Signal Processing

Costas Loop

Q: Why does a Costas loop need both an I and a Q arm?

A suppressed-carrier BPSK signal has no discrete spectral line at the carrier, so a simple phase-locked loop has nothing to lock to and the 180 degree data modulation would drag any single mixer around randomly. The Costas loop solves this by splitting the input into an in-phase (I) arm and a quadrature (Q) arm using local-oscillator references 90 degrees apart. The I arm demodulates the data; the Q arm carries the phase error. Multiplying I times Q removes the data modulation because the data sign appears in both arms and cancels, leaving a clean S-curve error term proportional to sin(2 times phase error). That product drives the loop filter and VCO, so the loop locks to the carrier phase even though no carrier tone is present. With only one arm there is no way to separate phase error from data transitions.

Q: What is the phase ambiguity of a Costas loop and how is it resolved?

A BPSK Costas loop has a stable lock point every 180 degrees, so it cannot tell a recovered bit stream from its logical complement; a QPSK Costas loop has four stable points spaced 90 degrees apart and so has a four-fold ambiguity. The loop will lock happily to any of these, but the demodulated data may come out inverted or rotated. Differential encoding (DBPSK or DQPSK), where information is carried in the transition between symbols rather than the absolute phase, makes the receiver immune to this ambiguity. Alternatively a known unique word or preamble in the frame lets the demodulator detect the rotation and apply the correct fixed phase correction before slicing.

Q: How do you set the loop bandwidth of a Costas loop?

Loop bandwidth is a compromise. A wide bandwidth gives fast acquisition and tolerates large frequency offset and oscillator drift, but it lets more thermal noise into the recovered carrier, raising phase jitter and degrading bit error rate. A narrow bandwidth cleans up the carrier and lowers the noise floor but acquires slowly and can lose lock under fading or Doppler. A common rule of thumb sets the single-sided loop noise bandwidth B_L between 0.5 and 5 percent of the symbol rate R_s. For a 1 Msym/s link that is roughly 5 to 50 kHz. The damping factor is usually chosen near 0.707 for a second-order loop to balance overshoot against settling time, and a frequency-aided or sweep acquisition stage is added when the expected offset exceeds the loop pull-in range.

/KOS-tuhs loop/

Invented by John P. Costas in 1956, this synchronous detector recovers a coherent local-oscillator reference from a suppressed-carrier signal that contains no usable carrier tone of its own. The input is mixed against two VCO outputs 90° apart to form in-phase (I) and quadrature (Q) baseband arms; multiplying the two arms strips the data modulation and yields a phase-error term proportional to sin(2θ_e) that drives a loop filter and a voltage-controlled oscillator. This makes it the workhorse carrier recovery circuit for BPSK and QPSK demodulators in satellite modems, telemetry receivers, and software-defined radios, typically operating at IF with loop bandwidths of 0.5 to 5 percent of the symbol rate.

Category: Signal Processing

Phase Error Slope: ∝ sin(2θ_e)

Lock Points: 2 (BPSK) / 4 (QPSK)

How the Costas Loop Recovers a Suppressed Carrier

Phase-shift keying intentionally suppresses the carrier: a BPSK waveform is the carrier multiplied by ±1, so its spectrum is a pair of data sidebands with no discrete line at the carrier frequency. A plain phase-locked loop has nothing to lock onto, and the 180° data transitions would whipsaw a single mixer's output. The Costas loop sidesteps this by performing the synchronous detection and the phase comparison at the same time. The received IF is split and mixed against two replicas of the VCO, one in phase and one shifted 90°. After lowpass filtering, the I arm carries cos(θ_e) times the data and the Q arm carries sin(θ_e) times the data, where θ_e is the phase error between the incoming carrier and the VCO.

The trick is the third multiplier. Forming the product I × Q gives a term proportional to sin(θ_e)cos(θ_e) = ½ sin(2θ_e), and because the data sign d = ±1 appears squared (d² = 1) it cancels out entirely. The result is a clean, data-free S-curve error voltage that is zero at lock and slopes through the origin, exactly what the loop filter needs to steer the VCO. For QPSK the same idea is extended with a four-quadrant arctangent or a hard-limited cross-coupled detector that produces a sin(4θ_e) characteristic with four stable lock points 90° apart.

Governing Equations

I and Q baseband arms (after LPF):
I = ½ A · d(t) · cos(θ_e) Q = ½ A · d(t) · sin(θ_e)

Phase detector output (BPSK):
e(t) = I × Q = ⅛ A² · d²(t) · sin(2θ_e) ≈ k · sin(2θ_e)

Second-order loop natural frequency & bandwidth:
ω_n = √(K_dK₀/τ₁) B_L = (ω_n/2)(ζ + 1/(4ζ))

Where A = signal amplitude, d(t) = ±1 data, θ_e = phase error, k = detector gain, ζ = damping factor (≈0.707 typical), B_L = single-sided loop noise bandwidth. Design target: B_L ≈ 0.005 to 0.05 × R_s (symbol rate).

Carrier Recovery Method Comparison

Method	Modulations	Phase Error Term	SNR @ Lock	Ambiguity	Notes
Costas loop	BPSK, QPSK	sin(2θ_e) / sin(4θ_e)	~5 to 8 dB E_b/N₀	180° / 90°	Self-noise rises at low SNR
Squaring loop	BPSK	From 2× multiplied tone	~6 to 9 dB	180°	Needs 2f₀ PLL + divide-by-2
Decision-directed	QPSK, QAM	Im(r · conj(d̂))	~7 to 10 dB	Per constellation	Degrades below decision threshold
Remodulator	BPSK, QPSK	Re-strip then PLL	~6 dB	180° / 90°	Lower self-noise than Costas
Pilot-tone PLL	Any (residual carrier)	Direct sin(θ_e)	~0 to 3 dB	None	Wastes power on the pilot

Common Questions

Frequently Asked Questions

Why does a Costas loop need both an I and a Q arm?

A suppressed-carrier BPSK signal has no discrete line at the carrier, so a plain PLL has nothing to lock to and the 180° data modulation would drag a single mixer around. Splitting into in-phase (I) and quadrature (Q) arms 90° apart lets the I arm demodulate data while the Q arm carries the phase error. The product I × Q removes the data because d² = 1, leaving a clean error term proportional to sin(2θ_e) that the loop locks on.

What is the phase ambiguity of a Costas loop and how is it resolved?

A BPSK Costas loop has stable lock points every 180°, and a QPSK loop has four points 90° apart, so the recovered bits may come out inverted or rotated. Differential encoding (DBPSK or DQPSK) carries information in the transition rather than absolute phase and is immune to the ambiguity. Alternatively a known unique word or preamble lets the demodulator detect the rotation and apply a fixed phase correction.

How do you set the loop bandwidth of a Costas loop?

It is a compromise: wide bandwidth acquires fast and tolerates frequency offset but admits more noise and raises jitter; narrow bandwidth cleans the carrier but acquires slowly and can drop lock under Doppler. A common rule sets the single-sided noise bandwidth B_L between 0.5 and 5 percent of the symbol rate (roughly 5 to 50 kHz at 1 Msym/s), with damping near 0.707 and a sweep stage when the offset exceeds pull-in range.

Related Terms

← Costa Precoding COTS →

Coherent Receivers

Specify Your Down-Conversion Chain

Building a BPSK or QPSK receiver that needs clean carrier recovery? Our frequency converters and integrated assemblies deliver the low-phase-noise IF your Costas loop depends on. Talk to our engineering team.

Get in Touch