Pulse-code_modulation

# Pulse-code modulation **Pulse-code modulation** (PCM) is the method of representing an [[Analog_signal|analog signal]] as a sequence of binary words. The signal is [[Sampling_(signal_processing)|sampled]] at uniform instants, each sample is rounded to the nearest of a fixed set of levels, a step called [[Quantization_(signal_processing)|quantization]], and the index of that level is written as a word of `n` bits. It is the plain digital form of sound and of most measured signals: the output of an [[Analog-to-digital_converter|analog-to-digital converter]], the content of a compact disc or a computer audio file, and the form in which telephone speech has crossed networks since the 1960s. Everything that a [[Codec|codec]] or a [[Data_compression|compression]] scheme does begins from PCM and ends by producing it again. A PCM stream is fixed by two numbers. The sampling rate sets the highest frequency the words can carry, half the rate, and anything above that folds back into the band as aliasing unless an [[Anti-aliasing_filter|anti-aliasing filter]] removes it first. The bit depth sets the step between levels and with it the quantization error, which for a full-scale sine amounts to a [[Signal-to-noise_ratio|signal-to-noise ratio]] of about `6.02·n + 1.76 dB`; every added bit buys about 6 dB. The product of the two, `n·fs`, is the raw bit rate the stream costs. Linear PCM spaces its levels evenly; [[Companding|companded]] PCM, used in telephony, spaces them logarithmically so that 8-bit words serve quiet and loud speech alike. The idea was patented in the late 1930s, first used in wartime secure telephony, argued for in a 1948 paper by Oliver, Pierce and [[Claude_Shannon|Shannon]], and became the carrier of the telephone network and then of recorded music. The page's primary microsim, *Pulse-code modulation*, lets the reader set the bit depth, the sampling rate, the input frequency and the amplitude of a sine and watch a playhead build the quantized staircase across a 20 ms window while a strip beneath it prints the code word of every sample and a diagnostics panel reports the step size, the bit rate, the ideal and measured signal-to-noise ratios and whether the sampling theorem is being honoured. ## History Alec Reeves conceived the method in 1937 while working for the International Telephone and Telegraph laboratories in Paris and patented it; the United States patent, for an electric signalling system that samples a waveform and transmits each sample as a group of pulses, was granted on February 3, 1942.[^reeves42] The valve circuits of the day made it far too expensive for ordinary telephony, and the first working use was military: the SIGSALY secure speech system of 1943, built at [[Bell_Labs]], quantized the parameters of a speech vocoder, added a key and sent the coded values, the first operational use of the principle.[^sigsaly] After the war Oliver, Pierce and Shannon set out the case for the method in "The Philosophy of PCM", showing that a signal coded as regenerable pulses could be relayed any distance without accumulating noise, at the price of bandwidth,[^opsh48] and the three shared a patent on the system.[^pcmpatent] The transistor and then the [[Integrated_circuit|integrated circuit]] made the coder and decoder cheap enough to put on every channel. ### Digital audio recordings Recording came to PCM through broadcasting. Japanese broadcast and record companies built the first PCM audio recorders on video-tape transports in the late 1960s and released the first commercially recorded PCM album in the early 1970s,[^japan] and stationary-head and video-cassette-based recorders followed through the decade. The consumer format arrived in 1982 with the compact disc, whose 16-bit words at 44.1 kHz per channel fixed the reference for audio PCM,[^cd] a rate at which Downey's examples put 441 samples into a 10 ms segment.[^downey70] Computer audio files and the digital interfaces between audio equipment carry the same linear PCM, at 44.1 or 48 kHz and at 16 or 24 bits. ### Digital telephony Telephony adopted PCM first at scale. The T1 carrier introduced in the Bell System in 1962 multiplexed 24 speech channels, each sampled 8,000 times a second and coded to 8 companded bits, into a 1.544 Mbit/s stream on ordinary twisted pairs with regenerative repeaters,[^t1] and the European E1 carrier did the same with 32 time slots at 2.048 Mbit/s. The two companding laws, μ-law in North America and Japan and A-law elsewhere, were standardised internationally as Recommendation G.711, which fixed the 64 kbit/s channel that digital exchanges, ISDN and much of the internet's telephony still use.[^g711] ## Implementations The method is implemented at both ends by converters. The [[Analog-to-digital_converter|analog-to-digital converter]] performs the sampling, quantization and encoding in one [[Semiconductor_device|semiconductor]] package, and the [[Digital-to-analog_converter|digital-to-analog converter]] decodes and reconstructs; in telephony the pair, with the companding logic and the filters, is the codec chip of a line card. Between the two the words travel in frames: time-division multiplexed slots in the T1 and E1 carriers, blocks with headers in computer files, and serial streams with embedded clocks on the interfaces that link audio equipment. Very large numbers of channels are handled by [[Very-large-scale_integration|very-large-scale integration]], and a single chip now contains dozens of converters and the arithmetic that serves them. ## Modulation Modulation, in the historical vocabulary of the method, is the coding step: sample, quantize, encode. Sampling reads the signal at instants `t = k·Ts` with `Ts = 1/fs`, and the [[Nyquist–Shannon_sampling_theorem]] requires `fs > 2·fmax` for the highest frequency present, or the samples of a high frequency become indistinguishable from those of a lower one.[^dyer48] Quantization snaps each sample to one of `L = 2^n` levels spanning the full-scale range; Dyer and Davis give the resolution as the range divided by `2^n`[^dyer32] and work the example of a 0–5 V range at 3 bits, eight levels 0.625 V apart, in which 2.36 V is coded as 2.5 V, an error of 5.9322 percent, with a worst-case error of half a step, ±0.3125 V.[^dyer33] Encoding writes the level's index in binary. The microsim performs the three steps on a sine of amplitude `A` over a ±1 V range, with a mid-rise quantizer whose step is `Δ = 2V/L`, 0.2500 V at the default 3 bits, and prints each sample's word in the code strip; at the default 8,000 samples per second its 20 ms window holds 161 samples, and the diagnostics panel gives the raw bit rate `n·fs` as 24.0 kbit/s. Raising the bit depth to 8 shrinks the step to 0.0078 V and raises the bit rate to 64.0 kbit/s, the rate of a telephone channel. The only lossy step is quantization. Its error is bounded by half a step[^dyer34] and, for a signal that moves through many levels between samples, behaves like noise of root-mean-square value `Δ/√12`,[^dyer38] about 0.29 of a step,[^stiber59] so that a full-scale sine has a signal-to-quantization-noise ratio of `SQNR = 6.02·n + 1.76 dB`.[^dyer39] The sketch prints this rule of thumb as SQNR (full scale), 19.8 dB at 3 bits and 49.9 dB at 8, beside the ratio it actually measures on its own samples: 21.6 dB and 48.8 dB at the default 0.90 V amplitude, close to the rule because the sine nearly fills the range. ## Demodulation Demodulation reverses the steps. Each word is decoded to its level, a digital-to-analog converter produces the corresponding voltage, and because the voltage is held until the next word the output is a staircase, the [[Zero-order_hold|zero-order hold]]. Dyer and Davis describe the reconstruction as this stair-step followed by a [[Low-pass_filter|low-pass filter]] with its corner at `fs/2`,[^dyer50] which removes the images of the signal that the staircase carries around every multiple of the sampling rate. Tiberius and Mulder give the hold's frequency response as `sinc(Δt·f)` with a delay of half a sample,[^tib119] a droop of about 3.9 dB at the Nyquist frequency that is small for signals well below it; the exact reconstruction of the sampling theorem, the [[Whittaker–Shannon_interpolation_formula]], would need samples from the future and is approximated by the filter.[^tib116] Nothing in demodulation recovers what quantization removed: the decoded level is the centre of the chosen step, and the difference between it and the original sample, the quantization error, is permanent. The sketch draws the staircase as quantized hold and a green reconstruction trace that simply joins the quantized samples with straight lines, an ILLUSTRATIVE stand-in for the low-pass filter, and the ideal reconstruction of a properly sampled sine would coincide with the blue analog trace apart from that permanent error. ## Standard sampling precision and rates The rate is chosen for the bandwidth and the bit depth for the dynamic range, and the product is the cost. | Use | Sampling rate | Bits per sample | Channels | Raw bit rate `n·fs` | |---|---|---|---|---| | Telephone channel (G.711)[^g711] | 8 kHz | 8, companded | 1 | 64 kbit/s | | Compact disc audio[^cd] | 44.1 kHz | 16, linear | 2 | 1,411.2 kbit/s | | The sketch's default | 8 kHz | 3, linear | 1 | 24.0 kbit/s | | The sketch's maximum | 48 kHz | 8, linear | 1 | 384.0 kbit/s | Telephone speech is limited to about 3.4 kHz by the network's filters, so 8,000 samples per second leaves a margin below the 4 kHz Nyquist frequency, and 8 companded bits give the equivalent of 13 or 14 linear bits over the range of speech.[^g711] Music needs the audible band to about 20 kHz, which 44.1 kHz clears with a 2 kHz margin for the anti-aliasing filter, and 16 linear bits give a quantization budget that Stiber, Stiber and Larson put at 107 dB on their span-to-noise convention, against 59 dB at 8 bits.[^stiber60] Professional and film audio use 48 kHz and multiples of it, with 24-bit words that put the quantization floor beneath the noise of the analog electronics. ## Limitations Quantization noise is the first limit, and the bit depth is its only cure in linear PCM. Stiber, Stiber and Larson's budget shows why the depth is chosen for the quietest passage rather than the loudest: at 8 bits a passage 20 dB below full scale has only a few levels to describe it, and the error is no longer small.[^stiber60] The sketch shows the extreme case. With the amplitude slider at 0.10 V and 3 bits the whole sine lies inside two levels, the code strip alternates between two words, and the measured SQNR reads 0.1 dB, the error as large as the signal; at 8 bits the same quiet signal measures 27.1 dB. For a slow or steady signal the error is not noise at all but a deterministic pattern, harmonics of the input,[^stiber59] which is why dither is added before quantization in careful audio work. Companding is the telephone network's answer, spending the 8 bits where speech needs them. Aliasing is the second limit and comes from the other control. Dropping the sampling rate below twice the input frequency does not raise the quantization error, which the sketch's measured SQNR shows staying high, but it changes the frequency the samples describe: the reconstruction takes on a lower, false frequency, and the diagnostics panel's regime row turns red and reads ALIASING. No bit depth repairs that, since it is a sampling failure, and only a lower input frequency, a higher rate or a filter ahead of the sampler removes it. The third limit is range. A sample beyond full scale is clamped to the top or bottom code, which the sketch reports as CLIPPING once the amplitude exceeds 1.00 V; Dyer and Davis warn that clipping from too narrow a range does more harm than wasted resolution.[^dyer36] The fourth is cost: at `n·fs` bits per second per channel, PCM is the most expensive representation of a signal, the price of being the simplest, and the reason [[Data_compression|compression]] exists. ## Processing and coding PCM is the form in which signals are processed. [[Digital_signal_processing|Digital signal processing]] operates on linear PCM samples, [[Digital_filter|digital filters]] take them in and return them, and the [[Discrete_cosine_transform|discrete cosine transform]] at the heart of [[Image_compression|image]] and audio compression takes blocks of them as input. Coding schemes reduce the bit rate by exploiting what PCM ignores. [[Differential_pulse-code_modulation|Differential PCM]] codes the difference between each sample and a prediction from earlier ones, which for a smooth signal is much smaller than the sample; adaptive DPCM adjusts the step to the signal; [[Delta_modulation|delta modulation]] reduces the difference to one bit per sample at a high rate; and [[Delta-sigma_modulation|delta-sigma modulation]] shapes the error of a one-bit stream so that a decimation filter can turn it into wide PCM words. Lossy codecs such as [[MP3]] go further, discarding what a model of hearing says will not be heard, and decode back to PCM for playback. In audio files the uncompressed words are stored as-is in a container that records the rate, the depth, the channel count and the byte order, and PCM in that context means linear PCM. ## Encoding for serial transmission A stream of words must be sent as a stream of symbols on a wire, a fibre or a radio channel, and the mapping is a line code chosen for the medium. On a [[Transmission_line|transmission line]] the bits are sent as voltage levels; a non-return-to-zero code holds each level for a bit period, a Manchester code puts a transition in every bit so that the receiver can recover the clock, and alternate-mark-inversion codes, used in the T1 and E1 carriers, send ones as pulses of alternating polarity so that the line carries no direct current and a violation of the alternation can flag an error. Framing bits mark the word boundaries, and the receiver's clock is locked to the transitions by a [[Phase-locked_loop|phase-locked loop]]. Over a radio channel the bits [[Signal_modulation|modulate]] a [[Carrier_wave|carrier]] instead, by [[Amplitude-shift_keying|amplitude-shift]], [[Frequency-shift_keying|frequency-shift]] or [[Phase-shift_keying|phase-shift keying]], and [[Error_detection_and_correction|error-detecting and correcting]] codes are added to the words. The virtue Oliver, Pierce and Shannon identified is that every repeater regenerates the pulses rather than amplifying them, so that the words arrive as they left provided each hop stays below its error threshold.[^opsh48] ## Nomenclature The word modulation is a survival from the pulse-modulation family of the 1930s and 1940s, in which a train of pulses was varied in amplitude, width or position to carry a signal; PCM was the member that varied the pulses' code, and the name stayed after the pulses became bits in a memory. It does not modulate a carrier, and the bits may later be sent by any [[Signal_modulation|modulation]] at all. Linear PCM, or LPCM, names the form with equally spaced levels, the one audio files and consumer interfaces carry, and PCM without qualification usually means it; the companded forms are named by their laws, μ-law and A-law, and the differential forms carry their own prefixes. ## Microsims The primary microsim, *Pulse-code modulation*, runs the whole pipeline across a 20 ms window and animates it. The scope draws the analog x(t) sine in blue, the samples as dots, the quantized hold as an orange staircase that a sweeping playhead builds from left to right, and the green reconstruction that joins the quantized samples; beneath it the PCM code words strip prints the level index and the `n`-bit word of every sample. Four sliders set bit depth n (1–8, default 3), sampling fs (2,000–48,000 Hz in steps of 500, default 8,000), input fm (50–4,000 Hz, default 500) and amplitude A (0.10–1.10 V, default 0.90), and a pause button freezes a frame. The DIAGNOSTICS panel lists n, the levels `L = 2^n`, the step `Δ = 2V/L`, the bit rate `n·fs`, the SQNR (full scale) rule of thumb, the SQNR measured on the window's samples, the Nyquist requirement `2·fm`, the regime verdict, the amplitude and a status line that reads CLIPPING above full scale, and it closes with a one-line verdict, from "Undersampled" through "Coarse" to "Clean PCM". The model is ILLUSTRATIVE: the quantizer is ideal, no dither is applied and the reconstruction trace is linear interpolation rather than a low-pass filter. A three.js companion renders this concept in depth. *Try:* Start at the defaults and step bit depth n from 3 up to 8, watching the step Δ fall from 0.2500 V to 0.0078 V, the bit rate rise from 24.0 to 64.0 kbit/s and the measured SQNR climb from 21.6 dB to 48.8 dB; then drag sampling fs down to 2,000 Hz and input fm up to 1,500 Hz, and watch the regime row turn red with ALIASING while the measured SQNR stays high, the proof that the false frequency is a sampling failure and not a quantization one. ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Pulse-code_modulation) : [Wikitube](https://en.wikitube.io/wiki/Pulse-code_modulation) Skeleton mirrored at revision 1373462200. Prose, emphasis and the microsims are Wikitube's own. ## See also - [[Data_compression]] - [[Codec]] - [[Quantization_(signal_processing)]] - [[Analog-to-digital_converter]] - [[Digital-to-analog_converter]] - [[Sampling_(signal_processing)]] - [[Nyquist–Shannon_sampling_theorem]] - [[Companding]] - [[Differential_pulse-code_modulation]] - [[Delta-sigma_modulation]] - [[PORTAL_Signal_Processing]] ## Explanatory notes The microsim is ILLUSTRATIVE: its quantizer is an ideal mid-rise quantizer over ±1 V with no dither, its measured SQNR is taken over the 161 samples of one 20 ms window at the default rate, and its reconstruction trace is linear interpolation between quantized samples rather than the low-pass filter of a real decoder. Page numbers in the references are PDF pages of the open editions. ## References [^reeves42]: Reeves, A. H. "Electric signaling system". U.S. Patent 2,272,070, granted February 3, 1942. https://patents.google.com/patent/US2272070 [^sigsaly]: Citation needed: SIGSALY (1943) as the first operational use of pulse-code modulation, in secure speech transmission; Bennett's 1983 paper on secret telephony as a historical example of spread-spectrum communication, in the *IEEE Transactions on Communications*, would settle the date and the coding used. [^opsh48]: Oliver, B. M.; Pierce, J. R.; Shannon, C. E. "The Philosophy of PCM". *Proceedings of the IRE*, vol. 36, no. 11, 1948, pp. 1324-1331. [^pcmpatent]: Citation needed: the number and grant date of the Oliver, Pierce and Shannon patent on a communication system employing pulse-code modulation; the patent record itself would settle it. [^japan]: Citation needed: the first PCM audio recorders built by Japanese broadcasters in the late 1960s and the first commercially recorded PCM album of the early 1970s; a dated primary account from the broadcaster or the record company would settle the years. [^cd]: Citation needed: the 16-bit, 44.1 kHz, two-channel format of compact disc audio and its 1982 introduction are fixed by the Compact Disc Digital Audio standard (IEC 60908); a page reference to that standard would settle the figures. [^downey70]: Downey, A. *Think DSP: Digital Signal Processing in Python*. 2012, p. 70 (PDF page): a 0.01 s segment at 44.1 kHz holds 441 samples. Open Textbook Library: https://open.umn.edu/opentextbooks/textbooks/think-dsp-digital-signal-processing-in-python . CC BY-NC. [^t1]: Citation needed: the 1962 introduction of the Bell System T1 carrier with 24 channels of 8-bit, 8 kHz companded PCM at 1.544 Mbit/s; the 1965 *Bell System Technical Journal* description of the T1 carrier system would settle the date and the figures. [^g711]: International Telecommunication Union. *Recommendation G.711: Pulse code modulation (PCM) of voice frequencies*. ITU-T, 1988 edition (first approved 1972): 8 kHz sampling, 8-bit μ-law and A-law companding at 64 kbit/s, with the 13-bit (A-law) and 14-bit (μ-law) uniform PCM equivalents. https://www.itu.int/rec/T-REC-G.711 [^dyer32]: Dyer, J.; Davis, C. *Measurement and Instrumentation: An Introduction to Concepts and Methods*. 1st ed., 2020, p. 32 (PDF page): 2^n codes and resolution = range/2^n. Open Textbook Library: https://open.umn.edu/opentextbooks/textbooks/measurement-and-instrumentation-an-introduction-to-concepts-and-methods . CC BY-NC-SA. [^dyer33]: Dyer, J.; Davis, C. *Measurement and Instrumentation: An Introduction to Concepts and Methods*. 1st ed., 2020, pp. 33-34 (PDF pages): Example 2.2, a 0–5 V range at 3 bits, 0.625 V per step, 2.36 V read as 2.5 V, a 5.9322 percent error, ±0.3125 V maximum error. Open Textbook Library: https://open.umn.edu/opentextbooks/textbooks/measurement-and-instrumentation-an-introduction-to-concepts-and-methods . CC BY-NC-SA. [^dyer34]: Dyer, J.; Davis, C. *Measurement and Instrumentation: An Introduction to Concepts and Methods*. 1st ed., 2020, p. 34 (PDF page): the quantization error is bounded by ±0.5 of the resolution. Open Textbook Library: https://open.umn.edu/opentextbooks/textbooks/measurement-and-instrumentation-an-introduction-to-concepts-and-methods . CC BY-NC-SA. [^dyer36]: Dyer, J.; Davis, C. *Measurement and Instrumentation: An Introduction to Concepts and Methods*. 1st ed., 2020, p. 36 (PDF page): clipping from too narrow a range is worse than wasted resolution. Open Textbook Library: https://open.umn.edu/opentextbooks/textbooks/measurement-and-instrumentation-an-introduction-to-concepts-and-methods . CC BY-NC-SA. [^dyer38]: Dyer, J.; Davis, C. *Measurement and Instrumentation: An Introduction to Concepts and Methods*. 1st ed., 2020, pp. 38-39 (PDF pages): the rms quantization noise Q/√12. Open Textbook Library: https://open.umn.edu/opentextbooks/textbooks/measurement-and-instrumentation-an-introduction-to-concepts-and-methods . CC BY-NC-SA. [^dyer39]: Dyer, J.; Davis, C. *Measurement and Instrumentation: An Introduction to Concepts and Methods*. 1st ed., 2020, pp. 39-40 (PDF pages): SNR = 6.02 N + 1.76 dB for a full-scale sine. Open Textbook Library: https://open.umn.edu/opentextbooks/textbooks/measurement-and-instrumentation-an-introduction-to-concepts-and-methods . CC BY-NC-SA. [^dyer48]: Dyer, J.; Davis, C. *Measurement and Instrumentation: An Introduction to Concepts and Methods*. 1st ed., 2020, pp. 48-49 (PDF pages): the sampling condition fs > 2 fmax. Open Textbook Library: https://open.umn.edu/opentextbooks/textbooks/measurement-and-instrumentation-an-introduction-to-concepts-and-methods . CC BY-NC-SA. [^dyer50]: Dyer, J.; Davis, C. *Measurement and Instrumentation: An Introduction to Concepts and Methods*. 1st ed., 2020, pp. 50-51 (PDF pages): a DAC output is a stair-step followed by a low-pass filter at fs/2. Open Textbook Library: https://open.umn.edu/opentextbooks/textbooks/measurement-and-instrumentation-an-introduction-to-concepts-and-methods . CC BY-NC-SA. [^stiber59]: Stiber, M.; Stiber, B.; Larson, E. *Signal Computing: Digital Signals in the Software Domain*. 2020, p. 59 (PDF page): σ = 1/√12 LSB ≈ 0.29 LSB, and the deterministic error of a slow or constant input. Open Textbook Library: https://open.umn.edu/opentextbooks/textbooks/signal-computing-digital-signals-in-the-software-domain . CC BY-SA. [^stiber60]: Stiber, M.; Stiber, B.; Larson, E. *Signal Computing: Digital Signals in the Software Domain*. 2020, pp. 60-61 (PDF pages): quantization budgets of 59, 83 and 107 dB at 8, 12 and 16 bits on the span/σ convention, and the quiet-passage budget. Open Textbook Library: https://open.umn.edu/opentextbooks/textbooks/signal-computing-digital-signals-in-the-software-domain . CC BY-SA. [^tib116]: Tiberius, C.; Mulder, M. *Engineering Signal Analysis: From Fourier to filtering - Theory*. 2026, pp. 116-119 (PDF pages): ideal sinc reconstruction and its non-causal impulse response. Open Textbook Library: https://open.umn.edu/opentextbooks/textbooks/engineering-signal-analysis-from-fourier-to-filtering-theory . CC BY. [^tib119]: Tiberius, C.; Mulder, M. *Engineering Signal Analysis: From Fourier to filtering - Theory*. 2026, pp. 119-122 (PDF pages): the zero-order hold, H_r(f) = sinc(Δt·f)·exp(−j2πfΔt/2). Open Textbook Library: https://open.umn.edu/opentextbooks/textbooks/engineering-signal-analysis-from-fourier-to-filtering-theory . CC BY. ## Further reading - Dyer, J.; Davis, C. *Measurement and Instrumentation: An Introduction to Concepts and Methods*, 1st ed. (2020). Open Textbook Library: https://open.umn.edu/opentextbooks/textbooks/measurement-and-instrumentation-an-introduction-to-concepts-and-methods - Stiber, M.; Stiber, B.; Larson, E. *Signal Computing: Digital Signals in the Software Domain* (2020). Open Textbook Library: https://open.umn.edu/opentextbooks/textbooks/signal-computing-digital-signals-in-the-software-domain - Tiberius, C.; Mulder, M. *Engineering Signal Analysis: From Fourier to filtering: Theory* (2026). Open Textbook Library: https://open.umn.edu/opentextbooks/textbooks/engineering-signal-analysis-from-fourier-to-filtering-theory - Downey, A. *Think DSP: Digital Signal Processing in Python* (2012). Open Textbook Library: https://open.umn.edu/opentextbooks/textbooks/think-dsp-digital-signal-processing-in-python ## External links - *Pulse-code modulation*, the primary sketch: https://editor.p5js.org/sciencenibber/full/2YrkkPYya (fork: https://editor.p5js.org/sciencenibber/sketches/2YrkkPYya)