Phase-shift_keying - Wikiparency

# Phase-shift keying **Phase-shift keying** is a digital modulation method that carries data by switching the phase of a fixed-frequency, fixed-amplitude [[Carrier_wave|carrier wave]] among a small set of allowed angles, rather than by varying its amplitude or its frequency. Each allowed phase is a symbol, and every symbol stands for one or more bits, so demodulation is a matter of measuring the received phase and deciding which allowed angle it lies closest to. The microsim on this page lets the reader raise the number of phases from two to sixteen and add channel noise, watching the [[Constellation_diagram|constellation]] points crowd together and start crossing into the wrong decision region. Phase-shift keying is one of three classical ways of keying a sinusoidal carrier, alongside [[Amplitude-shift_keying|amplitude-shift keying]] and [[Frequency-shift_keying|frequency-shift keying]], and it is the one used most widely because its symbols hold a constant envelope while still carrying several bits each. The simplest form, binary phase-shift keying, uses two opposite phases and one bit per symbol; quadrature phase-shift keying doubles that to two bits by using four phases, and higher orders trade a shrinking margin between symbols for still more bits per symbol. A separate branch, differential phase-shift keying, encodes each bit in the *change* of phase from one symbol to the next so that the receiver never needs a phase reference of its own. As a member of the wider family of [[Signal_modulation|digitally modulated]] carriers, phase-shift keying needs a receiver that can either track the carrier's phase directly, called coherent detection, or sidestep that tracking problem by comparing successive symbols, as differential detection does. Which family a system chooses shapes everything from receiver cost to the [[Signal-to-noise_ratio|signal-to-noise ratio]] it needs to hold a target error rate, and both threads run through the sections below. ## Introduction PSK signals share one fixed carrier frequency and amplitude, so every symbol sits on the same circle in the in-phase/quadrature plane and only its angular position changes; expanding the carrier into in-phase and quadrature components turns each phase into a pair of numbers that a receiver can recover by multiplying the incoming wave against two local references 90 degrees apart. A modulator with `M` equally spaced phases carries `log2 M` [[Binary_number|bits]] in every symbol, and assigning adjacent phases to bit patterns that differ in only one bit — Gray coding — keeps a single noise-induced slip to the nearest neighbour from flipping more than one bit at a time.[^ell134] Recovering the data requires the receiver to know where "zero phase" is, which is either tracked continuously from the incoming carrier, called coherent detection, or is replaced entirely by comparing one symbol's phase against the last — differential detection, treated in its own section below. The cost of adding more phases is geometric: on a circle of fixed radius, `M` evenly spaced points are separated by an angle of `2*pi/M`, so each doubling of `M` roughly halves the gap between neighbouring symbols and, with it, the margin against [[Noise_(electronics)|noise]]. ## Binary phase-shift keying (BPSK) Binary phase-shift keying (BPSK) is the simplest case, with `M = 2`: the carrier's phase sits at 0 degrees for one bit value and 180 degrees for the other, so the two symbols are opposite points on the constellation circle and every symbol carries exactly one bit. A common way to build the signal is to multiply the carrier by +1 or −1 according to the bit stream, which is exactly a phase flip of 180 degrees; demodulating is the reverse multiplication against a locally generated copy of the carrier, recovered by a [[Phase-locked_loop|phase-locked loop]] or an equivalent tracking circuit locked to the incoming signal rather than sent as a separate pilot tone. Because the two symbols sit as far apart as two points on a circle can be, BPSK tolerates more noise for a given symbol energy than any other PSK order, and its bit-error probability under additive white Gaussian noise reduces to the closed form `P_b = Q(sqrt(2*Eb/N0))`, where `Eb/N0` is the energy per bit divided by the noise power spectral density and `Q` is the Gaussian tail function.[^ell145] That formula is the reference curve the microsim compares every other order against: raising the order packs more bits into each symbol but always costs some of this margin back. ## Quadrature phase-shift keying (QPSK) Quadrature phase-shift keying (QPSK) raises the order to `M = 4`, spacing four phases 90 degrees apart so that every symbol carries two bits. The usual way to generate it is to run two independent binary streams through two [[Carrier_wave|carriers]] a quarter-cycle apart — one on the in-phase axis, one on the quadrature axis — and add them; decoding is the same split back into I and Q components followed by two independent one-bit decisions. With Gray-coded bit pairs, QPSK reaches the same bit-error probability as BPSK, `P_b = Q(sqrt(2*Eb/N0))`, while moving twice the data through the same symbol rate, which is why it is the workhorse of digital radio links built to fit a fixed channel bandwidth.[^ell145] The price is that a symbol error, when it happens, more often flips only one of the two bits, and the constellation is more crowded than BPSK's, so a timing or carrier-phase error costs more. Variants trade some of that theoretical efficiency for a gentler waveform: offset QPSK delays the quadrature stream by half a symbol so the two components never switch at the same instant, keeping the envelope from passing through zero and easing the linearity demanded of a power amplifier, and π/4-QPSK rotates the constellation by 45 degrees between symbols for a similar effect with a smaller peak-to-average envelope swing. ## Higher-order PSK Beyond four phases, M-ary PSK spaces `M` points evenly around the same circle: eight phases for 8-PSK, sixteen for 16-PSK, and so on, with each symbol carrying `log2 M` bits. Because the points share one circle of fixed radius, the distance between neighbours falls with `sin(pi/M)`, so doubling the order does not double the error rate gently — it shrinks the safety margin every symbol has to fall back on, and the usual approximation for the symbol-error probability captures that by scaling the noise margin's Gaussian-tail probability by the same sine factor.[^ell153] Reported bit-error curves across the common PSK, ASK and QAM families need anywhere from about 5 dB to 16 dB of Eb/N0 to hold a bit-error rate of one in a thousand, with the higher-order, more crowded constellations sitting at the expensive end;[^ell161] uncoded QPSK itself needs roughly 11 dB to reach one error in a million,[^ell163] which is the kind of figure system designers weigh against how many bits per symbol they actually need. An 8-PSK link moving fifty percent more data per symbol than QPSK must find several more decibels of [[Signal-to-noise_ratio|signal-to-noise ratio]] somewhere to keep the same reliability, which is the central trade the microsim's order control is built to show. ## Differential phase-shift keying (DPSK) Differential phase-shift keying (DPSK) puts the data in the *change* of phase from one symbol to the next instead of in the absolute phase, so the bit stream is first turned into a sequence of phase steps and then modulated exactly as ordinary PSK would be. The gain is at the receiver: instead of maintaining a coherent phase reference recovered from the carrier itself, a differential receiver only has to compare each received symbol against the one immediately before it, which can be done with a fixed delay line and a multiplier rather than a [[Phase-locked_loop|phase-locked loop]]. Differentially encoded and non-coherently detected BPSK trades some performance for that simplicity: its bit-error probability is `P_b = (1/2)*exp(-Eb/N0)`, roughly 1.5 dB worse than coherent BPSK at a one-percent bit-error rate,[^ell155] a gap that is often worth paying in a receiver that cannot afford the extra circuitry and lock time a coherent design needs, or in a channel whose phase drifts too quickly for a carrier-recovery loop to track. ## Applications Because it keeps a constant envelope while still carrying several bits per symbol, phase-shift keying turns up wherever a link has to trade throughput against how much [[Signal-to-noise_ratio|signal-to-noise ratio]] it can count on. Early Wi-Fi hardware falls back to differentially encoded PSK at its lowest, most reliable data rates, using DBPSK for 1 megabit per second and DQPSK for 2; Bluetooth's higher-throughput Enhanced Data Rate mode leaves plain frequency-shift keying behind for π/4-DQPSK and 8-DPSK to roughly double and triple its basic rate over the same channel. Satellite broadcasting standards offer QPSK and 8-PSK as selectable modes, so a broadcaster can dial in more bits per symbol on a clear day and fall back to a more robust, lower-order constellation when rain or a marginal dish eats into the link's margin. Deep-space and other very-low-power telemetry links sit at the opposite end of that same trade-off, using binary or quadrature PSK not for their bit rate but for the widest possible symbol-error margin per bit of received energy. ## Mutual information with additive white Gaussian noise Bit-error rate says how often a symbol is decoded wrong; mutual information asks a related but different question — how many bits, on average, does the channel actually let a receiver learn about the transmitted symbol at a given noise level? For an M-ary constellation corrupted by [[Noise_(electronics)|Gaussian noise]], mutual information rises from close to zero bits per symbol at very poor signal-to-noise ratio toward a ceiling of `log2(M)` bits per symbol as the noise falls, because no constellation can reveal more than the number of bits it was built to carry, however quiet the channel becomes. That ceiling is a specific case of the channel-capacity framework [[Claude_Shannon|Claude Shannon]] set out in 1948 for a bandwidth-limited channel carrying a signal corrupted by Gaussian noise,[^shannon48] and it explains why pushing a constellation's order past what the channel can support wastes symbols rather than delivering the extra bits it promises on paper. The gap between an uncoded scheme's bit-error curve and this information-theoretic ceiling is exactly the margin an [[Error_correction_code|error-correcting code]] is built to close, which is why modern high-order PSK links are essentially never sent without one. ## Microsims The primary microsim draws a live time-domain scope of the modulated carrier above a [[Constellation_diagram|constellation]] plot of the same symbols in the in-phase/quadrature plane. **ORDER** steps the constellation through two, four, eight and sixteen phases; **NOISE** sets the standard deviation of Gaussian noise added independently to each received sample's in-phase and quadrature components; **SYMBOLS/SCREEN** sets how many symbols the time scope displays at once. A "randomize symbols" button refills the bit pattern, and an "animate" toggle re-rolls the noise every frame so the cloud of received points visibly shimmers instead of sitting still. Samples that cross a decision boundary into the wrong wedge are marked in a different colour, and a diagnostics readout compares the noise's typical three-sigma spread against the gap between neighbouring constellation points. A three.js companion renders the same order-versus-noise trade-off for QPSK and 8-PSK side by side in more depth. *Try:* Set ORDER to sixteen phases, then raise NOISE until the diagnostics readout shows the noise's three-sigma reach has grown past the gap between neighbouring points, and watch symbols start flipping to the error colour. ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Phase-shift_keying) : [Wikitube](https://en.wikitube.io/wiki/Phase-shift_keying) Skeleton mirrored at revision 1374164261. Prose, emphasis and the microsims are Wikitube's own. ## See also - [[Constellation_diagram]] - [[Signal_modulation]] - [[Amplitude-shift_keying]] - [[Frequency-shift_keying]] - [[Phase-locked_loop]] - [[Error_correction_code]] - [[Signal-to-noise_ratio]] ## Notes The microsim's noise model adds independent Gaussian noise to each received sample's in-phase and quadrature components and assumes perfect symbol timing and carrier recovery; it does not model burst errors, fading, or a phase-tracking loop that has not yet locked, all of which raise the effective error rate above the closed-form curves quoted above. Page numbers in the footnotes are the PDF page numbers of the cited open editions. ## References [^ell134]: Ellingson, S. *Radio Systems Engineering – Revised First Edition*. 2023, p. 134 (PDF pages). Open Textbook Library: https://open.umn.edu/opentextbooks/textbooks/radio-systems-engineering . CC BY-NC. [^ell145]: Ellingson, S. *Radio Systems Engineering – Revised First Edition*. 2023, pp. 145–151 (PDF pages). Open Textbook Library: https://open.umn.edu/opentextbooks/textbooks/radio-systems-engineering . CC BY-NC. [^ell153]: Ellingson, S. *Radio Systems Engineering – Revised First Edition*. 2023, pp. 153–154 (PDF pages). Open Textbook Library: https://open.umn.edu/opentextbooks/textbooks/radio-systems-engineering . CC BY-NC. [^ell155]: Ellingson, S. *Radio Systems Engineering – Revised First Edition*. 2023, pp. 155–156 (PDF pages). Open Textbook Library: https://open.umn.edu/opentextbooks/textbooks/radio-systems-engineering . CC BY-NC. [^ell161]: Ellingson, S. *Radio Systems Engineering – Revised First Edition*. 2023, pp. 161–162 (PDF pages). Open Textbook Library: https://open.umn.edu/opentextbooks/textbooks/radio-systems-engineering . CC BY-NC. [^ell163]: Ellingson, S. *Radio Systems Engineering – Revised First Edition*. 2023, p. 163 (PDF pages). Open Textbook Library: https://open.umn.edu/opentextbooks/textbooks/radio-systems-engineering . CC BY-NC. [^shannon48]: Shannon, C. E. "A Mathematical Theory of Communication." *Bell System Technical Journal* 27 (1948): pp. 379–423, 623–656.