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:

$$\text{SNR} \;=\; \frac{P_\text{signal}}{P_\text{noise}} \;=\; \frac{|V_\text{signal}|^2}{|V_\text{noise}|^2},$$

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

$$\text{SNR}_\text{dB} \;=\; 10 \log_{10}\!\left(\frac{P_\text{signal}}{P_\text{noise}}\right) \;=\; 20 \log_{10}\!\left(\frac{V_\text{signal}}{V_\text{noise}}\right).$$

Standard SNR levels and their implications.

SNR (dB)	SNR (linear)	Quality / use
60	$10^6$	High-quality audio, instrumentation
50	$10^5$	Studio-quality recording
40	$10^4$	Typical communication standard
30	$10^3$	Acceptable for BER $< 10^{-12}$ at 10 Gb/s
20	100	Threshold for many digital communications
17	50	Q-factor 7; BER $\sim 10^{-12}$
13	20	BER threshold with forward error correction
10	10	Detection threshold for many radar / lidar
6	4	Near sensitivity limit for typical optical receivers
0	1	Signal equal to noise; not useful

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

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

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 $I_s = RP$:

$$\text{SNR} \;=\; \frac{I_s^2}{2q(I_s + I_d) \Delta f + 4 k_B T \Delta f / R_L + i_\text{TIA}^2 \Delta f},$$

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.

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:

$$\text{BER} \;=\; \frac{1}{2} \text{erfc}\!\left( \frac{Q}{\sqrt{2}} \right), \qquad Q \;=\; \sqrt{\text{SNR}}.$$

Specific BER-vs-Q values:

BER	Q (linear)	SNR (dB)
$10^{-3}$	3.09	9.8
$10^{-6}$	4.75	13.5
$10^{-9}$	6.0	15.6
$10^{-12}$	7.04	16.9
$10^{-15}$	7.96	18.0

For modern coherent receivers with PAM4 modulation, OSNR is the more useful metric than electrical SNR; relationships are 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 $\sqrt{N}$ 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.

For shot-noise-limited detection, OSNR and electrical SNR are simply related; for thermal-noise-limited detection, they are decoupled.

SNR improvement techniques.

Technique	Mechanism	SNR improvement
Optical amplification	Reduce relative impact of thermal noise	up to ~25 dB
Coherent detection	LO amplification	Up to ~ 20 dB vs direct
Lock-in detection	Narrowband filtering	$\sqrt{BW_\text{wide}/BW_\text{narrow}}$, often 30 – 50 dB
Boxcar averaging	Time-gated integration	Like lock-in for pulsed signals
Spatial filtering	Reject off-axis noise	10 – 30 dB depending on geometry
Polarization filtering	Reject orthogonal polarization noise	up to 30 dB
Cooling the detector	Reduce thermal/dark current noise	5 – 30 dB depending on temperature

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. To reach 60 dB SNR ($10^6$), you need $10^6$ photons — 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.