Questions: Nyquist Criterion for Zero Intersymbol Interference

Question 1

A digital communication system transmits at 2000 symbols per second through a bandlimited channel. According to the Nyquist criterion, what is the minimum channel bandwidth required to achieve zero ISI?

• A. 2000 Hz — bandwidth must match the symbol rate
• B. 4000 Hz — the channel must support twice the symbol rate
• C. 1000 Hz — the minimum bandwidth is half the symbol rate
• D. 500 Hz — raised cosine filtering halves the required bandwidth

Answer: (C)

The Nyquist minimum bandwidth for zero ISI is f_min = R_s/2, where R_s is the symbol rate. For 2000 symbols/second, f_min = 1000 Hz. A channel of 1000 Hz bandwidth can support a sinc-pulse system at 2000 symbols/second with zero ISI. This is the direct analog of the sampling theorem: just as 2B samples/second can reconstruct a B-Hz signal, a B-Hz channel can support 2B symbols/second with Nyquist pulse shaping.

Question 2

A system designer increases the raised cosine rolloff factor from α = 0.2 to α = 0.8, while keeping the symbol rate constant. What is the effect on the system?

• A. Bandwidth efficiency improves because the filter is more selective at α = 0.8
• B. ISI increases at α = 0.8 because the pulse tails decay more slowly
• C. Bandwidth increases by a factor of (1 + 0.8)/(1 + 0.2) = 1.5, but the pulse is more robust to timing errors
• D. The symbol rate must be reduced to compensate for the wider bandwidth at α = 0.8

Answer: (C)

Raised cosine bandwidth is (1 + α)/(2T), so increasing α from 0.2 to 0.8 increases bandwidth from 1.2/(2T) to 1.8/(2T) — a 50% increase. The tradeoff is that higher α produces pulses whose tails decay faster (as 1/t³), making them far more tolerant of timing jitter. At α = 0.2, the pulse tails are very slow-decaying and small timing errors cause severe ISI; at α = 0.8, timing errors have little effect. Real systems choose α as a tradeoff: more bandwidth for more robustness.

Question 3

ISI is a structural problem caused by pulse shape and cannot be eliminated by simply increasing the transmitted signal power.

• T. True
• F. False

Answer: True

ISI arises because the tails of a pulse spill into adjacent symbol intervals, and every other symbol in the stream contributes interference to any given sample. Increasing power amplifies both the desired symbol and all the interfering tails equally — the signal-to-ISI ratio does not improve. This is in contrast to additive noise, where more power improves the SNR. ISI must be addressed by designing the pulse shape to satisfy the Nyquist zero-crossing condition, or by equalizers that undo the channel's ISI at the receiver.

Question 4

The ideal sinc pulse is widely used in practical digital communication systems because it achieves the theoretical minimum bandwidth for zero ISI.

• T. True
• F. False

Answer: False

The sinc pulse satisfies the Nyquist condition perfectly in theory but is completely impractical for two reasons. First, its tails decay as 1/t — they are very slow to die away — so any timing error in sampling causes contributions from many neighboring symbols, producing catastrophic ISI. Second, implementing an ideal sinc filter requires an infinitely long impulse response (it is noncausal). Real systems use the raised cosine pulse, which sacrifices some spectral efficiency (using more bandwidth than the minimum) to achieve tails that decay as 1/t³, making the system robust to realistic timing errors.

Question 5

Why is the raised cosine pulse preferred over the ideal sinc pulse in practice, even though both satisfy the Nyquist zero-ISI condition at the correct sampling instants?

Model answer: Both pulses have zero crossings at all nonzero multiples of the symbol period T, so both produce zero ISI when sampled at exactly the right moments. The problem is that perfect sampling timing is impossible in real systems — there is always some jitter. The sinc pulse has tails that decay as 1/t: a small timing error ε means the pulse sampled at T + ε still has a large contribution from adjacent symbols. The raised cosine pulse's tails decay as 1/t³: the same timing error ε causes much smaller ISI, because the pulse value at T + ε is negligible. The raised cosine trades extra bandwidth (1+α times the Nyquist minimum) for this faster decay, making the system practical.

The core insight is that satisfying the Nyquist condition at exact sample points is necessary but not sufficient for reliable communication — you also need the pulse to be approximately zero near those sample points (not just exactly at them) to tolerate realistic timing imperfections. This is why rolloff factor α is a fundamental design parameter: α = 0 is theoretical perfection, and α > 0 is practical engineering.