SICE Annual Conference in Fukui, August 4-6, 2003
Fukui University, Japan

Audio Signal Compression
via Sampled-Data Control Theory
Shinjiro Ashida, Masaaki Nagahara, Yutaka Yamamoto
Graduate School of Informatics, Kyoto University
Kyoto 606-8501, JAPAN{sashida, nagahara, yy}@acs.i.kyoto-u.ac.jp
Abstract: The paper proposes a new digital audio coding method. This method combinessubband-coding with an interpolater which is designed by using sampled-data control theory.
Moreover, a new bit-allocation method is proposed, which is simpler than the conventional method,
MPEG-1 Audio. A design example is presented to show the advantages of the present method.
Keywords: Sampled-data control, subband-coding, bit allocation, MPEG-1 Audio

1. Introduction

account of the original analog characteristic. Therefore,
we propose a new method for audio signal compression

In many ﬁelds, digital systems have taken the placeof analog systems due to the rapid development of digital equipment. In digital communication, audio/speech
signals are transmitted or stored digitally. Because of alimit on channel capacity or memory size, digital signals
must be compressed economically and eﬃciently.

by using the sampled-data control theory, which cantake the analog characteristic into account, and natural
reconstruction is possible by our method. Moreover, weadopt a simple bit allocation method without the psychoacoustic model. A design example shows that ourmethod is superior to the conventional method.

A well-known compression method is subbandcoding7) . In subband-coding, a digital signal is divided
into multiple subbands of frequency by a multirate ﬁl-2. MPEG-1 Audio

terbank, and then quantized by allocating bits to eachsubband signal with respect to its energy. By using theWe here consider a conventional method, MPEG-1

fact that the frequency distribution of audio signals isgenerally uneven, we decide how to apportion the to-Audio. MPEG-1 Audio has three layers, Layer I, II,and III, and there is a trade-oﬀ between complexity oftal number of code bits available for the quantizationof the subband signals to minimize the audibility of thecomputation and accuracy of reconstruction. Table. 1summarizes this trade-oﬀ. Layer III, usually called MP3,quantization noise. MPEG-1 (Moving Picture Experts
Group) Audio2) is a kind of subband-coding, and widely

can reconstruct audio signals better than Layer I or II,
while the computation is involved, and hence in real-used in digital communication or digital audio.
However, MPEG-1 Audio has the following problems:

time computation, Layer I or II is often used.
We will now discuss Layer II in detail.

• At high compression ratio, the high-frequency component is cut terribly, and hence the restored soundmay be blurred.
• Coding based on psychoacoustic model4, 5) takes
much time, which may make trouble in real-timetransmission in the network.

The system

LayerIIIIII

complexity

simplesimplecomplicatedaccuracypoor

averagegoodTable. 1: Comparison of three Layers: complexity ofcomputation and accuracy of reconstruction

We propose a new method of audio coding which improves these problems.

of MPEG-1 Audio Layer II is shown in Fig. 1. First, theIn general, the original audio signal is an analog one,which has frequency component beyond the Nyquist fre-input audio signal with sampling frequency 44.1 [kHz] isdecomposed into 32 subbands of frequency by the anal-quency. Conventional methods, however, do not takeysis ﬁlterbank (Discrete Cosine Transform (DCT) ﬁlter--1182-

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Scalefactor
information
Analysis
Filterbank

Quantization

Bit
Allocation

FFT

referencegain(dB)

input

transmission

psychoacoustic
model

outputSynthesis
Filterbank

Dequantization

Fig. 1: Signal compression system of MPEG-1 Audiobank). Secondly, the bit-allocation system decides thenumber of bits of each subband with respect to the FFTof the input signal, the scalefactor information, and psychoacoustic model. Each of the decomposed signals is

In this system, the complicated procedures are fre-1, 6, 7)

proposed, and DCT ﬁlterbank is one of them. On
the other hand, the total number of code bits are allocated to minimize the audibility of the quantizationnoise according to psychoacoustic model. It is true that

310

2

310

10

210

310

frequency(Hz)

410

410

410

In the next section, we propose a new method which
improves these problems.

crete Cosine Transform (IDCT) ﬁlterbank), and ﬁnallythe signal is restored.
quency decomposition and bit allocation. For decomposing the frequency, various eﬃcient ﬁlterbanks are

210Fig. 2: FFT of signal (above: original, middle: 64[kbps], below: 48 [kbps])

then quantized and transmitted.
The transmitted signal passes through the dequantization block and the synthesis ﬁlterbank (Inverse Dis-60
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3. Proposed Method
3.1 Subband-Coding
The subband-coding system we propose is shown inFig. 3. In this ﬁgure, u is the input audio signal withsampling frequency 44.1 [kHz]. First, we downsample
u by the downsampler ↓ 2, and the data size is reduced

this procedure improves the quality of the restored sig-by half. Then, the ﬁlterbank in the encoder (analysisﬁlterbank) decomposes the signal into 16 subbands ofnal, but it costs much time, which results in transmission delay. Layer I and III also have this problem.

frequency1 , and each of the decomposed signals is quantized by the quantizer Q. The quantized signal is thenIn addition, Layer III has a lot of complicated proce-transmitted to the decoder.

dures such as adaptive ﬁlterbank, nonlinear quantization, or Haﬀman coding. Therefore Layer III requires

In the decoder, the quantized signal passes throughthe synthesis ﬁlterbank to produce a restored signal v̂.

much more time for encoding than Layer I and II.

Finally, the signal v̂ is upsampled by the upsampler ↑ 2
and ﬁltered by the digital ﬁlter Y (z), and we have theWe next consider the reconstruction characteristic ofMPEG-1 Audio Layer II. The FFT (fast Fourier transformation) of the restored signal is shown in the middleof Fig. 2, where the bit rate for transmission is 64 [kbps].

restored audio signal û with sampling frequency 44.1
[kHz].
The ﬁlter Y (z) is designed by sampled-data control

We can see that this FFT is similar to that of the original signal (shown above in Fig. 2) up to around 10theory 3) . Fig. 4 illustrates the error system for designing the ﬁlter Y (z). We design Y (z) which minimizes[kHz]. But when the bit rate is 48 [kbps] (shown below in Fig. 2), the FFT of the restored signal is sharply

the H ∞ norm of the sampled-data system Tew from theanalog input wc to the analog error output ec . Our

cut at the frequency around 6 [kHz]. In other words,
the high frequency spectrum is not reconstructed ade-

problem is formulated as follows:

quately, which often makes the restored sound blurred.

1 Each bandwidth is around 690 [Hz], which is the same as
MPEG-1 Audio.

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↓2

H0

↓ 16

Q

↑ 16

F0

H1

↓ 16

Q

↑ 16

F1v

step3: Allocate bi bits to the i-th subband in propor
tion to log2 Si under the constraint that 16i=1 bi =B, where B is the total number of bits.

++v̂

↑2Y (z)û

step4: Quantize the i-th subband data according to thequantization bit bi .

H15

↓ 16

F15

↑ 16

Q

encoder

Although this method is very simple, a problem
arises; no bit can be allocated to the low or middle frequency signals when the scale factor of a high-frequency

decoderFig. 3: Proposed audio signal compression system
Problem 1 Given a stable and strictly proper W (s),stable and proper P (s), delay step m and sampling timeh, ﬁnd Y (z) which minimizes
Tew ∞ :=wcW (s)supwc∈L2 [0,∞)Tew wc L2.
wc L2e−m(2h)sw
S2h↑2Y (z)Hh+P (s)subband is as large as that of a low or middle frequency
one. An example of this situation is shown in Fig. 5.
The left graph in Fig. 5 shows scale factors of the subbands, and the right graph shows the number of bits
allocated to each subband. Because the human auditory system is sensitive to low or middle frequency signals, the emptiness at the middle frequency results inan uncomfortable noise. In order to solve this problem,
we preferentially allocate 2 or 3 bits to low or middle
frequency subbands. Then we reallocate the remains ofecthe bits in proportion to the logarithm of the amplitude.

-Step 3 in the algorithm mentioned above is modiﬁed asfollows:

Fig. 4: Error system

step3-1: Allocate 2 bits to the i-th subbands (i =1, . . . , 9). Then divide the scale factor Si by 22 ,i = 1, . . . , 9, and subtract 2 × 9 from total bit num-3.2 Bit Allocation Algorithm

ber B;

It is necessary for eﬃcient compression to take account of the frequency characteristic when we allocate
bits to each subband signal. The fundamental way is toallocate bits in proportion to the logarithm of the amplitude of each subband signal. MPEG introduces adaptive bit allocation with psychoacoustic model4) . However, this method requires a lot of time for computation

:= 2,

Si

:= Si /2bi ,B:= B − 2 × 9.

i = 1, . . . , 9,

step3-2: Reallocate bi bits to i-th subband with(e.g., computation of FFT), and the delay is not desirable for real-time audio signal transmission.

bi :=b + c log S ,i2

c log S ,2Instead, we propose a simple algorithm for bit allocation. To begin with, let us consider a conventionaliii = 1, . . . , 9,

i = 10, . . . , 16,

16where c is a factor decided such that i=1 bi = B,and a denotes the greatest integer equal to or

algorithm of the logarithm bit allocation.
step1: Store M samples

smaller than a.

vi [0], vi [1], . . . , vi [M − 1],

The eﬀectiveness of this algorithm appears in Fig.

of the i-th subband (i=1,. . . , 16).
step2: Deﬁne scale factor Si as the maximum of theabsolute value of the stored samples:
Si := max {|vi [0]|, |vi [1]|, . . . , |vi [M − 1]|}.

bi

(1)

6, from which we can see that to the low or middle
frequency bands, 2 or 3 bits are allocated adequately.
This process may be explained as a psychoacoustic
model, which is simpler because it only requires comparison of the amplitude.

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the number of bits

403020321gain(dB)

Scale Factor (Code value)60

100510015

Subband Number

510

Subband Number

15Fig. 5: Basic bit allocation450

the number of bits

Scale Factor (Code value)

60

40
30
20

31015

3102

31010

210

310frequency(Hz)

410

410

410Fig. 7: FFT of signal (above: original, middle: proposed, below: MPEG-1 Audio)

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100

Subband Number

510

Subband Number

15ReferencesFig. 6: Proposed bit allocation

4. Design Example

1) R. E. Crochiere and L. R. Rabiner, Multirate Digital Signal Processing, Prentice Hall, 1983.
2) ISO/IEC JTCI/SC29, “Information technology –Coding of moving pictures and associated audio for
digital storage media at up to about 1.5 Mbit/s –We here present a design example. We compare theproposed method mentioned above with the conventional method (MPEG-1 Audio Layer II). The bit rateIS 11172 (Part 3: Audio),” 1992.

for transmission is 48 [kbps]. The FFT (fast Fouriertransformation) of the original signal is shown above3) M. Nagahara and Y. Yamamoto, “Sampled-data
H ∞ design of interpolators,” Systems, Informationin Fig. 7. Then we show the FFT of restored signalby the proposed method (in the middle of Fig. 7) andand Control, vol. 14, No. 10, pp. 483–489, 2001.

by MPEG-1 Audio (below in Fig. 7). We can see that
the FFT of the restored signal by MPEG-1 Audio issharply cut at the frequency about 6 [kHz], while theFFT by our method has some frequency component tothe Nyquist frequency (22.05 [kHz]). The reason is thatthe interpolation ﬁlter Y (z) is designed with analog performance. Practically, the sound by MPEG-1 Audio isblurred, while our sound is clearer.

4) P. Noll, “MPEG Digital Audio Coding,” IEEE Signal Processing Magazine, vol. 14, No. 5, pp. 59–
81, September 1997.
5) D. Pan, “A Tutorial on MPEG/Audio Compression,” IEEE Trans. on Multimedia, vol. 2, No. 2,
pp. 60–74, 1995.
6) J. Princen, J. Johnston and A. Bradley, “Subband/Transform Coding Using Filter Bank Designs
Based on Time Domain Alias Cancellation,” Proc.
ICASSP ’87, pp. 2161–2164, 1987.

5. Conclusion
We present a new method for audio signal compression by using the sampled-data H ∞ control theory. The7) P. P. Vaidyanathan, Multirate Systems and Filter
Banks, Prentice Hall, 1993.

proposed method is not only simpler to compute thanthe conventional one, but also takes account of the analog performance.

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