Ferkans — Interactive Telecom Tutor

Why Signal Space?

Continuous-time passband signals $\{s_m(t)\}_{m=0}^{M-1}$ live in $L^2(\mathbb{R})$ — an infinite-dimensional Hilbert space. Detection in such a space looks hopelessly abstract. The key structural result of this section is that only an $N \leq M$ -dimensional subspace matters. By projecting the received waveform onto an orthonormal basis of this subspace, we reduce detection of waveforms in function space to detection of vectors in $\mathbb{R}^N$ (or $\mathbb{C}^N$ ) — the latter is exactly the setting of Section 3.1. This reduction is the theoretical backbone of every digital demodulator.

Definition:
M-ary Signal Detection in AWGN

Let $\{s_m(t)\}_{m=0}^{M-1}$ be a finite set of finite-energy real (or complex baseband) waveforms on $[0,T]$ with $E_m = \|s_m\|^2 = \int_0^T |s_m(t)|^2 dt$ . Under $\mathcal{H}_m$ the received signal is $Y(t) \;=\; s_m(t) + Z(t), \qquad t\in[0,T],$ where $Z(t)$ is white Gaussian noise with two-sided PSD $N_0/2$ . The average symbol energy is $E_s = \frac{1}{M}\sum_{m=0}^{M-1} E_m$ and the SNR is $\text{SNR} = E_s/N_0$ .

The signals $s_m$ need not be orthogonal and need not have equal energy; we will handle the general case below.

Theorem: Existence of an Orthonormal Signal-Space Basis

For any set of $M$ finite-energy waveforms $\{s_m(t)\}$ there exist orthonormal functions $\{\phi_n(t)\}_{n=1}^{N}$ with $N \leq M$ such that each $s_m$ can be written as $s_m(t) \;=\; \sum_{n=1}^{N} x_{m,n}\, \phi_n(t), \qquad x_{m,n} = \int_0^T s_m(t)\overline{\phi_n(t)}\,dt,$ and the map $s_m \mapsto \mathbf{x}_m = (x_{m,1},\ldots,x_{m,N})^\top$ is an isometry: $\|s_m - s_j\|^2 = \|\mathbf{x}_m - \mathbf{x}_j\|^2$ . The dimension $N$ equals the rank of the Gram matrix $G_{mj} = \langle s_m, s_j\rangle$ .

This is exactly the Gram-Schmidt procedure from finite-dimensional linear algebra, applied in the Hilbert space $L^2[0,T]$ . The isometric embedding means that inner products, norms, and distances all survive the reduction from waveforms to vectors.

Proof

Construct basis by Gram-Schmidt

Set $\phi_1 = s_1 / \|s_1\|$ (assuming $s_1 \neq 0$ ; otherwise skip). For $m = 2, 3, \ldots, M$ define $u_m(t) = s_m(t) - \sum_{k < m} \langle s_m, \phi_k\rangle \phi_k(t).$ If $u_m \neq 0$ , let $\phi_{\text{next}} = u_m / \|u_m\|$ ; otherwise skip (this $s_m$ lies in $\text{span}\{s_1,\ldots,s_{m-1}\}$ ). The collection $\{\phi_n\}_{n=1}^N$ with $N \leq M$ is orthonormal by construction.

Expand each $s_m$ in the basis

By construction $s_m \in \text{span}\{\phi_1,\ldots,\phi_N\}$ , so $s_m = \sum_n x_{m,n}\phi_n$ with coefficients $x_{m,n} = \langle s_m, \phi_n\rangle$ . The $M\times N$ matrix of coefficients has rank $N = \text{rank}(\mathbf{G})$ .

Isometry

By Parseval, $\|s_m - s_j\|^2 = \sum_n |x_{m,n} - x_{j,n}|^2 = \|\mathbf{x}_m - \mathbf{x}_j\|^2$ . Likewise inner products are preserved: $\langle s_m, s_j \rangle = \langle \mathbf{x}_m, \mathbf{x}_j\rangle$ . $\blacksquare$

Theorem: The Projection Vector is a Sufficient Statistic

Define the projection coefficients $Y_n = \langle Y, \phi_n\rangle$ for $n = 1,\ldots,N$ and the "tail" process $\tilde Y(t) = Y(t) - \sum_{n=1}^N Y_n \phi_n(t)$ . Then $\mathbf{Y} = (Y_1,\ldots,Y_N)^\top = \mathbf{x}_m + \mathbf{Z},\qquad \mathbf{Z}\sim \mathcal{N}(\mathbf{0}, \tfrac{N_0}{2}\mathbf{I}_N)\;\;(\text{real case})$ and $\tilde Y$ is independent of $\mathbf{Y}$ and has the same distribution under every $\mathcal{H}_m$ . Consequently $\mathbf{Y}$ is a sufficient statistic for detection: no decision rule using the full waveform $Y(\cdot)$ can outperform the optimal rule based on $\mathbf{Y}$ .

White noise has independent components in any orthonormal basis. The $N$ "in-subspace" components carry all the signal information; the infinitely many "out-of-subspace" components carry only noise whose distribution is identical under every hypothesis — hence irrelevant.

Proof

Gaussianity of projections

The map $Y \mapsto Y_n$ is a continuous linear functional on a Gaussian process, so $(Y_1,\ldots,Y_N)$ is jointly Gaussian under each hypothesis. Its mean under $\mathcal{H}_m$ is $(\langle s_m, \phi_n\rangle)_n = \mathbf{x}_m$ and its covariance is $\mathbb{E}[Z_nZ_k] = \mathbb{E}[\langle Z,\phi_n\rangle \langle Z, \phi_k\rangle] = \tfrac{N_0}{2}\langle \phi_n,\phi_k\rangle = \tfrac{N_0}{2}\delta_{nk}$ .

Independence from the tail

Similarly, $\tilde Y(t)$ is a Gaussian process and for any $n\in\{1,\ldots,N\}$ and $t\in[0,T]$ , $\mathbb{E}[Y_n \tilde Y(t)] = \mathbb{E}[\langle Z,\phi_n\rangle (Z(t) - \sum_k \langle Z,\phi_k\rangle \phi_k(t))] = 0$ after straightforward expansion, using the white-noise covariance $\mathbb{E}[Z(s)Z(t)] = \tfrac{N_0}{2}\delta(s-t)$ . Jointly Gaussian and uncorrelated implies independent.

Sufficiency via Neyman-Fisher factorization

The density of $Y(\cdot)$ (interpreted via Radon-Nikodym with respect to the reference white-noise measure) factors as $p(y(\cdot)\mid\mathcal{H}_m) = g(\mathbf{y};\mathbf{x}_m) \cdot h(\tilde y(\cdot))$ where $g$ depends on $\mathbf{x}_m$ only through $\mathbf{y}=(y_1,\ldots,y_N)$ and $h$ does not depend on $m$ . The Neyman-Fisher theorem then gives sufficiency of $\mathbf{Y}$ . $\blacksquare$

Definition:
Minimum-Distance (ML) Decoder

Under the sufficient statistic $\mathbf{Y} = \mathbf{x}_m + \mathbf{Z}$ with $\mathbf{Z} \sim \mathcal{N}(\mathbf{0}, \tfrac{N_0}{2}\mathbf{I}_N)$ , the ML decision rule is $g_{\text{ML}}(\mathbf{y}) \;=\; \arg\min_{m\in\{0,\ldots,M-1\}} \|\mathbf{y} - \mathbf{x}_m\|^2$ — i.e., decide in favor of the constellation point closest in Euclidean distance to the received vector. With equiprobable signals of equal energy $E_m = E$ , this is also the MAP rule.

Expanding the squared distance gives an equivalent correlator form: $\arg\max_m \bigl( \text{Re}\langle \mathbf{y},\mathbf{x}_m\rangle - \tfrac{1}{2}\|\mathbf{x}_m\|^2 \bigr)$ . When energies are unequal the $-\tfrac{1}{2}\|\mathbf{x}_m\|^2$ term is the energy correction.

Example: QPSK: Decision Regions and Equivalent Scalar Statistic

Apply the minimum-distance decoder to QPSK with $\mathbf{x}_m = \sqrt{E_s}(\cos\theta_m, \sin\theta_m)$ and $\theta_m = \pi/4 + m\pi/2$ for $m = 0,1,2,3$ . What are the decision regions, and how does the decoder simplify?

Solution

Equal-energy constellation

All four points have $\|\mathbf{x}_m\|^2 = E_s$ , so the decoder reduces to $\hat m = \arg\max_m \langle \mathbf{y}, \mathbf{x}_m\rangle$ , which is an angular decision: the angle of $\mathbf{y}$ is quantized to the nearest $\theta_m$ .

Voronoi cells

The four perpendicular bisectors meet at the origin and split the plane into four quadrants: $\mathcal{R}_0 = \{(y_1,y_2) : y_1 > 0, y_2 > 0\}$ , etc.

Separable I/Q detection

Because the boundary lines are the coordinate axes, the decision reduces to sign decisions on $y_1$ and $y_2$ separately. QPSK is therefore two parallel BPSK channels, and its symbol error probability follows from the BPSK result by a union bound (tight up to cross-errors).

Voronoi Decision Regions for 16-QAM — The Voronoi cells for a standard 16-QAM constellation: decision regions are axis-aligned squares (boundaries at half-integer multiples of the grid spacing). Corner points have two neighbors, edge points have three, interior points have four.

⚠️Engineering Note

QAM Demapping Complexity and the I/Q Shortcut

A naive ML demapper for $M$ -QAM requires $M$ squared-distance computations per received symbol. For 1024-QAM (used in 5G NR with high-rank MIMO) this becomes prohibitive — $10^6$ multiplications per symbol at hundreds of millions of symbols per second. The key optimization: a square QAM constellation is the Cartesian product of two independent PAM constellations, so I and Q can be demapped separately in $O(\sqrt M)$ operations each, bringing the total to $O(\sqrt M)$ rather than $O(M)$ . Modern soft-output demappers additionally use the max-log approximation to avoid exponentials.

Practical Constraints

•
3GPP NR supports constellations up to 1024-QAM (Rel-17) — demapping must fit in microseconds per transport block
•
Soft LLR output requires the second-nearest point per bit, not only the nearest

📋 Ref: 3GPP TS 38.211, Section 5.1

Received Samples and Minimum-Distance Decoder

Transmit random constellation symbols through an AWGN channel, color received samples by their ML decision, and compare against the true (transmitted) label to count errors. Adjust the SNR to see how the noise cloud shrinks and errors become rare.

Parameters

Constellation

\text{SNR}

(dB)15

Samples500

Common Mistake: Forgetting the Energy-Correction Term

Mistake:

A student writes the ML decoder as $\hat m = \arg\max_m \langle \mathbf{y}, \mathbf{x}_m\rangle$ — correlation — and applies it to a non-uniform constellation like a PSK subset with amplitude-modulated symbols.

Correction:

Pure correlation is optimal only when $\|\mathbf{x}_m\|^2$ is constant across $m$ . For unequal-energy constellations (e.g. APSK, amplitude-and-phase-shift keying) you must include the energy correction: $\hat m = \arg\max_m \bigl(\text{Re}\langle \mathbf{y}, \mathbf{x}_m\rangle - \tfrac{1}{2}\|\mathbf{x}_m\|^2\bigr)$ . The geometric intuition: correlation finds the point with the largest projection, which biases decisions toward high-energy symbols.

Voronoi Cell

For a finite point set $\{\mathbf{x}_m\}$ in Euclidean space, the Voronoi cell of $\mathbf{x}_m$ is $\mathcal{V}_m = \{\mathbf{y} : \|\mathbf{y} - \mathbf{x}_m\| \leq \|\mathbf{y} - \mathbf{x}_j\|\,\forall j \neq m\}$ — the set of points closer to $\mathbf{x}_m$ than to any other point in the set. ML decision regions for AWGN with equiprobable, equal-energy symbols are exactly Voronoi cells.

Quick Check

Three orthogonal waveforms $s_0, s_1, s_2$ of unit energy are used for 3-ary signaling in AWGN. What is the dimension $N$ of the signal space, and what is the Euclidean distance between every pair?

$N = 3$ , $d = \sqrt{2}$

$N = 3$ , $d = 2$

$N = 2$ , $d = \sqrt{2}$

$N = 2$ , $d = 1$

Correction:

N = 3

,

d = \sqrt{2}

Three orthogonal unit-energy vectors span a 3-D space, and the distance between any pair is $\sqrt{1+1} = \sqrt{2}$ .

Why This Matters: Signal-Space View of OFDM and CDMA

OFDM and CDMA are both instances of the signal-space framework with different basis choices. OFDM picks $\phi_n(t) = e^{j2\pi n t/T} \mathbb{1}[0\leq t \leq T]$ — complex exponentials separated in frequency. CDMA picks $\phi_n(t)$ as time-translates of a PN sequence — separated in a code domain. The minimum-distance decoder of this section is then matched-filter demodulation in each case, and the analysis below applies directly.

See full treatment in Chapter 14

Signal-Space Detection and the Minimum-Distance Decoder

Why Signal Space?

Definition: M-ary Signal Detection in AWGN

Theorem: Existence of an Orthonormal Signal-Space Basis

Construct basis by Gram-Schmidt

Expand each $s_m$ in the basis

Isometry

Theorem: The Projection Vector is a Sufficient Statistic

Gaussianity of projections

Independence from the tail

Sufficiency via Neyman-Fisher factorization

Definition: Minimum-Distance (ML) Decoder

Example: QPSK: Decision Regions and Equivalent Scalar Statistic

Equal-energy constellation

Voronoi cells

Separable I/Q detection

Voronoi Decision Regions for 16-QAM

QAM Demapping Complexity and the I/Q Shortcut

Received Samples and Minimum-Distance Decoder

Parameters

Common Mistake: Forgetting the Energy-Correction Term

Voronoi Cell

Quick Check

Why This Matters: Signal-Space View of OFDM and CDMA

Definition:
M-ary Signal Detection in AWGN

Definition:
Minimum-Distance (ML) Decoder