--- title: "Figure of merit of non-coherent FM; threshold effect" subject: "Communication Systems" chapter: 9 key: "semester_6/communication_systems:fom-noncoherent-fm-threshold" source: https://www.bctengineer.com/q/semester_6/communication_systems/fom-noncoherent-fm-threshold site: BCT Engineering --- # Figure of merit of non-coherent FM; threshold effect Communication Systems · Chapter 9: Noise in Communication Systems - Asked in: 2074 Asoj (CS II), 6+2 marks - Video: [CT#39 Noise in the FM receiver / Figureof merit of FM receiver || EC Academy](https://www.youtube.com/watch?v=kc_Ixnmai2s) — EC Academy - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_6/communication_systems/fom-noncoherent-fm-threshold - Whole chapter: https://www.bctengineer.com/md/semester_6/communication_systems/chapter-9 ## Question Compute the figure of merit of non coherent FM System and explain the threshold effects. ## Short answer Output SNR is 3A_c²k_f²P/(2N_0W³) and reference SNR is A_c²/(2N_0W), giving FOM = 3k_f²P/W². For a tone this equals (3/2)β², improving quadratically with deviation ratio. Threshold effect occurs below ~10 dB carrier-to-noise ratio, where noise phase spikes cause clicks and output SNR drops abruptly. ## Answer #### Figure of merit of non-coherent FM **FM signal and receiver.** $s(t)=A_c\cos\left[\omega_ct+2\pi k_f\int m(t)dt\right]$, instantaneous frequency deviation $k_fm(t)$. Receiver: BPF ($B_T$) → limiter → discriminator (output $=\frac{1}{2\pi}\frac{d\phi}{dt}$) → LPF ($W$). **Noise in phasor form.** Write the narrowband noise as $n(t)=r(t)\cos[\omega_ct+\psi(t)]$. At high carrier-to-noise ratio ($A_c\gg r$), the resultant phase is $$ \theta(t)\approx\phi(t)+\frac{r(t)}{A_c}\sin[\psi(t)-\phi(t)] $$ ```text r(t) .------> resultant = carrier + noise / ^ small noise phasor rotates Ac / | r sin(psi-phi) the carrier phasor slightly ------>----+ ``` The noise term is statistically equivalent to $\frac{n_s(t)}{A_c}$ (the phase $\phi$ can be dropped for noise calculation). Discriminator output: $$ v(t)=\frac{1}{2\pi}\frac{d\theta}{dt}=k_fm(t)+\frac{1}{2\pi A_c}\frac{dn_s(t)}{dt} $$ **Output signal power:** $S_o=k_f^2P$. **Output noise power.** $n_s$ has PSD $N_0$ for $\lvert f\rvert\le B_T/2$. Differentiation multiplies the spectrum by $j2\pi f$, so the noise PSD at discriminator output is $$ S_{n_d}(f)=\frac{(2\pi f)^2}{(2\pi A_c)^2}N_0=\frac{N_0f^2}{A_c^2},\qquad \lvert f\rvert\le\frac{B_T}{2} $$ This **parabolic** noise spectrum is the key property of FM. After the LPF ($\lvert f\rvert\le W$): $$ N_o=\int_{-W}^{W}\frac{N_0f^2}{A_c^2}df=\frac{2N_0W^3}{3A_c^2} $$ ```text S(f) |\ /| | \ / | parabolic noise | \ / | after discriminator | \. ./ | (only |f| f -W 0 W ``` **Output SNR:** $$ \text{SNR}_o=\frac{k_f^2P}{2N_0W^3/3A_c^2}=\frac{3A_c^2k_f^2P}{2N_0W^3} $$ **Input SNR.** Received power $\frac{A_c^2}{2}$ (constant envelope); noise in $B_T$ is $N_0B_T$: $$ \text{SNR}_i=\frac{A_c^2}{2N_0B_T},\qquad \text{SNR}_{ref}=\frac{A_c^2}{2N_0W} $$ **Gain and figure of merit:** $$ \begin{aligned} \text{FOM} &= \frac{\text{SNR}_o}{\text{SNR}_{ref}}=\frac{3k_f^2P}{W^2} \\ \frac{\text{SNR}_o}{\text{SNR}_i} &= \frac{3k_f^2P\,B_T}{W^3} \end{aligned} $$ For a **single tone** $m(t)=A_m\cos\omega_mt$: $P=A_m^2/2$, $\Delta f=k_fA_m$, $\beta=\Delta f/W$, $B_T=2(\beta+1)W$ (Carson): $$ \text{FOM}=\frac{3}{2}\beta^2,\qquad \frac{\text{SNR}_o}{\text{SNR}_i}=3\beta^2(\beta+1) $$ So FM noise performance improves with the **square of the deviation ratio**: FM trades bandwidth for SNR. FM beats AM ($\text{FOM}=1/3$) when $\frac32\beta^2>\frac13$, i.e. $\beta>0.471$ (about 0.5). #### Threshold effect **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) ``` ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. What is the noise PSD at the FM discriminator output (high CNR)? A. Flat, N0 B. Parabolic, N0 f^2 / A_c^2 C. Inversely proportional to f D. Triangular, N0 |f| Answer: B. Differentiation multiplies the spectrum by j2 pi f, so noise PSD is proportional to f^2. (Key point: Parabolic noise) 2. What is the FOM of single-tone non-coherent FM? A. (3/2) beta^2 B. 3 beta^2 (beta + 1) C. beta^2 / 3 D. 1/3 Answer: A. FOM = 3 k_f^2 P / W^2 = (3/2) beta^2 for a single tone. (Key point: Figure of merit) 3. For what deviation ratio does FM beat AM (FOM = 1/3)? A. beta > 1 B. beta > 0.471 (about 0.5) C. beta > 2 D. beta > 0.1 Answer: B. (3/2) beta^2 > 1/3 gives beta^2 > 2/9, beta > 0.471. (Key point: FM versus AM) 4. What is the output noise power after the LPF of bandwidth W? A. N0 W / A_c^2 B. 2 N0 W^3 / (3 A_c^2) C. N0 W^2 / A_c^2 D. N0 B_T / A_c^2 Answer: B. Integrate N0 f^2/A_c^2 from -W to W. (Key point: Output noise integral) 5. What happens below the FM threshold of about 10 dB CNR? A. Output SNR improves faster B. The 2 pi phase jumps produce click spikes and SNR falls much faster than linearly C. Bandwidth decreases D. The limiter removes all noise Answer: B. Noise phasor exceeding the carrier causes clicks, and FM can be worse than AM below threshold. (Key point: Threshold effect) ## 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 - [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 - [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× - [Short note: Threshold effect in FM](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Ashort-note-threshold-effect-fm) — Communication Systems, chapter 9, asked 4× - [SNR gain of non-coherent FM detector](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Anoncoherent-fm-gain) — Communication Systems, chapter 9, asked 2× - [AM vs FM efficiency and complexity; coherent ASK error](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aam-fm-bw-efficiency-ask-error) — Communication Systems, chapter 9, asked 2×