Nyquist Rate and Symbol Rate | WRL-101

Shannon told us the ceiling on bits per second. This lesson is about the symbols that carry them: how many distinct symbols per second a band of a given width can hold before they start smearing into one another. The answer is another number from the early days of the field — Nyquist’s, from 1928 — and it, together with the bits-per-symbol of Module 5, sets the data rate of every real link.

The Nyquist Rate

Send symbols too fast through a band-limited channel and each one’s tail spills onto its neighbours — inter-symbol interference (ISI). Nyquist found the exact boundary: a baseband channel of bandwidth W can carry at most 2W symbols per second free of ISI. Equivalently, to send R_s symbols per second you need at least R_s/2 of baseband bandwidth:

Nyquist Signalling Rate

R_s \le 2W \qquad\Longleftrightarrow\qquad W_{\min} = \tfrac{R_s}{2}

R_s is the symbol rate (symbols/s, or baud) and W the baseband bandwidth in Hz. The minimum bandwidth to carry R_s baud is therefore W_min = R_s/2. (A passband/RF channel of width B carries R_s = B baud at this ideal.)

Symbol Rate Is Not Bit Rate

This is where Module 5 rejoins the story. The Nyquist rate limits symbols, but each symbol can carry several bits — that was the whole point of QAM. The bit rate is the symbol rate multiplied by the bits each symbol carries:

Bit Rate from Symbol Rate

R_b = R_s \times \log_2 M

A 1 Mbaud link sends 1 million symbols/s regardless of scheme — but that is 1 Mbps with BPSK, 2 Mbps with QPSK, and 6 Mbps with 64-QAM. Baud and bits per second are different units; confusing them is the classic mistake.

Why the Ideal Is Impossible

Nyquist’s 2W limit is reached only by a perfect brick-wall filter — a spectrum that is flat to R_s/2 and then vertical. In the time domain that filter is a sinc pulse: it rings on forever in both directions and is non-causal, so it cannot be built, and even a truncated version is disastrously sensitive to timing error. Real systems trade a little extra bandwidth for a pulse that is actually realizable.

Raised-Cosine Pulse Shaping

The standard fix is the raised-cosine pulse. It keeps Nyquist’s crucial property — the pulse is exactly zero at every other symbol instant, so a matched receiver sees no ISI — while softening the brick wall into a gentle cosine roll-off. A single parameter, the roll-off factor α (between 0 and 1), sets how gentle:

Raised-Cosine Bandwidth

B = \frac{R_s}{2}\,(1 + \alpha), \qquad 0 \le \alpha \le 1

The occupied baseband bandwidth grows from the Nyquist minimum R_s/2 by the factor (1 + α). The excess bandwidth — the price of realizability — is α × R_s/2, i.e. α expressed as a percentage over Nyquist.

The roll-off trade-off

α = 0 — the brick wall: minimum bandwidth (R_s/2), but long, slowly-decaying pulses that are fragile to timing jitter. Unbuildable in the limit.
α = 1 — 100% excess bandwidth (double the minimum), but short, fast-decaying pulses that are easy to filter and forgiving of timing error.
In between — real systems pick a small α: about 0.22 in 3G WCDMA, 0.1–0.25 in LTE, and even lower in modern systems, buying tight spectrum while keeping the filter practical.

Root-raised-cosine and the matched filter. In practice the raised-cosine shape is split in half — a root-raised-cosine (RRC) filter at the transmitter and an identical one at the receiver. Cascaded, the two multiply to a full raised cosine, giving zero ISI; and because the receive filter is a copy of the transmit pulse, it is also the matched filter that maximizes SNR (the best possible detector in Gaussian noise). One design choice does two jobs at once.

Putting It Together

A real link’s data rate now falls out of two independent choices. The available RF bandwidth B and the roll-off fix the symbol rate, R_s = B / (1 + α); the modulation order fixes the bits per symbol, log₂M. Multiply them for the bit rate:

Bandwidth B	Roll-off α	Symbol rate R_s	Modulation	Bit rate
1 MHz	0.25	0.8 Mbaud	QPSK (2 b)	1.6 Mbps
1 MHz	0.25	0.8 Mbaud	64-QAM (6 b)	4.8 Mbps
20 MHz	0.1	18.2 Mbaud	256-QAM (8 b)	145 Mbps

Notice the two levers are separate: bandwidth and roll-off buy symbols, modulation order buys bits per symbol. The next lesson combines them into a single figure of merit — spectral efficiency, in bits per second per hertz — and measures every scheme against Shannon’s ceiling.

Key Takeaways

Nyquist: a baseband channel of bandwidth W carries at most 2W symbols/s without ISI; equivalently W_min = R_s/2.
Symbol rate (baud) ≠ bit rate: R_b = R_s × log₂M — the same baud carries more bits with higher-order QAM.
The ideal brick-wall filter is a non-causal sinc pulse — unbuildable and timing-fragile — so real links use pulse shaping.
Raised-cosine keeps zero ISI at symbol instants; roll-off α widens the band to (R_s/2)(1 + α), the excess being α over Nyquist.
Small α = tight spectrum but fragile; large α = wide but robust. Real systems use α ≈ 0.1–0.25.
Root-raised-cosine split across TX and RX gives zero ISI and the SNR-optimal matched filter in one design.

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