Pulse position modulation is the process
by which the position of the pulse is varied in accordance
with the information contained in the sampled waveform.
Because the pulse width remains unchanged, the bandwidth
required for transmission of the pulse information remains
stationary. Narrow pulse systems require greater bandwidths,
but their information capacity is increased.
PPM Analysis
As the
illustrations below show, PPM can be obtained
from PDM with trailing–edge modulation
by inverting and differentiating so that the modulated
edges are changed into position–modulated positive
spikes.
Let us now explain the
marked superiority of PPM over PDM.
In fact, principal use of PDM is for the generation
and detection of PPM.
We should know that the information resides in the time
location of the pulse edges and not in the pulses themselves.
It means that the pulse power of pulse-duration modulation
is wasted power and for higher efficiency, the pulses
should be suppressed as mush as possible and transmit
only their edges. Therefore, only very short pulses
indicating the position of the edges need to be sent.
This corresponds to PPM.
Since non-uniform sampling is the most efficient type
of PDM for transmission purposes, we would
analyze PPM with non-uniform sampling.
Let tm be the time location or centre of
the mth pulse. If the sampling is uniform,
the mth pulse carries the sample value at t
= mTs, and
Where t0
= modulation constant representing maximum displacement
relative to t = mTs. But with non-uniform sampling,
the sample value is actually extracted at tm,
and not at mTs. That is
Since, PPM wave is a summation of constant-amplitude
position–modulated pulses, we can represent it
as,
Where A = Pulse amplitude, and p(t) = pulse
shape. Since, P(t) will have a very small duration
as compared to Ts, the pulse shape
can be taken as impulse, i.e.
For further analysis, we eliminate the positive
term tm by using the following technique:
If g(t) is a function having first-order zero
at
and
then it can be shown that
and then it can be shown that
or
Clearly, the right–hand side is independent
of l. To remove tm from d(t
– tm), we find a function g(t)
which satisfies g(tm)=0 and other
conditions, but does not contain tm.
Let which
is zero at
Also, for a given value of m, there can be only one
PPM pulse which occurs at
Thus,
Therefore, putting l = t ms and
we get,
Substituting in above e.q. we get:
We need not take absolute value since
for most signals of interest if .
We write the sum of impulses in the form of exponentials
using the following relation:
Therefore,
From the above equation we observe:
PPM with non-uniform sampling consists of linear
and exponential carrier modulation. Each harmonic
of fs is phase-modulated by the message
x(t)
and amplitude-modulated by x'(t).
Therefore, the PPM spectrum consists of AM and PM
sidebands centered at all multiples of fs
plus a dc component and a spectrum of x'(t). Clearly,
the
spectrum cannot be sketched with any ease at all even
for single-frequency modulation.
then it can be shown that
or 
which
is zero at 
we get,
for most signals of interest if 
.
