Results from the Receiver
Error module reveals several properties about digital
communication systems.
-
As the received signal becomes increasingly noisy, whether
due to increased distance from the transmitter (smaller
αα) or to increased noise
in the channel (larger
N
0
N
0
), the probability the receiver makes an error
approaches 1/212. In such
situations, the receiver performs only slightly better than
the "receiver" that ignores what was transmitted and merely
guesses what bit was transmitted. Consequently, it becomes
almost impossible to communicate information when digital
channels become noisy.
- As the signal-to-noise ratio increases, performance
gains--smaller probability of error
p
e
p
e
-- can be easily obtained. At a signal-to-noise
ratio of 12 dB, the probability the receiver makes an error
equals
10-8
10
-8
. In words, one out of one hundred million bits
will, on the average, be in error.
- Once the signal-to-noise ratio exceeds about 5 dB, the
error probability decreases dramatically. Adding 1 dB
improvement in signal-to-noise ratio can result in a factor
of 10 smaller
p
e
p
e
.
- Signal set choice can make a significant difference in
performance. All BPSK signal sets, baseband or modulated,
yield the same performance for the same bit energy. The
BPSK signal set does perform much better than the FSK signal
set once the signal-to-noise ratio exceeds about 5 dB.
Derive the expression for the probability of error that
would result if the FSK signal set were used.
The noise-free integrator output difference now equals
αA2T=α
E
b
2
α
A
2
T
α
E
b
2
. The noise power remains the same as in the BPSK
case, which from the probability of error equation yields
p e=Qα2EbN0
p e
Q
α
2
Eb
N0
.
The matched-filter receiver provides impressive performance once
adequate signal-to-noise ratios occur. You might wonder whether
another receiver might be better. The answer is that the
matched-filter receiver is optimal: No other receiver
can provide a smaller probability of error than the matched
filter regardless of the SNR. Furthermore, no signal
set can provide better performance than the BPSK signal set,
where the signal representing a bit is the negative of the
signal representing the other bit. The reason for this result
rests in the dependence of probability of error
p
e
p
e
on the difference between the noise-free integrator outputs: For a given
E
b
E
b
, no other signal set provides a greater difference.
How small should the error probability be? Out
of NN transmitted bits, on the
average
N
p
e
N
p
e
bits will be received in error. Do note the phrase "on
the average" here: Errors occur randomly because of the noise
introduced by the channel, and we can only predict the
probability of occurrence. Since bits are transmitted at a rate
RR, errors occur at an average
frequency of
R
p
e
R
p
e
. Suppose the error probability is an impressively
small number like
10-6
10
-6
. Data on a computer network like Ethernet is
transmitted at a rate
R=100Mbps
R
100
Mbps
, which means that errors would occur roughly 100 per
second. This error rate is very high, requiring a much smaller
p
e
p
e
to achieve a more acceptable average occurrence rate for errors
occurring. Because Ethernet is a wireline channel, which means
the channel noise is small and the attenuation low, obtaining
very small error probabilities is not difficult. We do have some
tricks up our sleeves, however, that can essentially reduce the
error rate to zero without resorting to
expending a large amount of energy at the transmitter. We need
to understand digital
channels and Shannon's Noisy Channel Coding Theorem.
"Electrical Engineering Digital Processing Systems in Braille."