A pulse‑shape design condition ensuring that, after transmit filtering, channel effects and receive filtering, the combined sampling‑rate impulse response g(t) sampled at symbol intervals nT yields g(nT)=0 for all integer n≠0; equivalently, the overall system produces zero intersymbol interference (ISI) at the symbol sampling instants. In frequency domain, Nyquist's first criterion requires that periodic summation of the pulse spectrum across shifts by 1/T be constant across the band so that aliasing between adjacent symbol spectra cancels at sampling instants.

Principle

If the combined transmit‑plus‑channel‑plus‑receive response satisfies the Nyquist condition (g(nT)=δ[n]), symbols sampled at rate 1/T are orthogonal in time and suffer no ISI, enabling symbol‑rate sampling and symbol‑wise detection without intersymbol interference assuming ideal timing and noise conditions.

Demonstration

Situation: Binary transmission using raised‑cosine transmit pulse with roll‑off α and matched receive filter sampled at t=nT. Recognition: Raised‑cosine pulses satisfy the Nyquist criterion for any α in [0,1] and yield g(nT)=0 for n≠0. Action: Sample the received waveform at symbol times and detect symbols without requiring an ISI‑canceling equalizer (neglecting channel distortion). Consequence: Achieves zero ISI at sampling instants; practical tradeoffs include pulse time‑domain tails (longer duration for low α) versus occupied bandwidth.

Misapplication

Designing a pulse to satisfy Nyquist criterion for an ideal channel but failing to account for channel distortion, timing offsets, sampling jitter or imperfect filtering; another error is assuming zero ISI under asynchronous sampling or in multipath without equalization.

Consequence

Proper application eliminates ISI at receiver sampling instants and simplifies symbol detection and receiver design; misapplication results in residual ISI, symbol errors and the need for equalization or timing recovery mechanisms.

Reversal

If the channel introduces linear distortion (frequency‑selective fading) or timing errors, the Nyquist property of the transmitted pulse no longer guarantees zero ISI at the receiver unless compensated (equalized) or unless matched filtering and timing recovery preserve the g(nT)=δ[n] condition.

Boundary

Clearly within: linear, time‑invariant transmission chains with symbol‑synchronous sampling and known sampling interval T where transmit and receive filters can be designed. Boundary case: channels with mild dispersion where pulse design plus modest equalization suffices. Clearly outside: asynchronous sampling, strongly time‑variant multipath without adaptive equalization, or nonlinear channels where the zero‑ISI criterion does not apply directly.

Semantic Tension

Tension between spectral efficiency (minimizing occupied bandwidth via small roll‑off) and time‑domain compactness (short pulse duration); satisfying Nyquist ISI criterion forces a tradeoff between bandwidth and intersymbol leakage (time‑domain tails).

Synthesis

The Nyquist ISI criterion is an explicit constraint that enforces temporal orthogonality at symbol instants by shaping spectrum; it permits symbol‑rate sampling without ISI under ideal timing and channel conditions but requires complementary measures (timing recovery, equalization) when practical channel impairments are present.