To emphasize the fact that every periodic signal has both a time
and frequency domain representation, we can exploit both to
encode information into a signal. Refer to
the Fundamental Model of Communication. We have
an information source, and want to construct a transmitter that
produces a signal
xt
x
t
. For the source, let's assume we have information to
encode every TT seconds. For
example, we want to represent typed letters produced by an
extremely good typist (a key is struck every
TT seconds). Let's consider the
complex Fourier series formula in the light of trying to encode
information.
We use a finite sum here merely for simplicity (fewer parameters
to determine). An important aspect of the spectrum is that each
frequency component
c
k
c
k
can be manipulated separately: Instead of finding the Fourier
spectrum from a time-domain specification, let's construct it in
the frequency domain by selecting the
c
k
c
k
according to some rule that relates coefficient values to the
alphabet. In defining this rule, we want to always create a
real-valued signal
xt
x
t
.
Because of the Fourier spectrum's
properties,
the spectrum must have conjugate symmetry. This requirement
means that we can only assign positive-indexed coefficients
(positive frequencies), with negative-indexed ones equaling the
complex conjugate of the corresponding positive-indexed ones.
Assume we have
NN letters to encode:
a
1
…
a
N
a
1
…
a
N
.
One simple encoding rule could be to make a single Fourier
coefficient be non-zero and all others zero for each letter. For
example, if
a
n
a
n
occurs, we make
c
n
=1
c
n
1
and
c
k
=0
c
k
0
,
k≠n
k
n
.
In this way, the
n
th
n
th
harmonic of the frequency
1T
1
T
is used to represent a letter. Note that the
bandwidth—the range of frequencies required for
the encoding—equals
NT
N
T
. Another possibility is to consider the binary
representation of the letter's index. For example, if the
letter
a
13
a
13
occurs, converting 1313 to
its base 2 representation, we have
13=11012
13
1101
.
We can use the pattern of zeros and ones to represent directly
which Fourier coefficients we "turn on" (set equal to one) and
which we "turn off."
Compare the bandwidth required for the direct encoding
scheme (one nonzero Fourier coefficient for each letter) to
the binary number scheme. Compare the bandwidths for a
128-letter alphabet. Since both schemes represent
information without loss -- we can determine the typed
letter uniquely from the signal's spectrum -- both are
viable. Which makes more efficient use of bandwidth and
thus might be preferred?
NN signals directly encoded
require a bandwidth of
NT
N
T
.
Using a binary representation, we need
log
2
NT
2
N
T
.
For
N=128
N
128
,
the binary-encoding scheme has a factor of
7128=0.05
7
128
0.05
smaller bandwidth. Clearly, binary encoding is superior.
Can you think of an information-encoding scheme that makes
even more efficient use of the spectrum? In particular, can
we use only one Fourier coefficient to represent
NN letters uniquely?
We can use NN different
amplitude values at only one frequency to represent the
various letters.
We can create an encoding scheme in the frequency domain to
represent an alphabet of letters. But, as this
information-encoding scheme stands, we can represent one letter
for all time. However, we note that the Fourier coefficients
depend only on the signal's characteristics
over a single period. We could change the signal's spectrum
every TT as each letter is
typed. In this way, we turn spectral coefficients on and off as
letters are typed, thereby encoding the entire typed
document. For the receiver (see the Fundamental Model of
Communication) to retrieve the typed letter, it would
simply use the Fourier formula for the complex Fourier spectrum
for each TT-second interval to
determine what each typed letter was. Figure 1 shows such a signal in the time-domain.
In this Fourier-series encoding scheme, we have used the fact
that spectral coefficients can be independently specified and
that they can be uniquely recovered from the time-domain signal
over one "period." Do note that the signal representing the
entire document is no longer periodic. By understanding the
Fourier series' properties (in particular that coefficients are
determined only over a TT-second
interval, we can construct a communications system. This
approach represents a simplification of how modern modems
represent text that they transmit over telephone lines.
"Electrical Engineering Digital Processing Systems in Braille."