CM · Digital modulation

Bandwidth and Symbol Rate

Each symbol takes time and band. In a given bandwidth B you cannot send more than 2B symbols per second, however clever you are. Learn this Nyquist ceiling, and why sending more bits means making each symbol heavier.

How tightly can you pack them?

Sweep the symbol rate Rs. While the pulses stay apart the symbols read clearly (gold); once Rs passes 2B the pulses overlap and smear into one blob (red, intersymbol interference).

Symbol rate RsRs = 1.50 × 2B

Sweep the rate. Rs = 2B is the Nyquist ceiling.

Symbol rate versus the Nyquist ceiling

Rs > 2B → ISI

ISI · overlap

Symbols take time

Sending one symbol needs one slot of time Ts. The number of symbols per second — the symbol rate — is Rs = 1/Ts. To go faster you narrow the slot and push the pulses closer together. But you cannot push them arbitrarily close, because the narrower a pulse is in time, the wider the band of frequencies it occupies.

Nyquist — bandwidth sets the ceiling

The Nyquist signaling theorem nails the ceiling down. In a channel of bandwidth B you can send at most 2B symbols per second without their interfering. Raise Rs above that and neighboring pulses spill into each other’s time slots, so when the receiver reads one symbol the next one bleeds in. This smearing is intersymbol interference (ISI), and once the pulses merge into one blob the bits cannot be recovered.

ObserveRs ≤ 2B

In bandwidth B the symbol rate cannot exceed 2B.

ChooseRb = Rs · log₂ ?

Multiply by log₂ M bits per symbol.

More bits make heavier symbols

The symbol rate is capped at 2B. So in the same band there is only one way to add bits: load more bits onto each symbol. With an M-ary scheme each symbol carries log₂ M bits, so the bit rate is Rb = Rs log₂ M. The efficiency per hertz, η = Rb/B, rises to 2 log₂ M. But the larger M is, the more crowded the constellation points become, demanding a higher signal-to-noise ratio — and at the very end stands the absolute ceiling Shannon drew.

Fill inη = Rb / ? (bit/s/Hz)

Efficiency is the bit rate over the bandwidth.

On your ownC = B log₂(1 + ?)

The higher ceiling is Shannon — in the next group.

Back to the first screen

As you raised the symbol rate the pulses crept closer, and the instant Rs passed 2B they overlapped, smeared into one blob, and turned red with ISI. Just below 2B the pulses stayed clearly resolved while filling the band, and turned gold; push too low and the gaps between pulses wasted the band. The one thing to hear in bandwidth and symbol rate is this: bandwidth sets the symbol-rate ceiling 2B, so more bits come not from sending symbols faster but from making each one heavier.

By the Nyquist signaling rate, the symbol rate of a channel of bandwidth B cannot exceed Rs ≤ 2B. Go faster and pulses overlap into intersymbol interference (ISI). The bit rate multiplies by bits per symbol, Rb = Rs log₂ M, and the spectral efficiency is η = Rb/B ≤ 2 log₂ M. To add bits in the same band you raise M to make symbols heavier, paying with higher SNR, and the absolute limit is the Shannon capacity.

In the next unit

The heavier the symbols and the harder you fight ISI, the more one thing kept appearing: the signal-to-noise ratio, SNR. The next group starts with what this noise actually is — where thermal noise comes from, what the ratio of signal power to noise power means, and why it decides every limit in communication.