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In 1948, the year when the Audio Engineering Society was founded, the life of an electronics engineer was relatively simple as there were not many practical options for analog telecommunications or recording. However, a few key developments would come to change that simple life. Those were the days in which shellac gramophone disks and AM radio ruled as the primary media for consumer audio distribution. The 12-inch LP (long-playing) vinyl disc had just been brought to market. Spinning at 331/3 rpm, this new high-density disc could play for up to 25 minutes per side. RCA had also subsequently introduced the 7-inch, 45-rpm vinyl disc, but it took many years for the early 78-rpm disks to disappear because the new vinyl formats needed new equipment on which to play them. Despite a slow beginning, the two new vinyl formats were dominant until the CD (Compact Disc) took over in 1982. Professional recording studios used magnetic tape audio recorders, which enabled considerable creative flexibility. Les Paul, for example, developed the technique of audio dubbing, bringing it to very high artistic and engineering standards. Video recorders, using helical scan heads and magnetic tape, were first brought to market by Ampex. Telecommunications, voice and telegraphy, were mostly done by cable or radio, while AM and FM broadcast radio brought sound directly from studio to listener. All the signals employed in such systems were analog, In 1948 digital technology was still in its infancy. Times were changing, however, and the keen observer could have noticed a number of signs of change.
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J. Audio Eng. Soc., Vol. 5, No.,
In 1948, the year when the Audio Engi-
neering Society (AES) was founded,
the life of an electronics engineer was
relatively simple as there were not
many practical options for analog
telecommunications or recording.
However, a few key developments
would come to change that simple life.
Those were the days in which shellac
gramophone disks and AM radio ruled
as the primary media for consumer
audio distribution. The 12-inch LP
(long-playing) vinyl disc had just been
brought to market. Spinning at 331/3
rpm, this new high-density disc could
play for up to 25 minutes per side.
RCA had also subsequently introduced
the 7-inch, 45-rpm vinyl disc, but it
took many years for the early 78-rpm
disks to disappear because the new
vinyl formats needed new equipment
on which to play them. Despite a slow
beginning, the two new vinyl formats
were dominent until the CD (Compact
Disc) took over in 1982. Professional
recording studios used magnetic tape
audio recorders, which enabled consid-
erable creative flexibility. Les Paul, for
example, developed the technique of
audio dubbing, bringing it to very high
artistic and engineering standards.
Video recorders, using helical scan
heads and magnetic tape, were first
brought to market by Ampex.
Telecommunications, voice and teleg-
raphy, were mostly done by cable or
radio, while AM and FM broadcast
radio brought sound directly from studio
to listener. All the signals employed in
such systems were analog, In 1948 digi-
tal technology was still in its infancy.
Times were changing, however, and the
keen observer could have noticed a
number of signs of change.
THE INVENTION OF PCM
The British engineer, Alec Reeves
(1902–1971) invented pulse-code modu-
lation (PCM) in 1937 when he worked
for the International Telephone and
Telegraph Company. This was a very
early invention in the history of electron-
ics, since it was only a few years after
Edwin Armstrong invented wideband
FM for high-quality radio broadcasting.
Reeves, instead of following the
traditional method of representing an
audio signal by using an electrical
current proportion to the sound level,
proposed that the electrical sound
signal should be sampled and quan-
tized at regular intervals. Then the
analog value of each sample would be
rounded to the nearest integer value,
which, in turn, would be represented
by a binary number and transmitted as
unequivocal on-off pulses. In principle,
such binary (two-level) signaling was a
return to the simple, robust technique
used by the telegraph (Morse code, for
example, uses a binary signal).
Because PCM is a method of repre-
senting an analog signal in digital
form, it is particularly well adapted to
work directly with digital data-process-
ing equipment. A notable disadvantage
of PCM is the high (analog) bandwidth
required of the transmission or storage
system. Sending recognizable speech
meant networks and radios would have
to carry millions of pulses a second.
Though Reeves’ extraordinary patent
showed how this might be done in
theory, the valve-based technology of
the time could not do the job.
In his 1937 patent, Reeves formu-
lated the major advantages of digital
PCM transmission: quality depends on
conversion steps only; quality is inde-
pendent of transmission media;
compatibility with different media and
traffic (video, audio, and data); low
cost; new features can easily be
embedded
These are highly significant and also
visionary conclusions. Reeves showed
his enormous engineering foresight, as
there are two essential assumptions that
are implicit in the above characteristics.
First, each quantized sample can be
transmitted with an arbitrarily small
probability of error. It was certainly not
clear in 1937 that this could be accom-
plished in theory, and neither he nor
others knew about practical methods
for achieving error-free transmission.
There was no research on the topic of
error-correcting codes. It would take
another ten years and a world war
before research on error-free digital
communication would take off.
Second, he assumed that conversion
from the analog to the digital domain,
and vice versa, could be done, either in
theory or practice, with arbitrarily fine
accuracy by use of sufficiently frequent
sampling, and by quantizing each
sample with a sufficiently large number
of levels. Early theoretical work by
mathematicians had been published,
but we may well assume that Reeves
was unaware of that literature.
By Kees A. Schouhamer Immink
A
Any Song,Anytime,Anywhere
Audio history is examined to identify the seminal contributions to the
digital audio revolution. Clues are discovered that could make it possible to
discern the likely direction of digital audio engineering in the future.
This article is based on a Richard C.
Heyser Memorial Lecture given by the
author in Berlin, April 2004, and a
Lunchtime Keynote at the AES 127th
Convention (an MP3 can be purchased at
www.softconference.com/aes). Imminck
received the AES Gold Medal in 1999
and was AES president, 2002/2003.
J. Audio Eng. Soc., Vol. 5, No.,
SIGSALY SYSTEM
Before taking a look at the two basic
laws that underlie the existence of dig-
ital sound, however, it is worthwhile
to make a short detour to consider a
great engineering achievement,
namely the Sigsaly digital encryption
system.
Pulse-code modulation could not be
implemented economically until the
invention of the transistor decades
later. For military operations, however,
economy is not always the primary
driving force. As is well known, the
war was very fruitful for technology. A
transmission system based on fully
fledged PCM was first developed by a
team of top engineers at Bell Labs for
the complex and very advanced
Sigsaly radio voice encipherment
system on which Churchill and
Roosevelt talked in total secrecy
during World War II (see Fig. 1) to
discus war matters. Terminals were in
London and at the Pentagon, in Sydney
and other places. An older analog
scrambling technology was not suffi-
ciently secure, and indeed by 1941 it
was broken by the Germans.
Sigsaly’s technology is absolutely
fascinating. The basic technologies
employed were not simply improve-
ments upon prior art; they were funda-
mentally new and absolutely necessary
for the system to work. The concepts
were proven in the laboratory, but
before the system was ready for final
development and deployment there were
several important system features that
needed refinement. Just one of the many
major problems that the Bell team faced
was the generation of a secret key. The
basic requirement for the key was that it
should be completely random, and must
not repeat, but could still be replicated at
both the sending and receiving ends of
the system. To that end, random wide-
band noise was generated by a mercury-
vapor rectifier vacuum tube. The noise
signal was sampled every twenty
milliseconds, and the samples were then
quantized into six levels. The level
information was converted into channels
of a frequency-shift-keyed (FSK) audio
tone signal, which could then be
recorded on standard vinyl phonograph
records. Two exact copies of the master
record were made: one for the sender
and one for the receiver. The random
digitized signal from the record was
combined, scrambled, with the digital
speech signals, and the scrambled audio
signal was transmitted to the receiver.
The receiver subtracted the same
random signal from the received signal,
and obtained the original, unscrambled,
speech signal. However, in order to do
so, both random signals had to be gener-
ated in exact synchronization at both the
sender’s and receiver’s sites. The play-
ing of the two records with the random
signal was accomplished by the use of
turntables driven very precisely by a
stable electric motor. For stability
reasons, the motor was kept in constant
operation, and the frequency for it was
derived directly from the terminal’s
frequency standard, a crystal clock oscil-
lator. For security reasons, the records
were used only once, and immediately
destroyed following a session.
The block diagram of Sigsaly’s
voice speech encoder (vocoder) resem-
bles the block diagram of modern
audio compression technology, such as
the one used in a mobile telephone.
The voice spectrum from 250 Hz to
3000 Hz was divided into ten channels,
and all ten channels were sampled.
Two channels sampled voice pitch and
unvoiced background hiss.
The hardware of that early voice
encoder and decoder did not fit into a
standard living room. Each terminal
used forty racks of equipment weigh-
ing about fifty tons, and filled a large
room. The installation in London was
so large that it required the basement
of Selfridges Department store. The
Bell engineers could be proud of their
product—the Germans did in fact
monitor Sigsaly for two years, but
were never able to unscramble the
voice communications.
CLAUDE SHANNON
The period of time during and immedi-
ately following World War II was a
fruitful one for technology. In 1948,
Claude Shannon (1916–2001, see Fig.
2) published his landmark article “A
Mathematical Theory of Communica-
tion,” which laid the foundation of
information theory. Information theory
is primarily concerned with the
mechanics of information handling.
The fundamental problem of communi-
cation is that of reproducing at one
point either exactly or approximately a
Any Song, Anytime, Anywhere
Fig. 2. Kees Immink, left, with Claude
Shannon in 1985, when Shannon
received the AES Gold Medal and
Immink received an AES Fellowship.
Fig. 1. The SIGSALY encryption system used during the Second World War
J. Audio Eng. Soc., Vol. 5, No.,
message selected at another point. Fre-
quently the messages have a meaning;
for example, they represent an audio or
video signal. This “content” aspect of
communication is irrelevant to the
engineering problem of error free com-
munication. By the way, Shannon
coined the term “bit” for an elementary
unit of information. Shannon proved
mathematically that communication
over noisy channels is possible with an
arbitrarily small probability of error. He
did not give a clue as to how engineers
could reach the promised land of error-
free communication in practice.
Real error-correcting codes were
first invented by Hamming in 1948;
his code could correct single bit errors.
The mathematicians Irvin Reed and
Gus Solomon formulated the error-
correcting capacity of their renowned
codes in a paper in 1960; they needed
only five pages to present their results.
They rigorously showed that Reed-
Solomon codes can correct any
number of errors within their designed
error-correction capacity. But it took
fifteen more years before mathemati-
cians found an efficient algorithm for
decoding such codes. It then took
another ten years before the codes
were practically applied, first in
magnetic audio tape recorders, later in
CD players, and thereafter in essen-
tially all other digital communication
and storage products. Today, for
example, the error correction of a $25
DVD player can correct about 100
byte errors per data block.
SAMPLING THEOREM
The sampling theorem was indepen-
dently formulated by at least four scien-
tists, namely Whittaker, Shannon, Kotel-
nikov, and Nyquist. Whittaker was
probably the very first who presented
the theorem in 1915. Shannon gave this
theorem as a kind of bonus in his 1948
information-theory article. The sampling
theorem is the most eminent equation in
digital audio engineering, as it mathe-
matically proves that an audio or other
continuous-time signal can be com-
pletely restored if it is observed (sam-
pled) at a rate at least twice the band-
width of that audio signal. The
mathematical elegance of the sampling
theorem might suggest that analog-to-
digital (AD) conversion or vice versa is
a straightforward procedure in practice.
Reliable converters, however, are diffi-
cult to construct, and they are still the
subject of academic and industrial study.
The conversion steps have always been
very difficult. While it has been under-
stood for a long time that, for example,
dither can be used to eliminate distortion
in AD converters, it has not always been
employed. As a result, high-quality
PCM audio systems have received criti-
cism over the years.
It is of some interest to observe that
engineers used the sampling of elec-
tronic signals very early on. The
sampling of signals goes back almost
100 years to when multiple simultane-
ous telegraphy signals were transported
over a single telegraph line. Clearly,
long submarine telegraphy cables are
very expensive, and simultaneous usage
of that cable by a group of independent
telegraphers would increase the profit
proportionally. Synchronous rotators
at the sending and
receiving end sam-
pled the telegraphy
signals. The engineers
observed that if the
rotation frequency
was sufficiently high
the telegraphy signals
could be transmitted
without distortion.
And now we know
that it took almost
fifty years before that rule of thumb was
mathematically proved.
ONWARD TO THE ROARING
SEVENTIES
We now take a big step in time from the
1950s to the 1970s. Practical progress
was made at Bell Labs and other places
to practically construct PCM transmis-
sion. Progress was slow, as the key com-
ponent of PCM transmission was very
young and just invented. The transistor,
born in 1947, would make PCM possi-
ble in practice. Large mainframe com-
puters were transistorized, and, later in
the 1960s, small so-called mini comput-
ers, for example the famous PDP-8 and
PDP-10, were brought to market. Such
mini computers were cheap, and their
proliferation triggered research in
low-cost data storage. In the early
1970s, Thomas Stockham (see Fig. 3)
used mini computers and regular com-
puter tape storage for his pioneering dig-
ital recording of sound. In 1975, he and
Malcolm Low founded Soundstream,
where they developed a 16-bit digital
audio recorder using a high-speed
instrument magnetic tape recorder.
Soundstream was the first commercial
digital recording company in the United
States, located in Salt Lake City. Com-
panies such as Denon had been experi-
menting with digital recording since
1971, but Stockham was the first to
make a commercial digital recording
using his own recorder in 1976, and
demonstrated the recordings at the 1976
AES convention. In the mid 1970s,
digital audio made its way into the
recording studios. Table 1 lists the char-
acteristics of some of the PCM systems
from early manufactures and their key
parameters.
High-quality PCM audio requires a
significantly larger bandwidth than a
regular FM audio signal. For exam-
Any Song, Anytime, Anywhere
Fig. 3. Thomas Stockham pioneered
the digital recording of sound using
computer tape storage
Table 1: Characteristics of some early
PCM systems
Manufacturer Type fs (kHz) Resolution (bits)
BBC Microwave 32 13
Soundstream Tape 37.5/42.5 16
Many PCM adaptor 44.056 13
Denon Tape 47.25 13
J. Audio Eng. Soc., Vol. 5, No.,
ple, a 16-bit PCM signal requires an
analog bandwidth of about 1–1.5 MHz,
and clearly a standard analog audio
recorder could not meet that require-
ment. The obvious answer, at that time,
was to use a video tape recorder, which
is capable of this high bandwidth, to
store the information. Such an audio
recording system therefore included two
machines, namely the PCM adaptor and
the video tape recorder. A PCM adaptor
took the analog audio (stereo) signal as
its input, and translated it into a series of
binary digits, which, in turn, was modu-
lated into a pseudo-video signal. The
pseudo-video signal could be stored on
any ordinary analog video tape
recorder, since these were the only
widely available devices with sufficient
bandwidth. This helps to explain the
choice of sampling frequency for the
CD, because the number of video lines,
frame rate, and bits per line dictate the
sampling frequency one can achieve if
wanting to store two channels of audio.
The sampling frequency of 44.1 kHz
was adopted in the Compact Disc
because at that time there was no other
practical way of storing digital sound
than by using a PCM converter and
video recorder combination. The
sampling frequencies of 44.1 and
44.056 kHz were the result of a need for
compatibility with the NTSC and PAL
video formats used for audio storage at
the time. The Sony 1600 was the first
commercial video-based 16-bit
recorder, and it continued in its 1610
and 1630 incarnations. These employed
16-bit quantization and a sampling
frequency of 44.1 (or 44.056 for NTSC)
kHz. The PCM adaptors could only
store a stereo signal and could not be
used for multitrack recording.
Much later we witnessed the advent
of dedicated professional digital multi-
track recorders, such as those based on
Mitsubishi’s ProDigi format and
Sony’s DASH format. These recording
machines accommodated the obliga-
tory 44.1 kHz, but also 48 and 32 kHz
as sampling rates. Digital mixing desks
and other digital equipment, such as
reverberation units, were introduced
and have become indispensable tools
in modern studios.
THE BIG BANG: COMPACT DISC
From 1973 to 1976, two Philips engi-
neers in Eindhoven were given a man-
date to develop an audio-only disc
based on optical videodisc technology.
They started by experimenting with an
analog approach using wide-band fre-
quency modulation. The problem with
this was that it was not really much
more immune to dirt and scratches
than an analog LP record, although
there was a certain improvement in
sound quality. So they decided to look
for a digital solution as electronic tech-
nology had become ripe for such a
step. They were successful, and around
1977 Philips, Sony, and also other
companies demonstrated the first pro-
totypes of a digital sound system using
a laser disc. In 1979, Sony and Philips
decided to work together, and they set
up a joint group of engineers whose
mission was to design the standard of
the new digital audio disc. Philips had
lost the market for the videodisc, but
had considerable optical expertise, as
well as expertise in servo systems and
digital and analog modulation systems.
Sony’s extensive expertise in error cor-
rection, PCM adapters, and channel
coding complemented this ideally. A
reasonable summary would be that
most of the physics was provided by
Philips and the digital audio experi-
ence by Sony. The Compact Disc Digi-
tal Audio System was first brought to
market in 1982. Sony was the first
company to develop a portable CD
player in 1985 (see Fig. 4). The new
audio disc was enthusiastically
received and its handling quality
received particular praise. Also in
1985, the CD-ROM (read-only mem-
ory) was introduced. With this it was
now possible to disseminate massive
amounts of computer data instead of
digital sound. A user-recordable CD
for data storage, CD-R, was introduced
in the early 1990s, and it became the
de facto standard for exchange and
archiving of computer data and music.
The CD and its later extensions have
been extremely successful: in 2004 the
annual worldwide sales of CD-Audio,
CD-ROM, and CD-R reached about 30
billion discs. By 2007, Philips reported
that 200 billion CDs had been sold
worldwide since the start in 1982.
25 YEARS AFTER THE CD
We are now in a very exciting time.
On the one hand we see systems with
extremely high audio quality demand-
ing super high-density recording sys-
tems such as DVD and Blu-ray, the
successors to the CD. On the other
hand, we see systems offering low-
quality compressed sound requiring
low-density storage.
An audio data-reduction system such
as MP3 (short for MPEG-1 or MPEG-
2, Layer-3) compresses sound files by
removing the inaudible information
from the recording, and by using
psychoacoustic information to “hide”
increased quantizing noise under the
masking threshold. The result is a
smaller audio file, which can be trans-
Any Song, Anytime, Anywhere
Fig. 5. Two examples of modern consumer digital audio devices, a USB
memory stick player (left) and an Apple iPod
Fig. 4. An early Sony Compact Disc player
ferred more efficiently without substan-
tially affecting sound quality.
Data-reduced audio files can be a
factor of ten to twenty smaller than the
original CD file. Any fears the record
industry had about digital recording in
1992 now seem insignificant with the
advent of compressed audio, MP3, and
the explosion of the Internet. Students
with an Internet connection and a
computer first used compressed audio.
The situation changed completely
when the general public started to use
mobile players using solid state or hard
disk drives (such as the Apple iPod),
such as those shown in Fig. 5. These
devices have completely replaced
Sony’s cassette-based Walkman.
FROM HI-FI TO WI-FI
The record companies have been using
the same business model for almost 100
years. Edison and Berliner introduced
this classical business model, when they
invented the phonograph and the gramo-
phone. In this model, to which everyone
has become accustomed, the distribution
of electronic sound has been accom-
plished by radio or gramophone. You go
to a shop and you buy a carrier of music,
ranging from the cylinder, gramophone,
LP, compact cassette, to CD. You take it
home and store it at home or in your car.
You can play the song you bought any
time and anywhere you want. Alterna-
tively, you can listen to music by radio
or TV, but here you can choose neither
the song nor the time. The digital CD
replaced the analog LP in 1982, but at
that time it did not change the above
business model.
We are rapidly moving toward an
era in which each home and car will
have a digital music library, which is
placed in the broom closet, basement,
trunk, or attic. Networked audio play-
ers are connected to the music library
by standard Internet technology such
as UTP cable or Wi-Fi radio.
Networked players are in essence dedi-
cated computers that have an Internet
connection and some software for
playing and selecting the songs. The
music is not necessarily of low-capac-
ity compressed quality. A digital music
library will have a storage capacity of,
say, 1 Terabyte (1000 Gigabyte),
which is equivalent to the storage
capacity of 2000 standard audio CDs.
Alternatively, such a digital music
library can store 50,000 CDs in
compressed quality, which amounts to
5 years of unrepeated, around-the-
clock music. All the above electronic
equipment is essentially state of the art,
and there are no inventions to be made.
For a reference, at the time of publica-
tion of this article, commodity PCs sell
with disk drives of about 320
Gigabytes, and an extra disk drive of 1
Terabyte will cost about $100 more.
Thus, the central storage of one hour of
music costs a fraction of a cent, while a
plastic CD jewel case (without CD)
costs more.
Alternatively, we may see the
creation of music-on-demand services.
A digital networked player can also
play songs directly taken from libraries
somewhere on the Internet using
streaming audio. This network will be
the biggest jukebox in the universe,
with no distribution costs, no pressing
costs, no returns, no out-of-stock items.
The primary question will be how
much storage will be done at home,
and how much will be done elsewhere.
In other words, what will be the
balance between the audio-on-demand
services and locally stored music?
This will depend on the price of the
services and the availability of those
services. It is not clear where the
balance between local storage and
streaming will be. But one thing is
ultimately clear, in this new business
paradigm there is no place for CDs or
super CDs. I predict that they will
become obsolete relatively quickly. So
in a few years, after storing the
content on your CDs onto your central
music library, compressed or uncom-
pressed, you will bring all your CDs to
the attic, and hopefully there is some
place next to your dusty collection of
old vinyl LPs that you put there some
twenty years ago.
Digital technology, specifically the
Internet, is at this very moment
completely changing the classic distri-
bution paradigm. As more and more
media content becomes available in
digitized form the fear of the content
providers about illegal means of dupli-
cation and redistribution increases. This
is particularly relevant when the disk
drives used to store the content are
external and therefore easily “portable.”
As for music piracy, or downloading
Any Song, Anytime, Anywhere
J. Audio Eng. Soc., Vol. 5, No.,
J. Audio Eng. Soc., Vol. 5, No.,
music for free from the Internet, every-
one believes it's a problem. At the end
of the day the technology will help
songwriters and performers more than it
will hurt them, since the cost of music
distribution itself is considerably less
than in the old model. The difficult task
has been for record companies to catch
up to changing technology and
consumer demand, but that process of
adaptation to a new business model is
now well under way with successful
online services such as Apple’s iTunes.
CONCLUSIONS
Digital audio has a rich history, and the
membership of the Audio Engineering
Society has played a key role in its
development. Considering the long
history of digital sound, it seems to be
incorrect to use the term digital audio
“revolution.” The term evolution
would be more appropriate, unless one
specifically deals with the short time
span, say ten years, following the
introduction of the Compact Disc.
For me it is clear that the future of
digital audio will be rich as well. But
we have to be very careful that we come
up with new products that will attract
the main base of consumers and not be
confined to the niche market of audio
buffs.
FURTHER READING
J. V. Boone and R. R. Peterson,
“The Start of the Digital Revolution,
SIGSALY Secure Digital Voice
Communications in World War II,”
Fort George G. Meade, Md., Center for
Cryptologic History, National Security
Agency (2000).
K. A. S. Immink, “The Compact
Disc Story.” J. Audio Eng. Soc., vol.
46, pp. 458–465 (1998 May).
K. A. S. Immink, “Beethoven,
Shannon, and the Compact Disc,”
IEEE Information Theory Society
Newsletter, vol. 57, pp. 42-46 (2007
Dec).
D. Kahn “Cryptology and the
Origins of Spread Spectrum,” IEEE
Spectrum (1984 Sep.).
S. P. Lipshitz, “Dawn of the Digital
Age,” J. Audio Eng. Soc., vol. 46, pp
37–42 (1998 Jan./Feb.).
C. E. Shannon, “A Mathematical
Theory of Communication,” Bell
System Technical Journal (1948).
Any Song, Anytime, …
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This paper attempts to give the reader a concise survey of the history of the development of digital audio technology in the Audio Engineering Society, as seen through its publications.
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This paper presents an account of the current state of sampling, 50 years after Shannon's formulation of the sampling theorem. The emphasis is on regular sampling, where the grid is uniform. This topic has benefitted from a strong research revival during the past few years, thanks in part to the mathematical connections that were made with wavelet theory. To introduce the reader to the modern, Hilbert-space formulation, we reinterpret Shannon's sampling procedure as an orthogonal projection onto the subspace of band-limited functions. We then extend the standard sampling paradigm for a presentation of functions in the more general class of “shift-in-variant” function spaces, including splines and wavelets. Practically, this allows for simpler-and possibly more realistic-interpolation models, which can be used in conjunction with a much wider class of (anti-aliasing) prefilters that are not necessarily ideal low-pass. We summarize and discuss the results available for the determination of the approximation error and of the sampling rate when the input of the system is essentially arbitrary; e.g., nonbandlimited. We also review variations of sampling that can be understood from the same unifying perspective. These include wavelets, multiwavelets, Papoulis generalized sampling, finite elements, and frames. Irregular sampling and radial basis functions are briefly mentioned
Dawn of the Digital AgeA mathematical theory of communicationSampling – 50 Years after Shannon
  • S P C E Lipshitz
  • Shannon
S.P. Lipshitz, ‘Dawn of the Digital Age’, AES Journal, pp 37-42, vol. 46, 1998. C.E. Shannon, ‘A mathematical theory of communication’, Bell System Technical Journal, 1948. M. Unser, ‘Sampling – 50 Years after Shannon’, Proceedings IEEE, vol. 88, pp. 569-587, 2000. 10
The Start of the Digital Revolution, SIGSALY Secure Digital Voice Communications in World War II
  • Reading J V Boone
  • R R Peterson George
  • G Meade
READING J. V. Boone and R. R. Peterson, "The Start of the Digital Revolution, SIGSALY Secure Digital Voice Communications in World War II," Fort George G. Meade, Md., Center for Cryptologic History, National Security Agency (2000).
  • K A S Immink
K. A. S. Immink, "Beethoven, Shannon, and the Compact Disc," IEEE Information Theory Society Newsletter, vol. 57, pp. 42-46 (2007