Additive White Gaussian Noise (AWGN) is a fundamental model used in information theory, signal processing, and communications engineering to represent the influence of random, unavoidable noise on a signal. Each component of its name Additive, White, and Gaussian describes a core mathematical property of the noise.

1) Properties of AWGN

i. Additive: The Noise Combines by Addition

The term "additive" implies that the noise, $N(t)$, is simply added to the original signal, $S(t)$, to produce the received signal, $Y(t)$.

$$\begin{equation} Y(t) = S(t) + N(t) \end{equation}$$

This linear combination is the simplest way a signal can be corrupted. The model assumes that the noise is independent of the signal and is not influenced by it in any multiplicative or more complex way.

ii. Gaussian: The Amplitude Follows a Normal Distribution

The "Gaussian" property describes the statistical distribution of the noise's amplitude in the time domain. At any given moment, the value of the noise voltage or current follows a Gaussian (or Normal) probability distribution.

The Probability Density Function (PDF) for a Gaussian random variable with a mean ($\mu$) of zero and a variance of $\sigma^2$ is given by:

$$\begin{equation} f_N(n) = \frac{1}{\sqrt{2\pi\sigma^2}} \exp\left(-\frac{n^2}{2\sigma^2}\right) \end{equation}$$

In this context:

• Mean ($\mu = 0$): Since the noise is random, its amplitude is equally likely to be positive or negative, resulting in an average value of zero over time.
• Variance ($\sigma^2$): This represents the average power of the noise. A higher variance means greater noise power and a wider, flatter bell curve.

iii. White: Uniform Power in the Frequency Domain

The term "white" is an analogy to white light, which contains all frequencies of the visible spectrum in equal proportion. For a signal, "white" noise means its power is uniformly distributed across the entire frequency spectrum. This crucial property is described by the Power Spectral Density (PSD).

2) Autocorrelation of AWGN

For white noise, the signal is completely uncorrelated with its value at any other instant.

• For any time lag $\tau \neq 0$, the correlation is zero.
• For a time lag of $\tau = 0$, the correlation is perfect, and the value is the average power of the noise, $\sigma^2$.

This is mathematically modeled using the Dirac delta function, $\delta(\tau)$, which is an infinitely sharp spike at zero and zero everywhere else.

$$\begin{equation} R_N(\tau) = E[N(t)N(t+\tau)] = \frac{N_0}{2} \delta(\tau) \end{equation}$$

The Wiener-Khinchin Theorem establishes the link between the two domains: it states that the PSD is the Fourier Transform of the autocorrelation function. Taking the Fourier Transform of the delta function in the time domain results in a constant value in the frequency domain, perfectly describing the flat PSD of white noise.

3) Power Spectral Density (PSD) of White Noise

The Power Spectral Density (PSD), denoted $S_N(f)$, describes how the power of a signal is distributed as a function of frequency. For ideal white noise, the PSD is a constant for all frequencies, from negative infinity to positive infinity.

The PSD is formally given by:

$$\begin{equation} S_N(f) = \frac{N_0}{2} \quad (\text{for all } f) \end{equation}$$

Here, $N_0$ is a fundamental constant representing the power spectral density in Watts per Hertz (W/Hz). The factor of 2 in the denominator accounts for the two-sided nature of the frequency spectrum (both positive and negative frequencies).

Significance of a Constant PSD:

• Uniform Power Distribution: A flat PSD means that the noise contributes an equal amount of power at every frequency. This is why filtering is effective: by restricting the frequency range (the bandwidth), we can reduce the total amount of noise power affecting the signal.
• Infinite Power (Theoretical): An ideal white noise signal with a perfectly flat PSD over an infinite bandwidth would have infinite total power. In reality, all physical systems have a finite bandwidth, so the noise power is always finite.

Calculating Noise Power in a Bandwidth: The total noise power, $P_N$, within a specific frequency bandwidth, $B$, can be calculated by integrating the PSD over that bandwidth. For a system with a bandwidth $B$, the noise power is:

$$\begin{equation} P_N = \int_{-B/2}^{B/2} S_N(f) \, df = \int_{-B/2}^{B/2} \frac{N_0}{2} \, df = \frac{N_0}{2} \times B \end{equation}$$

For a passband system (using only positive frequencies), the power is often expressed as:

$$\begin{equation} P_N = N_0 \times B \end{equation}$$

This simple, linear relationship shows that the total noise power in a system is directly proportional to its bandwidth.

4) Where is AWGN Found?

AWGN is not just a theoretical convenience; it is an accurate model for many real-world sources of random noise, including:

• Thermal Noise (Johnson-Nyquist Noise): Arises from the thermal agitation of charge carriers (usually electrons) inside an electrical conductor at temperatures above absolute zero. This is a primary source of noise in electronic components like resistors, amplifiers, and sensors.
• Shot Noise: Occurs due to the discrete nature of electric charge in electronic devices like diodes and transistors. The random arrival of electrons or holes at a junction creates a fluctuating current.
• Atmospheric and Deep Space Noise: Electromagnetic waves radiated by the Earth's atmosphere, the sun, and other celestial objects (like black-body radiation) are often modeled as AWGN, especially in satellite and deep-space communication links.