INTRODUCTION TO DATA COMMUNICATION 5

density of (rl,r2,...,rD) given i, i.e. decide i such that

20r-#

2 (1)

7- 1

is minimum. That sum is also equal to Writ) — sl(f)||2, where II II is derived from the

innerproduct defined above, so that the demodulator essentially performs minimum distance

decoding.

The resulting probability of error can be shown to depend only on the distances \\sl — sJ\\2

between signals at the channel output, i.e. on

f\S'(f) -

SHf)\2 \H(f)\2/K(f)df

,

leading to the obvious conclusion that the signals sl (t) should differ at frequencies where H(f) is

large (i.e. in the channel passband).

K(f) is often assumed to be constant in that band, so that the innerproduct x,y is simply

the usual innerproduct

fx(t)y(t)dt .

The previous channel model is not accurate. Non linear and time varying distortions occur and

noise is not always Gaussian. Nonetheless modems typically exhibit the structure just derived:

first "filtering" elements and samplers compute the projection of the received waveform. They are

followed in turn by a non linear decision device that chooses the signal closest to the received

waveform.

The signal space S has usually 1 or 2 dimensions, the basis functions being of the form x(t)

cos(27rfct) and x(t) sm(2irfct), where x(t) has a Fourier transform confined to the low

frequencies. The usual telephone lines have a passband between 300 and 3000 Hz, so that fc is

typically 1650 or 1800 Hz. n is 2, 3, 4, or 6 depending on whether data is transmitted at 2400,

4800, 9600 or 14400 bits per second.

3. TRANSMISSION OF SEQUENCES. The number N of binary digits that must be

transmitted by the modem is usually very large so that there is an enormous (2^) number of

waveforms, and the theory developed in the previous section cannot be implemented without more

structure being added.