Ferkans — Interactive Telecom Tutor

Noise and Randomness Are Unavoidable

Real communication signals are corrupted by noise (thermal, interference) and transmitted through channels with random characteristics (fading). To analyse system performance, we need the tools of random signal theory: power spectral density, filtering of random processes, and optimal detection. This section connects the signal-processing framework of this chapter to the probability theory of Chapter 2.

Definition:
Wide-Sense Stationary (WSS) Process

A random process $X(t)$ is wide-sense stationary (WSS) if:

$\mathbb{E}[X(t)] = \mu_X$ (constant mean)
$R_X(t_1, t_2) = R_X(t_1 - t_2) = R_X(\tau)$ (autocorrelation depends only on the lag $\tau = t_1 - t_2$ )

where $R_X(\tau) = \mathbb{E}[X(t + \tau) X^*(t)]$ .

Properties of the autocorrelation:

$R_X(0) = \mathbb{E}[|X(t)|^2] = P_X$ (average power)
$R_X(-\tau) = R_X^*(\tau)$ (Hermitian symmetry)
$|R_X(\tau)| \le R_X(0)$ (maximum at zero lag)

Definition:
Power Spectral Density (PSD)

The power spectral density of a WSS process $X(t)$ is the Fourier transform of its autocorrelation:

$S_X(f) = \int_{-\infty}^{\infty} R_X(\tau)\,e^{-j2\pi f\tau}\,d\tau = \mathcal{F}\{R_X(\tau)\}.$

The inverse relation is

$R_X(\tau) = \int_{-\infty}^{\infty} S_X(f)\,e^{j2\pi f\tau}\,df.$

Properties:

$S_X(f) \geq 0$ for all $f$ (non-negative)
$\int_{-\infty}^{\infty} S_X(f)\,df = R_X(0) = P_X$ (total power)
For real $X(t)$ : $S_X(f) = S_X(-f)$ (even function)

This is the Wiener–Khinchin theorem.

Theorem: Output PSD of a WSS Process Through an LTI System

If a WSS process $X(t)$ with PSD $S_X(f)$ is the input to an LTI system with transfer function $H(f)$ , the output $Y(t)$ is also WSS with:

Output PSD: $S_Y(f) = |H(f)|^2\,S_X(f)$
Output autocorrelation: $R_Y(\tau) = \mathcal{F}^{-1}\{|H(f)|^2 S_X(f)\}$
Cross-PSD: $S_{XY}(f) = H(f)\,S_X(f)$
Output mean: $\mu_Y = H(0)\,\mu_X$

The LTI system shapes the input spectrum by $|H(f)|^2$ — it passes power at frequencies where $|H(f)|$ is large and attenuates where $|H(f)|$ is small. This is how filters work on random signals.

Proof

Output autocorrelation

$R_Y(\tau) = \mathbb{E}[Y(t+\tau)Y^*(t)]$ . Since $Y(t) = \int h(\alpha) X(t-\alpha)\,d\alpha$ :

$R_Y(\tau) = \int\!\!\int h(\alpha) h^*(\beta)\, \underbrace{\mathbb{E}[X(t+\tau-\alpha)X^*(t-\beta)]}_{R_X(\tau-\alpha+\beta)}\, d\alpha\,d\beta$ .

Fourier transform

Taking the Fourier transform of $R_Y(\tau)$ :

$S_Y(f) = H(f) H^*(f) S_X(f) = |H(f)|^2 S_X(f)$ . $\blacksquare$

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Example: White Noise Through a Low-Pass Filter

White noise $X(t)$ with PSD $S_X(f) = N_0/2$ passes through an ideal LPF with bandwidth $W$ : $H(f) = \mathrm{rect}(f/(2W))$ . Find the output PSD, total output power, and autocorrelation.

Solution

Output PSD

$S_Y(f) = |H(f)|^2 S_X(f) = \frac{N_0}{2}\,\mathrm{rect}\!\left(\frac{f}{2W}\right).$ $The output has flat PSD within$ [-W, W]$ and zero outside.

Output power

$P_Y = \int_{-W}^{W} \frac{N_0}{2}\,df = N_0 W.$ $

Autocorrelation

$R_Y(\tau) = \mathcal{F}^{-1}\left\{\frac{N_0}{2}\,\mathrm{rect}\!\left(\frac{f}{2W}\right)\right\} = N_0 W\,\mathrm{sinc}(2W\tau).$ $The output samples are uncorrelated (zero crossings of$ \mathrm{sinc} $) when spaced$ 1/(2W) $apart — consistent with the Nyquist rate.$ \blacksquare$

Theorem: Matched Filter

Consider the detection problem: a known signal $s(t)$ of duration $T$ is received in additive white Gaussian noise (AWGN):

$r(t) = s(t) + n(t), \qquad S_n(f) = N_0/2.$

The matched filter $h(t) = s^*(T - t)$ (time-reversed conjugate of the signal) maximises the output signal-to-noise ratio (SNR) at time $t = T$ :

$\text{SNR}_{\max} = \frac{2E_s}{N_0}$

where $E_s = \int_0^T |s(t)|^2\,dt$ is the signal energy.

Equivalently, $H(f) = S^*(f) e^{-j2\pi fT}$ .

The maximum SNR depends only on the signal energy, not on the signal shape.

The matched filter "weights" each frequency by how much signal energy is present there relative to the (flat) noise. Since the noise is white (flat PSD), the optimal strategy is to emphasise frequencies where the signal is strongest — which means correlating with the signal itself.

Proof

SNR expression

At $t = T$ , the signal component of the output is $y_s(T) = \int S(f) H(f)\,df$ and the noise power is $P_n = \frac{N_0}{2}\int |H(f)|^2\,df$ .

$\text{SNR} = \frac{|y_s(T)|^2}{P_n} = \frac{\left|\int S(f) H(f)\,df\right|^2} {\frac{N_0}{2}\int |H(f)|^2\,df}.$

Apply Cauchy–Schwarz

By the Cauchy–Schwarz inequality:

$\left|\int S(f) H(f)\,df\right|^2 \le \int |S(f)|^2\,df \cdot \int |H(f)|^2\,df$

with equality iff $H(f) = \alpha S^*(f)$ for some $\alpha$ . Substituting:

$\text{SNR} \le \frac{\int |S(f)|^2\,df \cdot \int |H(f)|^2\,df} {\frac{N_0}{2}\int |H(f)|^2\,df} = \frac{2}{N_0}\int |S(f)|^2\,df = \frac{2E_s}{N_0}.$

The delay factor $e^{-j2\pi fT}$ ensures causality. $\blacksquare$

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Definition:
Correlator Receiver

An equivalent implementation of the matched filter is the correlator: multiply $r(t)$ by $s(t)$ and integrate:

$z = \int_0^T r(t)\,s^*(t)\,dt.$

The correlator output $z$ equals the matched filter output sampled at $t = T$ . The correlator is often preferred in digital implementations because it does not require storing the impulse response, only the template signal $s(t)$ .

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Matched Filter Detecting a Pulse in Noise

A rectangular pulse is hidden in strong additive noise. The matched filter output shows a clear peak at the correct time, demonstrating that the maximum SNR depends only on signal energy

E_s

, not on the noise realisation.

Top: received signal

r(t) = s(t) + n(t)

. Bottom: matched filter output with peak at

t = T

.

Matched Filter Demonstration

See how the matched filter maximises SNR. The input is a known pulse $s(t)$ buried in white noise. Compare the matched filter output with a mismatched filter to see the SNR difference.

Parameters

Pulse shape

Input SNR (dB)0

Why This Matters: The Matched Filter in Every Receiver

The matched filter principle underlies every digital communication receiver. In a CDMA system, the RAKE receiver is a bank of matched filters, one per multipath. In OFDM (Wi-Fi, LTE, 5G), the DFT operation at the receiver is equivalent to a bank of matched filters, one per subcarrier. The correlator form appears directly in GPS receivers, where the incoming signal is correlated with local replicas of the spreading code to achieve processing gain against noise.

Definition:
Additive White Gaussian Noise (AWGN)

The standard noise model in communications:

Additive: $r(t) = s(t) + n(t)$ (noise adds to signal)
White: $S_n(f) = N_0/2$ for all $f$ (flat PSD, implying $R_n(\tau) = \frac{N_0}{2}\delta(\tau)$ — uncorrelated at any nonzero lag)
Gaussian: $n(t)$ is a Gaussian random process (samples are jointly Gaussian)

The parameter $N_0$ is the noise power spectral density (watts/Hz) and equals $k_B T_{\text{sys}}$ where $k_B$ is Boltzmann's constant and $T_{\text{sys}}$ is the system noise temperature.

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Key Takeaway

Key Results for Random Signals Through Linear Systems. (1) WSS input through an LTI system produces a WSS output with $S_Y(f) = |H(f)|^2 S_X(f)$ . (2) The matched filter $h(t) = s^*(T-t)$ maximises output SNR to $2E_s/N_0$ — depending only on signal energy, not shape. (3) AWGN ( $S_n(f) = N_0/2$ ) is the fundamental noise model; its flat PSD makes the matched filter simply a correlator with the signal template. (4) These results form the foundation for receiver design (Chapter 9) and capacity analysis (Chapter 11).

Common Mistake: White Noise Has Infinite Power

Mistake:

Treating white noise $n(t)$ as a physically realisable signal with finite power.

Correction:

$P_n = \int_{-\infty}^{\infty} S_n(f)\,df = \int_{-\infty}^{\infty} N_0/2\,df = \infty$ . White noise is a mathematical idealisation. In practice, noise is always bandlimited by the receiver's front-end filter, giving finite power $P_n = N_0 W$ . The white noise model is accurate within the system bandwidth and greatly simplifies analysis.

Quick Check

A WSS process with PSD $S_X(f) = 1$ (white, unit density) passes through a filter with $|H(f)|^2 = 4$ for $|f| < 100$ and $0$ otherwise. What is the output power $P_Y$ ?

$4$ W

$400$ W

$800$ W

$200$ W

Correction:

800

W

Correct. $P_Y = \int_{-100}^{100} 4 \cdot 1\,df = 4 \times 200 = 800$ W.

Quick Check

A rectangular pulse of amplitude $A$ and duration $T$ is detected by a matched filter in AWGN with PSD $N_0/2$ . What is the output SNR?

$A^2 T / N_0$

$2A^2 T / N_0$

$A^2 / (N_0 T)$

$A^2 / (2N_0 W)$

Correction:

2A^2 T / N_0

Correct. The signal energy is $E_s = A^2 T$ , so $\text{SNR} = 2E_s/N_0 = 2A^2T/N_0$ .

Deeper Treatment in the FSP Book

The WSS processes and PSD material here provides the minimum needed for wireless channel analysis. The FSP book (Foundations of Stochastic Processes) develops the full theory: strict-sense stationarity, ergodic theorems, Markov chains, point processes, and the Karhunen–Loève expansion. Readers who need to prove convergence results or analyse queueing systems should consult FSP Chapters 6–8.

Why This Matters: From Matched Filter to Optimal Detection Theory

The matched filter is the simplest case of optimal detection: a known signal in AWGN. The FSI book (Foundations of Statistical Inference) extends this to unknown signals, multiple hypotheses, and composite hypothesis testing. The Neyman–Pearson, Bayesian, and GLRT frameworks generalise the matched filter to scenarios where the signal or channel parameters are unknown — the foundation for MIMO detection (Chapter 16) and channel estimation (Chapter 9).

Power Spectral Density (PSD)

$S_X(f) = \mathcal{F}\{R_X(\tau)\}$ . Describes how the average power of a WSS process is distributed across frequency.

Matched Filter

The filter $h(t) = s^*(T-t)$ that maximises output SNR for detection of $s(t)$ in AWGN. $\text{SNR}_{\max} = 2E_s/N_0$ .

AWGN

Additive White Gaussian Noise. The standard noise model in communications: additive, flat PSD $N_0/2$ , Gaussian distributed.

Wide-Sense Stationary (WSS)

A random process with constant mean and autocorrelation that depends only on the time lag $\tau$ .

Random Signals Through Linear Systems

Noise and Randomness Are Unavoidable

Definition: Wide-Sense Stationary (WSS) Process

Definition: Power Spectral Density (PSD)

Theorem: Output PSD of a WSS Process Through an LTI System

Output autocorrelation

Fourier transform

Example: White Noise Through a Low-Pass Filter

Output PSD

Output power

Autocorrelation

Theorem: Matched Filter

SNR expression

Apply Cauchy–Schwarz

Definition: Correlator Receiver

Matched Filter Detecting a Pulse in Noise

Matched Filter Demonstration

Parameters

Why This Matters: The Matched Filter in Every Receiver

Definition: Additive White Gaussian Noise (AWGN)

Key Takeaway

Common Mistake: White Noise Has Infinite Power

Quick Check

Quick Check

Deeper Treatment in the FSP Book

Why This Matters: From Matched Filter to Optimal Detection Theory

Power Spectral Density (PSD)

Matched Filter

AWGN

Wide-Sense Stationary (WSS)

Definition:
Wide-Sense Stationary (WSS) Process

Definition:
Power Spectral Density (PSD)

Definition:
Correlator Receiver

Definition:
Additive White Gaussian Noise (AWGN)