Signal to Noise Ratio

Signal-to-Noise Ratio (SNR) is the most direct measure of signal quality in any communication system. It quantifies how much stronger the desired signal is compared to the background noise at a given point in the system. Every design decision in a receiver, from filter choice to amplifier gain, ultimately aims to maximize SNR at the detector.

In GATE and university exams, SNR appears in problems on modulation comparison, noise figure, demodulator performance, and link budget calculations. A clear understanding of input SNR, output SNR, and their dB representation is essential.

Core Concept Explanation

SNR is defined as the ratio of signal power to noise power at the same point in a system. At the input of a receiver, the signal arrives from the channel along with channel noise. At the output of the receiver, the demodulated signal exists alongside noise contributed by both the channel and the receiver hardware.

The input SNR is calculated using signal and noise powers at the receiver input port. The output SNR is calculated after demodulation, at the point where the final baseband signal is recovered. These two quantities are generally different because the receiver's processing and internal noise change the balance between signal and noise power.

In analog modulation systems such as AM and FM, the output SNR depends on the modulation type and the demodulation method. FM demodulators, for example, can actually improve SNR compared to input when the carrier-to-noise ratio exceeds a threshold, due to the FM capture effect and threshold behavior.

Mathematical Expression

Linear SNR is defined as the power ratio: SNR = S/N = Signal Power / Noise Power. Both S and N must be in the same units (watts or milliwatts) and measured at the same point in the circuit.

The logarithmic form is: SNR(dB) = 10 log10(S/N). This is always expressed in decibels (dB). A 3 dB SNR corresponds to a power ratio of 2:1, meaning signal is twice as strong as noise. A 10 dB SNR means the signal is 10 times the noise power.

For communication systems comparison, the figure of merit used is often the output SNR as a function of the transmit power or Eb/N0, where Eb is energy per bit and N0 is one-sided noise PSD. The relationship between SNR and bit error rate (BER) determines the practical usefulness of a modulation scheme.

Practical Understanding

For a standard DSB-SC (double sideband suppressed carrier) AM system, the output SNR equals the input SNR, giving no SNR advantage. For SSB-AM, the same result holds. For conventional AM with modulation index m, the output SNR is lower than DSB-SC because power is wasted in the carrier.

FM provides a significant SNR advantage over AM above the threshold. The output SNR for FM is 3A^2 mf^2 (mf + 1) times the baseband SNR for a sinusoidal message, where mf is the modulation index. This is why FM is preferred for high-quality audio broadcasting despite requiring more bandwidth.

Given:
Signal power Si = 2 mW, Noise power Ni = 0.02 mW at receiver input

Why this formula applies:
SNR is the ratio of signal to noise power; dB form uses log10

Formula:
SNR = Si / Ni
SNR(dB) = 10 log10(SNR)

Substitution:
SNR = 2 mW / 0.02 mW = 100

Calculation:
SNR(dB) = 10 log10(100) = 10 x 2 = 20 dB

Final Answer with units:
SNR = 100 (linear) = 20 dB

Exam Tip: When comparing AM and FM in GATE questions, remember that FM output SNR = 3 mf^2 x (input SNR for baseband) for sinusoidal modulation. This FM SNR improvement comes at the cost of bandwidth, consistent with the bandwidth-SNR tradeoff principle.

Key Points on SNR Analysis

• Input SNR = Si/Ni, where Si and Ni are signal and noise powers at the receiver input port.
• Output SNR = So/No, measured after demodulation at the baseband output stage.
• SNR(dB) = 10 log10(SNR). Each 3 dB improvement doubles the SNR ratio. Each 10 dB is a factor of 10.
• DSB-SC and SSB have the same SNR performance; DSB-SC uses twice the bandwidth of SSB for the same message.
• FM threshold: below a certain input SNR, FM output SNR collapses sharply — this is a key GATE exam topic.

Quick Revision

• SNR = S/N (linear) = 10 log10(S/N) in dB.
• Input SNR: measured at receiver input. Output SNR: measured after demodulation.
• DSB-SC, SSB: SNRo/SNRi = 1. Conventional AM: SNRo/SNRi < 1. FM: SNRo = 3mf^2(mf+1) x SNRi.
• 3 dB = factor of 2; 10 dB = factor of 10; 20 dB = factor of 100 in linear SNR.
• FM improves SNR above threshold but degrades catastrophically below it.
• Exam trap: SNR in dB uses 10 log10 (power ratio), not 20 log10 — 20 log10 is used only for voltage or amplitude ratios.
• Higher SNR means lower bit error rate in digital systems and better audio quality in analog systems.