Every communication system exists in a battle against noise. Random fluctuations—thermal agitation in electronics, atmospheric disturbances, electromagnetic interference—continuously corrupt transmitted signals. But not all modulation schemes suffer equally in this battle.

Amplitude Modulation presents a fundamental vulnerability: because information is encoded in the signal's amplitude, any noise that alters that amplitude directly corrupts the information. This stands in stark contrast to frequency modulation (FM) and phase modulation (PM), where amplitude variations can often be removed without losing information.

Understanding AM's noise sensitivity is not merely academic—it explains why FM replaced AM for music broadcasting, why digital systems moved toward phase-based modulation, and why modern high-speed links use sophisticated amplitude correction techniques.

Before analyzing how noise affects AM, we must understand what noise is and where it comes from.

Noise: A Definition

Noise is any unwanted signal that combines with the desired signal, obscuring or corrupting the information. In communication systems, we distinguish between:

1. Internal Noise (Random, Fundamental)

• Thermal Noise (Johnson-Nyquist): Random electron motion in resistive materials $$P_n = kTB$$ Where k = Boltzmann's constant, T = temperature (Kelvin), B = bandwidth (Hz)

• Shot Noise: Discrete nature of electron/photon flow $$i_n^2 = 2qI_dB$$ Where q = electron charge, I_d = DC current

2. External Noise (Environmental)

• Atmospheric noise (lightning, static)
• Cosmic noise (sun, galactic sources)
• Man-made interference (motors, switches, other electronics)
• Interference from other communication systems

Additive White Gaussian Noise (AWGN):

For mathematical analysis, we typically model noise as AWGN:

• Additive: Noise adds to signal: r(t) = s(t) + n(t)
• White: Flat power spectral density: S_n(f) = N_0/2 watts/Hz
• Gaussian: Amplitude follows normal distribution with zero mean

$$p(n) = \frac{1}{\sqrt{2\pi\sigma^2}} e^{-\frac{n^2}{2\sigma^2}}$$

Where σ² is the noise variance (power).

Why AWGN Matters:

AWGN is not just a convenient model—it represents the fundamental noise floor of any communication system. Thermal noise in receivers is genuinely AWGN up to very wide bandwidths. Even when other noise sources dominate, AWGN analysis provides bounds on achievable performance.

Now let's analyze precisely how noise corrupts an AM signal through the demodulation process.

The Received AM Signal with Noise:

$$r(t) = s_{AM}(t) + n(t)$$

Where: $$s_{AM}(t) = A_c[1 + \mu m_n(t)]\cos(2\pi f_c t)$$

and n(t) is bandlimited AWGN around the carrier frequency.

Representing Narrowband Noise:

Noise near the carrier can be decomposed into in-phase and quadrature components:

$$n(t) = n_I(t)\cos(2\pi f_c t) - n_Q(t)\sin(2\pi f_c t)$$

Both n_I(t) and n_Q(t) are independent Gaussian processes with the same power as n(t).

The Complete Received Signal:

$$r(t) = [A_c(1 + \mu m_n(t)) + n_I(t)]\cos(2\pi f_c t) - n_Q(t)\sin(2\pi f_c t)$$

This can be written in envelope-phase form:

$$r(t) = E(t)\cos(2\pi f_c t + \theta(t))$$

Where the envelope is:

$$E(t) = \sqrt{[A_c(1 + \mu m_n(t)) + n_I(t)]^2 + n_Q^2(t)}$$

The Critical Insight:

An envelope detector outputs E(t), which is supposed to equal A_c[1 + μm(t)] but actually includes noise terms. The noise distorts both the amplitude (through n_I) and introduces spurious phase fluctuations (through n_Q).

High SNR Approximation:

When signal power >> noise power, n_Q terms become negligible:

$$E(t) \approx A_c(1 + \mu m_n(t)) + n_I(t)$$

The noise simply adds to the signal—a 1:1 degradation.

Let's quantify the noise performance of AM through rigorous SNR calculations.

Definitions:

• Channel SNR (SNR_c): Ratio of total signal power to noise power in the channel bandwidth
• Output SNR (SNR_o): Ratio of recovered message power to noise power after demodulation
• Figure of Merit: Ratio of output SNR to input SNR, measuring modulation efficiency

For Standard AM (DSB-FC):

Transmitted Signal Power: $$P_t = P_c\left(1 + \frac{\mu^2}{2}\right) = \frac{A_c^2}{2}\left(1 + \frac{\mu^2}{2}\right)$$

For sinusoidal modulation at 100% modulation (μ = 1): $$P_t = 1.5 P_c$$

Power in Message (Sidebands Only): $$P_{message} = \frac{\mu^2 P_c}{2} = \frac{\mu^2 A_c^2}{4}$$

Output SNR Calculation:

After envelope detection (high SNR case):

$$SNR_o = \frac{\mu^2 A_c^2 / 2}{2N_0 B_m} = \frac{\mu^2 P_c}{N_0 B_m}$$

Where B_m is the message bandwidth.

The Figure of Merit for AM:

$$\eta_{AM} = \frac{SNR_o}{SNR_c} = \frac{\mu^2}{1 + \mu^2/2}$$

Critical Finding: Even at 100% modulation, AM's figure of merit is only 0.667—meaning output SNR is 1.76 dB worse than input SNR. At lower modulation indices, the penalty is severe.

The transition from AM to FM broadcasting for music was driven largely by noise performance. Let's quantify the difference.

FM's Fundamental Advantage:

In FM, information is in the frequency (or phase), not amplitude. A hard limiter at the receiver clips the amplitude to a constant level—removing all amplitude noise before demodulation.

FM Figure of Merit:

$$\eta_{FM} = 3\beta^2\left(\frac{B_m}{B_n}\right) \times SNR_c$$

Where:

• β = modulation index (frequency deviation / message frequency)
• B_m = message bandwidth
• B_n = noise bandwidth

For wideband FM with β = 5 (typical for broadcasting):

$$\eta_{FM} = 3 \times 25 = 75$$

FM provides SNR improvement of 75 times (18.75 dB!) over a baseband system with the same power.

The Bandwidth Trade-Off:

FM's noise advantage comes at the cost of bandwidth. Carson's rule gives FM bandwidth:

$$B_{FM} = 2(\beta + 1)B_m$$

For β = 5, B_m = 15 kHz (music): $$B_{FM} = 2(5 + 1)(15) = 180 \text{ kHz}$$

Compared to AM's 30 kHz, FM uses 6× more spectrum. This is why:

• AM broadcasting occupies 540-1700 kHz (MF band) with 10 kHz channel spacing
• FM broadcasting occupies 88-108 MHz (VHF band) with 200 kHz channel spacing

The Shannon Connection:

This trade-off is fundamental. Shannon's capacity theorem shows that bandwidth and SNR can be traded:

$$C = B\log_2(1 + SNR)$$

FM trades bandwidth for SNR improvement—perfectly valid within Shannon's framework.

Digital ASK (including OOK) has a different noise analysis because we're concerned with bit errors rather than output SNR.

The Decision Mechanism:

At each sampling instant, the receiver measures the signal amplitude and compares against a threshold:

$$\hat{b} = \begin{cases} 1 & \text{if } V_{sample} > V_{th} \ 0 & \text{if } V_{sample} \leq V_{th} \end{cases}$$

Error Occurs When:

1. A '1' was sent but noise pushed the sample below threshold (miss)
2. A '0' was sent but noise pushed the sample above threshold (false alarm)

For Gaussian Noise:

$$P(error|1) = Q\left(\frac{V_1 - V_{th}}{\sigma_n}\right)$$ $$P(error|0) = Q\left(\frac{V_{th} - V_0}{\sigma_n}\right)$$

Where σ_n is the noise standard deviation.

Optimal Threshold:

For equal prior probabilities and equal noise variance (Gaussian, no fading):

$$V_{th,opt} = \frac{V_1 + V_0}{2}$$

But in real systems:

• Fading changes V_1 dynamically
• DC wander shifts V_0
• Non-uniform bit probabilities may apply

Adaptive Thresholding Techniques:

While Gaussian noise degrades all modulations, impulse noise creates a particularly severe problem for AM.

Impulse Noise Characteristics:

• Very short duration (microseconds to milliseconds)
• Very high amplitude (10-100× normal noise)
• Broadband spectrum (affects all frequencies simultaneously)
• Sources: lightning, switching transients, engine ignition, digital circuit transitions

Impact on AM:

An impulse superimposed on an AM signal creates a spike in the envelope:

• Heard as a sharp "click" or "pop" in audio
• Directly audible since amplitude = information
• No natural mechanism to remove it

Impact on FM:

The same impulse in FM:

• Briefly affects envelope (clipped by limiter)
• Causes momentary phase/frequency disturbance
• Heard as brief "tick" (much less annoying)
• Can be largely suppressed by blanking circuits

Shannon's information theory provides fundamental limits on reliable communication in noisy channels. These limits apply to all modulation schemes, including AM.

Shannon-Hartley Theorem:

$$C = B \log_2\left(1 + \frac{S}{N}\right)$$

Where:

• C = channel capacity (bits per second)
• B = bandwidth (Hz)
• S = signal power (watts)
• N = noise power (watts)

Implications for AM:

With fixed power budget P and bandwidth B, maximum achievable rate is limited by noise N.

Example:

AM broadcast channel: B = 10 kHz, SNR = 30 dB (1000:1)

$$C = 10,000 \times \log_2(1001) \approx 99,700 \text{ bps}$$

Theoretically, ~100 kbps could be transmitted over an AM broadcast channel—far exceeding the ~20 kbps needed for voice. But AM doesn't approach this limit because it's not designed for capacity efficiency.

The Efficiency Perspective:

Shannon capacity tells us what's theoretically possible. The ratio of actual data rate to capacity measures efficiency:

$$\eta = \frac{R_{actual}}{C}$$

Key Insight: Modern digital systems approach Shannon capacity because they use sophisticated coding and modulation. Traditional analog AM/FM are far from efficient—they trade capacity for simplicity.

We've thoroughly analyzed amplitude modulation's noise vulnerability from multiple perspectives. Let's consolidate the key insights:

What's Next:

Having understood AM's theoretical foundations, practical implementations, and fundamental limitations, we'll conclude with Applications of Amplitude Modulation—exploring where AM/ASK remains preferred despite its noise sensitivity, from AM broadcasting to fiber optics, and understanding why context determines modulation choice.