Content uploaded by Kees Schouhamer Immink

Author content

All content in this area was uploaded by Kees Schouhamer Immink on Oct 03, 2020

Content may be subject to copyright.

ENGINEERING REPORTS

GracefulDegradationof DigitalAudioTransmission

Systems*

K. A. SCHOUHAMER IMMINK

Philips Research Laboratories, 5600 JA,Eindhoven, The Netherlands

A new technique for encoding and decoding digital audio signals is presented. It

offers the advantage of a graceful degradation of the audio performance when the signal

is conveyed over channels with a wide range of bandwidths. The flexibility of the new

techniques also provides a key to the future evolution of the quality of digital audio

or video systems within the same digital format. The class of systems described may

well be of particular value in applications involving the storage or broadcast of digital

audio signals when the exact bandwidth of the communication system is unknown.

0 INTRODUCTIONfound in analog systems, which usually show a more

gradual or graceful degradation.

The quality and the reliability of digital audio storage Another basic, undesired property inherent in digital

or communication systems are so widely appreciated communications is the lack of flexibility of digital for-

that it is scarcely conceivable that one might dispute mats. Once a digital communication or storage system

their properties. In this report we will not question the has been designed, it is almost impossible to upgrade

perceptual quality of digital audio; we merely wish to or downgrade the quality of the digital audio signal in

impeach some of the fundamental characteristics of the a manner compatible with the predefined format.

digital communication channel itself. Changing to a higher sampling rate or a finer quanti-

It is well known that the quality of the decoded signal zation is impossible within the defined digital format.

is,up to certain system thresholds, almost ideal. As The magnificent improvement of the audio quality of

an illustrative example consider the Compact Disc sig- the Compact Cassette, all within the same format, would

nal. The EFM encoded signal requires a bandwidth of have been impossible when instead of the analog signal

at least 1.2 MHz for proper functioning [1]. If,for format a digital audio format would have been chosen.

some reason, the bandwidth is reduced to I MHz, for It can be argued that, for the Compact Disc system, in

example, the symbol error rate can quite abruptly be- the future there will be no need to improve the quality,

come so high that the decoding electronics cannot cope but the poor man's digital audio systems, which can

with it and will start muting. Basically this should be hardly be called hi-fi, are now on the drawing boards.

considered a catastrophic breakdown, while the system Actually this is not a real problem but for the fact that

should still be capable of supporting 80% of its normal many of these new formats have to last for decades.

capacity. Even when very smart error correction As time goes on and technology improves, more room

schemes are included, we will find that when the band- will become available. When that happens, we will be

width is below the channel's threshold value, the re- sorry, since extensions compatible with the current

ceiver will not be able to retrieve any useful information formats are out of the question. For a digital audio

at all. A similar phenomenon can be observed when broadcast standard it is important that the vast quantities

additive noise perturbs the communication channel, of existing consumer equipment should not suddenly

Up to a certain threshold level of the noise power the be made obsolete--existing equipment must be capable

system will function properly, but once the noise power of processing any improved signals togive the quality

goes beyond this threshold, the communication system now attainable.

will suddenly cease to exist. This effect is not normally By way of example, let us consider the MAC video

signal [2]. One of the notable properties of MAC video

* Presented at the 82nd Convention of the Audio Engi- transmission is the absence of any absolute limitation

neering Society, London, 1987 March 10-13.in bandwidth reduction. Unlike the PAL and NTSC

J,AudioEng.Soc.,Vol.36,No.1/2,1988January/February 27

IMMINK ENGINEERING REPORTS

video signals, one can freely convey the MAC video and detection technique. To focus our thoughts, let us

signal over transmission systems covering a wide range concentrate, for example, on a magnetic or optical re-

of bandwidths with the corresponding quality of the corder. It is well known that some parts of the available

received video signal. This property is lost as soon as frequency band of these recording systems are less re-

the MACvideo signal is time multiplexed with a digital liable than others. The low-frequency range of a mag-

component whose rate and coding lead to a minimal netic recorder has insufficient signal-to-noise ratio since

bandwidth greater than, or equal to, that of the MAC the inductive head does not respond to signals of low

video signal. The D2-MAC video format has been cho- frequency. An optical recorder also exhibits an unre-

sen to accommodate the audio plus video signal in the liable low-frequency response caused by noise which

existing 5-6-MHz cable bandwidth. Once more band- is induced by reflection variations or fingerprints on

width is available, the video quality will improve, but the optical disk. The frequency characteristics in the

unless the digital audio format is changed, the audio high-frequency range of both recorder types are rela-

quality will remain the same, thus again fueling the tively unreliable due to defocusing, tape-head loss,

notion that audio is a stepchild of television broad- and so on. We proceed by partitioning the available

casting, bandwidth Binto msmaller (virtual) subchannels with

In this report we study a new way of encoding and a bandwidth B/m. We order the subchannels according

decoding digital audio signals which possesses the virtue to their a priori reliability. The subchannel with the

of improved bandwidth adaptability compared with greatest reliability is assigned to the most significant

conventional digital transmission systems. In Sec. 1 bit of the digital audio signal. By modulation of the

we explain in qualitative terms the basic idea of the incoming data on orthogonal time functions,so-called

frequency domain detection technique with a simple carriers (and demodulation in the receiver), it is possible

example. In Sec. 2 we apply the new technique to the to fill the total available frequency range [3]. The signals

transmission of digital audio, generated in this way are multiple valued or even con-

tinuous. Unfortunately many communication or re-

1BASIC CONCEPT cording channels are "hard-limiting" and accept only

two-level full-T pulses. Examples are optical recording,

Prior to the development of a quantitative model for where only pits or lands can be recorded, and magnetic

the coding and detection technique we consider some recording,which is normally employed in such a way

basics of digital transmission. There are a number of that the resulting magnetic domains are positively or

ways in which digital symbols can be represented by negatively saturated. To overcome this difficulty we

physical parameters.All these involve assigning a range proceed as follows.

of waveforms of a continuously variable physical func- We choose candidate codewords of length nwith

tion to represent some digital symbol. Most digital elements from a binary alphabet. The codewords are

systems now in use are binary and synchronous, which also selected in such a way that they divide,according

means that in each symbol time interval or time slot a to a predefined frequency domain criterion, the available

condition such as current or no current, pit or no pit, frequency range of the channel into smaller independent

or positive or negative magnetization is transmitted (or subchannels. The frequency domain criterion will be

stored). (Note that we use the parlance of the corn- based on the (Walsh)-Hadamard transform, which has

munication engineer throughout.) The receiver, under the virtues of simplicity and analytical tractability. A

the control of its clock, properly phased with respect selection of codewords based on other frequency domain

to the incoming data, samples the received signal at criteria is certainly feasible. The major advantage of

the middle of each time slot. It is well known that a Hadamard domain processing compared with Fourier

single pulse transmitted over a bandwidth-limited sys- domain processing is simplified computation, since the

tern is smeared out in time due to convolution with the Hadamard transform requires only addition operations,

channel's impulse response. A sample at the center of whereas the Fourier transform requires complex arith-

a symbol interval is a weighted sum of amplitudes of metic.

pulses in several adjacent intervals. This phenomenon We now select the codewords, denoted by x,x i E

is called intersymbol interference (ISI). If the product {- 1, 1}, i= 1 ..... n, to have an odd parity, that is,

of symbol time interval and system bandwidth is re- the number of + 1's is odd. The codeword set is denoted

duced, the intersymbol interference becomes more se- by S. The reason behind this particular choice of code-

vere. This effect can become so severe that the receiver, word set will become apparent in the following example.

even in the absence of noise, can no longer distinguish It can readily be verified that 2 n-x codewords satisfy

between the symbol value and the intersymbol inter- this condition. We also define the frequency domain

ference and will start to make errors.The basic idea representation yof x, which results from the nxn

behind the technique to be discussed in the subsequent Hadamard transform,

sections is that the information is not contained in one

time interval but is spread out over a plurality of in- y=H,,x

tervals.

We now leave this typical time-domain approach and where H,,, Hn(i,j) E {- 1, 1}, is the Sylvester-type

turn to a frequency-domain discussion of the coding n×nHadamard matrix. This selection of the frequency

28 J. Audio Eng. Soc., Vol. 36, No. 1/2, 1988 January/February

ENGINEERING REPORTS DIGITALAUDIOTRANSMISSION SYSTEMS

domain transform restricts the codeword length to rially. The received codeword, denoted by r, generally

powers of 2 (n_ 4). Due to the particular choice of distorted and corrupted with noise, is now Hadamard

the codeword set Swe find (this can easily be verified) transformed in the receiver into n= 4 frequency com-

for any x_Sthat the elements Yi of the vector y= ponents, denoted by _i, 1 _< i_<n, or

nn2¢ are nonzero.

The basic concept of the new technique is now il- _ = Har. (1)

lustrated by a simple example.

The estimated values of zi, 1 <_ i<_ n, denoted by _i,

1.1 Example 1 are subsequently determined from the polarity of _,

Suppose, for the sake of simplicity, that the codeword that is, 2 i=0 if Yi _ 0; else Zi =1. Element 21 is

is of length 4. The Hadamard matrix is skipped; the other three decoded symbols _2..... Z4

are the received (estimated) source symbols. As a further

I+l +l +1 !ii1 clarification, the Hadamard transform is only imple-

merited in the receiver. It is now easy to understand

+1 +1 -1

H4 = . why the code set consists of codewords with an odd

+ 1 - 1 -1 parity; detection can be kept very simple by just taking

the signs of the frequency transform components _i-

+1 - 1 +1 -The reader may have noticed in Table 1that the source

word assignment to a specific codeword has been pre-

The coefficients of the Hadamard matrix are ordered

in correspondence with increasingly rapid variation of pared in such a way that the symbols z2 ..... z4 (the

the zero crossings (the usual notion of frequency in the three symbols furthest to the right in the column furthest

case of the Fourier transform). It is inappropriate here to the right) are equal to the three source symbols (the

to consider in detail the formal establishment of the column furthest to the left). The variable _1, which

Hadamard matrix or transform, a comprehensive dis- actually contains the unknown dc term of the received

codeword, is not used. Columns 2, 3, and 4 of the

cussion of which may be found in [3] and its many Hadamard matrix have an equal number of +__l's so

references, for example, that any unknown superimposed dc term will be canceled

The 2n-1 = 8 codewords with odd parity are (+ I,

- 1, -1, -1), (- I, +1, -I, -I), (-1, -i, +1, -1), here and will have no effect on the decoding.

It is also intriguing to observe that the power density

(- 1, - 1, -1, + 1) and their inverses. Table 1 shows function of the code discussed here does not exhibit a

the eight source words, their corresponding channel

representations x, according to a coding rule to be vanishing power at the low-frequency end, while this

is a property usually found to be mandatory in channel

explained later, with their associated frequency domain

representations y=H4xand the vector zwith elements codes to be applied in de-constrained channels [4].

zi = 0 ifyi < 0; else z_ = 1. It has been seen in this example that an unreliable

In this example we find that m= 3 source symbols subchannel situated at the low-frequency range can (by

an a priori assignment) be discarded from the available

can be mapped onto n= 4 channelbits, so we conclude

that the rate of this code is 3/4. How do we proceed? set. In a similar way we can, by an a priori assignment

We elucidate the new technique by assuming that the of source words/codewords, eliminate one of the other

subchannels. This is made possible by the remarkable

physical channel has an undesirable response at the

fact that all vectors zhave an odd parity and that they

low-frequency end. For example, the received signal

are all distinct (as can be verified in Table 1). In other

is superimposed on an unknown (quasi) dc current. We

also assume that transmitter and receiver are perfectly words, the eight codewords presented in Table 1 are

synchronized, that is, the beginnings and endings of universally applicable. Irrespective of the particular

the different codewords are known. Suppose we want subset of subchannels used, we employ the same set

of codewords. A set with this property is called a key

to transmit the source word 100. The corresponding set. We found by computer search that for codeword

channel representation found from Table 1, namely

(-1, + 1, -1, -1), is transmitted. It is tacitly assumed length n= 8 a key set can be assembled. A computer

verification showed that this is not possible for n=

that the symbols of the codewords are transmitted se-

16.

This introductory section has sought to outline in

Table 1. An example of a rate 3/4 dc insensitive code. qualitative terms the main idea of the frequency domain

detection technique. In the following section we take

Source x y z a closer look at the more quantitative effects of inter-

000 -1 +1 +1 +1 +2 -2 -2 -2 1000 symbol interference and additive noise.

001 -1 -1 +1 -1 -2 -2 -2 +2 0001

010 -1 -1 -1 +1 -2 -2 +2 -2 0010

011 +1 -1 +1 +1 +2 -2 +2 +2 1011 2 GRACEFUL DEGRADATION

100 -1 +1 -1 -1 -2 +2 -2 -2 0100

101 +1 +1 +1 -I +2 +2 -2 +2 1101 In the preceding section we showed that a codeword

110 +1 +1 -1 +1 +2 +2 +2 -2 1110 set can be assembled which can be detected in such a

111 +1 -I -1 -1 -2 +2 +2 +2 0111

way that an unknown dc component does not disturb

d. Audio Eng. Soc., Vol. 36,No. 1/2, 1988 January/February 29

IMMINK ENGINEERINGREPORTS

the detection. The same codeword set can be employed r(t+Tc) - r(t+ 2To) + r(t+ 3Tc)}/2 of subchannels

to transmit information over a bandwidth-limited chan- 2 and 3, respectively. The eye pattern of subchannel 4

channel with the resulting virtue that parts of the trans- is not shown.

mitted information can be retrieved when the bandwidth Figs. 2 and 3 show the effects of the reduction of

is reduced below a certain value. The codeword/source the relative bandwidth to 0.4 and 0.3, respectively.

word assignment is chosen is such a way that the upper (The other parameters are fixed.) Figs. 2(a) and 3(a)

channel does not contain user information. Table 2 correspond to the signal depicted in Fig. l(a). We ob-

shows the new assignment. Note that the same codeword serve that the higher a particular subchannel is situated

set is employed as that in Table 1. in the frequency band of the channel, the more it is

We leave the foregoing time-discrete analysis and affected by the intersymbol interference caused by the

turn to the time-continuous case. We assume that the relative bandwidth reduction.

channel has a low-pass spectral characteristic with si- Another general point worth mentioning is that of

nusoidal rolloff [5]. The channel waveform, denoted the measure of degradation, the audibility, of the re-

by g(t), can be written as constructed signal. How can we compare quantitatively

a channel with a given reliability with another channel

sin 2_rBt cos 213_rBt consisting of a number of subchannels each with a cer-

g(t) - 2_rBt 1 - 16132B2tz tain reliability? It is notoriously difficult to specify

quantitatively the degree of annoyance experienced by

where Bis the bandwidth of the transmission system a human observer. Dostis [6] arrived at the following

and 13, 0 _< 13_< 1, is the rolloff parameter. The term simple mathematical relation that takes account of the

sinusoidal rolloff becomes clear when we write the effects of both quantization and incorrect decoding of

corresponding frequency characteristic of the channel the received data,

G(f/B),

"1, 0 <_ f/B <_ (l - [3) SNR = 2-zN + 4 _ Pr(Ei)2 -2i

i=1

G(f/B) = 2 1 - sin _ (fiB) ,

where SNR is the signal-to-noise ratio of the recon-

(1 - 13) < f/B _ (1 + 13) structed audio signal and Pr(Ei) is the probability that

.0, fiB > (1 + 13) . the ith audio bit is erroneously received. It is assumed

that the sound is linearly quantized with 2Nlevels and

Fig. l(a)shows the eye patterns of thereceived signal that a natural representation is employed. For more

r(t), which is formed by convolution of the codewords details of the premises of the formula used the reader

and the channel waveform. The parameters are relative is referred to the original literature [6]. When Pr(Ei)

bandwidth B Tc = 0.5, Tc being the channel bit interval, is small, we find the well-known relation between SNR

and the number of quantization levels,

and rolloff parameter 13= 0.1, no additive noise. The

figure is drawn in such a way that a complete received SNR = 2_ -_ 6NdB .

codeword can be seen. The four eyes at sampling mo-

ments t= -1.5, -0.5, 0.5, and 1.5 are fully open. An illuminating experiment concerning the effec-

Fig. l(b) shows the eye pattern of the signal }](t) =

{r(t) + r(t+Tc) + r(t+ 2Tc) + r(t+ 3Tc)}/2 received tiveness of the new technique is to compare its per-

formance with that of the conventional, uncoded, digital

in the first subchannel; the sampling moments are at

t= 0. The scaling by a factor of 2 has been done in data transmission system. For purposes of illustration

we have computed the SNR of an N= 15-bit linear

order to be able to compare Fig. l(a) and (b) on the

same signal-to-noise basis. In a similar way Fig. l(c) quantization system based on a system with an n= 4

codeword length as a function of the bandwidth of the

and (d) shows the eye patterns of_2(t) = {r(t) + r(t+

Tc) - r(t+ 2Tc) - r(t+ 3Tc)}/2 and _3(t) = {r(t) - transmission system. The choice of the value of Nis

not arbitrary. We choose the codeword assignment in

such a way that five codewords accommodate a 15-bit

audio sample. We assume a data format where the five

Table 2. Rate 3/4 low-frequency code. most significant bits of the audio sample are placed in

Source x y z the lowest subchannel, and so on. We now find that

the probability Pr(Ei) of erroneously receiving an audio

000 -1 -1 +1 -1 -2 -2 -2 +2 0001

001 -1 -1 -1 +1 -2 -2 +2 -2 0010 bit is given by Pr(Ei) = Pr(el), 1 _< i_< 5, Pr(Ei) =

010 -1 +1 -I -1 -2 +2 -2 -2 0100 Pr(e2), 6 _< i_< 10, and so on. The probability of

011 + 1 - 1 - 1 - 1 - 2 + 2 + 2 + 2 01 l 1 receiving a symbol in error in the ith channel is denoted

100 -1 +1 +1 +1 +2-2-2-2 1000

101 +l -1 +1 +1 +2 -2 +2 +2 1011 by Pr(ei), i= 1.... ,4. The channel characteristic

110 + I + 1 + 1 - 1 +2 +2 -2 +2 1101 used in our computations has a sinusoidal rolloff with

111 +1 +1 -1 -1 +2 +2 +2 -2 II10 rolloff parameter 13= 0.1.

30 d. Audio Eng. Soc., Vol. 36, No. 1/2, 1988 January/February

ENGINEERINGREPORTS DIGITALAUDIOTRANSMISSIONSYSTEMS

Fig. 4 shows the SNR of both the new and the con- audio signal with an SNR of 40 dB. As can be seen,

ventional systems as a function of the relative bandwidth the cost of the new system is fairly low; it requires a

BT of the transmission system (no additive noise as- 10% larger bandwidth than the conventional system to

sumed). The relative bandwidth of the two systems has support maximum user capacity. In the next example

been scaled in such a way that they convey, per unit we give some numerical data to exemplify the previous

of time, equal amounts of user information 1/T (bits theory.

per second) at maximum bandwidth. Note that TJT =

3/4. The diagram shows better than words can explain 2.1 Example 2

the sudden loss of quality--the threshold effect--of The data bit rate of a digital audio signal is readily

the conventional system when the transmission band- found. Assume a stereo channel, 15-bit quantization,

width is reduced below a certain critical value. The and a 48-kHz sample frequency. The data bit rate (ex-

diagram reveals quite clearly that the new system is cluding error correction, subcode, synchronization, etc.)

partitioned into three subsystems. At a relative band- is 2 × 48 x 15 = 1.440 Mbit/s, or T= 694 ns. From

width BT = 0.2, which is a factor of 2 smaller than Fig. 4 we find that when conventional techniques are

the critical bandwidthof the conventional system, the used, the minimum requirement of the bandwidth of

new system is still capable of supporting a 5-bit digital the transmission channel is 0.42 x 1.44 = 600 kHz.

T-

-2-1 O -2 -1 0 I2

TIMEVTO TIME/TC

Ca) Ca)

- - 2 _ -1

TIME/T 0 TIME/T 0

(b) (b)

o _i_'_ _L"_ _),_"-/\-"--_..._ _ "_'_<'-_%"_,_._. _l_"k_ "='='=

-2 -1 o-2 o _

TIME/T0 TIME/T0

(c) (c)

-2 -1 0 -2_-- -- --_.. 0 -_---" "_"-4"'

_2

TIM_?/TC TIMC/T 0

(d) (d)

Fig. 1. Eye patterns; relative bandwidthBTc=0.5. Fig. 2. Eye patterns; relative bandwidth BT¢ = 0.4.

J. AudioEng. Sot., Vol. 36, No, 1/2, 1988 January/February 31

IMMINK ENGINEERING REPORTS

The diagram clearly reveals that when the channel 2

bandwidthis smallerthan this thresholdvalue, nore- ..l

= il:

ception at all is possible. When the new technique is _ ....... -..."

used, we find that the channel bit rate, that is, the bit _ g /:

rate after the channel encoder, is 4/3 x 1.440 = 1.92 z

Mbit/s. The minimum bandwidth for a full 15-bit re- :__ /-I--'l" ] i"J-"ception is 0.47×1.44--677 kHz. When the channel _...-i..--.-t- "?"

bandwidth is 0.36 x 1.44 = 518 kHz, we are still able J,.-.i-.:,_..................... "......

to decode a 10-bit audio signal. When the bandwidth o-, " "

0.1 0.2 0.3 0.4 0.8 O.

is further reduced to0.18 x 1.44 = 260 kHz, we can RelativeBandwidth8"1"

retrieve a 5-bit audio signal.

It is possible and indeed preferable in this coding Fig. 4. Signal-to-noise ratio as a function of transmission

bandwidth. Dashed curve--new system; dotted curve--

scheme for each of the created subchannels to have a conventional system.

separate error correction and detection control. The

error control mechanism can decide whether one or

more of the subchannels are beyond their operating

conditions and may decide not to use them, whereby it should be noted that this decision can be made entirely

in the receiver. Obviously, a subset of the available

subchannels can also be used as a tool for design flex-

2!

.._____ ibility. For example, we may decide to implement a

t .'_=_3_,._. _:_ r _ decoding of the full set or of part of the set of the

N

_I_-__-'--_ _ _:__ available subchannels, which leads to a corresponding

me / =-_-.. ------_.__.,__--.'_____ reduction of the required decoding hardware and of

,.onu ,it,of ae oaea aiosig a, ougwe

--_ coined the term "graceful" degradation for the new

-2

-a -1 0 1 2 technique, it can be understoodthat the new technique

TIME/Tc provides a key to future evolution, that is, upgrading,

(a) ofthe quality of digital audio systemswithin the same

2 digitalformat. Ifcurrenttechnologyallowsa relative

_-- _ =_ _ _'* bandwidth of B T = 0.2, we can receive a 5-bit digital

i _ _ _ _ audiosignal.Astimegoeson, technologywillprovide

more bandwidth, which allows the reception of higher

z0

_ _ _ quality audio in a manner compatible with the low-

aJ _ "_--"- - _ _ - _ _ bandwidth system.

-a -1 oi 2 3 CONCLUSIONS

TIME/TC

(b) A new coding and detecion technique was presented

which offers the virtues of graceful degradation and

2 _ _ /_ _ _ _ the flexibility to up- or downgrade the quality of the

._,_x,,, _.._.._-_-_.._ _ -- _ transmission system. It has been shown that residual

o _-_ _ _ _ _ _ _ intersymbol interference has a different effect on the_,e,// _ error probability of the least and most significant sym-

_ _ _ _ _ _ _ _ bolswhenfrequencydomaindetectionof a setofcode-

-I

_. -......_e_'- _--,...__ words is employed. The class of systems described

-2 o _ maywellbeofparticularvalueinapplicationsinvolving

TIME/TC the storage or transmission of ordered data, such as

digital video or audio,when the receiver or the trans-

(c) mitter has no exact knowledge of the channel bandwidth.

a Theflexibilityofthenewcodingtechniquealsoprovides

_ 1___-..__ ___ akeytothefutureevolutionofthequalityofdigital

zo _ _ _ audioor video signalswithinthesamedigitalformat.

a_ __ _ _ _ _ _ .I ><----'-_" 4 REFERENCES

-2 [1] J. P.J. HeemskerkandK.A. SchouhamerIm-

-2 -1 0 1 2

TIME/TCmink, "Compact Disc: System Aspects and Modula-

(d) tion,"Philips Tech.Rev., vol. 40, pp. 157-164 (1982).

[2] H. Mertens and D. Wood, "Standards Proposed

Fig. 3. Eye patterns; relative bandwidth BT= = 0.3. by the EBU for Satellite Broadcasting and Cable Dis-

32 O.Audio Eng. Soc., Vol. 36, No. 1/2, 1988 January/February

ENGINEERINGREPORTS DIGITALAUDIOTRANSMISSIONSYSTEMS

tribution," J.IERE, vol. 56, pp. 53-61 (1986). 238 (1983 Apr.).

[3] H. F. Harmuth, Transmission of Information by [5] J.G. Proakis, Digital Communications (McGraw-

Orthogonal Functions, 2nd ed. (Springer, Berlin and Hill, New York, 1983).

New York, 1972). [6] I. Dostis, "The Effectof Digital Errorson PCM

[4] T. T. Doi,"Channel Codings for Digital Audio Transmission of Compandored Speech," Bell Sys.Tech.

Recordings," J.Audio Eng.Soc., vol. 31, pp. 224- J., vol. 44, pp. 2227-2243 (1965).

THE AUTHOR

Kees A. Schouhamer Immink was born in Rotterdam, Dr. Immink is coauthor of a book entitled Principles

The Netherlands, in 1946. He received a B.S. degree of Optical Disc Systems. He was awarded a fellowship

from Rotterdam Polytechnic in 1967 and M.S. and bytheAESin 1985 for "his work in the area of optical

Ph.D. degrees from the Eindhoven University of laser disk and for detailed study on channel codes for

Technology in 1974 and 1985, respectively, all in the Compact Disc." He serves as a member of the Or-

electrical engineering. He joined the Philips Research ganizing Committee of the International Conference

Laboratories, Eindhoven, in 1968, where his work in- on Video, Audio, and Data Recording and is on the

volved the signalprocessing side of opticalrecording committee of the AES Netherlands Section andthe

systems. Since 1986, he has been a member of the Benelux Community on Information and Communi-

magnetic recording group at Philips Research. cation Theory. He is a fellowof the IERE.

J. Audio Eng. Soc., Vol. 36, No. 1/2, 1988 January/February 33