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Quadrature phase-shift keying (QPSK) encodes two bits in every symbol by selecting one of four carrier phases separated by 90 degrees. It offers twice the bits per symbol of BPSK while retaining the same ideal coherent-AWGN bit-error probability when both use Gray coding and are compared at the same Eb/N0. That does not mean QPSK always delivers twice the real-world data rate: coding, pulse shaping, pilots, synchronization overhead, channel conditions, and implementation losses also matter.
This guide explains how QPSK maps bits to constellation points, how its I/Q transmitter and receiver work, why phase ambiguity matters, how QPSK compares with BPSK, and when variants such as OQPSK, DQPSK, and π/4-QPSK are preferable.
What is QPSK?
QPSK is a four-state phase-shift-keying modulation scheme. A modulator groups the input stream into pairs of bits. Each pair selects one of four possible phases of a carrier, normally spaced 90 degrees apart.
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For example, with a 45-degree phase offset, the four constellation phases are 45°, 135°, 225°, and 315°. The carrier amplitude is ideally constant at the symbol decision points; the information is represented by phase.
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Because there are four possible symbols, each symbol represents:
log2(4) = 2 bits
Therefore, for an uncoded bitstream:
Rs = Rb / 2
where Rs is the symbol rate and Rb is the bit rate. A 1-Mbit/s QPSK stream therefore has a 500-ksymbol/s rate before accounting for coding and protocol overhead.
QPSK is often described as two orthogonal BPSK signals transmitted at the same time: one on the in-phase (I) carrier and one on the quadrature (Q) carrier. The carriers are 90 degrees apart, so their ideal components do not interfere with one another.
For an introduction to complex I/Q notation, see the GNU Radio IQ Complex Tutorial.
QPSK constellation and Gray-coded mapping
A QPSK constellation is a plot of the complex baseband symbols. The horizontal axis is the I component and the vertical axis is the Q component. A common normalized representation uses the four points:
(+1, +1)(-1, +1)(-1, -1)(+1, -1)
The corresponding complex symbols are written as I + jQ. The point (+1,+1), for example, has a positive I component and a positive Q component.
| Bits | I sign | Q sign | Example phase with 45° offset | Complex point |
|---|---|---|---|---|
| 00 | + | + | 45° | +1 + j |
| 01 | – | + | 135° | -1 + j |
| 11 | – | – | 225° | -1 – j |
| 10 | + | – | 315° | +1 – j |
This is one common Gray-coded mapping. Moving to an adjacent point changes only one bit: 00 becomes 01, 01 becomes 11, and so on. Gray coding does not alter the physical constellation; it changes the labels assigned to the points. Its advantage is that the most likely nearest-neighbor symbol error usually becomes only one bit error.
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1Scan for outdated or missing drivers - takes under a minute2Clear out junk files and repair common Windows errors3Fix the driver behind crashes, sound loss and screen glitchesThe mapping is not universal. An implementation specification must define the phase offset, the order in which points are enumerated, which bit controls I or Q, whether the mapping is absolute or differential, and the receiver’s bit ordering. For example, DVB-related specifications explicitly define their QPSK mapping conventions; a receiver using a different convention can produce apparently valid constellation decisions but incorrect recovered data. See the ETSI QPSK specification material.
How QPSK is represented mathematically
A passband QPSK waveform can be expressed as:
sm(t) = √(2Es/Ts) cos(2πfct + 2πm/4 + φ)
where:
m ∈ {0, 1, 2, 3}selects one of the four phase states;Esis the energy per symbol;Tsis the symbol duration;fcis the carrier frequency; andφis an initial phase offset, such as the 45° offset used in the example table.
In complex baseband notation, a symbol is proportional to:
xm = exp(j(2πm/4 + φ))
The baseband representation is useful because it separates the two orthogonal signal dimensions into I and Q. A real RF transmitter later combines those components with cosine and sine carriers.
Why QPSK carries twice the bits per symbol of BPSK
BPSK has two possible phase states and therefore carries one bit per symbol. QPSK has four states and carries two bits per symbol. If the two systems use the same symbol pulse shape and occupy comparable bandwidth, QPSK can carry twice the uncoded bit rate at the same symbol rate.
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That is a statement about modulation efficiency, not a guarantee about application throughput. Net throughput may be reduced by:
- forward-error-correction parity;
- pilot and synchronization symbols;
- preamble and frame headers;
- guard intervals and cyclic prefixes;
- scrambling and framing overhead;
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The relevant rate relationship is simply:
Rb = 2Rs
for uncoded QPSK payload bits. A coded system’s net information rate is lower and depends on its code rate and protocol structure.
QPSK transmitter chain
A practical QPSK transmitter normally performs these operations:
- Prepare the data. The system frames, scrambles, and channel-codes the source bits as required.
- Group bits into pairs. The coded stream is divided into two-bit words.
- Map each pair. Each word is converted to one complex constellation symbol using the specified mapping.
- Split into I and Q. The real part of each symbol drives the I branch and the imaginary part drives the Q branch.
- Apply pulse shaping. A raised-cosine or root-raised-cosine filter controls bandwidth and limits intersymbol interference.
- Combine the branches. The I and Q waveforms modulate orthogonal cosine and sine carriers, or are generated as a complex digital baseband signal.
- Convert and transmit. Digital-to-analog conversion, frequency conversion, amplification, and the antenna or optical transmitter produce the transmitted waveform.
A common design uses a root-raised-cosine filter at the transmitter and a matched root-raised-cosine filter at the receiver. Together, the pair approximates a raised-cosine response at the symbol-sampling point.
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The roll-off factor α controls excess bandwidth. A lower α generally produces a narrower spectrum, but it makes filter design and symbol-timing recovery more demanding. The GNU Radio PSK demodulation tutorial demonstrates constellation mapping, root-raised-cosine filtering, and channel modeling.
QPSK receiver and demodulation
A QPSK receiver is not just a four-point lookup table. Before it can decide which point was sent, it must establish where the symbols occur and how the received constellation is rotated and scaled.
A representative coherent receiver contains the following stages:
- RF downconversion and sampling: the received signal is converted to a suitable intermediate frequency or complex baseband and digitized.
- Matched filtering: the receiver applies a filter matched to the transmit pulse shape. This improves the signal-to-noise ratio at the sampling instant and reduces out-of-band noise.
- Symbol-timing recovery: a timing loop finds the correct sampling phase and symbol interval, then reduces the samples to approximately one sample per symbol.
- Frequency-offset correction: the receiver estimates and removes the difference between transmitter and receiver oscillator frequencies.
- Carrier-phase recovery: a Costas loop or another digital phase estimator rotates the constellation into the correct orientation.
- Constellation decisions: the receiver selects the nearest valid QPSK point.
- Bit de-mapping: the selected point is converted back to the correct two-bit label, including the agreed bit order.
- Protocol recovery: differential decoding, descrambling, frame synchronization, and channel decoding are applied when the system requires them.
The GNU Radio tutorial illustrates matched filtering, symbol synchronization, a fourth-order Costas loop, constellation decoding, and bit-order handling in a working QPSK flowgraph.
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QPSK has fourfold rotational symmetry. A carrier-recovery loop may lock to the correct phase, or to a phase that is rotated by 90°, 180°, or 270°. The resulting constellation can look clean even though every recovered label is rotated relative to the transmitter’s mapping.
Systems resolve this ambiguity using one or more of the following:
- known preambles or synchronization words;
- pilot symbols;
- differential encoding and decoding;
- testing the recovered data against a frame or code structure; or
- an explicit ambiguity-resolution algorithm.
This issue is documented in NASA’s Digital Communications Design for Deep Space Missions, which discusses phase relationships and ambiguity in QPSK and OQPSK receivers.
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QPSK BER compared with BPSK
For Gray-coded, coherently detected QPSK in an additive white Gaussian noise (AWGN) channel, the ideal bit-error probability is:
Pb = Q(√(2Eb/N0))
This is the same ideal BER expression as coherent BPSK. The reason is that QPSK’s I and Q decisions behave like two independent orthogonal BPSK decisions. QPSK puts two bits into each symbol, but the energy is also divided between the two dimensions; when performance is normalized by energy per bit, the ideal BER is the same.
Consequently:
| Comparison | BPSK | Gray-coded coherent QPSK |
|---|---|---|
| Constellation points | 2 | 4 |
| Bits per symbol | 1 | 2 |
| Ideal BER in AWGN | Q(√(2Eb/N0)) |
Q(√(2Eb/N0)) |
| Symbol rate for the same uncoded bit rate | Rb | Rb/2 |
This result assumes coherent detection, Gray mapping, AWGN, and a consistent Eb/N0 definition. Real links can perform worse because of phase noise, frequency offset, timing errors, multipath, amplifier nonlinearity, I/Q imbalance, imperfect filters, coding losses, and synchronization overhead. A measured BER should therefore not be compared with the ideal curve without documenting the test conditions. The ideal relationship is described in the MathWorks QPSK modulator documentation.
Symbol errors and bit errors are not the same
A symbol error means the receiver selected the wrong one of four constellation points. A bit error means one of the two bits represented by that symbol is wrong.
One incorrect QPSK symbol can cause one or two bit errors. Gray coding makes adjacent-point errors produce one bit error, while a jump to the opposite point changes both bits. In a noisy channel, adjacent decisions are generally more likely than opposite-point decisions, which is why Gray coding improves the BER-to-symbol-error relationship.
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Pulse shaping, bandwidth, and amplifier behavior
Unfiltered rectangular symbol pulses have broad spectral sidelobes. Pulse-shaping filters reduce those sidelobes and control the occupied bandwidth. Raised-cosine filtering is specified by a roll-off factor α, which determines the excess bandwidth above the ideal Nyquist bandwidth.
In simplified form, reducing α:
- narrows the occupied spectrum;
- requires sharper filter characteristics;
- increases sensitivity to timing errors; and
- can make practical synchronization more difficult.
QPSK is sometimes described as a constant-envelope modulation because its ideal symbol states have equal magnitude. That description needs care. With practical pulse shaping, the waveform between symbol decisions can have envelope variation. Conventional QPSK can also make large phase transitions, including 180-degree changes, and some transitions can pass through or near the origin. These excursions can stress a nonlinear power amplifier.
Offset QPSK (OQPSK) delays one branch relative to the other so that I and Q do not change at the same instant. The largest phase transitions are thereby limited, reducing envelope excursions and making the signal friendlier to some power amplifiers. NASA’s technical discussion of offset QPSK covers these spectral and amplifier considerations.
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QPSK variants
Differential QPSK (DQPSK)
In ordinary absolute QPSK, the receiver determines each symbol’s phase relative to a recovered carrier reference. In differential QPSK, the information is encoded in the phase change from one symbol to the next.
Differential operation can reduce dependence on an absolute carrier-phase reference and can help address the M-fold phase ambiguity of M-PSK. The trade-off is that differential detection generally has a different error-performance curve and can lose performance compared with ideal coherent detection. Transmitter and receiver must also agree on the initial reference and differential rule.
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Offset QPSK (OQPSK)
OQPSK uses the same basic I/Q idea as QPSK but offsets one branch by half a symbol period, or otherwise staggers branch transitions according to the system design. Because only one branch changes at a time, the phase does not make the largest instantaneous jumps of conventional QPSK.
OQPSK is useful when amplifier efficiency and controlled envelope behavior are important. It is not interchangeable with ordinary QPSK: the receiver must know the branch timing, filtering, mapping, and synchronization conventions.
π/4-QPSK and π/4-DQPSK
π/4-shifted schemes use alternating constellation sets or differential phase transitions. Their design can avoid particular transitions and support practical carrier-recovery behavior. The details differ among implementations, so “π/4-QPSK” does not by itself specify whether the system uses coherent or differential detection, which mapping is used, or how pulse shaping is implemented.
For a technical overview of the distinctions among these phase-shifted formats, see the Wiley reference entry on π/4-QPSK.
Dual-polarization QPSK
In coherent optical communications, QPSK can be sent independently over two orthogonal polarization states. This is called dual-polarization QPSK. It is a system-level extension, not simply a different four-point single-polarization constellation.
Optical receivers typically use coherent detection and digital signal processing for polarization separation, equalization, carrier recovery, and decoding. The two polarizations provide two parallel signal dimensions, subject to the optical channel and receiver’s ability to separate them. Research in the Journal of Lightwave Technology describes coherent optical QPSK detection and digital carrier-phase estimation.
Where QPSK is used
Satellite and space communications
QPSK is relevant to satellite links, spacecraft communications, and receiver research because it provides more bits per symbol than BPSK without the higher constellation density of higher-order QAM. DVB-related specifications include Gray-coded QPSK options, and DVB-S2 guidance uses QPSK in robust operating conditions alongside forward-error-correction choices.
NASA technical records document QPSK in satellite communications, spacecraft links, optical-modem technology, and receiver studies. That evidence supports calling QPSK important in space and satellite communications, but it does not justify claiming that every current satellite or spacecraft link uses QPSK.
Cellular systems
QPSK has been included among the modulation capabilities of cellular systems such as LTE, alongside 16QAM and 64QAM. The actual modulation selected depends on the standard release, link direction, channel, scheduling decision, coding, and radio conditions. It is inaccurate to say that every LTE or 5G transmission always uses QPSK.
The 3GPP LTE progress material shows QPSK as one of the available modulation formats.
Coherent optical communications
Optical networks use coherent QPSK, including dual-polarization implementations, when the system needs a balance between spectral efficiency, receiver complexity, and noise tolerance. These systems use sophisticated coherent receivers and digital signal processing rather than the simple detector used in an introductory textbook diagram.
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Software-defined radio and education
QPSK is a useful SDR exercise because a complete implementation exposes nearly every major digital-receiver task: pulse shaping, timing recovery, frequency correction, carrier-phase recovery, constellation decisions, mapping, and frame handling. GNU Radio provides a practical guided flowgraph for simulating and demodulating QPSK.
Independent reader supportYour contribution helps us test, update, and keep practical guides available for everyone.QPSK troubleshooting checklist
If the constellation is wrong or the recovered bits are unusable, inspect the receiver in stages rather than adjusting everything at once.
- Verify the symbol rate. Confirm the configured symbol rate and samples per symbol. A wrong rate prevents timing recovery even when the RF frequency is correct.
- Check the sample-rate relationship. Ensure the ADC or SDR sample rate supports the intended samples per symbol and that any resampler has the expected ratio.
- Inspect pulse-shaping filters. Confirm that transmit and receive filters use compatible root-raised-cosine or raised-cosine settings, including roll-off and span.
- Look for frequency offset. A steadily rotating constellation usually indicates residual carrier-frequency offset or an incorrect center frequency.
- Check timing recovery. A constellation that periodically opens, smears, or changes shape can indicate incorrect symbol timing, clock error, or an unsuitable loop bandwidth.
- Resolve phase ambiguity. A clean constellation with consistently incorrect bits may be rotated by 90°, 180°, or 270°. Test known words, pilots, differential decoding, or all allowed rotations.
- Validate the mapping. Compare the transmitter and receiver’s Gray map, phase offset, I/Q assignment, bit order, and absolute-versus-differential mode.
- Check I/Q impairments. Gain imbalance, quadrature phase error, DC offsets, and LO leakage can distort the four points into an ellipse or unevenly spaced clusters.
- Check the channel. Multipath and frequency-selective distortion may require equalization. Noise alone produces clusters; structured smearing often indicates synchronization or channel distortion.
- Check amplifier operation. Compression can rotate and spread points and may create nonlinear spectral regrowth. Reduce drive level or use an amplifier with adequate linearity for the waveform.
- Confirm SNR and decoding. A good constellation does not guarantee a valid frame if the SNR is inadequate or the frame synchronization and channel-decoding parameters are wrong.
A useful diagnostic order is: view the raw spectrum, verify frequency, view filtered I/Q, confirm timing, inspect the constellation before and after carrier recovery, test phase rotations, and only then examine decoded frames. The GNU Radio tutorial and NASA receiver references both illustrate why QPSK reception depends on coordinated synchronization and detection stages.
QPSK compared with related modulation choices
| Format | Core characteristic | Main practical consideration |
|---|---|---|
| BPSK | Two phases; one bit per symbol | Simple and robust, but lower bits per symbol than QPSK |
| QPSK | Four phases; two bits per symbol | Good throughput per symbol, but requires timing and carrier recovery |
| OQPSK | I and Q transitions are staggered | Reduces large phase transitions and envelope excursions |
| DQPSK | Information is represented by phase changes | Can simplify phase-reference requirements but usually changes BER performance |
| π/4-QPSK | Uses alternating shifted constellation sets or related transitions | Behavior depends strongly on the specified detection and filtering method |
| Higher-order QAM | Uses amplitude as well as phase | Can carry more bits per symbol but generally needs better channel quality and linearity |
How to simulate QPSK
A basic simulation can follow this sequence:
- Generate random bits.
- Group them into pairs.
- Apply a documented Gray-coded mapping.
- Upsample and apply a root-raised-cosine filter.
- Add frequency offset, phase offset, timing error, and AWGN as desired.
- Matched-filter the received samples.
- Recover timing and carrier phase.
- Make constellation decisions and de-map the bits.
- Compare transmitted and received bits to calculate BER.
For a controlled AWGN experiment, plot BER against Eb/N0 and compare the measured result with Q(√(2Eb/N0)). Add impairments one at a time so that their effects are distinguishable. Documentation for QPSK modulation in MATLAB is useful for a software-based model, while the GNU Radio guided tutorial provides an SDR-oriented flowgraph.
For readers who want a physical reference while studying constellations, BER, and receiver synchronization, an optional QPSK modulation textbook can complement the online tutorials. Choose a title that covers digital modulation, pulse shaping, carrier recovery, and error analysis rather than relying on a short modulation-only reference.
Frequently Asked Questions
Is QPSK better than BPSK?
Neither is universally better. QPSK carries two bits per symbol and can provide twice the uncoded bit rate at the same symbol rate, while Gray-coded coherent QPSK and coherent BPSK have the same ideal BER at a given Eb/N0 in AWGN. QPSK also has additional phase-ambiguity and synchronization considerations.
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Its ideal constellation points have equal magnitude, but practical pulse-shaped QPSK can have envelope variation between symbol decisions. Conventional QPSK can make large phase transitions and may stress a nonlinear amplifier. OQPSK limits those transitions.
What causes a rotating QPSK constellation?
A rotating constellation usually indicates residual carrier-frequency offset, an incorrect center frequency, or a phase-tracking loop that has not converged. A fixed but incorrectly oriented constellation is more commonly a phase-offset or phase-ambiguity problem.
Why does my QPSK constellation look correct but the bits are wrong?
Check the phase rotation, Gray-code mapping, I/Q assignment, bit order, differential versus absolute mode, and frame synchronization. A receiver can make geometrically correct decisions using a mapping that does not match the transmitter.
What is the bandwidth of QPSK?
It depends on the symbol rate, pulse-shaping filter, and roll-off factor. With raised-cosine-family filtering, the roll-off factor controls excess bandwidth: lower roll-off narrows the spectrum but makes timing recovery and filtering more demanding. There is no single bandwidth value for all QPSK systems.
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The Bottom Line
QPSK represents two bits with one of four carrier phases and can double bits per symbol compared with BPSK. Its ideal Gray-coded coherent BER in AWGN is the same as BPSK’s when measured at the same Eb/N0, but practical performance depends on pulse shaping, timing, frequency and phase recovery, mapping, channel conditions, coding, and amplifier behavior. Treat ordinary QPSK, OQPSK, DQPSK, π/4-QPSK, and dual-polarization QPSK as distinct implementations with distinct receiver requirements.
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