--- title: "SSB figure of merit and 32-QAM constellation" subject: "Communication Systems" chapter: 9 key: "semester_6/communication_systems:ssb-figure-of-merit-32qam" source: https://www.bctengineer.com/q/semester_6/communication_systems/ssb-figure-of-merit-32qam site: BCT Engineering --- # SSB figure of merit and 32-QAM constellation Communication Systems · Chapter 9: Noise in Communication Systems - Asked in: 2081 Chaitra, 5+3 marks - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_6/communication_systems/ssb-figure-of-merit-32qam - Whole chapter: https://www.bctengineer.com/md/semester_6/communication_systems/chapter-9 ## Question Derive the expression for figure of merit for the SSB receiver. Define an efficient constellation diagram of a 32-QAM. ## Short answer SSB coherent receiver figure of merit is FOM=SNR_o/SNR_c=1, same as DSB-SC. An efficient 32-QAM constellation is a cross shape: a 6×6 square grid with the four corners removed, reducing average power while preserving minimum distance between points. ## Answer #### Figure of merit of SSB receiver **Figure of merit** $=\dfrac{\text{SNR}_o}{\text{SNR}_c}$, where $\text{SNR}_c$ is the channel SNR: average received signal power divided by noise power in the message bandwidth $W$. Coherent SSB receiver: BPF of width $W$ → product modulator with $\cos\omega_ct$ → LPF ($W$). Noise PSD $N_0/2$. **Signal:** $s(t)=\frac{A_c}{2}[m(t)\cos\omega_ct-\hat m(t)\sin\omega_ct]$ (USB). Since $m$ and $\hat m$ have equal power $P$ and are orthogonal: $$ P_s=\frac{A_c^2}{4}\cdot\frac{P}{2}\cdot2=\frac{A_c^2P}{4},\qquad \text{SNR}_c=\frac{A_c^2P}{4N_0W} $$ **Noise** after BPF (width $W$): $n(t)=n_c\cos\omega_ct-n_s\sin\omega_ct$ with $\overline{n_c^2}=\overline{n_s^2}=N_0W$. **Detector output** (multiply by $\cos\omega_ct$, LPF): $$ y(t)=\frac{A_c}{4}m(t)+\frac12n_c(t) $$ $$ \begin{aligned} S_o &= \frac{A_c^2P}{16},\qquad N_o=\frac{N_0W}{4} \\ \text{SNR}_o &= \frac{A_c^2P/16}{N_0W/4}=\frac{A_c^2P}{4N_0W} \end{aligned} $$ $$ \text{FOM}_{SSB}=\frac{\text{SNR}_o}{\text{SNR}_c}=1 $$ SSB has the same noise performance as DSB-SC and baseband, while using only half the bandwidth. #### Efficient constellation of 32-QAM In 32-QAM each symbol carries $\log_232=5$ bits. 32 is not a perfect square, so a square grid cannot be used. The efficient arrangement is the **cross constellation**: a $6\times6$ square grid of points (36) with the **4 corner points removed**, giving 32 points. Removing the corners (which have the highest energy) lowers the average and peak power for the same minimum distance $d$. ```text Q | . * * * * . * * * * * * * * * * * * -----------+------------ I * * * * * * * * * * * * . * * * * . | * = point (32), . = removed corner ``` Points lie at $I,Q\in\{\pm1,\pm3,\pm5\}\cdot\frac d2$ except the four corners $(\pm5,\pm5)$. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. How is the figure of merit defined? A. SNR_o divided by the SNR of the carrier B. SNR_o divided by the channel SNR in the message bandwidth W C. Bandwidth of output over input D. SNR_i divided by SNR_o Answer: B. FOM = SNR_o / SNR_c where SNR_c is received power over N0 W. (Key point: Figure of merit definition) 2. What is the figure of merit of coherent SSB? A. 1 B. 2 C. 1/3 D. 1/2 Answer: A. SNR_o = SNR_c = A^2 P/(4 N0 W), so FOM = 1. (Key point: SSB FOM value) 3. After multiplying by cos(wc t) and low-pass filtering, what does the SSB output contain? A. (A_c/2) m(t) + n_s(t) B. A_c m(t) + n_c(t) C. (A_c/4) m(t) + (1/2) n_c(t) D. (A_c/4) m(t) + (1/2) n_s(t) Answer: C. The coherent detector keeps m(t) and the in-phase noise n_c, each scaled by the product with cos. (Key point: Detector output) 4. How is the efficient 32-QAM constellation formed? A. A square 8x4 grid of 32 points B. Two concentric rings of 16 points C. A 5x7 grid with 3 points removed D. A 6x6 grid with the 4 corner points removed Answer: D. 36 - 4 = 32 points; removing high-energy corners lowers average and peak power for the same minimum distance. 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