Ferkans — Interactive Telecom Tutor

Why Geometry?

Digital modulation maps bits to waveforms. Each waveform $s_m(t)$ lives in a function space — an infinite-dimensional vector space. The remarkable insight of Kotel'nikov, Shannon, and Wozencraft is that for detection purposes, we need only the projections of the received signal onto a finite-dimensional subspace spanned by the transmitted waveforms. This reduces optimal detection to a problem in Euclidean geometry: finding the closest point in $\mathbb{R}^N$ .

Definition:
Signal Space

The signal space for a set of $M$ waveforms $\{s_1(t), s_2(t), \ldots, s_M(t)\}$ is the smallest linear subspace of $L^2(\mathbb{R})$ containing all $M$ signals.

Equip $L^2(\mathbb{R})$ with the inner product

$\langle f, g \rangle = \int_{-\infty}^{\infty} f(t)\, g^*(t)\, dt$

and induced norm $\|f\| = \sqrt{\langle f, f \rangle}$ . The signal space has dimension $N \leq M$ , and each signal is represented by a coordinate vector $\mathbf{s}_m \in \mathbb{R}^N$ with respect to an orthonormal basis $\{\varphi_1(t), \ldots, \varphi_N(t)\}$ :

$s_m(t) = \sum_{n=1}^{N} s_{m,n}\, \varphi_n(t), \qquad s_{m,n} = \langle s_m, \varphi_n \rangle$

In most practical modulation schemes $N \leq 2$ (in-phase and quadrature), but FSK and OFDM naturally produce higher-dimensional signal spaces.

,

Theorem: Gram-Schmidt Orthonormalisation for Waveforms

Given any set of $M$ energy-finite waveforms $\{s_1(t), \ldots, s_M(t)\}$ , the Gram-Schmidt procedure produces an orthonormal set $\{\varphi_1(t), \ldots, \varphi_N(t)\}$ , $N \leq M$ , such that every $s_m(t)$ is a linear combination of $\{\varphi_n(t)\}_{n=1}^{N}$ .

The procedure is:

$\varphi_1(t) = s_1(t) / \|s_1\|$
For $m = 2, \ldots, M$ :

$u_m(t) = s_m(t) - \sum_{n=1}^{N_m} \langle s_m, \varphi_n \rangle\, \varphi_n(t)$

If $\|u_m\| > 0$ , set $\varphi_{N_m+1}(t) = u_m(t) / \|u_m\|$ and increment $N_m$ . Otherwise, $s_m(t)$ lies in the span of the existing basis functions.

This is exactly the Gram-Schmidt procedure from Chapter 1, applied to the function space $L^2$ instead of $\mathbb{C}^n$ . The inner product $\langle f, g \rangle = \int f\, g^*\, dt$ replaces $\mathbf{y}^H \mathbf{x}$ .

Proof

Well-posedness

Each $s_m(t)$ has finite energy $\|s_m\|^2 < \infty$ , so it belongs to $L^2(\mathbb{R})$ . The inner product is well-defined by the Cauchy-Schwarz inequality in $L^2$ .

Orthonormality

By construction, $\langle \varphi_i, \varphi_j \rangle = \delta_{ij}$ . Each new $\varphi_n(t)$ is orthogonal to all previous ones because the projection $\sum \langle s_m, \varphi_n \rangle \varphi_n(t)$ removes all components in the existing subspace.

Spanning property

Every $s_m(t)$ is expressed as a linear combination of the $\varphi_n(t)$ by construction. The dimension $N \leq M$ because at most one new basis function is added per signal. $\blacksquare$

Definition:
Sufficient Statistics

Consider the AWGN channel model:

$r(t) = s_m(t) + w(t)$

where $w(t)$ is white Gaussian noise with two-sided PSD $N_0/2$ . The sufficient statistics for deciding which signal $s_m(t)$ was transmitted are the projections of $r(t)$ onto the basis functions:

$r_n = \langle r, \varphi_n \rangle = \int_{-\infty}^{\infty} r(t)\, \varphi_n^*(t)\, dt, \qquad n = 1, \ldots, N$

The observation vector $\mathbf{r} = [r_1, \ldots, r_N]^T$ satisfies

$\mathbf{r} = \mathbf{s}_m + \mathbf{w}$

where $\mathbf{w} \sim \mathcal{N}(\mathbf{0}, (N_0/2)\mathbf{I}_N)$ .

The components of $r(t)$ orthogonal to the signal space are pure noise, independent of which signal was sent, and carry no information about the transmitted message.

This is why signal-space analysis works: the infinite-dimensional observation $r(t)$ is compressed to an $N$ -dimensional vector $\mathbf{r}$ with no loss of information relevant to detection.

,

Example: BPSK Signal-Space Representation

Binary phase-shift keying (BPSK) uses two signals:

$s_1(t) = \sqrt{\frac{2E_b}{T_b}} \cos(2\pi f_c t), \qquad s_2(t) = -\sqrt{\frac{2E_b}{T_b}} \cos(2\pi f_c t)$

for $0 \leq t \leq T_b$ . Find the signal-space representation.

Solution

Identify the basis

Both signals are scalar multiples of the same waveform, so $N = 1$ . Apply Gram-Schmidt:

$\varphi_1(t) = \frac{s_1(t)}{\|s_1\|} = \sqrt{\frac{2}{T_b}} \cos(2\pi f_c t), \qquad 0 \leq t \leq T_b$

Compute coordinates

$s_{1,1} = \langle s_1, \varphi_1 \rangle = \sqrt{E_b}KATEXPLACEHOLDER0ENDs_{2,1} = \langle s_2, \varphi_1 \rangle = -\sqrt{E_b}$ $The two signal points on the real line are$ \mathbf{s}_1 = +\sqrt{E_b} $and$ \mathbf{s}2 = -\sqrt{E_b} $, separated by$ d{\min} = 2\sqrt{E_b} $.$ \blacksquare$

Theorem: ML Detector Minimises Euclidean Distance

Under the AWGN channel model with equiprobable signals, the maximum-likelihood (ML) detector selects

$\hat{m} = \arg\min_{m \in \{1, \ldots, M\}} \|\mathbf{r} - \mathbf{s}_m\|^2$

That is, the ML detector chooses the signal point closest to the received vector $\mathbf{r}$ in Euclidean distance.

Equivalently, using the correlation metric:

$\hat{m} = \arg\max_{m} \left[ 2\,\mathbf{s}_m^T \mathbf{r} - \|\mathbf{s}_m\|^2 \right]$

For equal-energy constellations ( $\|\mathbf{s}_m\|^2 = E_s$ for all $m$ ), this simplifies to the maximum-correlation detector:

$\hat{m} = \arg\max_{m}\; \mathbf{s}_m^T \mathbf{r}$

The received vector $\mathbf{r} = \mathbf{s}_m + \mathbf{w}$ is the transmitted point corrupted by isotropic Gaussian noise. Isotropic noise is equally likely to push $\mathbf{r}$ in any direction, so the most likely transmitted point is simply the nearest one.

Proof

Likelihood function

Since $\mathbf{w} \sim \mathcal{N}(\mathbf{0}, (N_0/2)\mathbf{I})$ , the conditional PDF of $\mathbf{r}$ given $s_m$ was sent is

$p(\mathbf{r} \mid s_m) = \frac{1}{(\pi N_0)^{N/2}} \exp\!\left(-\frac{\|\mathbf{r} - \mathbf{s}_m\|^2}{N_0}\right)$

ML rule

Maximising $p(\mathbf{r} \mid s_m)$ is equivalent to minimising the exponent $\|\mathbf{r} - \mathbf{s}_m\|^2$ .

Correlation metric

Expanding: $\|\mathbf{r} - \mathbf{s}_m\|^2 = \|\mathbf{r}\|^2 - 2\mathbf{s}_m^T\mathbf{r} + \|\mathbf{s}_m\|^2$ . The term $\|\mathbf{r}\|^2$ is independent of $m$ , giving the correlation metric. $\blacksquare$

,

Definition:
Decision Regions (Voronoi Tessellation)

The decision region $\mathcal{R}_m$ for signal $\mathbf{s}_m$ is the set of all received vectors $\mathbf{r}$ that the ML detector assigns to message $m$ :

$\mathcal{R}_m = \{\mathbf{r} \in \mathbb{R}^N : \|\mathbf{r} - \mathbf{s}_m\| \leq \|\mathbf{r} - \mathbf{s}_k\| \;\; \forall\, k \neq m\}$

The collection $\{\mathcal{R}_1, \ldots, \mathcal{R}_M\}$ forms a Voronoi tessellation of $\mathbb{R}^N$ . The boundary between $\mathcal{R}_m$ and $\mathcal{R}_k$ is the perpendicular bisector of the segment $\overline{\mathbf{s}_m \mathbf{s}_k}$ .

An error occurs when the noise vector $\mathbf{w}$ pushes $\mathbf{r}$ outside $\mathcal{R}_m$ (the region of the actually transmitted signal). The error probability depends on $d_{\min}$ and the noise variance $N_0/2$ .

Signal-Space Visualization

Explore the signal-space representation of standard modulation schemes. Transmitted constellation points are shown as blue markers, received noisy observations as scattered dots, and Voronoi decision regions as coloured polygons.

Parameters

Modulation scheme

Show decision regions

SNR (dB)15

Key Takeaway

The signal-space representation reduces optimal detection of $M$ waveforms from an infinite-dimensional problem (processing $r(t)$ for all $t$ ) to a finite-dimensional one (finding the nearest point in $\mathbb{R}^N$ , $N \leq M$ ). The sufficient statistics $\mathbf{r} = \mathbf{s}_m + \mathbf{w}$ lose no information relevant to detection — the noise component orthogonal to the signal space is independent of which signal was sent.

Common Mistake: Signal Space is Not Frequency Domain

Mistake:

Confusing the signal-space axes $\varphi_1, \varphi_2$ with the frequency domain or the time domain.

Correction:

The signal space is an abstract inner-product space whose axes are orthonormal basis functions. For QPSK, the two axes happen to be the in-phase ( $\cos$ ) and quadrature ( $\sin$ ) carriers, but this is a coincidence of the basis choice, not a frequency-domain representation. For 4-FSK, the four axes are four different carrier frequencies — still a signal space, but now 4-dimensional.

Quick Check

A modulation scheme uses $M = 8$ equally spaced signals on a circle of radius $\sqrt{E_s}$ (8-PSK). What is the dimension $N$ of the signal space?

1

2

8

4

Correction:

2

All 8-PSK signals are linear combinations of $\cos(2\pi f_c t)$ and $\sin(2\pi f_c t)$ , so $N = 2$ .

Signal Space

A finite-dimensional subspace of $L^2(\mathbb{R})$ spanned by a set of transmitted waveforms. Each signal is represented as a vector in $\mathbb{R}^N$ , reducing detection to a geometric problem.

Sufficient Statistics

The projections of the received signal onto an orthonormal basis spanning the signal space. They contain all the information in the observation that is relevant for optimal detection.

Decision Region

The subset of the observation space assigned to a particular signal point by the detector. For ML detection under AWGN, the decision regions form a Voronoi tessellation around the constellation points.

Gram-Schmidt in Signal Space

Watch the Gram-Schmidt orthonormalisation procedure applied to three signal waveforms in a two-dimensional signal space. The animation shows how projections and residuals produce an orthonormal basis, and how the third signal lies in the span of the first two (

N = 2

, not 3).

The Gram-Schmidt procedure reduces

M

signals to an

N

-dimensional orthonormal basis (

N \leq M

).

Detection Theory in Greater Depth

The ML detector derived here is a special case of hypothesis testing in Gaussian noise. The general framework — Neyman-Pearson, likelihood ratios, sufficient statistics, and the connection to Bayesian decision theory — is developed rigorously in the Foundations of Statistical Inference (FSI) book. Readers interested in optimal detection for non-Gaussian noise, composite hypotheses, or sequential detection should consult that treatment.

Chapter 9 of this book provides a self-contained summary of the key detection results needed for digital communications.

Historical Note: Wozencraft, Jacobs, and the Signal-Space Approach

1947-1965

The signal-space approach to digital communications was popularised by John Wozencraft and Irwin Jacobs in their 1965 textbook Principles of Communication Engineering. Building on earlier work by Kotel'nikov (1947) and Shannon (1948), they showed that optimal detection of digital signals in AWGN reduces to a nearest-neighbour problem in Euclidean space. This geometric viewpoint unified the analysis of all linear modulation schemes and remains the foundation of modern digital communications textbooks.

Signal-Space Concepts