Ferkans — Interactive Telecom Tutor

Definition:
White Gaussian Noise (WGN)

A white Gaussian noise process $\{N(t)\}$ is a zero-mean Gaussian process with autocorrelation $R_N(\tau) = \frac{N_0}{2}\,\delta(\tau)$ where $N_0$ is the (one-sided) noise power spectral density and $\delta(\tau)$ is the Dirac delta function. Equivalently, its PSD is flat: $P_N(f) = \frac{N_0}{2}, \quad -\infty < f < \infty.$

The term "white" comes from optics: just as white light contains all frequencies equally, WGN has equal power at every frequency. The constant $N_0/2$ is the two-sided PSD; the one-sided PSD (for $f \geq 0$ only) is $N_0$ .

,

White Gaussian Noise Cannot Be Realized

The total power of WGN is $\int_{-\infty}^{\infty} P_N(f)\,df = \infty$ : infinite power in any finite interval of time. This means WGN has infinite variance at every time instant and its "sample paths" do not exist as ordinary functions. WGN is a generalized process (a random distribution in the sense of Schwartz).

In practice, we always encounter WGN through a filter. Any filtered version — even through an ideal lowpass with finite bandwidth — produces a valid (finite-variance) Gaussian process. WGN is the idealized input model; only its filtered outputs have physical meaning.

Theorem: WGN Samples at Distinct Times Are Independent

If $\{N(t)\}$ is white Gaussian noise, then for any $t_1 \neq t_2$ , $\text{Cov}(N(t_1), N(t_2)) = C_N(t_1 - t_2) = \frac{N_0}{2}\,\delta(t_1 - t_2) = 0.$ Since $N(t_1)$ and $N(t_2)$ are jointly Gaussian and uncorrelated, they are independent.

For Gaussian random variables, uncorrelated implies independent. WGN has zero correlation between any two distinct time instants, so all samples are mutually independent. This is the continuous-time analog of i.i.d. Gaussian noise.

Proof

Uncorrelated from definition

By definition, $R_N(\tau) = (N_0/2)\delta(\tau)$ . For $\tau \neq 0$ , $\delta(\tau) = 0$ , so $\text{Cov}(N(t_1), N(t_2)) = 0$ whenever $t_1 \neq t_2$ .

Independence from Gaussianity

The pair $(N(t_1), N(t_2))$ is jointly Gaussian (by definition of a GP). For jointly Gaussian RVs, zero covariance implies independence. $\blacksquare$

Definition:
Band-Limited White Gaussian Noise

Band-limited WGN is a zero-mean Gaussian process $\{N_B(t)\}$ with PSD $P_{N_B}(f) = \begin{cases} N_0/2, & |f| \leq W \\ 0, & |f| > W \end{cases}$ Its autocorrelation is $R_{N_B}(\tau) = N_0W\,\text{sinc}(2W\tau)$ and its power is $\mathcal{P}_{N_B} = N_0W$ (finite).

Unlike ideal WGN, band-limited WGN has continuous sample paths and finite power. It is the physically realizable model.

Example: Autocorrelation of Band-Limited WGN

Derive the autocorrelation $R_{N_B}(\tau)$ of band-limited WGN with bandwidth $W$ and verify that $R_{N_B}(0)$ gives the total power.

Solution

Inverse Fourier transform of the PSD

$R_{N_B}(\tau) = \int_{-\infty}^{\infty} P_{N_B}(f)\,e^{j2\pi f\tau}\,df = \frac{N_0}{2}\int_{-W}^{W} e^{j2\pi f\tau}\,df$ $

Evaluate the integral

$= \frac{N_0}{2} \cdot \frac{e^{j2\piW\tau} - e^{-j2\piW\tau}}{j2\pi\tau} = \frac{N_0}{2} \cdot \frac{2\sin(2\piW\tau)}{2\pi\tau} = N_0W\,\text{sinc}(2W\tau)$ $

Check total power

$R_{N_B}(0) = N_0W\,\text{sinc}(0) = N_0W$ , matching $\int_{-W}^{W} (N_0/2)\,df = N_0W$ . $\checkmark$

Definition:
Discrete-Time i.i.d. Gaussian Noise

The discrete-time analog of WGN is an i.i.d. sequence $\{N[n]\}$ where each $N[n] \sim \mathcal{N}(0, \sigma^2)$ independently. Its autocorrelation is $r_{NN}[m] = \sigma^2\,\delta[m]$ and its PSD is the constant $P_N(f) = \sigma^2$ for $|f| \leq 1/2$ .

This is the canonical noise model in digital communications and signal processing. Unlike continuous-time WGN, it is a perfectly well-defined random sequence with finite variance $\sigma^2$ at each sample.

White Gaussian Noise Through a Filter

Visualize the input WGN PSD (flat) and the output PSD after passing through an LTI filter. Observe how the filter shapes the noise spectrum and reduces total power.

Parameters

Filter type

f_c

(Hz)3

N_0

1

Common Mistake: Do Not Compute the Variance of WGN at a Single Point

Mistake:

Writing $\text{Var}(N(t)) = R_N(0) = (N_0/2)\delta(0) = \infty$ and then attempting to use this "variance" in computations like SNR calculations.

Correction:

WGN does not have a well-defined variance at a single point — it is infinite. Finite quantities emerge only after filtering: if you pass WGN through a filter with noise bandwidth $B_N$ , the output noise power is $N_0 B_N$ , which is finite. Always work with filtered noise when computing physical quantities.

Quick Check

What is the power spectral density of white Gaussian noise with parameter $N_0$ ?

$P_N(f) = N_0$ for all $f$

$P_N(f) = N_0/2$ for all $f$

$P_N(f) = N_0\,\delta(f)$

$P_N(f) = N_0/2$ for $|f| \leq W$

Correction:

P_N(f) = N_0/2

for all

f

By definition, WGN has a flat two-sided PSD equal to $N_0/2$ .

Why This Matters: Thermal Noise in Communication Receivers

The dominant noise in radio receivers is thermal (Johnson-Nyquist) noise, generated by random electron motion in resistive components. Its PSD is $P_N(f) = kT$ (W/Hz) for frequencies below $\sim 10^{12}$ Hz, where $k$ is Boltzmann's constant and $T$ is absolute temperature. This is effectively white over any communication bandwidth. Setting $N_0/2 = kT$ connects the abstract WGN model to the physical noise floor: for $T = 290$ K, $N_0/2 \approx 2 \times 10^{-21}$ W/Hz, or equivalently $-174$ dBm/Hz.

White Gaussian Noise (WGN)

A zero-mean Gaussian process with flat PSD $P_N(f) = N_0/2$ for all $f$ . It has infinite power and serves as the idealized noise model; physical noise is always a band-limited version.

Power Spectral Density (PSD)

The Fourier transform of the autocorrelation function of a WSS process: $P_x(f) = \int_{-\infty}^{\infty} r_{xx}(\tau)\,e^{-j2\pi f\tau}\,d\tau$ . Units: watts per hertz (W/Hz).

Related: White Gaussian Noise (WGN)

🔧Engineering Note

Noise Temperature and Noise Figure in Receiver Design

In receiver design, the noise floor is quantified by the equivalent noise temperature $T_e$ , which combines antenna temperature, amplifier noise, and cable losses. The noise figure $F = 1 + T_e/T_0$ (with $T_0 = 290$ K) measures how much the receiver degrades the input SNR.

For a cascade of stages with gains $G_1, G_2, \ldots$ and noise figures $F_1, F_2, \ldots$ , Friis's formula gives: $F_{\text{total}} = F_1 + \frac{F_2 - 1}{G_1} + \frac{F_3 - 1}{G_1 G_2} + \cdots$ The first-stage noise figure dominates when $G_1$ is large — a fundamental principle in low-noise amplifier (LNA) design.

Practical Constraints

•
LNA noise figures for 5G NR sub-6 GHz: typically 1.5--3 dB
•
Noise temperature of the cosmic microwave background: 2.7 K

Historical Note: Johnson and Nyquist: The Discovery of Thermal Noise

1928

In 1928, John B. Johnson at Bell Labs experimentally measured the tiny voltage fluctuations across resistors due to thermal agitation of electrons. Harry Nyquist, also at Bell Labs, immediately provided the theoretical explanation using thermodynamic arguments, showing that the noise power spectral density is $kT$ per unit bandwidth, independent of the resistance value. This Johnson-Nyquist noise sets the fundamental limit on the sensitivity of all electronic communication systems.

White Gaussian Noise

Definition: White Gaussian Noise (WGN)