The signal to be transmitted
is at first sampled and the samples are given to quantized
for rounding-off operation. The quantized pulses are
coded into groups using a binary code. In the binary
code only two levels are transmitted usually 1 and 0
corresponding to carrier ON and OFF. Each pulse group
transmitted represents the quantizing levels as a binary
number. The maximum number of pulse in the pulse groups
depends upon the total number of quantizing levels used
in the system. A 5 bit code has 32 quantizing levels.
In general, an 'n' bit code has 2 n quantizing levels.
However, the actual signals are most likely to have
both positive and negative values causing difficulty
in coding.
This problem is overcome by adding a
DC bias to the signal so that signal will always remain
positive. Another problem with speech signals is large
amplitude variation which then requires a large number
of quantizing levels. Amplitude compressor circuits
are employed to reduce large peaks in the signals and
this reduces the number of quantizing levels for a given
accuracy and also reduces the channel bandwidth. At
the receiver, is included the expander circuit to bring
the compressed signal back to its original form.
Modulating signal and DC bias {see figure - Sampling
instants}
Quantizing
levels and PCM pulse-trains for a typical signal. The
signal is biased by a DC voltage in such a way that
it does not become negative at any instant. This biased
signal is now sampled at fixed instants
and the signal amplitudes at these instants are converted
into binary. Thus, at the instant ,
sample amplitude equals ONE equalizing level. If a 4-bit
binary code is used, this will be represented by 0001
similarly at time , the
signal amplitude equals 10 units which is represented
by 1010 in the binary code. As these pulses are required
to be transmitted during the sampling interval allotted
for the channel, narrow pulse-widths are used for PCM
with resultant increase in the bandwidth. If positive
and negative pulses are employed for transmission of
1 and 0, the resulting ternary PCM pulse-train
is produced, as in function shown which has been illustrated
below.
Thus, PCM provides a communication
system in which the signal is converted into binary.
Because of this, the system is commonly referred to
as Digital Communication system. It is worthwhile
to note that a PCM receiver has just to recognize the
presence or absence of a pulse in ordinary PCM or the
polarity of the incoming pulses in bipolar to ternary
PCM and convert these pulses into equivalent analogue
signals. Exact shape or amplitude of these pulses does
not make any difference in the signal reproduced at
the receiver output. Thus, the system provides high
noise immunity.
A block schematic
of a PCM (a) transmission and (b) receiver
The figure above shows
the block schematic of a PCM transmitting and receiving
system. The transmission system consists of a low pass
filter with a cut-off frequency half of the sampling
frequency. The output of this filter is given to a sampler
by the quantized circuit and finally converted as a
PCM pulse-train by an encoder circuit. At the receiver,
these pulses are decoded and converted into equivalent
analogue signal. Higher frequency components present
in the output are attenuated by a low pass filter. The
receiver may include an expander circuit if a compressor
circuit has been employed in the transmitter.
THE S/N RATIO AND CHANNEL CAPACITY
OF PFM
Consider a baseband PCM system
in which the number of equally spaced coded pulse amplitudes
is m, and the transmission bandwidth is Br.
Since the entropy of the pulsed
signal is
and the pulse rate is ,
the information rate on
the channel is:
Therefore, the capacity
of the channel would be:
Now our task is to find
that value of S/N ratio which makes the decoding
errors negligible.
We assume that the noise on the channel never causes
a pulse to be lost or misinterpreted by the receiver.
We would, therefore, consider only the source of noise
to be due to the original quantization of the signal
which causes the receiver output to have error due to
this noise component. Let k be the quantization level
(i.e., volts steps).
We have already seen that the error probability Ps
is small if the voltage spacing between pulse amplitudes
is k
is the rms noise voltage.
We assume bipolar pulses because otherwise the signal
contains a dc component which does not contain any information
leading to wastage of power.
The pulse amplitudes of
the PCM signal would be
For,
For,
As seen as below the maximum
peak value of the signal
Now for maximum information
transfer, the
amplitudes should be equally likely and there should
not be any spacing between the pulses. Therefore, the
average signal power of the quantized signal will be:
Since all amplitudes are
equally likely,
Hence,
Since,
The above expression indicates the threshold
power requirement or the minimum channel S/N ratio as
a function of .
Therefore,
The parameter
determines the minimum allowable spacing between
pulse amplitudes for the specified decoding error probability.
As already seen, PCM is most
efficient just above threshold, so that
Therefore by above e.q.
Hence the channel capacity becomes:
if
Since the first term corresponds to
maximum and ideal capacity Cmax, therefore,
The departure form ideal value is governed
by the value of
, which, in turn, depends upon:
and the signal amplitudes at these instants are converted
into binary. Thus, at the instant 
,
sample amplitude equals ONE equalizing level. If a 4-bit
binary code is used, this will be represented by 0001
similarly at time 
, the
signal amplitude equals 10 units which is represented
by 1010 in the binary code. As these pulses are required
to be transmitted during the sampling interval allotted
for the channel, narrow pulse-widths are used for PCM
with resultant increase in the bandwidth. If positive
and negative pulses are employed for transmission of
1 and 0, the resulting ternary PCM pulse-train
is produced, as in function shown which has been illustrated
below. 
and the pulse rate is 
,
is the rms noise voltage.
amplitudes should be equally likely and there should
not be any spacing between the pulses. Therefore, the
average signal power of the quantized signal will be:
.
Therefore, 
determines the minimum allowable spacing between
pulse amplitudes for the specified decoding error probability.
