Signal-to-noise ratio (SNR)

The signal-to-noise ratio (SNR) is the dimensionless ratio of signal power (or amplitude squared) to noise power in a measurement. It is the universal figure of merit for any measurement, sensor, or communication channel. The larger the SNR, the better the system's ability to distinguish signal from noise.

Definitions. Two equivalent forms:

$$\text{SNR}=\frac{P_{\text{signal}}}{P_{\text{noise}}}=\frac{\mathrm{\mid }{V}_{\text{signal}}{\mathrm{\mid }}^2}{\mathrm{\mid }{V}_{\text{noise}}{\mathrm{\mid }}^2},$$

where powers are time-averaged. SNR is typically expressed in decibels:

$$\begin{array}{rl}{\displaystyle {\text{SNR}}_{\text{dB}}}& {\displaystyle =10{\log }_{10}\text{\hspace{0.17em}⁣}\left(\frac{P_{\text{signal}}}{P_{\text{noise}}}\right)}\\ {\displaystyle }& {\displaystyle =20{\log }_{10}\text{\hspace{0.17em}⁣}\left(\frac{V_{\text{signal}}}{V_{\text{noise}}}\right)\mathrm{.}}\end{array}$$

Standard SNR levels and their implications.

SNR vs noise figure. Noise figure (NF) of an amplifier or receiver describes how the system degrades input SNR:

$${\text{NF}}_{\text{dB}}={\text{SNR}}_{\text{in,dB}}-{\text{SNR}}_{\text{out,dB}}\mathrm{.}$$

A noise-free amplifier has NF = 0 dB; real amplifiers have NF $>$ 0 dB. EDFAs typically have NF = 4 – 6 dB; low-noise electrical amplifiers achieve NF < 1 dB.

SNR for an optical receiver. Combining the principal noise sources (shot, thermal, dark current, etc.) for a photodiode with photocurrent $I_s=RP$:

$$\text{SNR}=\frac{I_s^2}{[2q(I_s+I_d)+4k_{B}T\mathrm{/}{R}_L+i_{\text{TIA}}^2]\mathrm{\Delta }f},$$

where $I_d$ is dark current, $R_L$ is the load resistance, and $i_{\text{TIA}}$ is TIA input-referred current noise.

In two limiting regimes:

• Thermal-noise-limited (low signal): $\text{SNR}\propto P_s^2$ (doubling signal quadruples SNR)
• Shot-noise-limited (high signal): $\text{SNR}\propto P_s$ (doubling signal doubles SNR)

The crossover signal power is typically 10 – 100 μW for telecom receivers. The Photodiode Noise Calculator evaluates this expression, with avalanche gain and excess noise for an APD, and plots SNR against optical power alongside the shot-noise limit.

SNR and bit error rate. For binary on-off keying (OOK) and assuming Gaussian noise, the bit error rate (BER) relates to SNR through the Q-factor:

$$\text{BER}=\frac{1}{2}\text{erfc}\text{\hspace{0.17em}⁣}\left(\frac{Q}{\sqrt{2}}\right),Q=\sqrt{\text{SNR}}\mathrm{.}$$

Specific BER-vs-Q values:

For coherent receivers carrying QPSK or QAM, OSNR is the more useful metric than electrical SNR, and the relationship between the two is nonlinear.

SNR penalty and improvement. System-level engineering tracks how each element adds noise or degrades SNR:

• Pre-detection optical amplifier (EDFA): improves SNR by lifting signal above receiver thermal noise floor
• Multiple averaging: improves SNR by a factor $N$ in power ($\sqrt{N}$ in amplitude) for $N$ independent measurements
• Coherent detection: improves SNR by LO-amplification of weak signal
• Lock-in detection: improves SNR by narrowing measurement bandwidth
• DSP and equalization: can recover SNR lost to deterministic distortions (ISI, dispersion)

SNR vs OSNR. In optical communications, two complementary metrics exist:

• OSNR: optical signal-to-noise ratio, measured at the optical input of the receiver before detection. Quantifies the optical signal against ASE noise from upstream amplifiers.
• SNR (electrical): ratio at the electrical output, including detector and electronics noise.

When signal-ASE beat noise dominates, as in an amplified direct-detection link, the electrical SNR follows the OSNR, roughly ${\text{SNR}}_{\text{el}}\approx \text{OSNR}{B}_{\text{ref}}\mathrm{/}(2\mathrm{\Delta }f)$ for an electrical bandwidth $\mathrm{\Delta }f$; when thermal or shot noise dominates, the electrical SNR is set by the received power and the receiver, and the two are decoupled.

SNR improvement techniques.

SNR in single-photon counting. For dim signals, the signal "level" is the photon arrival rate $r_s$ [counts/s]; the noise consists of detector dark counts $r_d$ and any background light counts $r_b$. The SNR is:

$${\text{SNR}}_{\text{counting}}=\frac{r_s^2\cdot T}{r_s+r_d+r_b},$$

where $T$ is the integration time. For shot-noise-limited counting (dark/background negligible): $\text{SNR}=r_{s}T$ = total number of photons detected. Reaching 60 dB SNR (${10}^6$) requires ${10}^6$ photons, the same as in the bright limit.

References: Saleh & Teich, Fundamentals of Photonics (3rd ed., 2019), Ch. 18 (photodetection); Agrawal, Fiber-Optic Communication Systems (4th ed., 2010), Ch. 4; Razavi, Design of Integrated Circuits for Optical Communications (2nd ed., 2012) for the IC noise treatment.