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# Digital Communcation System Properties

Module by: Don Johnson. E-mail the author

Summary: Several properties of digital communication systems make them preferable to analog systems.

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.

## Exercise 1

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

### Solution

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 =Qα2 E b N 0 p e Q α 2 E b N 0 .

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.

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