Noise in AM Systems

Noise performance of an analog communication system is measured not just by how much noise is present, but by how much the system degrades the signal-to-noise ratio compared to a reference baseband system. The figure of merit, defined as the ratio of output SNR to input SNR, captures this degradation in a single number. For amplitude modulation systems, the figure of merit depends critically on the type of AM used and the detection method employed, making this topic a central area of analysis in analog communication theory and GATE examinations.

SNR Definitions and the Reference Baseband System

Before comparing AM systems, a common reference is needed. The baseband reference system is defined as a system that transmits the message m(t) directly over the channel with the same total transmitted power and same noise power spectral density N_0/2. Its output SNR is SNR_baseband = S_T / (N_0 W), where S_T is the transmitted power and W is the message bandwidth. All AM system figures of merit are expressed as the ratio of the system output SNR to this baseband SNR.

The figure of merit (FOM) is defined as SNR_o / SNR_baseband. A FOM of 1 means the system performs exactly as well as direct baseband transmission. A FOM less than 1 means the modulation scheme incurs a noise penalty. A FOM greater than 1 would indicate a gain over baseband, which is achievable by wideband systems like FM but not by AM.

Coherent Detection of DSB-SC

In double-sideband suppressed carrier (DSB-SC) modulation, the transmitted signal is s(t) = A_c m(t) cos(2 pi f_c t). With coherent detection (multiplying the received signal by a local carrier and lowpass filtering), the output SNR is SNR_o = A_c^2 P_m / (2 N_0 W), where P_m is the message power. The transmitted power is S_T = A_c^2 P_m / 2. Comparing with the baseband reference, the FOM for DSB-SC with coherent detection equals 1. This means DSB-SC coherent detection has no noise penalty relative to baseband, despite occupying twice the bandwidth.

Coherent Detection of DSB-FC (Conventional AM)

In conventional AM (DSB-FC), the transmitted signal is s(t) = A_c [1 + mu m_n(t)] cos(2 pi f_c t), where mu is the modulation index and m_n(t) is the normalized message with peak value 1. The carrier component A_c cos(2 pi f_c t) carries no information but consumes power. The useful signal power fraction is mu^2 P_mn / (1 + mu^2 P_mn), where P_mn is the normalized message power. For a sinusoidal message with mu = 1, P_mn = 0.5 and the efficiency is 0.5 / (1 + 0.5) = 1/3. The figure of merit for DSB-FC with coherent detection is FOM = mu^2 P_mn / (1 + mu^2 P_mn), which is always less than 1. This is the noise penalty paid for the wasted carrier power.

Coherent Detection of SSB

In single-sideband (SSB) modulation, only one sideband is transmitted, so the bandwidth is W instead of 2W. The noise collected at the receiver input is proportional to bandwidth, so SSB collects half the noise of DSB-SC for the same bandwidth. However, SSB also has half the signal power when compared on equal transmitted power. These effects cancel exactly, giving SSB a figure of merit of 1, identical to DSB-SC. SSB does not improve SNR over DSB-SC but it achieves the same SNR with half the bandwidth, making it spectrally efficient.

Envelope Detection and the Threshold Effect

Envelope detection of conventional AM avoids the need for a phase-locked local carrier. At high SNR above threshold, envelope detection performs identically to coherent detection, yielding the same FOM = mu^2 P_mn / (1 + mu^2 P_mn). However, below a certain input SNR threshold (typically around 10 dB), envelope detection breaks down abruptly. The envelope detector loses track of the signal and noise dominates, causing a rapid collapse of output SNR. This is analogous to, but less severe than, the FM threshold effect.

Numerical Example

A conventional AM (DSB-FC) system uses a sinusoidal message and a modulation index of 0.8. We calculate the figure of merit and compare it to DSB-SC to understand the noise penalty introduced by the carrier.

Given:
Modulation type: DSB-FC (Conventional AM)
Modulation index: μ = 0.8
Message: sinusoidal, so P_mn = 1/2 = 0.5

Why this formula applies:
FOM for DSB-FC coherent = useful signal power fraction of total transmitted power.

Formula:
FOM = μ² P_mn / (1 + μ² P_mn)

Substitution:
FOM = (0.8)² × 0.5 / (1 + (0.8)² × 0.5)
    = 0.64 × 0.5 / (1 + 0.64 × 0.5)
    = 0.32 / (1 + 0.32)

Calculation:
FOM = 0.32 / 1.32

Final Answer: FOM ≈ 0.242 (24.2%)
The DSB-FC system is 75.8% worse than DSB-SC or baseband for the same transmitted power.

Exam Tip: For GATE, remember the three figures of merit: DSB-SC = 1, SSB = 1, DSB-FC = mu^2 P_mn / (1 + mu^2 P_mn). For a sinusoidal message with mu = 1, FOM of DSB-FC = 1/3. A common trap is forgetting that both SSB and DSB-SC achieve FOM = 1, meaning SSB does not outperform DSB-SC in SNR terms.

Quick Revision

• Figure of merit = SNR_o / SNR_baseband. Reference: baseband system with same S_T and noise spectral density N_0/2.
• DSB-SC coherent detection: FOM = 1. No noise penalty despite using 2W bandwidth.
• SSB coherent detection: FOM = 1. Same noise performance as DSB-SC, half the bandwidth.
• DSB-FC coherent detection: FOM = mu^2 P_mn / (1 + mu^2 P_mn). Always less than 1 due to carrier power waste.
• For sinusoidal message, mu = 1 gives P_mn = 0.5 and FOM of DSB-FC = 1/3.
• Envelope detection matches coherent detection for DSB-FC above threshold. Below threshold, SNR collapses rapidly.
• Trap: SSB and DSB-SC have identical figures of merit = 1. SSB advantage is bandwidth efficiency, not SNR.