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Lifting Discrete Results to $L^2$

Real receivers operate on continuous-time waveforms $y(t)$ before sampling. The discrete vector-space story of Sections 2.1--2.3 translates verbatim to the Hilbert space $L^2([0,T])$ of square-integrable functions once we install an appropriate inner product. The payoff is a geometric picture of every digital modulation scheme as a constellation of signal vectors, with optimal detection performed by projection onto the span of those vectors --- the signal-space view that underlies all of coherent communications.

Definition:
The $L^2$ Inner Product

For real-valued signals $u(t), v(t)$ defined on the interval $[0,T]$ , the $L^2$ inner product is

$\langle u, v\rangle = \int_0^T u(t)\, v(t)\,dt,$

with induced norm $\|u\| = \sqrt{\langle u, u\rangle}$ . The space of functions with finite norm is $L^2([0,T])$ --- a Hilbert space.

All linear-algebra identities established for $\mathbb{R}^n$ --- Cauchy--Schwarz, projection onto a subspace, Gram--Schmidt --- transfer to $L^2([0,T])$ once sums over $i$ are replaced by integrals.

Definition:
Continuous-Time Detection in AWGN

Let $s(t)$ be a known deterministic waveform on $[0,T]$ with finite energy $E_s = \int_0^T s^2(t)\,dt$ . Let $w(t)$ be a zero-mean white Gaussian noise process with autocorrelation $\mathbb{E}[w(t)w(\tau)] = \tfrac{N_0}{2}\delta(t-\tau)$ . The continuous-time detection problem is

$\mathcal{H}_0: y(t) = w(t), \qquad \mathcal{H}_1: y(t) = s(t) + w(t), \qquad t\in[0,T].$

Definition:
Continuous-Time Matched Filter

The matched filter for signal $s(t)$ is the linear time-invariant filter with impulse response

$h(t) = s(T - t), \qquad t\in[0,T].$

The output of the filter when driven by $y(t)$ is $z(t) = (y*h)(t) = \int_0^T y(\tau)\,h(t-\tau)\,d\tau$ .

Sampling the output at $t = T$ gives $z(T) = \int_0^T y(\tau) s(\tau)\,d\tau = \langle y, s\rangle$ --- the $L^2$ correlator. The time-reversal $s(T-t)$ is precisely what makes the convolution reproduce a correlation at the sampling instant.

Theorem: The $L^2$ Correlator Is the Continuous-Time Sufficient Statistic

For the continuous-time problem in Definition DContinuous-Time Detection in AWGN, the sufficient statistic for testing $\mathcal{H}_0$ against $\mathcal{H}_1$ is

$T(y) = \langle y, s\rangle = \int_0^T y(t) s(t)\,dt.$

Under $\mathcal{H}_0$ , $T\sim\mathcal{N}(0, \tfrac{N_0}{2}E_s)$ ; under $\mathcal{H}_1$ , $T\sim\mathcal{N}(E_s, \tfrac{N_0}{2}E_s)$ . The detection performance satisfies

$P_d = Q\!\Bigl(Q^{-1}(P_f) - \sqrt{2E_s/N_0}\Bigr).$

This is the identical statement as the discrete case, with the deflection $d^2 = 2E_s/N_0$ once again emerging as the only quantity that matters. Sampling resolution, oversampling, and the particular discretisation grid all fall away.

Proof

Project onto the signal direction.

Define the unit-norm waveform $\phi(t) = s(t)/\sqrt{E_s}$ . Any observation $y(t)\in L^2$ decomposes as

$y(t) = Y_1\,\phi(t) + y_\perp(t), \qquad Y_1 = \langle y,\phi\rangle, \;\; \langle y_\perp, \phi\rangle = 0.$

Show that $y_\perp$ is independent of the hypothesis.

Under both hypotheses $y_\perp(t)$ equals the projection of $w(t)$ onto the orthogonal complement of $\mathrm{span}\{\phi\}$ --- its distribution is identical under $\mathcal{H}_0$ and $\mathcal{H}_1$ . By Fisher--Neyman factorisation ([?s05:thm-fisher-neyman]) $Y_1$ is sufficient.

Distribution of $Y_1$.

$Y_1 = \int_0^T y(t)\phi(t)\,dt$ is the integral of a Gaussian process against a deterministic function, hence Gaussian. Under $\mathcal{H}_0$ : mean $0$ ; variance $\int_0^T\int_0^T \phi(t)\phi(\tau)\cdot\tfrac{N_0}{2}\delta(t-\tau)\,dt\,d\tau = \tfrac{N_0}{2}$ . Under $\mathcal{H}_1$ : mean $\int_0^T s(t)\phi(t)\,dt = \sqrt{E_s}$ ; same variance.

Scale back to $T(y)$ and compute $P_D$.

$T(y) = \sqrt{E_s}\,Y_1$ , so $T$ has the stated Gaussian distributions. The ROC formula follows as in [?s01:thm-pd-awgn]. $\blacksquare$

Theorem: The Continuous-Time Matched Filter Maximises Output SNR

Among all linear time-invariant filters with impulse response $h(t)$ of finite energy, the filter $h^\star(t) = s(T-t)$ maximises the output signal-to-noise ratio at the sampling instant $t=T$ :

$\mathrm{SNR}_{\mathrm{out}}(h) = \frac{|(h*s)(T)|^2}{(N_0/2)\|h\|^2} \;\leq\; \frac{2E_s}{N_0},$

with equality iff $h(t) = c\,s(T-t)$ .

Proof

Rewrite the numerator.

$(h*s)(T) = \int_0^T h(T-\tau) s(\tau)\,d\tau = \langle \tilde{h}, s\rangle,$ $where$ \tilde{h}(\tau) = h(T-\tau) $. By change of variables,$ |\tilde h| = |h|$.

Apply Cauchy--Schwarz in $L^2$.

$|\langle \tilde h, s\rangle|^2 \leq \|\tilde h\|^2 \|s\|^2 = \|h\|^2 E_s,$ $with equality iff$ \tilde h(t) = c,s(t) $, i.e.,$ h(t) = c,s(T-t)$.

Conclude.

$\mathrm{SNR}_{\mathrm{out}}(h) \leq \frac{\|h\|^2 E_s}{(N_0/2)\|h\|^2} = \frac{2E_s}{N_0}. \quad\blacksquare$ $

Theorem: Signal-Space Representation via Gram--Schmidt

Let $\{s_1(t),\dots,s_M(t)\}\subset L^2([0,T])$ be $M$ waveforms. The Gram--Schmidt procedure produces an orthonormal basis $\{\phi_1,\dots,\phi_N\}$ with $N\leq M$ and coefficients $s_m(t) = \sum_{k=1}^N s_{mk}\phi_k(t)$ , $s_{mk} = \langle s_m,\phi_k\rangle$ . Under AWGN, the sufficient statistic for detecting which $s_m$ was transmitted is the projection vector $\mathbf{y} = (\langle y,\phi_1\rangle,\dots,\langle y,\phi_N\rangle)^{\mathsf{T}}\in\mathbb{R}^N$ , and the ML detector chooses $\widehat{m} = \arg\min_m \|\mathbf{y}-\mathbf{s}_m\|$ .

Any set of waveforms lives in at most an $M$ -dimensional subspace of $L^2$ . Gram--Schmidt finds that subspace, and the detector only needs the $N$ coordinates of $y(t)$ inside it. Everything else is noise orthogonal to every hypothesis --- informationally irrelevant.

Proof

Extract the relevant subspace.

Write $y(t) = \sum_{k=1}^N y_k\phi_k(t) + y_\perp(t)$ with $y_k = \langle y,\phi_k\rangle$ and $y_\perp\perp\phi_k$ for all $k$ . Under hypothesis $\mathcal{H}_m$ , $y_k = s_{mk} + w_k$ with $w_k = \langle w,\phi_k\rangle$ .

Distribution of the noise coordinates.

$\mathbf{w} = (w_1,\dots,w_N)^{\mathsf{T}}$ has $\mathbb{E}[w_k w_l] = \tfrac{N_0}{2}\langle\phi_k,\phi_l\rangle = \tfrac{N_0}{2}\delta_{kl}$ , so $\mathbf{w}\sim\mathcal{N}(\mathbf{0}, \tfrac{N_0}{2}\mathbf{I}_N)$ .

Irrelevance of $y_\perp$.

$y_\perp(t) = w_\perp(t)$ regardless of which signal is sent, since $s_m \in \mathrm{span}\{\phi_k\}$ . Furthermore $w_\perp$ is independent of $\mathbf{w}$ (orthogonal Gaussians are independent). Thus $\mathbf{y}$ is sufficient.

ML detection is minimum-distance.

Given $\mathbf{y}\sim\mathcal{N}(\mathbf{s}_m, \tfrac{N_0}{2}\mathbf{I}_N)$ , the ML rule is $\widehat{m} = \arg\max_m p(\mathbf{y}|\mathcal{H}_m) = \arg\min_m \|\mathbf{y}-\mathbf{s}_m\|^2$ . $\blacksquare$

Example: QPSK as a Two-Dimensional Constellation

The four QPSK waveforms are $s_m(t) = A\cos(2\pi f_c t + m\pi/2 + \pi/4)$ for $m\in\{0,1,2,3\}$ over $[0,T]$ , with $f_cT$ a large integer. Construct the signal-space representation and identify $N$ .

Solution

Expand via trigonometric identities.

$s_m(t) = A\cos(m\pi/2+\pi/4)\cos(2\pi f_c t) - A\sin(m\pi/2+\pi/4)\sin(2\pi f_c t)$ .

Choose basis.

Let $\phi_1(t) = \sqrt{2/T}\cos(2\pi f_c t)$ and $\phi_2(t) = -\sqrt{2/T}\sin(2\pi f_c t)$ . These are orthonormal on $[0,T]$ (using $f_cT\in\mathbb{Z}$ ).

Read off coordinates.

$s_m = (A\sqrt{T/2}\cos(m\pi/2+\pi/4), A\sqrt{T/2}\sin(m\pi/2+\pi/4))^{\mathsf{T}}$ . The four points lie on a circle of radius $\sqrt{E_s} = A\sqrt{T/2}$ at $45^\circ, 135^\circ, 225^\circ, 315^\circ$ . So $N=2$ , and ML detection is nearest-neighbour on a $2\times 2$ grid.

Example: Matched Filter for a Rectangular Pulse

Compute the matched-filter impulse response and plot its output when driven by $y(t)=s(t)+w(t)$ for the rectangular pulse $s(t) = \mathbf{1}_{[0,T]}(t)$ .

Solution

Impulse response.

$h(t) = s(T-t) = \mathbf{1}_{[0,T]}(T-t) = \mathbf{1}_{[0,T]}(t)$ . The matched filter is the integrator of width $T$ --- it simply averages the input over a $T$ -second window.

Output when $y(t)=s(t)$.

$(s*h)(t)$ is the triangular autocorrelation of the rectangular pulse: zero for $t<0$ , rising linearly to a peak of $T$ at $t=T$ , then falling linearly to zero at $t=2T$ . Peak is attained at $t=T$ , precisely where the detector samples.

Interpretation.

The integrator is the simplest matched filter. For this reason RC-integration is used throughout elementary receiver designs (chopper amplifiers, lock-in detectors) as a practical matched-filter surrogate.

Common Mistake: Causality of the Matched Filter

Mistake:

"The matched filter $h(t) = s(T-t)$ looks non-causal because $s(-t)$ would be non-causal."

Correction:

On the interval $[0,T]$ , $s(T-t)$ is the time-reversed signal shifted by $T$ so that it occupies $[0,T]$ again. This built-in delay is precisely what makes the filter causal: the output at time $t=T$ depends only on past samples of $y$ during $[0,T]$ .

Common Mistake: Sample at $t=T$ , Not at the Peak

Mistake:

"To maximise SNR I should sample the matched-filter output at its empirical peak."

Correction:

Peak-picking is a separate detector (the max-correlator for unknown delay --- a GLRT!) and has a different, generally worse, false-alarm behaviour than sampling at the known delay $t=T$ . The optimality of $2E_s/N_0$ is tied to sampling at the known delay.

Quick Check

For $M$ distinct signals in $L^2([0,T])$ , the signal-space dimension $N$ satisfies:

$N = M$ always.

$N \leq M$ , with equality iff the signals are linearly independent.

$N \geq M$ .

$N$ equals the number of orthogonal signals in the set.

Correction:

N \leq M

, with equality iff the signals are linearly independent.

Gram--Schmidt produces at most $M$ nonzero basis functions; duplicates collapse.

Quick Check

In the AWGN autocorrelation $\mathbb{E}[w(t)w(\tau)]=\tfrac{N_0}{2}\delta(t-\tau)$ , what is the physical interpretation of $N_0/2$ ?

Noise power in watts.

Two-sided noise PSD in W/Hz.

Noise energy.

Noise sample variance.

Correction:

Two-sided noise PSD in W/Hz.

The one-sided PSD is $N_0$ , the two-sided is $N_0/2$ .

Key Takeaway

Detecting any waveform in any finite-dimensional set against AWGN reduces to detecting a vector in $\mathbb{R}^N$ against white Gaussian noise, where $N$ is the dimension of the signals' span. Every digital modulation --- BPSK, QPSK, $M$ -QAM, $M$ -PSK, pulse-position modulation, orthogonal frequency-shift keying --- is an instance of this reduction.

Why This Matters: Signal Space and Modulation Taxonomy

The signal-space dimension $N$ determines a modulation's geometry: $N=1$ for $M$ -PAM, $N=2$ for $M$ -PSK / $M$ -QAM, $N=M$ for orthogonal $M$ -FSK, and $N=T/T_c$ for CDMA (where $T_c$ is the chip period). The minimum distance $d_{\min}$ within the constellation, combined with the AWGN deflection result, determines the symbol-error probability via the union bound --- a topic deferred to Chapter 3.

See full treatment in Chapter 3

Historical Note: Wozencraft and Jacobs (1965)

1960s

The signal-space view of digital modulation was systematised in Wozencraft & Jacobs' 1965 textbook Principles of Communication Engineering. They took the geometric picture --- inherited from the Karhunen--Loeve expansion and from Shannon's 1948 geometric capacity argument --- and made it the pedagogical centrepiece of modulation theory. Every subsequent digital-communications textbook has followed their lead; ours is no exception.

The Continuous-Time Matched Filter

Lifting Discrete Results to L2L^2L2

Definition: The L2L^2L2 Inner Product

Definition: Continuous-Time Detection in AWGN

Definition: Continuous-Time Matched Filter

Theorem: The L2L^2L2 Correlator Is the Continuous-Time Sufficient Statistic

Project onto the signal direction.

Show that $y_\perp$ is independent of the hypothesis.

Distribution of $Y_1$.

Scale back to $T(y)$ and compute $P_D$.

Theorem: The Continuous-Time Matched Filter Maximises Output SNR

Rewrite the numerator.

Apply Cauchy--Schwarz in $L^2$.

Conclude.

Theorem: Signal-Space Representation via Gram--Schmidt

Extract the relevant subspace.

Distribution of the noise coordinates.

Irrelevance of $y_\perp$.

ML detection is minimum-distance.

Example: QPSK as a Two-Dimensional Constellation

Expand via trigonometric identities.

Choose basis.

Read off coordinates.

Example: Matched Filter for a Rectangular Pulse

Impulse response.

Output when $y(t)=s(t)$.

Interpretation.

Common Mistake: Causality of the Matched Filter

Common Mistake: Sample at t=Tt=Tt=T, Not at the Peak

Quick Check

Quick Check

Key Takeaway

Why This Matters: Signal Space and Modulation Taxonomy

Historical Note: Wozencraft and Jacobs (1965)

Lifting Discrete Results to $L^2$

Definition:
The $L^2$ Inner Product

Definition:
Continuous-Time Detection in AWGN

Definition:
Continuous-Time Matched Filter

Theorem: The $L^2$ Correlator Is the Continuous-Time Sufficient Statistic

Common Mistake: Sample at $t=T$ , Not at the Peak