Binary pulse position modulation (PPM) is used by the Mode S transponder radar system at 1090 MHz. Every symbol time is divided into two slots. If the transmitter is instructed to send a binary "0", then it transmits a constant energy signal in the first half. If a "1" ought to be sent, then it transmits a signal in the second half. Equation (1) captures this idea. The message m(t) is a binary vector of ones and zeroes. The (baseband) signal that is sent converts the binary bit into a symbol with energy in the first half slot or the second half slot. T is the symbol time. Each slot, empty or full, is half of a symbol time. This baseband signal is upconverted to an RF or optical frequency, and the signal level **a** is the peak amplitude of a carrier wave. Binary PPM has a DC component, so it is generally not used over direct connection copper transmission systems, such as telephony lines or Ethernet wired LANs (remember those?), where DC requires a capacitively coupled front end, and a choke to bleed off charge build-up on the line.

The autocorrelation function of a binary-pulse-position modulation waveform having baseband signalling equations (1) is given by (2):

m(t) = 0 if s(t) = a for T <= t <= (T + 1/2 T)
and = 0 for (T + 1/2 T) <= t <= 2T
m(t) = 1 if s(t) = 0 for T <= t <= (T + 1/2 T) (1)
and = a for (T + 1/2 T) <= t <= 2T
for all time periods T = ...-2,-1,0,1,2...

The figure below illustrates how a binary data stream is converted into baseband PPM.

Binary message vector m(t) = { 1 1 0 1 0 0 0 1 }
--------- ----- ----
... | | | | | ...
---- ----- -------------
| | | | | | | | |
0 1 2 3 4 5 6 7 8
Binary pulse position modulation signalling waveform s(t)
-- --- ----- ----- --- --- -- a
... | | | | | | | | | | | | ...
----- --- ----- --- --- ----- 0
| | | | | | | | |
0 1 2 3 4 5 6 7 8

The autocorrelation function, R_{SS}(τ) is given by Equation (2).

R_{ss}(τ) = (0.5-0.375⋅(τ/T))⋅(T/2)⋅a^{2} for 0<=|τ|<=(T/2)
= (τ/(4⋅T))⋅(T/2)⋅a^{2} for (T/2)<=|τ|<= (T) (2)
= (1/4)⋅(T/2)⋅a^{2} when T<=|τ|

which looks like

Autocorrelation Function of B-PPM Signal
R_{ss}(τ)
| 1
* -
/|\ 2
/ | \
/ | \ 1
-------------------* / | \ *-- - ------------------
\ / | \ / 4
\ / | \ /
\ / | 1 \ /
* | - *
| 8
| | | | | ==> time diff τ
-T -T/2 0 T/2 T
NOTE: All of the amplitudes should be multiplied by a^{2}*(T/2)

We wanted to look at the autocorrelation function of this waveform, because the power spectral density of the waveform, S(f), is equal to the fourier transform of the autocorrelation function R(τ). (The Wiener-Khinchine theorem is a beautiful thing to communications engineers.) The power spectral density tells us many things, including the signal bandwidth, the concentration of signal energy around the RF center frequency, and how fast the PSD rolls off with frequency - and therefore what kind of filter we need to suppress energy sidelobes.

I see that there is no node for autocorrelation function yet. Will write that one when I have some time. This is just a reference example. Some signal waveforms have very interesting and surprising ACFs and PSDs. They're interesting to me, anyway. YMMV.

**
NOTES
**

- Remind me again why I keep writing nodes that have equations in them, and why these beautiful equations are reduced to infantile looking ASCII art because HTML doesn't support equations properly? HTML sucks. I can't believe it was invented by a guy at CERN whose primary customer base was physicists who love equations.
- HTML sucks for writing equations.
- Note to self: Stop this exercise in self-flagellation. Stop writing equations on E2. They're horrible and ugly.
- You guys seem to like these little notes the best.
- I don't know whether to be flattered or irritated.

**
References
**

- Ziemer & Tranter,
**Principles of Communications**, Houghton-Mifflin (c)1976

It's hard to believe that Prof. Owen R. Mitchell would put such a powerful book into the hands of his simpleton students. Incredibly well written. Can't sing the praises of the pedagogical writing style of Roger Ziemer & Bill Tranter loudly enough. Take note, MIT poseurs. The midwestern professors are kicking your asses.
- Ronald Bracewell,
**The Fourier Transform and Its Applications**, 2nd Ed., McGraw-Hill, (c)1978. A good friend had Bracewell for some PhD classes at Stanford. He said Bracewell was a wonderful teacher. Elegant and visual.
- Oh, there should be a delta function at f = 0, on account of the fact that I wrote the waveform to have a DC component. Oh well. Need to fix this some time.