Q4Digital Communication

(a) Matched Filter Impulse Response and Matched Signal

Given: H(f) = (1-e^(-j2πft))/(j2πf), where the parameter in the exponent should be understood as a specific pulse duration T (the notation in the problem uses 't', but in this context it represents the fixed pulse-duration parameter, not the time variable of h(t) itself — this is standard notation in this classic textbook problem, where H(f) = (1-e^(-j2πfT))/(j2πf)).

(i) Impulse response h(t): recognizing this frequency response as the Fourier transform of a rectangular pulse: the Fourier transform of a rectangular pulse of unit amplitude, existing from t=0 to t=T, is known to be:

which exactly matches the given H(f). Therefore, taking the inverse Fourier transform:

(ii) Matched signal waveform: the matched filter theorem states that a filter matched to a known signal pulse s(t) has an impulse response h(t) = s(T-t), i.e., a time-reversed and delayed replica of the signal it is matched to. Since the derived h(t) is itself a rectangular pulse of amplitude 1 over [0,T] (which is symmetric under the time-reversal-and-delay operation, since a constant-amplitude rectangular pulse over [0,T], when time-reversed and shifted by T, maps back onto itself), the filter is matched to a rectangular pulse signal s(t) of unit amplitude and duration T (identical in shape to h(t) itself, as expected for this simple, symmetric rectangular pulse case).

This result confirms the well-known general property that a matched filter for a rectangular baseband pulse of duration T and amplitude A is simply an integrate-and-dump filter over the same interval T — precisely the simplest and most commonly used correlator/matched-filter receiver structure for baseband binary/M-ary PAM signal detection, since a matched filter is guaranteed (by the matched filter theorem) to maximize the output signal-to-noise ratio at the correct sampling instant, for any given transmitted pulse shape corrupted by additive white Gaussian noise.

(b) Effects of Inter-Symbol Interference (ISI)

Inter-Symbol Interference (ISI) is the phenomenon in which the transmitted pulse for one symbol spreads out in time (due to the finite bandwidth of the transmission channel/filtering, which cannot pass an ideal, infinitely narrow pulse without temporal spreading) sufficiently to overlap with and corrupt the correct detection of adjacent symbol pulses at the receiver's sampling instants.

Cause: any physical, band-limited transmission channel or filter, when excited by an ideal rectangular (or other time-limited) pulse, produces an output pulse response that extends in time beyond the original pulse duration (the channel's impulse response necessarily has a finite 'tail' due to the bandwidth limitation, by the fundamental time-bandwidth trade-off inherent to all linear systems) — when successive symbol pulses are transmitted at a rate comparable to or faster than the channel's ability to fully settle between symbols, these spread-out pulse tails from earlier symbols add to (interfere with) the current symbol's pulse at the receiver's sampling instant, corrupting the correct detection decision.

Effect on digital transmission performance: ISI directly degrades the bit-error-rate (BER) performance of a digital communication system, since the interfering contributions from neighboring symbols effectively add extra, data-dependent noise/distortion on top of the usual channel additive noise, reducing the effective margin between the decision threshold and the actual received sample value — in severe cases, ISI can cause the 'eye diagram' (a standard visualization tool overlaying many successive received symbol waveforms) to fully 'close,' meaning no reliable decision threshold exists at all that can correctly distinguish between different transmitted symbol values, causing the system's error rate to become unacceptably high regardless of how much the channel's signal-to-noise ratio itself is improved (since ISI is a deterministic distortion effect, not a random-noise effect, and cannot be overcome simply by increasing transmit power).

Mitigation: ISI is controlled through careful pulse shaping at the transmitter (and/or equalization at the receiver) designed to satisfy Nyquist's zero-ISI criterion — the requirement that the overall combined pulse shape (transmit filter × channel × receive filter) produces a response that passes through zero at every symbol sampling instant except the current one being detected, ensuring that neighboring symbols contribute exactly zero interference at the correct sampling instant even though their pulse tails may still be non-zero at other times. The raised-cosine pulse shape (and its square-root-raised-cosine variant, split between transmitter and receiver) is the most widely used practical pulse shape satisfying this Nyquist criterion, offering a tunable excess-bandwidth (roll-off factor) trade-off between bandwidth efficiency and practical robustness to timing/filtering imperfections; adaptive channel equalizers are additionally used at the receiver in many practical systems to compensate for ISI introduced by channel characteristics that are not precisely known or that vary over time (such as in wireless multipath channels), further mitigating the ISI that inherent transmit/receive pulse-shaping filters alone cannot fully eliminate.

Relating the Matched Filter to the Correlation Receiver

It is useful to connect the matched-filter result of part (a) explicitly to the correlation-receiver structure used throughout this paper (for example, in the BFSK and PSK detection questions elsewhere in this exam). A matched filter with impulse response h(t)=s(T-t), sampled at time t=T, produces an output:

which is exactly the correlation integral of the received signal r(t) against the known transmitted pulse shape s(t) — i.e., sampling a matched filter's output at the end of the symbol interval is mathematically identical to computing the correlator output over that same interval. For the specific rectangular pulse case derived here (h(t)=1 over [0,T]), this correlation integral reduces to simply integrating (accumulating) the received signal over the symbol period and then 'dumping' (resetting) the integrator for the next symbol — precisely the well-known 'integrate-and-dump' receiver structure used in practice for baseband PAM/PCM line-code detection, since a running rectangular-window integration is trivial to implement with a simple analog integrator or, in modern systems, with a digital accumulator sampled at the symbol rate.

This equivalence between matched filtering and correlation detection is significant because it shows that the two design approaches — implementing a fixed analog filter with the time-reversed pulse shape as its impulse response, versus explicitly multiplying the received waveform by a locally generated reference copy of the pulse and integrating the product — are mathematically interchangeable ways of achieving the same, provably SNR-optimal detection performance for a known pulse shape in additive white Gaussian noise. The matched-filter/correlator equivalence generalizes directly to the coherent detection structures used for BFSK and M-ary PSK elsewhere in this paper: those receivers are simply banks of correlators (or, equivalently, matched filters), each matched to one of the possible transmitted waveforms, with the largest-output correlator determining the detected symbol.

ISI and the eye diagram in relation to the matched filter output: since the matched/correlator filter derived in part (a) is itself the receive-side element of the overall end-to-end pulse shape, ISI performance is directly assessed by examining the eye diagram formed by overlaying many successive matched-filter output waveforms — a wide-open eye indicates minimal ISI and a large noise margin at the optimum sampling instant, whereas increasing ISI (from insufficient pulse shaping, bandwidth-limited channels, or timing jitter) progressively narrows the eye opening both vertically (reducing noise margin) and horizontally (reducing the tolerable sampling-time error), directly connecting the qualitative ISI discussion in part (b) to the quantitative matched-filter output derived in part (a).