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The Nyquist criterion specifies conditions on pulse response p(t) for zero intersymbol interference (ISI) at sampling times: p(nTs) = 1 for n=0 and p(nTs) = 0 for n≠0. In frequency domain: Σ P(f + k/Ts) = Ts. This ensures adjacent symbols do not interfere, enabling reliable symbol recovery from noisy channels.
From the sampling theorem, you know that a bandlimited signal with bandwidth B Hz can be reconstructed from samples taken at 2B samples per second, the Nyquist rate. Now consider the inverse problem in digital communications: you want to transmit discrete symbols (bits, or higher-order constellation points) through a channel with a limited bandwidth, at the highest possible symbol rate. Each transmitted symbol must be represented by a pulse that fits within the channel bandwidth — but narrow-bandwidth pulses have long time-domain tails that extend into neighboring symbol intervals. When those tails overlap and corrupt the detection of adjacent symbols, the result is intersymbol interference (ISI).
Think concretely: you transmit symbol a₀ = +1 using a pulse p(t), then symbol a₁ at time T later, then a₂ at 2T, and so on. The received signal is the sum r(t) = Σ aₙ p(t − nT). When you sample r(t) at time t = 0 to recover a₀, you get not just p(0) · a₀ but also p(−T) · a₁ + p(−2T) · a₂ + …. If the pulse has nonzero values at those shifted sampling times, every other symbol leaks into your detection of a₀. ISI is the additive interference from every symbol in the sequence, and it cannot be removed by simply increasing signal power — it is a structural problem caused by the pulse shape.
The Nyquist criterion provides the exact condition on p(t) that guarantees zero ISI at the sampling instants: p(nT) = 1 for n = 0, and p(nT) = 0 for all nonzero integers n. In words, the pulse must pass through zero at every symbol period except its own. The sinc function sinc(t/T) = sin(πt/T)/(πt/T) satisfies this exactly — it equals 1 at t = 0 and crosses zero at every multiple of T. The sinc pulse corresponds to a rectangular spectrum of bandwidth 1/(2T), achieving the theoretical maximum symbol rate of 2B symbols per second over a channel of bandwidth B. This is the Nyquist rate for transmission, directly analogous to the sampling theorem you know.
In practice, the ideal sinc pulse is unusable: its tails decay as 1/t and never reach zero, so any timing error causes catastrophic ISI, and it requires an infinitely long filter. The raised cosine spectrum is the engineering solution. It modifies the rectangular spectrum with a smooth rolloff over an "excess bandwidth" Δf = α/2T, where α ∈ [0, 1] is the rolloff factor. The resulting pulse still satisfies the Nyquist zero-crossing condition, but its tails decay as 1/t³ instead of 1/t, making it robust to timing errors. The cost is reduced bandwidth efficiency: the raised cosine with rolloff α requires bandwidth (1+α)/(2T) instead of the minimum 1/(2T). Choosing α is a fundamental design tradeoff in every digital communication system — α = 0 maximizes spectral efficiency but demands perfect timing; α = 1 halves the spectral efficiency but tolerates practical timing jitter. Most real systems (cellular, satellite, cable modem) use α between 0.2 and 0.5 as a practical compromise.