A designer increases the raised-cosine roll-off factor from α = 0.2 to α = 0.8. What are the consequences for the communication link?
AISI increases because the wider spectrum causes more interference between adjacent symbols
BISI decreases because the extra bandwidth provides guard space; the Nyquist zero-crossing property is weakened
CBandwidth usage increases, but ISI is unchanged — the zero-crossing property is preserved; the benefit is faster time-domain decay that makes the system more robust to timing jitter
DThe symbol rate must decrease proportionally because the increased bandwidth consumes the available channel
The zero-ISI property (zero crossings at multiples of Ts) is preserved for all values of α — roll-off does not change whether there is ISI when sampling perfectly. What α controls is the decay rate of the impulse response: higher α gives 1/t³ decay instead of 1/t, making the system far more tolerant of timing errors. The tradeoff is that a wider roll-off uses more bandwidth (up to twice the minimum at α = 1). ISI robustness and bandwidth efficiency are what trade off, not ISI performance per se.
Question 2 Multiple Choice
In a real digital communication system, why is pulse shaping typically split between a root-raised-cosine (RRC) filter at the transmitter and another RRC filter at the receiver, rather than placing the full raised-cosine at the transmitter?
ABecause receivers cannot implement complex spectral shapes — only simple filters like square windows
BBecause applying the full filter at the transmitter would violate FCC spectral mask regulations in most bands
CBecause the receiver must apply a matched filter (the time-reverse of the transmit pulse) to maximize SNR, and for the raised-cosine this matched filter is an RRC — the transmit and receive RRC filters cascade to produce the full raised-cosine at the sampling instant, achieving both zero ISI and optimal noise performance
DBecause splitting the filter reduces total computational cost equally between transmitter and receiver hardware
Matched filtering is required to maximize SNR at the sampling instant. The matched filter is the time-reverse of the transmit pulse. For a symmetric raised-cosine, the matched filter is also a raised-cosine — but if the receiver applies a full raised-cosine on top of the full transmit raised-cosine, the cascade is a 'double raised-cosine' that does NOT satisfy the Nyquist criterion and reintroduces ISI. Splitting each end into the square root solves this: RRC_tx × RRC_rx = full raised-cosine, achieving zero ISI and matched filtering simultaneously.
Question 3 True / False
A raised-cosine filter with roll-off factor α = 0.5 has worse ISI performance at the sampling instant than the sinc pulse (α = 0), because it uses more bandwidth.
TTrue
FFalse
Answer: False
The zero-ISI property (the Nyquist criterion) holds for all values of α — zero crossings occur at exactly multiples of Ts regardless of roll-off. ISI performance at the correct sampling instant is equally zero for any α. In practice, ISI is actually better with higher α because the faster 1/t³ decay (versus 1/t for sinc) means neighboring symbol tails are much smaller when timing is imperfect. The tradeoff is bandwidth, not ISI at the ideal sampling time.
Question 4 True / False
The sinc pulse is theoretically optimal for bandwidth efficiency (minimum bandwidth for zero-ISI signaling) but impractical because small timing errors cause large ISI due to its slow 1/t amplitude decay.
TTrue
FFalse
Answer: True
The sinc pulse's rectangular spectrum uses the absolute minimum bandwidth (1/(2Ts)), but it decays only as 1/t. Any receiver timing offset causes energy from many neighboring symbols to pile up at the sampling instant — even a tiny fractional-symbol timing error produces significant ISI from the slowly-decaying tails of dozens of surrounding symbols. This sensitivity to timing jitter makes the theoretically optimal sinc pulse impractical, which is exactly the problem the raised-cosine filter was designed to solve by trading some bandwidth for much faster temporal decay.
Question 5 Short Answer
Explain why applying the full raised-cosine filter entirely at the transmitter (rather than splitting it as root-raised-cosine at both ends) would fail to achieve the goals of a well-designed digital communication link.
Think about your answer, then reveal below.
Model answer: Applying the full raised-cosine at the transmitter satisfies zero-ISI but fails noise optimization. The receiver must apply a matched filter — the time-reverse of the transmit pulse — to maximize signal-to-noise ratio. If the transmit pulse is a full raised-cosine, the matched receive filter is also a full raised-cosine, and the cascade is a double raised-cosine whose spectrum is the square of the raised-cosine spectrum. This does not satisfy the Nyquist zero-ISI criterion, reintroducing intersymbol interference. The root-raised-cosine split is the solution: transmit RRC × receive RRC = full raised-cosine, simultaneously achieving zero ISI and matched filtering.
This is the key system-level insight that goes beyond block-diagram understanding: the filter design is constrained by the need to jointly satisfy two requirements (zero ISI and matched filtering) that must be shared across the link. Neither requirement can be satisfied independently at one end without compromising the other.