--- title: "Threshold effect in FM; error probability of coherent PSK" subject: "Communication Systems" chapter: 9 key: "semester_6/communication_systems:threshold-effect-psk-error" source: https://www.bctengineer.com/q/semester_6/communication_systems/threshold-effect-psk-error site: BCT Engineering --- # Threshold effect in FM; error probability of coherent PSK Communication Systems · Chapter 9: Noise in Communication Systems - Asked in: 2076 Chaitra (CS II), 4+6 marks - Video: [L 6 | Probability of Error of Optimum Receiver | Optimum Filter | Performance Analysis Digital Comm](https://www.youtube.com/watch?v=Ibo_iB87ZuU) — Dopamine - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_6/communication_systems/threshold-effect-psk-error - Whole chapter: https://www.bctengineer.com/md/semester_6/communication_systems/chapter-9 ## Question Explain threshold effect in detection of FM signal. Derive the expression of error probability for coherent detection of Phase Shift Keying (PSK). ## Short answer Threshold effect in FM occurs when CNR drops below ~10 dB, causing noise phasor to exceed carrier and produce 2π phase jumps (clicks), rapidly degrading output SNR making FM worse than AM. Coherent BPSK error probability is P_e = 1/2 erfc(√(E_b/N_0)) = Q(√(2E_b/N_0)), derived from correlator output with Gaussian noise and zero threshold. ## Answer #### Threshold effect in FM detection **Threshold effect in FM.** The FM improvement $\text{SNR}_o\propto\beta^2$ holds only when the carrier is much stronger than the noise at the discriminator input. When the carrier-to-noise ratio (CNR) falls below a **threshold, about 10 dB** (typically 10–13 dB), the noise phasor sometimes becomes larger than the carrier and the resultant phasor **encircles the origin**, causing a sudden $2\pi$ phase jump. ```text high CNR: noise wobbles low CNR: resultant can swing the tip slightly around origin -> 2pi jump .-. .---. 0 -->( * ) 0 --( * ) '-' '---' ``` Each $2\pi$ jump produces an impulse (spike) in the discriminator output, heard as **clicks**. The click rate rises rapidly as CNR drops, so the output SNR falls much faster than linearly and the signal is mutilated. Above threshold, FM is excellent; below it, FM is worse than AM. Wider deviation (larger $\beta$) means larger $B_T$, more noise and a higher threshold. ```text SNRo(dB) . FM above threshold | . (slope 1, offset 3/2 beta^2) | . | . | . <- threshold knee (CNR ~ 10 dB) | .: | . : AM / baseband | . : +------------------> SNRi (dB) ``` The threshold can be lowered (threshold extension) by using FM feedback (FMFB) or PLL demodulators. Pre-emphasis/de-emphasis improves the output SNR above threshold but does not lower the threshold. #### Error probability of coherent PSK **BPSK signals** over $0\le t\le T_b$ (antipodal): $$ s_1(t)=A\cos\omega_ct\ \ (1),\qquad s_2(t)=-A\cos\omega_ct\ \ (0),\qquad E_b=\frac{A^2T_b}{2} $$ **Coherent receiver:** multiply by the locally generated $\cos\omega_ct$, integrate over $T_b$, sample, compare with threshold $\lambda=0$. ```text r(t) -->(x)--> integrate --> sample --> y > 0 ? 1 : 0 ^ 0..Tb t=Tb | cos(wc t) (from carrier recovery) ``` Correlator output $y=a_i+n$ with $$ a_1=+\frac{AT_b}{2},\qquad a_2=-\frac{AT_b}{2},\qquad \sigma^2=\frac{N_0T_b}{4} $$ The pdfs of $y$ are Gaussian centred at $\pm AT_b/2$. Error when 0 is sent ($y>0$): $$ \begin{aligned} P_e &= \int_0^{\infty}\frac{1}{\sqrt{2\pi}\sigma}e^{-(y+AT_b/2)^2/2\sigma^2}dy=\frac12\operatorname{erfc}\left(\frac{AT_b/2}{\sqrt2\,\sigma}\right) \\ &= \frac12\operatorname{erfc}\left(\frac{AT_b/2}{\sqrt{N_0T_b/2}}\right)=\frac12\operatorname{erfc}\left(\sqrt{\frac{A^2T_b}{2N_0}}\right) \end{aligned} $$ By symmetry the error for a sent 1 is the same, so $$ P_e(\text{BPSK})=\frac12\operatorname{erfc}\left(\sqrt{\frac{E_b}{N_0}}\right)=Q\left(\sqrt{\frac{2E_b}{N_0}}\right) $$ BPSK is the best binary scheme: it needs 3 dB less $E_b/N_0$ than coherent ASK or FSK for the same $P_e$. Example: $E_b/N_0=10$ dB gives $P_e=Q(\sqrt{20})\approx3.9\times10^{-6}$. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. At roughly what carrier-to-noise ratio does the FM threshold effect appear? A. About 10 dB (typically 10 to 13 dB) B. About 0 dB C. About 30 dB D. About 3 dB Answer: A. Below roughly 10 dB CNR the noise phasor can exceed the carrier and cause phase jumps. (Key point: Threshold level) 2. What produces the clicks below the FM threshold? A. The limiter saturating B. Pre-emphasis C. The resultant phasor encircling the origin, causing a 2 pi phase jump that gives a spike D. Carrier frequency drift Answer: C. Each 2 pi jump gives an impulse at the discriminator output, heard as a click. (Key point: Cause of clicks) 3. What are the correlator outputs and noise variance for BPSK? A. +/- A Tb/2 and sigma^2 = N0 Tb/4 B. +/- A Tb and sigma^2 = N0 Tb/2 C. A Tb/2 and 0 and sigma^2 = N0 Tb/4 D. +/- A Tb/4 and sigma^2 = N0 Tb Answer: A. The integral of A cos^2 over Tb is A Tb/2 and the noise variance is (N0/2)(Tb/2). (Key point: BPSK decision variable) 4. What is the Pe of coherent BPSK and how does it compare with ASK/FSK? A. (1/2) erfc sqrt(Eb/2N0); the same as ASK B. (1/2) erfc sqrt(Eb/N0) = Q(sqrt(2Eb/N0)); 3 dB better than coherent ASK or FSK C. (1/2) exp(-Eb/N0); 3 dB worse D. (1/2) erfc sqrt(Eb/N0); 6 dB better Answer: B. Antipodal signals have twice the distance energy, so BPSK needs 3 dB less Eb/N0. (Key point: BPSK error probability) ## Related in Communication Systems - [Threshold effect in non-coherent FM detection and its correction](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Athreshold-noncoherent-fm-correction) — Communication Systems, chapter 9 - [Binary error probability; threshold effect in FM and mitigation](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aerror-prob-threshold-effect) — Communication Systems, chapter 9, asked 2× - [Error probability of coherent ASK](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aask-coherent-error-probability) — Communication Systems, chapter 9, asked 2× - [Figure of merit of non-coherent FM; threshold effect](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Afom-noncoherent-fm-threshold) — Communication Systems, chapter 9 - [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 - [Need for shift keying; ASK vs FSK vs PSK](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Ashift-keying-ask-fsk-psk) — Communication Systems, chapter 7