the solution
achieves rate-distortion optimality, it promises a useful frame-
work for performance evaluation of other search schemes (e.g.,
see [33] and [34]). The performance benefit is achieved at the
expense of computational complexity as compared to the TLS
and it is incurred only at the encoder. It is important to empha-
sizethat theproposed schemeleavesthebit stream syntax intact
and the AAC decoder unaltered. The method is hence standard-
compatible. Preliminary results of this work have been reported
in [35] and [36].
The organization of the paper is as follows. Section II
provides a brief background to the problem. The proposed
trellis-based search method is derived in Section III. The im-
plementation of the proposed search within AAC is described
in Section IV, and results are summarized in Section V.
II. BACKGROUND
A. Objective Measures in Audio Coding
Most objective measures employed in rate-distortion opti-
mization of the encoder are designed to model subjective, per-
ceptual distortion. On the one hand, simple metrics such as the
mean-squared error (MSE) fail to model perceptual distortion
accurately. On the other, metrics with relatively good modeling
accuracy, such as PAQM [37] and PEAQ [38], [39], are too
complex to be used in run-time optimization of the encoder.
While a suitable objective metric that accurately models the
subjective quality remains an unsolved problem, most widely
used objective measures involve the NMR [15], [16], which
is the ratio of the quantization noise energy to the masking
threshold in the given critical band [7]–[10]. The NMR in the
critical band may equivalently be viewed as a weighted squared
error (WSE) whereby the weights are simply the inverse of the
masking threshold in the critical band. NMR below unity in
a critical band indicates that quantization noise in that band
is imperceptible. At low rates it is often impossible to main-
tain the NMR below unity in all the critical bands. Hence, the
NMR values obtained from the various critical bands are com-
bined into a scalar distortion metric. Two common metrics are:
ANMR, which is the NMR averaged over all the critical bands
in the frame, and MNMR, which is the maximum NMR of all
the critical bands in a frame [17], [18].
Let
be the squared quantization error,
critical band , and
be the total number of bands. ANMR is
given by
be the weight of
(1)
and MNMR by
(2)
Subjective listening tests performed by us [36] and in [17]
and [18] indicate substantial differences in the quality of audio
signal resulting from optimization of the two metrics. At low
rates,optimizationofMNMRmetricresultedinfewerannoying
artifacts such as clicks, but the average quality was perceived to
be inferior to ANMR. However, there was no general consistent
preference for either.

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AGGARWAL et al.: TRELLIS-BASED OPTIMAL PARAMETER VALUE SELECTION FOR AUDIO CODING625
Fig.1.
are applied prior to quantization and coding (QC). The psychoacoustic model
outputs the masking threshold which is used for rate-distortion optimization.
BlockdiagramoftheAACencoder.Transformandpreprocessingtools
B. MPEGs Advanced Audio Coding
This section focuses on the quantization module of AAC.
A simplified, high-level block diagram of the AAC encoder is
shown in Fig. 1. The quantization and coding (QC) module,
which is central to this work, is shown in greater detail. The
time domain signal is grouped into overlapping frames and
transformed into the spectral domain using the modified dis-
crete cosine transform (MDCT). The transform yields a set of
1024 coefficients that are then quantized using the QC module.
In the QC module, the transform coefficients are grouped into
nonuniform frequency bands, termed scale factor band (SFB),
and all coefficients within a given SFB are quantized using the
same nonuniform scalar quantizer which is characterized using
a compander (see [1] and [2], for further details). The quantizer
is a scaled version of the generic quantizer, and is determined
by the scale factor (SF) parameter, which is selected for each
SFB and controls the desired noise level in the band. The time
domain signal is also input to the psychoacoustic model, whose
output is the masking threshold for each SFB.
Statistical redundancy in the quantized coefficient indices
is exploited by the use of entropy and run-length coding tech-
niques. AAC offers a set of 12 predesigned Huffman codebooks
(HCB), from which one is selected for each SFB for encoding
the quantized coefficients indices. In addition to the quantized
coefficient indices, side information must be transmitted to
specify SF and HCB selections for each SFB. SF values are
differentially encoded using a variable length code, and HCB
selection is encoded using a run-length code. The rate-distor-
tion optimization at the encoder involves the choice of SF and
HCB values for each SFB.
C. Parameter Value Selection in Current Audio Encoders
Recall thatremovalof perceptually irrelevantinformation via
quantization noise shaping is implemented in audio coding by
appropriately selecting the values of the encoding parameters
forthevariousfrequencybands.Thisproblemis,inturn,closely
related to the problem of bit allocation, which has been exten-
sively covered in the signal compression literature. A compre-
hensive coverage of this topic is beyond the scope of this paper,
and can be found in [14] and [19]. We will only briefly outline
here the relevant portions of the classic problem of bit alloca-
tion and its known water-filling solution [20]–[22] which stems
from high-resolution quantization theory.
The bit allocation problem is one where a fixed bit budget
needs to be distributed among different coefficients in order
to minimize the distortion (e.g., NMR) at hand. Let
number of bits allocated to, and
coefficient . Let
be the target rateand
of coefficients in the frame. The problem of bit allocation may
be stated as
be the
be the resulting distortion of,
be the total number
(3)
where
represents the bit allocation vector and
is the optimal allocation. Early solutions to the problem of
bit allocation use high-resolution (quantization) approximation
[20]–[22] to model the distortion as
(4)
where
depends on the slope of the probability density function of the
coefficients. All coefficients are typically assumed to have the
same . This model is the basis of the celebrated solution to the
problem of independent bit allocation
is the variance of coefficient , andis a constant that
(5)
where
allocationisoptimal,itiseasytoseethattheresultingdistortion
is the same for all coefficients, i.e.,
(for proof see [14]). When the bit
(6)
Hence, the optimal bit allocation can be implemented by a
simple water-filling algorithm, where the same level of distor-
tion is maintained at all coefficients, and this level is varied to
meet the target rate. Note that in the context of audio, (3) cor-
responds to the ANMR measure. An interesting (and perhaps
surprising) observation is made when one analyzes the bit allo-
cation problem for minimizing the MNMR distortion metric. It
turns out that the same water-filling solution optimizes MNMR
metric as well, at high resolution (see the Appendix for details).
Variants of the basic water-filling algorithm are typically
employed for selection of parameter values in audio coding.
Consider, for example, TLS [23], which consists of two nested
loops. The task of the inner iteration loop is to uniformly
change the SF values of all the SFBs by a constant amount,
and determine the HCB values so that the given spectral data
may be encoded while satisfying the rate constraint. The outer
loop changes the SF values of individual SFBs, and thus shapes
the quantization noise to best match the psychoacoustic model.
In a nutshell, TLS tries to maintain the NMR in each SFB
below a given level, and then adjusts this level to meet the rate
constraint.
One major drawback of the approach is the use of the dis-
tortion model given in (4). The model makes it difficult, and
often impossible, to account for the side-information rate when
performing dynamic bit allocation. Shoham and Gersho pro-
posed an alternative Lagrangian-based solution to account for
the side-information rate [13], [40], without recourse to high-
resolution approximation or other analytical models of the
distortion. However, they assumed coefficient independence in

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626IEEE TRANSACTIONS ON AUDIO, SPEECH, AND LANGUAGE PROCESSING, VOL. 14, NO. 2, MARCH 2006
Fig. 2.
using VM-TLS andTB-ANMR (proposed).Theside-informationrate isplotted
versus the total rate for a single channel 44.1-kHz sampled audio signal. Side-
information includes bits consumed to transmit SF and HCB values.
Side-information employed by the TLS for a AAC implementation
calculating the side-information rate. Similar results were also
reported in [24] and [25]. The more general case of dependent
bit allocation was addressed in [12].
D. Problem Motivation and Challenges
The encoder’s problem is to select the values of the encoding
parameters so as to minimize the distortion metric for the given
targetrate.Thisproblemiscomplicatedbyseveralfactors.Asthe
statistical characteristics of the audio signal vary considerably
with time, parameter values must be chosen dynamically. A
trade-off emerges wherein dynamic selection helps reduce the
rate required to transmit the quantized coefficients but must be
transmittedasside-informationandhenceincreasestherate.Fur-
ther, thereexist dependenciesacrossthespectral coefficients(or
criticalbands)whichaffectthetotalbitrate.Thesedependencies
are, in fact, the motivation behind AACs use of run-length and
differential coding of HCB and SF values, respectively. Thus,
the side-information rate (and hence the total rate) is a joint
function of all parameter values used to encode the coefficients
in the frame. It cannot be expressed as a simple sum of the bits
independently optimized for encoding individual parameter
values. This observation points to a major shortcoming of the
conventional water-filling approach, which relies critically on
the invalid assumption of parameter independence. Yet another
drawback of the conventional approach is due to the underlying
rate-distortion model, which is derived from high-resolution
quantization theory. The model not only breaks down when
encoding rates are low, but also fails to accurately account for
the(timevarying)raterequiredtotransmittheside-information.
Conventional schemes do not take parameter dependencies into
account and fail to explicitly optimize the side-information rate.
TLS, in particular, accounts for the side-information rate only
by counting the side-information bits in the inner (rate) loop.
However,itdoes notexplicitlyoptimizetheencoding parameter
values while accounting for their contribution to the side-infor-
mationrate.Athighrates,thepriceofignoringexplicitoptimiza-
tion of the side-information rate may be tolerable because the
side-information rate forms a relatively small percentage of the
total rate. Fig. 2 shows the rate consumed in transmission of SF
and HCB values versus the total rate. It is evident that, at low bit
rates,side-informationmayconsumeasmuchas30%–40%ofthe
totalrate.Attheserates,ignoringside-informationandparameter
dependencies often results in a severe performance penalty.
Theproblemisfurthercomplicatedinthecaseofaudiobythe
factthatdifferentobjectivecriteria,suchasANMRandMNMR,
may be used for encoder optimization. Note that this compli-
cation disappears whenever the assumptions of high-resolution
and parameterindependence are valid. Recall furtherthatin this
case the same water-filling algorithm optimizes both criteria.
Thus, the TLS-based search method can afford to be agnostic of
the distortion metric. However, these assumptions fail to hold in
practical audio coding. In fact, subjective tests [17], [18], [36]
indicate that in practice (when the above assumptions do not
hold) the perceived output quality of the optimum solution for
thetwomeasuresdiffersignificantly,especiallyatlowrates.The
goal of efficient audio compression makes it imperative to opti-
mize a correctly chosen distortion metric.
III. JOINT SELECTION OF PARAMETER VALUES:
PROBLEM FORMULATION
In this section, we tackle the problem in the context of gen-
eral audio coding. To concretize the presentation, we employ
the AAC framework for illustrating the relevant concepts. For
thegeneral formulation, we continueto use terminology consis-
tentwiththeonecommonlyemployedinclassicalbitallocation,
wherein the parameter values are selected for each coefficient.
The formulation is specialized in a straightforward manner to
the case of AAC, where parameter values are selected per SFB.
It should perhaps be reemphasized that this approach is not re-
stricted to AAC but is, in fact, applicable to a wide variety of
audio coding standards including AC-3 [41] and G.722.1 [6].
A. Parameter Space
The quantization and encoding of each spectral coefficient is
determined by a limited set of encoding parameters. In the spe-
cificcaseofAAC,theencoderselectsvaluesfortwoparameters,
SFandHCB,foreachSFBintheframe.Oncethischoiceismade,
thequantizationandcodingoperationsmaybeperformedforall
the coefficients in that SFB. Hence, SF and HCB, whose values
arechosenperSFB,constitutetheencodingparametersforAAC.
The parameter space of a coefficient (or a band) is the set of all
permissible values of all the parameters for the coefficient (or
band).Apointintheparameterspaceisgivenbythecombination
ofvaluesfor the(typically multiple)encodingparameters inuse
by the specific compression algorithm. Note that AAC sets re-
strictiveboundsonthequantizationindexvaluesandthedynamic
range of the quantized coefficients that may employ a given
HCB. These restrictions effectively reduce the parameter space.
B. Cost Function Formulation
Let
where
possible parameters. Without loss of generality, we assume
for simplicity the same parameter space for each coefficient.
represent the parameter for the th coefficient,
is the parameter space with

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AGGARWAL et al.: TRELLIS-BASED OPTIMAL PARAMETER VALUE SELECTION FOR AUDIO CODING 627
Let the number of coefficients in the frame be
the set of parameters for the
. For the case of AAC, let us denote the set
of all possible SF values by
values by
distinct SF values anddistinct HCB values. Further, let
be the SF value andbe the HCB value for the th SFB in the
frame. Vectors
andare used to denote the selected SF and
HCB values for all the SFBs in the frame, i.e.,
and . The combined parameter space for each
SFB in AAC is the product space
elements:
. We denote
coefficients by the vector
and HCB
. Note that we allow for
and has
,.
C. Total Rate and Distortion
The total rate,
rameter vector
, and distortion,, are functions of the pa-
(7)
In order to make the formulation applicable to all scenarios of
potential interest, neither the distortion nor the rate is assumed
additive over individual coefficients.
To illustrate this rate and distortion calculation we return to
the example of AAC. The total rate required for quantization in
AAC can be divided into three parts: bits required to transmit
the quantized coefficient indices; bits required to transmit the
SF values; and bits required to transmit the HCB values.
• Let
be the number of bits required to encode
the quantized coefficient indices of the th SFB using the
SF value of
and HCB value of
spectral coefficients,
is completely determined by the
two parameters).
• Let
denote the number of bits specifying SF for a SFB.
Since AAC employs differential coding of the SFs,
function of two parameters,
and we write explicitly
• Similarly, let
represent the number of bits needed to en-
code the HCB value of the SFB. The run-length coding of
HCB produces 9 bits whenever
erwise. Hence,
is a function of
explicitly
.
Combining the three functions, the number of bits,
transmitting the th SFB is given by
. (Note that given the
is a
and
.
, for the th SFB,
and no bits oth-
and we write and
, for
(8)
The total number of bits produced for the entire frame is then
(9)
where
Given the spectral coefficients, to calculate the distortion in
SFB
we need only the band’s SF value
the quantized coefficients, and the corresponding quantization
andare initialized to zero.
, which determines
noise. Let
weight (inverse of the masked threshold) of the th SFB, the
NMRoftheSFBequals
be used as the metric to combine the NMRs from the different
SFBs. ANMR and MNMR for AAC can be calculated by sub-
stituting
forin (1) and (2), respectively.
Theproblemofparametervaluesselectionmaynowbestated
mathematically as
represent the quantization noise. If is the
.EitherANMRorMNMRcan
(10)
where
case of AAC,
depending on the criterion in use.
is the target bit rate for the frame. Note that, in the
is given by (9), and by (1) or (2),
IV. TRELLIS-BASED OPTIMIZATION
Let us now consider the solution of the optimization problem
of(10).Thereare
possiblechoicesateachstageandthereare
such stages. A straightforward brute-force solution to (10)
has complexity in the order of
theremaybeasmanyas49SFBs,60SFs,and12HCBs,andthe
complexity of the brute-force search is
clearly impractical. We outline next an alternative approach to
thisoptimizationproblem,whichisbasedondynamicprogram-
ming [30]. First, standard Lagrangian formulation is employed
toconvert(10)intoanunconstrainedoptimizationproblem.The
Lagrangian cost function so obtained, is then demonstrated to
exhibit the property of dynamic programming optimality [30].
The well-known Viterbi search [42], [43] through a trellis is ap-
plied to achieve the optimal solution at highly reduced com-
plexity. Detailed algorithmic description of the proposed so-
lution’s application to AAC is presented for the two objective
measures. (A general description of the Viterbi algorithm is
available at the above references).
The standard Lagrangian procedure to reformulate the con-
strainedoptimizationproblemof(10)yieldstheLagrangiancost
. In the case of AAC,
, which is
(11)
where
is the Lagrange multiplier. Clearly
(12)
is the unconstrained minimization problem whose solution
is also the solution of (10), once
constraint
. The original constrained minimization
problem is hence solved by iterating over the different values
of
so as to achieve the target rate.
is adjusted to satisfy the
A. Dynamic Programming Solution
We construct a trellis with
ulate the states with the parameter values
simple three-stage trellis is shown in Fig. 3. With every branch
in this trellis we associate a cost corresponding to its contribu-
tion to the overall Lagrangian cost. The cost associated with
the branch connecting
and
Clearly,everypaththroughthetrellisgivesaparticularchoiceof
stages and
states and pop-
,. A
is denoted by.