Nyquist’s Transmission Research: Signaling Rate and Bandwidth

In 1928, Harry Nyquist presented “Certain Topics in Telegraph Transmission Theory” to the American Institute of Electrical Engineers, which published it in its Transactions. Working in communications research within the AT&T and Bell Laboratories organization, he confronted a practical difficulty in long-distance telegraphy: a transmitter could change the current rapidly, but those changes were not necessarily as clear at the far end of the line.

A line has resistance, inductance, and capacitance, as well as equipment connected at its terminals. After passing through these elements, a short current pulse may become lower, longer, or oscillatory. When the next pulse arrives, the tail of its predecessor may still affect the receiver. Increasing telegraph speed was therefore not merely a matter of timing, but also of whether waveforms could still be distinguished after passing through the line.

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Directly calculating every transient after each switching operation was cumbersome for lines with complex terminal equipment. Nyquist instead began with steady-state characteristics: he represented a signal as a superposition of components at different frequencies, examined how the line changed each component’s amplitude and phase, and then reconstructed the received waveform. Even as connected equipment became more complex, analysis could still center on the same set of frequency characteristics.

Bandwidth describes the range of frequencies a system can effectively transmit; it is not the strength of the current. Rapidly changing waveforms require a richer range of frequency components. If a line attenuates some of them, the received waveform changes. Nyquist therefore considered the transmitted pulse, line response, and receiving requirements together, rather than demanding that every line reproduce an ideal square wave without alteration.

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The paper first adopted a criterion for distortionless transmission: the current at the midpoint of each signal element should correspond to the transmitted value. If the receiver read the signal at agreed instants, the waveforms did not have to remain separate at every moment. In modern terms, a pulse could extend into neighboring time slots, provided its contribution was zero at the decision instants for other symbols, so that it caused no intersymbol interference there.

For example, in a sequence of high and low levels, if the tail of a preceding high level remained positive at the next reading, it might raise what should have been a low level. Appropriately designed pulses and filter responses could make the tails cross zero at those designated instants. This example shows that a waveform’s loss of a square shape does not automatically indicate a transmission error. What matters is which quantity the receiving device uses to make its decision.

Nyquist also discussed receivers that made decisions using other criteria, such as pulse duration or the area over a specified interval. Different telegraph instruments required different conditions for correct reception. The paper therefore offered a way to translate equipment requirements into mathematical constraints, rather than merely prescribing a universal waveform.

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For an ideal low-pass channel under the corresponding interference-free conditions, there is a definite relationship between bandwidth and the number of independently assignable pulses per second. The limit later came to be commonly expressed as at most two real-valued symbols per second for each hertz of bandwidth. The quantity being counted is symbols: a symbol may have two or more permitted amplitudes, so symbols per second cannot be treated unconditionally as bits per second.

Hartley’s research, published in the same year, further discussed information in terms of the logarithm of the number of possible signals. For a given number of symbols, increasing the number of distinguishable levels increased the range of choices that could be expressed. Real receivers, however, face noise and limits on amplitude resolution; levels cannot be packed arbitrarily closely. Nyquist’s 1928 work focused on line frequency bands and distortion conditions. Its ideal transmission analysis alone could not justify promising unlimited information rates on real lines.

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The paper analyzed not only direct-current telegraphy but also carrier telegraphy, phase discrimination, single-sideband operation, and other cases. It outlined how transmitted waveforms could be altered so that a given line produced the desired received waveform. Bandwidth became more than a number in a line handbook: it was a shared design constraint for pulses, filters, and terminal equipment. Engineers could first define the receiving objective and then examine whether the system as a whole could meet it.

In “A Mathematical Theory of Communication” in 1948, Shannon explicitly built on Nyquist’s and Hartley’s work, incorporating noise and source statistics into capacity analysis. Later sampling theory and digital communications continued to use Nyquist’s name, but the event anchored here is his 1928 telegraph paper. It transformed the problem of overlapping waveforms in high-speed transmission into calculable, testable conditions, providing a foundation for subsequent design.

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