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MULTI-RATE EXTENSION OF THE SCALABLE TO LOSSLESS PSPIHT AUDIO CODER
Mohammed Raad, Ian Burnett
School of Electrical,Computer and
Telecommunications Engineering,
University Of Wollongong, Australia
Alfred Mertins
University of Oldenburg
School of Mathematics and Natural Sciences
D-26111 Oldenburg, Germany
ABSTRACT
This paper extends a scalable to lossless compression scheme to
allow scalability in terms of sampling rate as well as quantization
resolution. The scheme presented is an extension of a perceptu-
ally scalable scheme that scales to lossless compression, producing
smooth objective scalability, in terms of SNR, until lossless com-
pression is achieved. The scheme is built around the Perceptual
SPIHT algorithm, which is a modification of the SPIHT algorithm.
An analysis of the expected limitations of scaling across sampling
rates is given as well as lossless compression results showing the
competitive performance of the presented technique.
1. INTRODUCTION
The aim of Lossless compression is the reduction of bandwidth
or memory required to transmit or store the original audio sig-
nal. That is, the error between the original Pulse Code Modulated
(PCM) signal and the compressed version is zero.
Lossless audio coding may be approached from a signal mod-
eling perspective [1],[2],[3] where the signal is typically modeled
using a linear predictor, which may either be FIR or IIR [2]. The
aim of using a linear predictor is to decorrelate the audio sam-
ples in the time domain and to reduce the signal energy that must
be entropy coded [1]. As these coding schemes rely on entropy
compression, the statistics of the signal being coded have a strong
influence on the performance of the coder. Compression ratios re-
ported range between 1.4 and 5.3 [1].
Lossless audio compression has also been approached from
a transform coding perspective [4][5][6]. This approach employs
time to frequency transforms as de-correlation engines instead of
linear predictors. In the cases of [5] and [6] the integer MDCT
(IMDCT) is employed to allow the integer representation of the
transform coefficients, and hence streamlining the lossless com-
pression of these coefficients.
Recently, there has been a growing interest in the develop-
ment of scalable compression schemes as well as scalable to loss-
less compression schemes [7][5][6][8]. Considering the advances
in the bandwidth availability for cellular telephone and internet
users, it is clear that a compression scheme that combines both
scalability and lossless compression is of interest and potential
use. The ability to smoothly scale from narrower bandwidth sig-
nals to wider bandwidth signals with different quantization res-
olution is also of interest, as pointed out in [8]. In this paper,
we present an extension of the scalable audio coder presented in
[9] to allow for the expansion of bandwidth as well as increase in
quantization resolution. The scheme presented in [9] allows very
fine granular scalability as well as competitive compression at the
lossless stage across different bandwidths and quantization resolu-
tions. The compression scheme is built around transform coding of
audio, similar to [4], [6] and [5]. Particularly, a modified version
of the Set Partitioning In Hierarchical Trees (SPIHT) algorithm
[10], named Perceptual SPIHT (PSPIHT), is used to allow scala-
bility as well as perfect reconstruction. The use of PSPIHT and
SPIHT allows the coder to quantize the transform coefficients in
such a manner that only the input audio segment’s statistics are
required, avoiding the necessity to design dedicated entropy code
books. Scalability in bandwidth is obtained through the scalable
transmission of the error between the wider bandwidth signal and
the narrower bandwidth signal in the frequency and time domains.
This paper is organized as follows. Section 2 describes the
different components of the scalable-to-lossless scheme. Section
3 gives an outline of the bandwidth scalable scheme as well as an
analysis of expected performance. Section 4 presents the lossless
results obtained across four bandwidths, and Section 5 provides a
brief conclusion.
2. THE SCALABLE TO LOSSLESS SCHEME
The scalable to lossless scheme presented in [9] is the basis upon
which the bandwidth scalable coder is built, as such it will be de-
scribed first. Figure 1 illustrates the PSPIHT scalable to lossless
scheme. It consists of the combination of the lossy coder pre-
sented in [11], which is based on the Modulated Lapped Transform
(MLT) and SPIHT, and a lossless coder for transmitting the error
incurred from the lossy part. The lossy part is given by the right
half of the structure in Fig. 1, and the error coding (if present)
takes place in the left half. Note that both parts of the coder are
based on the SPIHT algorithm. In this section we mainly focus on
the lossy part of the structure, referred to as MLT-PSPIHT.
The input signal is transformed using the MLT where floating
point calculations are used. The transform coefficients are encoded
using PSPIHT, and the bitstream is transmitted to the decoder. We
will refer to this bitstream as bst1which is further divided into
bst1aand bst1bby PSPIHT. This second stage division aims to
separate perceptually significant coefficients from perceptually in-
significant coefficients such that bst1acontains the perceptually
significant coefficients and is transmitted before bst1b.bst1is de-
coded at the encoder and the synthesized audio is subtracted from
the original audio to obtain the output error. Here integer opera-
tions are used, so that the error output is integer and has a dynamic
range that is typically less than that of the original integer signal.
The time-domain error signal is then encoded into Bitstream bst2,
using SPIHT. At the decoder, both bitstreams are received as part
of one global bitstream, with bst1making up the first part of the
total bitstream for this section of the scheme. The decoder may
decode up to any rate desired.
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xn()
xn()
xn()
xn()
xn()
sn()
e()n
e()n
+
+
-
-
~
~
^
^
Round
to integer
Round
to integer
PSPIHT Encode
SPIHT Encode
Bitstream two Bitstream one
PSPIHT Decode
SPIHT Decode PSPIHT Decode
IMLT
MLTIMLT
Fig. 1. The scalable-to-lossless scheme based on SPIHT and
PSPIHT
2.1. The PSPIHT algorithm
SPIHT [10] is a coding algorithm that allows the transmission of
coefficients in a pseudo-sorted fashion where the most significant
bits of the largest coefficients are sent first. The sorting is carried
out according to the magnitudes of the coefficients. The generated
bitstream is fully embedded, allowing optimal reduction of coding
noise with every additional bit sent [10]. It can be truncated at any
point to achieve the best reconstruction for the actual number of
bits sent. The original design of SPIHT was aimed at image com-
pression, and the intent was to use the algorithm in the frequency
domain [10]. However, the algorithm may also be used in the time
domain.
PSPIHT is a modification of SPIHT in the frequency domain
that allows the transmission of perceptually significant coefficients
ahead of perceptually insignificant coefficients whilst quantizing
both sets of coefficients with the same resolution. Such an algo-
rithm can maintain the potential for lossless synthesis as energy
significant spectral components, that are perceptually insignificant,
are not distorted more than perceptually significant spectral com-
ponents. The modification focuses on introducing a perceptual
significance test to allow the required bitstream formatting. The
perceptual significance test is based on the perceptual entropy of
the given coefficient as determined in [12]. For PSPIHT a few
new definitions are added to those used by SPIHT and listed in
[10]. Firstly, vpe is defined to be a binary vector with perceptual
significance information for the sub-band coefficients. That is, if
vpe(n)=1then coefficient nis perceptually significant, other-
wise it is perceptually insignificant. Also, LP I SP is defined to
be the list of perceptually insignificant, but energy significant co-
efficients. That is, LP I SP contains pointers to coefficients that
are significant in terms of energy (or magnitude) but lie in spectral
bands that contain other more significant coefficients which have
masked them. Finally, the perceptually significant component of
Bitstream one is labelled as bst1aand the perceptually insignifi-
cant component as bst1b. Fixed limits can be set for the size of
bst1aand bst1b. The complete algorithm is listed in [9] and so
will not be listed here. The operation of PSPIHT differs primar-
ily from SPIHT in the application of a perceptual significance test
after the energy significance test.
In the sorting pass, the energy significance test is maintained
as the first test. Sorting bits are sent to bst1auntil an energy sig-
nificant coefficient is encountered. This coefficient is then tested
for perceptual significance by checking the corresponding entry in
vpe, if the coefficient is found to be significant (and bst1ais not
full) then the sign bit and further refinement bits are sent to bst1a,
otherwise these bits are sent to bst1b. The perceptual significance
test is only applied to individual coefficients and not to whole sets
as is the energy significance test. The same process is followed at
the decoder which obtains the test results (the sorting information),
sign bits and significant bits from the bitstream. Note that the ma-
jor task of the algorithm is to re-arrange the bitstream produced
so that it reflects perceptual significance, allowing more percep-
tually accurate synthesis at lower rates. Some extra overhead is
encountered in the bitstream formatting as a pointer must also be
transmitted indicating the length of bst1a. Thisisnecessaryfor
the decoder to be able to divide the total bitstream correctly and to
allow bst1ato be less than its hard-coded maximum length, should
the signal contain fewer significant components than expected. Al-
though the listed algorithm outputs perceptual significance infor-
mation it does so only for energy significant components and even
then only when there is space in bst1a, hence it would be rare to
encounter a situation where all of vpe is transmitted.
3. EXTENDING THE SCALABLE TO LOSSLESS CODER
The scalable to lossless coder presented in Section 2 may be ex-
panded to achieve scalability in terms of sampling frequency (and
thus bandwidth) as well as quantization resolution. In this paper
we consider the sampling frequencies 44.1, 48, 96 and 192 kHz
with quantization resolutions of 16, 20 and 24 bits. It should be
noted that the 192 kHz sampled signals may be considered syn-
thetic data as they are up-sampled versions of the 96 kHz sam-
pled signals. One may approach this form of scalability from two
opposing perspectives; a top-down approach which dictates that
the highest sampled signal be coded losslessly hence allowing the
lower sampled, coarser quantized signals to be extracted from it
through the use of bandlimiting and quantization resolution reduc-
tion. This approach is adopted in [6]. On the other hand, one
EUROSPEECH 2003 - GENEVA
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may take a continuous refinement approach that codes the lower
sampled, coarser quantized signals first and scales (continuously
refining) until the higher sampled signals are also losslessly repre-
sented.
Since the scalable to lossless coding scheme proposed in [9]
produced lossless compression results competitive with the state
of the art, it is adopted as the compression scheme for signals with
a sampling frequency of 44.1 kHz and quantization resolution of
16 bits per sample. In other words, the continuous refinement ap-
proach is adopted in this paper.
3.1. An analysis of bandwidth scalable compression
Before describing how bandwidth scalability may be achieved, it
is useful to consider the possible limitations that scaling in fre-
quency may place on the compression ratios achieved. Let b01 be
the number of bits per sample used to quantize a given signal (x1)
that is sampled at f1Hz. Similarly, let b02 be the number of bits
per sample used for another version of the signal (x2) sampled at
f2Hz with f2>f
1and b02 ≥b01 . Also, let the compressed ver-
sions of these signals be represented at b11 and b12 bits per sample.
The compression ratios of each signal are given by α1=b01
b11 and
α2=b02
b12 .
Using the continuous refinement approach means that the total
number of bits used to compress the higher sampled signal is a sum
of the total number of bits used to compress the lower sampled
signal as well as the number of bits spent coding the resulting error
signal. The total number of bits used to code the higher sampled
signal, per second, may thus be expressed as:
B2=B1+Be(1)
where B1,B2,andBeare the total number of bits used to code
x1,x2and the residual signal erequired to expand x1to x2.Now,
B1=b11 ×f1and hence,
b12 ×f2=b11 ×f1+be×f2
b02
α2
×f2=b01
α1
×f1+be×f2
Assuming that the compression ratio is to be kept constant at α,
then
(b12 −be)f2=b11f1
b02
α−be=b01
α
f1
f2
∴be=1
α(b02 −b01
f1
f2
)(2)
The foregoing analysis indicates that as the differences in the
sampling frequency and quantization resolution increase, more bits
may be spent on the residual signal whilst maintaining a constant
compression ratio. That is, if x1is sampled at a significantly lower
frequency to x2and the quantization resolution between the two
signals is also significant then one may spend more bits on be
whilst maintaining a constant compression ratio.
3.2. The bandwidth scalable coder
Figure 2 shows the proposed technique for scaling in frequency
and quantization resolution. Signal x1is losslessly coded before
any bits are transmitted that allow the higher sampled frequency
to be reconstructed. Thus, both the encoder and decoder have the
complete x1signal. The MLT (popularly known as the MDCT)
Fig. 2. The proposed scaling in bandwidth and resolution scheme
is used to transform x1, the obtained coefficients are then zero
padded and scaled. The scaling factor may be one if the MLT co-
efficients are originally normalized, otherwise it is given by k=
f2
f1. Whilst this results in a good approximation of the frequency
domain representation of x2, there is some error that must be ac-
counted for. Figure 3 shows an example frame that has been sam-
pled at 96 kHz. The MLT coefficients of the frame are approxi-
mated by zero padding and scaling the coefficients of the 48 kHz
sampled version of the frame. The error is also shown on two
scales, the left hand scale shows the detail in the error whilst the
right hand scale compares the magnitude of the approximation er-
ror to that of the actual MLT coefficients. It is notable that the
error is considerably smaller in magnitude than the original set of
coefficients which means it is easier to code losslessly (due to the
smaller dynamic range required).
The error coefficients are coded using SPIHT and then added
to the approximated coefficients. This new set of coefficients then
provides a better approximation of the actual MLT coefficients and
so when the inverse MLT is applied, the time-domain error signal
is small. The time domain error signal is calculated in integer form
and transmitted via the use of SPIHT leading to the lossless repre-
sentation of x2.
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0
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2500
−0.2
0
0.2
0
200
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1200
−2
−1
0
1
0
200
400
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−2
−1
0
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−0.01
0
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1200
−2
−1
0
1
Sample number
(a)
(b)
(c)
(d)
Fig. 3. (a) The original 96 kHz sampled signal (b) The MLT co-
efficients at 96 kHz (c) The approximated 96 kHz coefficients (d)
The error in approximation
Table 1. objective results for 96 kHz sampled files
Content
(compression ratio, bits per sample)
Bits, Sampling (kHz)
violin
vocal
16, 44.1
2.67, 6.0
3.70, 4.33
20, 48
1.64, 12.2
2.01, 9.93
24, 96
1.84, 13.05
1.99, 12.05
24, 192
2.13, 11.29
2.27, 10.53
4. RESULTS
Lossless compression results are presented in this section at sam-
pling rates 44.1, 48, 96 and 192 kHz. The subjective performance
of the PSPIHT coder has been previously reported to be compara-
ble with that of the MPEG AAC (VM) at 16, 32 and 64 kbps [9].
Similarly, the smooth objective scalability that may be achieved
when SPIHT is employed was described in that paper.
The lossless results presented here are for two files originally
recorded at 96 kHz sampling rate and 24 bits resolution. These
files were then resampled to obtain higher and lower sampled ver-
sions. As such, the 192 kHz signal may only be considered a syn-
thetic test signal. The content of the test signals as well as the
obtained lossless compression results are listed in Table 1. The
lossless compression results are competitive with the state of the
art at 16 bits, 44.1 kHz (the CD standard) [1]. Also, as the band-
width increases the compression ratio for both signals varies in
agreement with the behaviour predicted by the analysis in Section
3. Namely, the reduction in compression ratio decreases as the
difference in sampling frequency (i.e. f
2
−f
1
)and quantization
resolution (b
02
−b
01
) increases. It is notable that the compression
ratios obtained are also competitive with the state of the art.
5. CONCLUSION
A scalable to lossless compression scheme that scales in band-
width and quantization resolution has been presented. The scheme
is based on the PSPIHT and SPIHT algorithms as well as the MLT.
The PSPIHT algorithm is a modified version of the SPIHT algo-
rithm that allows the transmission of perceptually significant coef-
ficients in a set sorted manner whilst maintaining the same quanti-
zation resolution for perceptually insignificant coefficients. This
allows energy significant components of the signal to be main-
tained at higher rates. Bandwidth scalability was achieved by ap-
proximating the MLT coefficients of the higher sampled signal
from the losslessly decoded lower sampled signal and then trans-
mitting the resulting error in the frequency domain as well as the
time domain. The lossless performance was shown to be better
maintained if the difference in the sampling rate and quantization
resolutions of the signals was not trivial. The lossless compression
results presented showed this to be true. The presented results also
showed that competitive lossless compression may be achieved us-
ing the proposed scheme at a number of sampling frequencies and
quantization resolutions.
6. REFERENCES
[1] M. Hans and R.W. Schafer, “Lossless compression of digital
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AES preprint 4358, November 1996.
[4] T. Liebchen, M. Purat, and P. Noll, “Lossless transform cod-
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[7] T.S. Verma, A perceptually based audio signal model with
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partment of Electrical Engineering, Stanford university, Oc-
tober 1999.
[8] T. Moriya, “Report of AHG on issues in lossless audio cod-
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[9] M. Raad, A. Mertins, and I. Burnett, “Scalable to lossless
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[10] Amir Said and William A. Pearlman, “A new, fast, and ef-
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