Signal-to-Noise Ratio (SNR)

The Signal-to-Noise Ratio (SNR) quantifies the quality of a signal by comparing its power to the power of background noise. It is expressed in decibels (dB), with higher values indicating cleaner, less noisy signals.

Calculation of SNR

SNR is defined as:

SNR = P_signal / P_noise

In decibels:

SNR (dB) = 10 × log₁₀(P_signal / P_noise)
or
SNR (dB) = 10 × log₁₀(P_signal) – 10 × log₁₀(P_noise)

For voltage-based measurements:

SNR (dB) = 20 × log₁₀(V_signal / V_noise)
(The factor 20 accounts for the squared relationship between voltage and power.)

SNR vs. Carrier-to-Noise Ratio (C/N)

Although related, SNR and Carrier-to-Noise Ratio (C/N) are distinct:

• SNR considers the full signal power (including modulation) relative to noise and is essential in digital communication and data integrity.

• C/N compares only the carrier wave’s power to noise, commonly used in RF and satellite systems.

If signal and noise occupy different bandwidths, the two are related by:

SNR = C/N × (B_n / B_s)

where:

• B_n = noise bandwidth

• B_s = signal bandwidth

Example Applications

• SNR = 20 dB → signal power is 100× stronger than noise

• Digital TV requires ≥15 dB for stable reception

• CD-quality audio typically achieves an SNR of ~90 dB, enabling high-fidelity playback

Significance of SNR

SNR plays a critical role in various technical domains:

• Telecommunications: Determines maximum channel capacity according to the Shannon-Hartley theorem

• Wireless Systems: Defines coverage range and data throughput

• Audio Engineering: Determines clarity in music and voice transmission

• Imaging & Sensing: Affects resolution in medical imaging, radar, and sensors

In practice, SNR is often the key factor determining system performance, signal clarity, and measurement reliability.