--- title: "Costas loop detector and phase error correction" subject: "Communication Systems" chapter: 2 key: "semester_6/communication_systems:costas-loop-phase-error" source: https://www.bctengineer.com/q/semester_6/communication_systems/costas-loop-phase-error site: BCT Engineering --- # Costas loop detector and phase error correction Communication Systems · Chapter 2: Amplitude Modulation - Asked in: 2073 Bhadra (CS I), 4+4+2 marks - Video: [DSBSC detection using Costas loop and Squaring loop.](https://www.youtube.com/watch?v=Z3PXUt14qro) — Dr.Aarthi. G - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_6/communication_systems/costas-loop-phase-error - Whole chapter: https://www.bctengineer.com/md/semester_6/communication_systems/chapter-2 ## Question Draw the block diagram of Costas Loop detector and explain how it demodulates DSB-SC AM and corrects for phase error. ## Short answer The Costas loop multiplies DSB‑SC by cos(ωct+φ) and sin(ωct+φ). Low‑pass filtering gives I = (Ac/2)m(t)cosφ and Q = (Ac/2)m(t)sinφ. Their product drives a loop filter that adjusts the VCO to force φ→0, so the I‑channel yields the demodulated message without a pilot carrier. ## Answer #### Block diagram of Costas loop ```text I-channel +----->( X )-----> LPF ---+-------> output | ^ | (Ac/2)m(t)cos(phi) | | cos(wc t+phi) v s(t) ----+ [VCO] <-- LPF <--( X ) | | (loop ^ phase | [-90 deg] filter) | discriminator | | sin(wc t+phi) | +----->( X )-----> LPF --+ Q-channel (Ac/2)m(t)sin(phi) ``` Parts: two product modulators (I and Q), two LPFs, a VCO with a $90^\circ$ phase shifter, and a phase discriminator (multiplier + loop filter). #### How it demodulates DSB-SC Input $s(t) = A_cm(t)\cos\omega_ct$; VCO output $\cos(\omega_ct + \phi)$. $$ \begin{aligned} y_I(t) &= \text{LPF}\{A_cm(t)\cos\omega_ct\cos(\omega_ct+\phi)\} = \frac{A_c}{2}m(t)\cos\phi \\ y_Q(t) &= \text{LPF}\{A_cm(t)\cos\omega_ct\sin(\omega_ct+\phi)\} = \frac{A_c}{2}m(t)\sin\phi \end{aligned} $$ When the VCO is locked ($\phi = 0$), $y_I = \tfrac{A_c}{2}m(t)$: the **I channel output is the demodulated message**, and $y_Q = 0$. #### How it corrects phase error The phase discriminator multiplies the I and Q outputs: $$ e(t) = y_Iy_Q = \frac{A_c^2}{4}m^2(t)\sin\phi\cos\phi = \frac{A_c^2}{8}m^2(t)\sin2\phi $$ The loop filter averages this to a slowly varying control voltage $v_c \propto \overline{m^2}\sin2\phi \approx 2\overline{m^2}\phi$ for small $\phi$. - If $\phi > 0$, $v_c > 0$: the VCO is pulled back to reduce $\phi$. - If $\phi < 0$, $v_c < 0$: the VCO is pushed the other way. - At $\phi = 0$: $v_c = 0$ and the loop is in lock. Because $m^2(t) \ge 0$, the correction direction does not depend on the polarity of the message. So the Q channel acts as an error detector, and the loop keeps the local carrier coherent with the received one, avoiding the $\cos\phi$ loss and quadrature null. #### Key points - Automatic phase and frequency tracking without a pilot carrier. - 180° phase ambiguity (lock at $\phi = \pi$ gives $-m(t)$), harmless for audio. - Needs $m(t)$ to be non-zero most of the time. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Which set of blocks forms the Costas loop? A. Envelope detector and AGC B. Two product modulators (I and Q), two LPFs, a VCO with 90-degree shifter, and a phase discriminator C. A single multiplier and a high-pass filter D. A mixer, IF amplifier and detector Answer: B. The I and Q coherent detectors share one VCO, and the product of their outputs drives the VCO. (Key point: Costas loop parts) 2. Which channel gives the demodulated message when locked? A. Q channel B. The I channel C. The phase discriminator D. The VCO Answer: B. yI = (Ac/2) m(t) cos(phi) becomes the message at phi = 0, while yQ becomes zero. (Key point: Which channel gives output) 3. If phi > 0, what does the loop do? A. Control voltage vc goes positive and pulls the VCO phase back to reduce phi B. It increases phi C. It switches off the VCO D. It inverts m(t) Answer: A. vc is proportional to sin(2 phi), so its sign tells the direction and the negative feedback drives phi toward 0. (Key point: Phase correction) 4. What is the 180 degree phase ambiguity of the Costas loop? A. The loop never locks B. The loop can lock at phi = pi, giving -m(t), harmless for audio C. The message is lost D. The loop locks only at 90 degrees Answer: B. sin(2 phi) is zero at both 0 and pi, so the output may be inverted. (Key point: Phase ambiguity) ## Related in Communication Systems - [Costas loop as synchronous receiver for DSB-SC](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Acostas-loop-dsbsc) — Communication Systems, chapter 2 - [Phase error in coherent DSB-SC detection; AM demodulation by PLL](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aphase-error-dsbsc-am-pll) — Communication Systems, chapter 2 - [Phase error in synchronous DSB-SC detection; synchronized receiver](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aphase-error-dsbsc-costas) — Communication Systems, chapter 2 - [Compare DSB-AM, DSB-SC, SSB, VSB; SSB by phase discrimination](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Acompare-am-variants-ssb-phase) — Communication Systems, chapter 2 - [Synchronous detection of DSB-SC; phase and frequency error](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Asynchronous-dsbsc-phase-freq-error) — Communication Systems, chapter 2 - [Envelope detector; can it demodulate DSB-SC?](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aenvelope-detector-dsbsc) — Communication Systems, chapter 2, asked 2× ## Related in other subjects - [Closed loop control system block diagram explained](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Awhat-is-cs-bd-closed-loop) — Control Systems, chapter 1, asked 4× - [Open and closed loop systems with block diagram](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Aol-cl-bd-role) — Control Systems, chapter 1, asked 2× - [Control system block diagram and closed loop advantages](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Awhat-is-cs-bd-cl-advantages) — Control Systems, chapter 1 - [Block diagram reduction: G1–G4, H1, H2, unity loop](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Abdr-g1-g4-h1h2-2075a) — Control Systems, chapter 3, asked 4×