--- title: "Binary error probability; threshold effect in FM and mitigation" subject: "Communication Systems" chapter: 9 key: "semester_6/communication_systems:error-prob-threshold-effect" source: https://www.bctengineer.com/q/semester_6/communication_systems/error-prob-threshold-effect site: BCT Engineering --- # Binary error probability; threshold effect in FM and mitigation Communication Systems · Chapter 9: Noise in Communication Systems - Asked in: 2079 Bhadra (CS II), 7+3 marks; 2070 Chaitra (CS II), 7+3 marks (asked 2 times) - Video: [Probability of error in binary PCM](https://www.youtube.com/watch?v=v9AuroCrdzg) — EC Video - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_6/communication_systems/error-prob-threshold-effect - Whole chapter: https://www.bctengineer.com/md/semester_6/communication_systems/chapter-9 ## Question Derive the expression for evaluating error probability in binary communication system. What is threshold effect in FM? What are the techniques for the mitigation of threshold effect in FM? ## Short answer For a binary system in AWGN, the error probability is P_e = (1/2)erfc(√(E_d/(4N_0))) where E_d is the energy of the difference signal. The threshold effect in FM occurs when the carrier-to-noise ratio falls below about 10 dB, causing clicks and rapid SNR deterioration. Techniques to mitigate it include phase-locked loop demodulators and FM feedback. ## Answer #### Error probability in a binary communication system **Model.** In a binary system, symbol 1 is sent as $s_1(t)$ and 0 as $s_2(t)$, each lasting $T_b$, with equal probability. Channel adds white Gaussian noise $w(t)$ of PSD $N_0/2$. The receiver filter output is sampled at $t=T_b$: ```text s_i+w +--------+ y(t) sample y +---------+ ------>| filter |------>-/ ------->| y > lam?|--> 1/0 +--------+ t=Tb +---------+ ``` The sample is $y=a_i+n$, where $a_1, a_2$ are the noise-free outputs ($a_1>a_2$) and $n$ is Gaussian with zero mean and variance $\sigma^2$. Conditional pdfs: $$ f(y\mid1)=\frac{1}{\sqrt{2\pi}\sigma}e^{-(y-a_1)^2/2\sigma^2},\qquad f(y\mid0)=\frac{1}{\sqrt{2\pi}\sigma}e^{-(y-a_2)^2/2\sigma^2} $$ ```text f(y|0) f(y|1) .--. .--. / \ | / \ / \ | / \ / ...\...|.../... \ --------a2---lam---a1----------> y Pe0 = area right of lam under f(y|0) Pe1 = area left of lam under f(y|1) ``` **Decision rule:** choose 1 if $y>\lambda$, else 0. For equal priors the optimum threshold is midway, $\lambda=\frac{a_1+a_2}{2}$. **Error when 0 is sent** ($y>\lambda$): with $u=(y-a_2)/\sqrt2\sigma$, $$ \begin{aligned} P_{e0} &= \int_{\lambda}^{\infty}\frac{1}{\sqrt{2\pi}\sigma}e^{-(y-a_2)^2/2\sigma^2}dy=\frac{1}{\sqrt{\pi}}\int_{(\lambda-a_2)/\sqrt2\sigma}^{\infty}e^{-u^2}du \\ &= \frac12\operatorname{erfc}\left(\frac{a_1-a_2}{2\sqrt2\,\sigma}\right) \end{aligned} $$ By symmetry $P_{e1}$ (1 sent, $y<\lambda$) is the same. The average error probability is $$ P_e=\tfrac12P_{e0}+\tfrac12P_{e1}=\frac12\operatorname{erfc}\left(\frac{a_1-a_2}{2\sqrt2\,\sigma}\right)=Q\left(\frac{a_1-a_2}{2\sigma}\right) $$ where $\operatorname{erfc}(x)=\frac{2}{\sqrt\pi}\int_x^{\infty}e^{-u^2}du$ and $Q(x)=\frac12\operatorname{erfc}(x/\sqrt2)$. **With a matched filter** (matched to $s_1-s_2$), the maximum of $\frac{(a_1-a_2)^2}{\sigma^2}$ is $\frac{2E_d}{N_0}$, where $E_d=\int_0^{T_b}[s_1(t)-s_2(t)]^2dt$. Therefore $$ P_e=\frac12\operatorname{erfc}\left(\sqrt{\frac{E_d}{4N_0}}\right) $$ $P_e$ depends only on the **energy of the difference signal** relative to $N_0$, not on pulse shape. Making the two signals as different as possible (antipodal) gives the lowest $P_e$. | Scheme | $s_1(t)$, $s_2(t)$ | $E_d$ | $P_e$ | |---|---|---|---| | Unipolar / ASK (OOK) | $A\cos\omega_ct$, $0$ | $2E_b$ | $\frac12\operatorname{erfc}\sqrt{E_b/2N_0}$ | | Polar / PSK | $\pm A\cos\omega_ct$ | $4E_b$ | $\frac12\operatorname{erfc}\sqrt{E_b/N_0}$ | | Coherent FSK | $A\cos\omega_1t$, $A\cos\omega_2t$ | $2E_b$ | $\frac12\operatorname{erfc}\sqrt{E_b/2N_0}$ | ($E_b$ = average energy per bit.) #### Threshold effect in FM **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 can perform 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) ``` #### Mitigation of threshold effect **Techniques to reduce (extend) the threshold:** 1. **FM feedback (FMFB) demodulator:** the VCO output tracks the signal, so the IF bandwidth needed is reduced (compressed deviation); less noise enters, threshold lowered by about 5–7 dB. 2. **Phase-locked loop (PLL) demodulator:** the loop's narrow bandwidth tracks the instantaneous frequency and rejects noise outside it; gives a few dB threshold extension. 3. **Pre-emphasis and de-emphasis:** boost high message frequencies before modulation and cut them after detection; this reduces the high-frequency (parabolic) noise and improves the output SNR by about 10–13 dB, although it does not itself lower the threshold. 4. **Reducing IF bandwidth / modulation index** where possible, or increasing transmitted power (raises CNR above threshold). 5. **Diversity reception**, which helps keep the CNR above threshold during fading. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. How is the sample at the decision device modelled? A. y = a_i * n B. y = a_i - a_j C. y is uniform D. y = a_i + n with Gaussian zero-mean noise of variance sigma^2 Answer: D. The noise-free level plus Gaussian noise. (Key point: Noise model) 2. For equal priors, what is the optimum threshold and the resulting Pe? A. lambda = a1 and Pe = Q((a1 - a2)/sigma) B. lambda = 0 and Pe = 1/2 C. lambda = (a1 + a2)/2 and Pe = Q((a1 - a2)/(2 sigma)) D. lambda = a2 and Pe = erfc(a1) Answer: C. The midpoint threshold gives Pe = (1/2) erfc((a1-a2)/(2 sqrt2 sigma)). (Key point: Optimum threshold and Pe) 3. With a matched filter, on what does Pe depend? A. Pulse shape alone B. Energy of the difference signal Ed relative to N0 C. Carrier frequency alone D. Bit rate alone Answer: B. Pe = (1/2) erfc(sqrt(Ed/4N0)). (Key point: Matched filter Pe) 4. Why is PSK better than ASK/OOK? A. Ed = 4Eb versus 2Eb B. Ed is smaller C. They have equal Ed D. PSK uses no noise Answer: A. Larger difference-signal energy gives a lower error probability. (Key point: PSK vs ASK) 5. What is the FM threshold effect and when does it appear? A. SNR rises below 10 dB B. It appears only for small beta C. It appears at high CNR D. Output SNR drops sharply when CNR is below about 10 dB, due to clicks from 2-pi phase jumps Answer: D. Noise exceeding the carrier makes the phasor encircle the origin. (Key point: Threshold effect) 6. Which is a threshold-extension technique? A. Increase beta B. Wider IF filter C. FM feedback or PLL demodulator D. Remove the limiter Answer: C. These track the signal with narrower effective bandwidth, lowering the threshold by several dB. (Key point: Mitigation) ## Related in Communication Systems - [Threshold effect in FM; error probability of coherent PSK](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Athreshold-effect-psk-error) — Communication Systems, chapter 9 - [Error probability in a binary communication system](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aerror-probability-binary-system) — Communication Systems, chapter 9 - [Error probability in binary baseband system](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aerror-probability-binary-baseband) — Communication Systems, chapter 9, asked 2× - [Binary baseband error probability; comparison with M-ary](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aerror-prob-binary-vs-mary) — Communication Systems, chapter 9, asked 3× - [Error probability of binary ASK extended to M-ary](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aask-error-prob-mary) — Communication Systems, chapter 9 - [M-ary error probability; binary vs M-ary](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Amary-error-compare-binary) — Communication Systems, chapter 9 ## Related in other subjects - [Multiple access and near-far effect mitigation](https://www.bctengineer.com/question.md?key=semester_7%2Fwireless_communication%3Amultiple-access-near-far) — Wireless Communication, chapter 7