--- title: "SNR gain of non-coherent FM detector" subject: "Communication Systems" chapter: 9 key: "semester_6/communication_systems:noncoherent-fm-gain" source: https://www.bctengineer.com/q/semester_6/communication_systems/noncoherent-fm-gain site: BCT Engineering --- # SNR gain of non-coherent FM detector Communication Systems · Chapter 9: Noise in Communication Systems - Asked in: 2073 Shrawan (CS II), 8 marks; 2071 Chaitra (CS II), 8 marks (asked 2 times) - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_6/communication_systems/noncoherent-fm-gain - Whole chapter: https://www.bctengineer.com/md/semester_6/communication_systems/chapter-9 ## Question Derive the expression for evaluation the gain parameter (SNR₀/SNRᵢ) of non-coherent FM detector. ## Short answer For non-coherent FM detection, noise after discriminator has parabolic PSD ∝ f². The SNR improvement SNR_o/SNR_i = 3β²(β+1) for a single-tone modulating signal with deviation ratio β, demonstrating noise reduction proportional to β². ## Answer A **non-coherent FM detector** (limiter + frequency discriminator) responds to the instantaneous frequency of the received signal; no local carrier is needed. The gain parameter $\text{SNR}_o/\text{SNR}_i$ is found assuming high carrier-to-noise ratio (above threshold). **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). ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. What assumption is used for the non-coherent FM noise analysis? A. Noise much larger than the carrier B. Zero message power C. Carrier amplitude much larger than noise amplitude (above threshold) D. Narrow deviation only (beta much less than 1) Answer: C. With A_c >> r the phase error is approximately n_s/A_c, which linearises the analysis. (Key point: High-CNR approximation) 2. What is the shape of the noise PSD at the discriminator output? A. Parabolic, N0 f^2 / A_c^2 B. Flat, N0/2 C. Linear in f D. Rectangular Answer: A. Differentiation multiplies the noise spectrum by j2 pi f, so the power spectrum goes as f^2. (Key point: Discriminator noise spectrum) 3. What is the output noise power after the LPF of bandwidth W? A. 2 N0 W / A_c^2 B. N0 W^2 / A_c^2 C. N0 W^3 / A_c^2 D. 2 N0 W^3 / (3 A_c^2) Answer: D. Integrating N0 f^2/A_c^2 from -W to W gives 2 N0 W^3/(3 A_c^2). (Key point: Output noise power) 4. For single-tone FM, what is the figure of merit? A. (3/2) beta B. (3/2) beta^2 C. 1/3 D. 3 beta^2 (beta + 1) Answer: B. FOM = 3 k_f^2 P / W^2 with P = A_m^2/2 and beta = k_f A_m/W gives 1.5 beta^2. (Key point: Figure of merit) 5. Why does FM improve noise performance? A. SNR improves with the square of beta, trading bandwidth for SNR B. It uses less bandwidth than AM C. Its noise PSD is flat D. It has no threshold effect Answer: A. Output SNR grows as beta^2 at the cost of larger bandwidth B_T = 2(beta+1)W. (Key point: FM versus AM) ## Related in Communication Systems - [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 - [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 - [Short note: Coherent detector](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Ashort-note-coherent-detector) — Communication Systems, chapter 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× - [General FM expression; FM using phase modulator](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Afm-general-expression-phase-modulator) — Communication Systems, chapter 3 ## Related in other subjects - [Ideal op-amp properties; non-inverting gain](https://www.bctengineer.com/question.md?key=semester_2%2Fbasic_electronics_engineering%3Aopamp-props-noninverting-gain) — Basic Electronics Engineering, chapter 4, asked 3× - [Voltage gain of non-inverting op-amp](https://www.bctengineer.com/question.md?key=semester_2%2Fbasic_electronics_engineering%3Aopamp-noninverting-gain) — Basic Electronics Engineering, chapter 4, asked 2× - [Non-inverting op-amp voltage gain](https://www.bctengineer.com/question.md?key=semester_1%2Ffundamental_of_electrical_and_electronics_engineering%3Anon-inverting-amplifier-gain) — Fundamental of Electrical and Electronics Engineering, chapter 6 - [Derive expression for tile drain spacing](https://www.bctengineer.com/question.md?key=semester_6%2Firrigation_and_drainage_engineering%3Atile-drain-spacing-derivation) — Irrigation and Drainage Engineering, chapter 9, asked 8×