Why can't a noisy channel carry unlimited data?

Sending more bits through a channel requires the receiver to distinguish more possibilities in the signal it observes. Noise blurs those possibilities, while limited bandwidth restricts how rapidly useful signal features can vary. These physical constraints explain why a communication link has a finite reliable information rate even when the transmitter and receiver use sophisticated software.

Bandwidth here means a range of frequencies available to the signal, measured in hertz. It is related to but not identical with a data rate measured in bits per second. A narrow channel smooths rapid changes, causing neighboring symbols to overlap in time. A wider channel can support faster variation, but the receiver must still distinguish those variations from noise and interference. Merely naming a larger number of symbols does not make them reliably separable.

One approach sends several bits per symbol by using more possible signal levels or combinations of amplitude and phase. That increases the information each ideal symbol can represent. However, if the total signal range stays fixed, the alternatives move closer together. Noise can then push a received signal across the decision boundary between alternatives. The illustration contrasts two widely separated levels with several closely packed levels under the same disturbance size.

For an ideal channel with additive white Gaussian noise and an average power constraint, Shannon's capacity formula is C = B log₂(1 + S/N). B is bandwidth and S/N is the signal-to-noise power ratio expressed as an ordinary ratio, not decibels. The formula gives an upper bound on arbitrarily reliable communication under its assumptions. It does not promise that any practical code reaches the limit at any delay or complexity.

For example, with a 1,000-hertz bandwidth and S/N equal to 15, the ideal bound is 4,000 bits per second because log₂(16) equals four. Real usable rates also reflect coding overhead, protocol information, interference, hardware limits and required reliability. Increasing signal power has diminishing returns in this model because the logarithm grows slowly. Expanding bandwidth and improving the noise environment can offer different benefits, with their own engineering costs.

Error-correcting codes use redundancy to make messages more distinguishable across sequences of symbols. They can greatly improve reliability but cannot evade the channel's information limit. Likewise, compression can reduce the bits needed to represent a source without increasing the physical channel capacity itself. The central insight is that information is carried by distinguishable physical states. Noise and bandwidth determine how many such distinctions the receiver can reliably recover over time.