Ferkans — Interactive Telecom Tutor

Why Gaussian Processes and White Noise Are Central

Every communication system operates in noise. The Gaussian process and its idealised spectral limit, white Gaussian noise, together constitute THE noise model of telecommunications theory:

Physical origin. Thermal noise arises from random charge-carrier motion; by the CLT (Section 2.5), the superposition of many independent microscopic contributions is Gaussian.
Analytical tractability. A GP is completely specified by its mean and autocorrelation --- the same second-order statistics that define WSS. Gaussianity is preserved through all linear operations.
Worst-case optimality. Among all noise processes with a given variance, the Gaussian maximises entropy and thus minimises channel capacity --- AWGN provides a universal performance bound.

This section formalises these ideas: GP definition and properties, white noise and its band-limited version, complex baseband AWGN, matched filtering, noise figure, and channel capacity.

Definition:
Gaussian Process

A stochastic process $\{X(t),\; t \in T\}$ is a Gaussian process if, for every $n \geq 1$ and every $t_1, \ldots, t_n \in T$ ,

$\mathbf{X} = \bigl(X(t_1),\, \ldots,\, X(t_n)\bigr)^T$

is jointly Gaussian with density

$f_{\mathbf{X}}(\mathbf{x}) = \frac{1}{(2\pi)^{n/2}\,|\mathbf{C}|^{1/2}} \exp\!\Bigl(-\tfrac{1}{2} (\mathbf{x} - \boldsymbol{\mu})^T \mathbf{C}^{-1} (\mathbf{x} - \boldsymbol{\mu})\Bigr),$

where $\boldsymbol{\mu} = E[\mathbf{X}]$ and $\mathbf{C}$ has entries $C_{ij} = R_X(t_i,t_j) - \mu_X(t_i)\mu_X(t_j)$ .

Consequence: A GP is completely determined by $\mu_X(t)$ and $R_X(t_1,t_2)$ . No higher-order statistics carry additional information.

Theorem: Fundamental Properties of Gaussian Processes

Let $\{X(t)\}$ be a Gaussian process.

(a) WSS Gaussian $\Rightarrow$ SSS.

(b) Closure under linear operations. Any linear operation (filtering, sampling, integration, differentiation) on a GP yields a GP.

(c) Uncorrelated $\Leftrightarrow$ independent. For jointly Gaussian $X(t_1), X(t_2)$ : $\mathrm{Cov}(X(t_1), X(t_2)) = 0 \;\Longleftrightarrow\; X(t_1), X(t_2) \text{ are independent.}$

(a) Since a Gaussian distribution is determined by its first two moments, WSS shift-invariance of these moments forces all finite-dimensional distributions to be shift-invariant --- SSS.

(b) Every component in a linear receiver chain preserves Gaussianity.

(c) For general RVs, uncorrelated $\not\Rightarrow$ independent. For Gaussians, zero covariance makes the joint PDF factor.

Show Hint

For (c), write the joint Gaussian PDF when the off-diagonal covariance is zero.

Proof

Proof of (c): Uncorrelated $\Leftrightarrow$ independent

( $\Leftarrow$ ) Independence always implies zero covariance.

( $\Rightarrow$ ) Suppose $X_1, X_2$ are jointly Gaussian with $\mathrm{Cov}(X_1,X_2)=0$ . The covariance matrix is $\mathbf{C} = \mathrm{diag}(\sigma_1^2,\sigma_2^2)$ , so

$f_{X_1,X_2}(x_1,x_2) = \frac{1}{2\pi\sigma_1\sigma_2} \exp\!\left( -\frac{(x_1-\mu_1)^2}{2\sigma_1^2} -\frac{(x_2-\mu_2)^2}{2\sigma_2^2}\right) = f_{X_1}(x_1)\,f_{X_2}(x_2).$

The joint PDF factors $\Rightarrow$ independence. $\blacksquare$

Remark on (a)

WSS gives shift-invariant mean and autocorrelation. Since these fully determine all finite-dimensional Gaussian distributions, every joint distribution is shift-invariant --- the definition of SSS. $\square$

Definition:
White Gaussian Noise (WGN)

A white Gaussian noise process $\{n(t)\}$ is zero-mean, stationary Gaussian with:

$R_n(\tau) = \frac{N_0}{2}\,\delta(\tau), \qquad S_n(f) = \frac{N_0}{2}\;\;\forall f.$

The PSD is flat ("white"). $N_0$ has units W/Hz (equivalently J), often $N_0 = k_B T$ . White noise has infinite total power --- $R_n(0) = (N_0/2)\,\delta(0) \to \infty$ --- because the flat PSD extends over infinite bandwidth. Distinct samples $n(t_1), n(t_2)$ ( $t_1 \neq t_2$ ) are uncorrelated and, by property (c), independent.

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White Noise Is an Idealisation

No physical noise has a flat PSD to infinite frequency --- thermal noise rolls off above $k_B T/h \approx 6 \times 10^{12}$ Hz at room temperature. But signal bandwidths satisfy $W \ll k_B T/h$ , so within the passband the PSD is effectively flat. Rule of thumb: model noise as white, but always filter before computing power.

Theorem: Band-Limited White Noise

Pass WGN ( $S_n(f)=N_0/2$ ) through an ideal lowpass filter of bandwidth $W$ . The output $n_W(t)$ satisfies:

(a) $S_{n_W}(f) = N_0/2$ for $|f|\leq W$ , zero otherwise.

(b) Output power: $P_{n_W} = N_0 W.$

(c) Autocorrelation: $R_{n_W}(\tau) = N_0 W\,\mathrm{sinc}(2W\tau)$ .

(d) Nyquist-rate samples ( $\tau = k/(2W)$ , $k$ integer) are independent: $R_{n_W}(k/(2W)) = N_0 W\,\delta_{k0}$ .

Band-limiting tames the infinite power to a finite $N_0 W$ . The sinc autocorrelation shows correlation over timescales $\sim 1/(2W)$ . Nyquist-rate sampling converts the continuous-time AWGN channel into a sequence of iid Gaussian scalar channels.

Proof

Output PSD and power

$S_{n_W}(f)=|H(f)|^2 S_n(f)$ gives the rectangular PSD. $P_{n_W}=\int_{-W}^{W}(N_0/2)\,df = N_0 W$ . $\square$

Autocorrelation

$R_{n_W}(\tau) = \int_{-W}^{W}\frac{N_0}{2}\,e^{j2\pi f\tau}df = N_0 W\,\mathrm{sinc}(2W\tau).$ $At$ \tau=k/(2W) $:$ \mathrm{sinc}(k)=\delta_{k0} $.$ \blacksquare$

Definition:
Complex Baseband (AWGN) Noise

$n(t) = n_I(t) + j\,n_Q(t),$ $where$ n_I, n_Q $are independent, real, zero-mean WGN each with PSD$ N_0/2 $. The complex noise has$ R_n(\tau) = N_0,\delta(\tau) $, PSD$ S_n(f)=N_0 $, and is **circularly symmetric** ($ E[n(t+\tau),n(t)]=0 $). This is the standard AWGN channel:$ r(t) = s(t) + n(t)$.

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Example: Matched Filter Output in AWGN

Signal $s(t)$ with energy $E_s = \int |s(t)|^2\,dt$ is received as $r(t) = s(t) + n(t)$ in complex AWGN (PSD $N_0$ ). The matched filter output is $y = \int r(t)\,s^*(t)\,dt$ . Find the decomposition, noise distribution, and output SNR.

Solution

Step 1: Decompose

$y = \underbrace{\int |s(t)|^2\,dt}_{E_s} + \underbrace{\int n(t)\,s^*(t)\,dt}_{\eta} = E_s + \eta.$ $

Step 2: Noise distribution

$\eta$ is a linear functional of the GP $n(t)$ , hence Gaussian. $E[\eta]=0$ . Variance: $E[|\eta|^2] = \int\!\!\int s^*(t)\,s(t')\, N_0\delta(t-t')\,dt\,dt' = N_0 E_s.$ So $\eta \sim \mathcal{CN}(0, N_0 E_s)$ .

Step 3: Output SNR

$\text{SNR}_{\mathrm{out}} = \frac{E_s^2}{N_0 E_s} = \frac{E_s}{N_0}.$ $This is the **maximum** SNR achievable by any linear filter:$ \boxed{\text{SNR}_{\mathrm{MF}} = E_s / N_0.}$

Theorem: Matched Filter Output as a Sufficient Statistic

For $M$ -ary detection in AWGN ( $r(t) = s_i(t) + n(t)$ ), the matched filter outputs $y_i = \int r(t)\,s_i^*(t)\,dt$ form a sufficient statistic: $\mathbf{y}=(y_0,\ldots,y_{M-1})^T$ contains all information in $r(t)$ relevant to deciding which signal was sent. The residual $n'(t) = r(t) - \sum_i y_i\phi_i(t)$ is independent of $\mathbf{y}$ and of the transmitted signal.

The matched filter projects $r(t)$ onto the signal subspace, extracting all useful information and discarding irrelevant noise. This reduces an infinite-dimensional problem to a finite-dimensional one. Full proof deferred to the detection theory chapter (Chapter 3).

Definition:
Noise Figure and Equivalent Noise Temperature

Noise figure $F$ : input-to-output SNR ratio at reference temperature $T_0 = 290$ K:

$F = \frac{\text{SNR}_{\mathrm{in}}} {\text{SNR}_{\mathrm{out}}} \geq 1.$

Equivalent noise temperature: $T_e = T_0(F-1)$ , giving system noise $T_{\mathrm{sys}} = T_a + T_e$ and $P_n = k_B T_{\mathrm{sys}} W$ .

Friis formula for $K$ cascaded stages:

$F_{\mathrm{total}} = F_1 + \frac{F_2-1}{G_1} + \frac{F_3-1}{G_1 G_2} + \cdots + \frac{F_K-1}{G_1 \cdots G_{K-1}}.$

The first stage dominates --- hence the importance of a low-noise amplifier (LNA) at the receiver front end.

Why This Matters: AWGN: The Baseline Channel

The AWGN channel capacity is

$C = W\log_2\!\left(1 + \frac{P}{N_0 W}\right) \quad\text{bits/s}.$

Why AWGN is the universal baseline:

No real channel with the same bandwidth and SNR can exceed $C$ .
The Shannon limit gives the minimum energy per bit: $E_b/N_0 \geq \ln 2 \approx -1.59$ dB (as $C/W \to 0$ ).
Fading channel ergodic capacity $C_{\mathrm{erg}} = E_h[W\log_2(1+|h|^2 P/(N_0 W))]$ satisfies $C_{\mathrm{erg}} \leq C_{\mathrm{AWGN}}$ by Jensen's inequality.

Every BER curve, throughput plot, and outage analysis is benchmarked against the AWGN limit.

Historical Note: Johnson and Nyquist: Thermal Noise (1928)

In 1928 at Bell Labs, John B. Johnson measured voltage fluctuations across a resistor, and Harry Nyquist derived from thermodynamics that the available noise power in bandwidth $\Delta f$ is

$P_n = k_B T\,\Delta f,$

giving $N_0 = k_B T$ . At $T = 290$ K: $N_0 \approx 4.00 \times 10^{-21}$ W/Hz $= -174$ dBm/Hz --- the thermal noise floor in every link budget. The Planck correction $S_n(f) = hf/(e^{hf/(k_BT)}-1)$ is negligible below $\sim 6 \times 10^{12}$ Hz.

Quick Check

WGN with $S_n(f)=N_0/2$ passes through an ideal bandpass filter (centre frequency $f_c=2$ GHz, one-sided bandwidth $W=20$ MHz). The output noise power is:

$N_0 \times 20\times10^6$

$(N_0/2)\times 20\times10^6$

$N_0 \times 2\times10^9$

Correction:

N_0 \times 20\times10^6

$P_n = (N_0/2)\times 2W = N_0 W = N_0\times 20\times 10^6$ . The centre frequency is irrelevant for white noise.

Quick Check

Let $\{X(t)\}$ be a zero-mean Gaussian process. Which statement is TRUE?

Uncorrelated samples may still be dependent.

WSS automatically implies SSS.

Nonlinear processing preserves Gaussianity.

Correction:

WSS automatically implies SSS.

A Gaussian distribution is determined by its first two moments. WSS shift-invariance of these moments forces all joint distributions to be shift-invariant --- SSS.

Quick Check

Signal energy $E_s = 10^{-6}$ J, noise $N_0 = 10^{-9}$ W/Hz. Matched filter output SNR?

30 dB

60 dB

Cannot determine without knowing the bandwidth.

Correction:

30 dB

$\text{SNR}=E_s/N_0 = 10^{-6}/10^{-9}=10^3=30$ dB.

Gaussian Process

A process whose every finite-dimensional distribution is jointly Gaussian. Fully specified by $\mu_X(t)$ and $R_X(t_1,t_2)$ . WSS $\Rightarrow$ SSS; uncorrelated $\Leftrightarrow$ independent; closed under linear operations.

White Noise

Zero-mean stationary process with flat PSD $S_n(f)=N_0/2$ and $R_n(\tau)=(N_0/2)\delta(\tau)$ . Infinite total power; always band-limited by the receiver in practice.

Power Spectral Density (PSD)

$S_X(f)=\mathcal{F}\{R_X(\tau)\}$ . For WGN: $N_0/2$ . For complex baseband noise: $N_0$ . Always nonneg.; integrates to average power.

Additive White Gaussian Noise (AWGN)

Channel model $r(t)=s(t)+n(t)$ with $n(t)$ zero-mean complex WGN. Capacity: $C = W\log_2(1+\text{SNR})$ .

Noise Figure

$F = \text{SNR}_{\mathrm{in}}/\text{SNR}_{\mathrm{out}} \geq 1$ . Equivalent noise temperature: $T_e = T_0(F-1)$ . First stage dominates via Friis formula.

Common Mistake: "White Noise Has Infinite Power"

Mistake:

"White noise has infinite power, so the model is unphysical and unusable."

Correction:

White noise does have infinite total power. But we never observe raw white noise --- every receiver has finite bandwidth. The correct procedure: model noise as white, apply $|H(f)|^2$ , then integrate to get finite output power $P = N_0 W$ . The infinite-power "paradox" arises only from computing power before filtering, which never happens in a real system. White noise is an idealisation like the Dirac delta: defined by what it does inside integrals.

⚠️Engineering Note

ADC Quantisation Noise vs Thermal Noise

The AWGN model assumes continuous-amplitude observations, but every practical receiver digitises the signal with an analog-to-digital converter (ADC) of finite resolution $b$ bits.

Quantisation noise model. For a uniform $b$ -bit quantiser with full-scale range $[-V_{\mathrm{fs}}, V_{\mathrm{fs}}]$ , the quantisation step is $\Delta = 2V_{\mathrm{fs}} / 2^b$ and the quantisation noise power is approximately

$\sigma_q^2 = \frac{\Delta^2}{12}.$

The signal-to-quantisation-noise ratio (SQNR) for a sinusoidal input is the well-known rule of thumb:

$\mathrm{SQNR} \approx 6.02\,b + 1.76 \;\text{dB}.$

When does quantisation noise matter?

High-SNR regime: When the thermal noise power $\sigma_n^2 = N_0 W$ is comparable to or smaller than $\sigma_q^2$ , the ADC becomes the bottleneck. For a 10-bit ADC ( $\mathrm{SQNR} \approx 62$ dB), this occurs at received SNR above $\sim 60$ dB --- rare in wireless but common in wired (fibre, cable) and in self-interference channels of full-duplex systems.
Low-resolution ADCs (1--4 bits): Actively researched for massive MIMO base stations to reduce power consumption (each ADC consumes power $\propto 2^b \cdot f_s$ where $f_s$ is the sampling rate). A 256-antenna BS at 1 GHz bandwidth with 12-bit ADCs would consume $> 10$ W in ADCs alone. Reducing to 3--4 bits cuts this by $100{\times}$ -- $250{\times}$ but introduces significant nonlinear distortion that the Gaussian noise model no longer captures.
5G NR mmWave: Typical implementations use 8--10 bit ADCs per antenna per I/Q rail. The quantisation noise is $> 30$ dB below thermal noise at cell-edge SNR and can usually be ignored.

Design rule: The AWGN model is valid when the ADC resolution satisfies $b > (\text{SNR}_{\mathrm{dB}} - 1.76)/6.02 + 2$ (providing $\sim 12$ dB of margin). Below this threshold, the Bussgang decomposition should be used to model the quantisation distortion as an equivalent linear channel plus uncorrelated non-Gaussian noise.

Practical Constraints

•
ADC power scales as 2^b * f_s (Walden figure of merit)
•
mmWave arrays: 8-10 bit ADCs per I/Q per antenna
•
Low-resolution (1-4 bit) ADCs require Bussgang linearisation

📋 Ref: 3GPP TS 38.104 (NR BS receiver characteristics)

⚠️Engineering Note

Noise Figure Cascading and the Friis Formula

The single noise PSD parameter $N_0$ in the AWGN model abstracts away a chain of amplifiers, mixers, and filters, each contributing its own noise. The Friis formula determines the effective system noise temperature:

$T_{\mathrm{sys}} = T_1 + \frac{T_2}{G_1} + \frac{T_3}{G_1 G_2} + \cdots,$

where $T_k$ and $G_k$ are the noise temperature and available power gain of the $k$ -th stage. Equivalently, for noise figures:

$F_{\mathrm{sys}} = F_1 + \frac{F_2 - 1}{G_1} + \frac{F_3 - 1}{G_1 G_2} + \cdots.$

Practical consequences:

LNA dominance: A low-noise amplifier (LNA) with $F_1 = 1$ dB and $G_1 = 20$ dB placed first suppresses all subsequent stages by $20$ dB. Swapping the LNA with a lossy filter ( $F = 3$ dB, $G = -3$ dB) first degrades the system noise figure by $> 2$ dB.
Typical values (5G NR sub-6 GHz BS): LNA noise figure $0.5$ -- $1.5$ dB, mixer $8$ -- $12$ dB, overall receiver NF $\leq 5$ dB per 3GPP TS 38.104.
mmWave: Higher LNA noise figures ( $2$ -- $4$ dB) due to transistor limitations at 28/39 GHz, partially compensated by array gain in phased-array receivers.

Link to theory: The one-sided noise PSD used throughout this chapter is $N_0 = k_B T_{\mathrm{sys}}$ , where $k_B = 1.381 \times 10^{-23}$ J/K is Boltzmann's constant. At room temperature ( $T_0 = 290$ K), $N_0 = -174$ dBm/Hz. Adding a receiver noise figure of $F$ dB gives an effective noise floor of $-174 + F$ dBm/Hz.

Practical Constraints

•
3GPP specifies max receiver NF: 5 dB (BS), 7 dB (UE) for FR1
•
LNA must be placed before lossy elements to preserve sensitivity
•
k_B T_0 = -174 dBm/Hz at T_0 = 290 K

📋 Ref: 3GPP TS 38.104, Section 7 (Receiver characteristics)

Key Takeaway

Gaussianity is preserved by linearity. Noise stays Gaussian through filters, mixers, samplers, correlators --- making BER and threshold calculations tractable.
White noise gives clean formulas. $P_n = N_0 W$ and $\text{SNR}_{\mathrm{MF}} = E_s/N_0$ .
AWGN is the universal benchmark. $C = W\log_2(1+\text{SNR})$ bounds every channel with the same bandwidth and power.
Bridge to detection theory. The matched filter output is a sufficient statistic, reducing infinite-dimensional waveform observation to finite-dimensional Gaussian statistics (Chapter 3).

In one sentence: The GP model combined with AWGN gives tractable analysis because Gaussianity is preserved through all linear operations.

Gaussian Processes and White Noise

Why Gaussian Processes and White Noise Are Central

Definition: Gaussian Process

Theorem: Fundamental Properties of Gaussian Processes

Proof of (c): Uncorrelated $\Leftrightarrow$ independent

Remark on (a)

Definition: White Gaussian Noise (WGN)

White Noise Is an Idealisation

Theorem: Band-Limited White Noise

Output PSD and power

Autocorrelation

Definition: Complex Baseband (AWGN) Noise

Example: Matched Filter Output in AWGN

Step 1: Decompose

Step 2: Noise distribution

Step 3: Output SNR

Theorem: Matched Filter Output as a Sufficient Statistic

Definition: Noise Figure and Equivalent Noise Temperature

Why This Matters: AWGN: The Baseline Channel

Historical Note: Johnson and Nyquist: Thermal Noise (1928)

Quick Check

Quick Check

Quick Check

Gaussian Process

White Noise

Power Spectral Density (PSD)

Additive White Gaussian Noise (AWGN)

Noise Figure

Common Mistake: "White Noise Has Infinite Power"

ADC Quantisation Noise vs Thermal Noise

Noise Figure Cascading and the Friis Formula

Key Takeaway

Definition:
Gaussian Process

Definition:
White Gaussian Noise (WGN)

Definition:
Complex Baseband (AWGN) Noise

Definition:
Noise Figure and Equivalent Noise Temperature