Results from the Receiver Error module (Digital Communication in the Presence of Noise) reveals several properties about digital communication systems.

- As the received signal becomes increasingly noisy, whether due to increased distance from the transmitter (smaller α) or to increased noise in the channel (larger
*N*_{0}), the probability the receiver makes an error approaches**1/2**. 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*can be easily obtained. At a signal-to-noise ratio of 12 dB, the probability the receiver makes an error equals 10_{e }^{−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} - 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.

**Exercise 6.18.1**

Derive the expression for the probability of error that would result if the FSK signal set were used.

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} on the difference between the noise-free integrator outputs: For a given *E*_{b}, no other signal set
provides a greater difference.

How small should the error probability be? Out of *N* transmitted bits, on the average *Np _{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 R, errors occur at an average
frequency of

*Rp*. Suppose the error probability is an impressively small number like 10

_{e}^{−6}. Data on a computer network like Ethernet is transmitted at a rate

*R*= 100Mbps, which means that errors would occur roughly 100 per second. This error rate is very high, requiring a much smaller

*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 (Digital Channels ) and Shannon's Noisy Channel Coding Theorem (Noisy Channel Coding Theorem ).

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