Several factors of error in digital receivers are discussed.
When we incorporate additive noise into our channel model, so
that
, errors can creep in. If the transmitter sent bit
0 using a
BPSK
signal set , the integrators' outputs in the
matched filter
receiver would be:
It is the quantities containing the noise terms that cause
errors in the receiver's decision-making process. Because theyinvolve noise, the values of these integrals are random
quantities drawn from some probability distribution that varyerratically from bit interval to bit interval. Because the noise
has zero average value and has an equal amount of power in allfrequency bands, the values of the integrals will hover about
zero. What is important is how much they vary. If the noise issuch that its integral term is more negative than
, then the receiver will make an error, deciding that
the transmitted zero-valued bit was indeed a one. Theprobability that this situation occurs depends on three factors:
Signal Set Choice — The difference
between the signal-dependent terms in the integrators'outputs
(equations
[link] ) defines how
large the noise term must be for an incorrect receiverdecision to result. What affects the probability of such
errors occurring is the energy in the difference of the received signals incomparison to the noise term's variability. The signal-difference energy equals
For our BPSK baseband signal set, the difference-signal-energy term is
.
Variability of the Noise Term — We
quantify variability by the spectral height of the white noise
added by the channel.
Probability Distribution of the Noise
Term — The value of the noise terms relative
to the signal terms and the probability of their occurrencedirectly affect the likelihood that a receiver error will
occur. For the white noise we have been considering, theunderlying distributions are Gaussian. Deriving the following expression for the probability the
receiver makes an error on any bit transmission is complicated but can be found at
[link] and
[link] .
Here
is the integral
.This integral has no closed form expression, but it can be
accurately computed. As
[link] illustrates,
is a decreasing, very nonlinear function.
The term
equals the energy expended by the transmitter in sending the
bit; we label this term
. We arrive at a concise expression for the
probability the matched filter receiver makes a bit-receptionerror.
[link] shows how the receiver's
error rate varies with the signal-to-noise ratio
.
Derive the probability of error expression for the modulated
BPSK signal set, and show that its performance identicallyequals that of the baseband BPSK signal set.
The noise-free integrator outputs differ by
, the factor of two smaller value than in the
baseband case arising because the sinusoidal signals haveless energy for the same amplitude. Stated in terms of
, the difference equals
just as in the baseband case.
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