B. P. Lathi, Zhi Ding - Modern Digital and Analog Communication Systems-Oxford University Press (2009)

10.7 General Expression for Error Probability of Optimum Receivers 567
From this discussion, one very interesting fact emerges: whenever the optimum receiver
is used, the error probability does not depend on specific signal waveforms; it depends only
on their geometrical configuration in the signal space.
Bandwidth and Power Trade-offs of M -ary Orthogonal Signals
As illustrated by Landau and Pollak, 3 the dimensionality of a signal is 2BT M + 1, where T M is
the signal duration and B is its essential bandwidth. It follows that for an N -dimensional signal
space (N ::: M ), the bandwidth is B = (N - l )/2TM. Thus, reducing the dimensionality N
reduces the bandwidth.
We can verify that N-dimensional signals can be transmitted over (N - l )/2T M Hz by
constructing a specific signal set. Let us choose the following orthonormal signals:
1
<po ( t) = ,./T;i
(10.123)
(l)k- 1 (t) = ff sin ( w 0 t)
(l)k(t) = ff COS ( W 0 t)
These k + 1 orthogonal pulses have a total bandwidth of (k/2)(w 0 /2n) = k/2T M Hz. Hence,
when k + 1 = N, the bandwidth* is (N - l)/2TM. Thus, N = 2TMB + 1.
To attain a given error probability, there is a trade-off between the average energy of the
signal set and its bandwidth. If we reduce the signal space dimensionality, the transmission
bandwidth is reduced. But the distances among signals are now smaller, because of the reduced
dimensionality. This will increase P eM . Hence, to maintain a given low P eM , we must now
move the signals farther apart; that is, we must increase energy. Thus, the cost of reduced
bandwidth is paid in terms of increased energy. The trade-off between SNR and bandwidth
can also be described from the perspective of information theory (Sec. 13.6).
M-ary signaling provides us with additional means of exchanging, or trading, the transmission
rate, transmission bandwidth, and transmitted power. It provides us flexibility in designing
a proper communication system. Thus, for a given rate of transmission, we can trade the transmission
bandwidth for transmitted power. We can also increase the information rate by a
* Here we are ignoring the band spreading at the edge. This spread is about I /T M Hz. The actual bandwidth exceeds
(N - l) /2TM by this amount.