Ferkans — Interactive Telecom Tutor

Why Known-Signal Detection Matters

Every coherent digital receiver and every active sensing system --- radar, sonar, GNSS, ultrasound imaging --- faces the same elementary inference problem: is a known waveform $\mathbf{s}$ present in the observation, or is the receiver hearing only noise? The answer, when the noise is Gaussian and white, is exceptionally clean: the optimal test is a linear functional of the data, the correlation between what we observe and what we would expect. Out of this elementary observation grow the matched filter, the signal-space view of modulation, and the information-theoretic $E_s/N_0$ scaling that we will meet again in Part II of this book.

We develop the theory in discrete time --- the reader should verify that every statement extends to the continuous-time $L^2$ setting we treat formally in Section 2.4.

Definition:
Known-Signal Detection Problem in AWGN

Let $\mathbf{s} = (s_1, \dots, s_n)^{\mathsf{T}} \in \mathbb{R}^n$ be a known deterministic signal and let $\mathbf{w} \sim \mathcal{N}(\mathbf{0}, \sigma^2 \mathbf{I})$ be an additive white Gaussian noise vector with per-sample variance $\sigma^2 = N_0/2$ . The binary hypothesis testing problem

$\mathcal{H}_0: \mathbf{y} = \mathbf{w}, \qquad \mathcal{H}_1: \mathbf{y} = \mathbf{s} + \mathbf{w},$

is called the known-signal detection problem in AWGN. The signal energy is $E_s = \|\mathbf{s}\|^2 = \sum_{i=1}^{n} s_i^2$ .

The assumption that $\mathbf{s}$ is known corresponds, in communications, to a coherent receiver that has acquired phase and timing; in radar, to a clairvoyant detector that knows the target range and velocity. Composite generalisations are taken up in Section 2.2.

Definition:
Correlator Statistic

The correlator statistic for the detection problem above is the inner product of the observation and the signal template,

$T(\mathbf{y}) \;=\; \langle \mathbf{y}, \mathbf{s}\rangle \;=\; \sum_{i=1}^n s_i \, y_i \;=\; \mathbf{s}^{\mathsf{T}} \mathbf{y}.$

Geometrically, $T(\mathbf{y})$ is (up to the factor $\|\mathbf{s}\|$ ) the orthogonal projection of $\mathbf{y}$ onto the one-dimensional subspace spanned by $\mathbf{s}$ .

Definition:
Deflection Coefficient

For a test statistic $T(\mathbf{y})$ with finite mean and variance under both hypotheses, the deflection coefficient is

$d^2 \;=\; \frac{\bigl(\mathbb{E}_1[T] - \mathbb{E}_0[T]\bigr)^2}{\text{Var}_{0}(T)}.$

It measures the separation of the hypothesis means in units of the noise-only standard deviation of $T$ .

The deflection coefficient is the signal-to-noise ratio of the test statistic. For Gaussian observations and a linear statistic it determines performance exactly; more generally it governs the leading-order behaviour of $P_D$ at low SNR.

Theorem: LRT for Known Signal in AWGN Reduces to the Correlator

For the detection problem in Definition DKnown-Signal Detection Problem in AWGN, the log-likelihood ratio is

$\ell(\mathbf{y}) \;=\; \frac{1}{\sigma^2}\,\mathbf{s}^{\mathsf{T}}\mathbf{y} \;-\; \frac{E_s}{2\sigma^2},$

so the Neyman--Pearson test at level $\alpha$ is equivalent to comparing the correlator statistic $T(\mathbf{y}) = \mathbf{s}^{\mathsf{T}}\mathbf{y}$ to a threshold $\gamma$ :

$g(\mathbf{y}) = 1 \iff T(\mathbf{y}) > \gamma.$

Only the component of $\mathbf{y}$ along the signal direction carries information about which hypothesis is true; noise in orthogonal directions is irrelevant. The LRT extracts exactly this component.

Proof

Write the two Gaussian densities.

Under $\mathcal{H}_0$ , $\mathbf{y}\sim\mathcal{N}(\mathbf{0},\sigma^2\mathbf{I})$ , so

$p_0(\mathbf{y}) = (2\pi\sigma^2)^{-n/2} \exp\!\Bigl(-\tfrac{1}{2\sigma^2}\|\mathbf{y}\|^2\Bigr).$

Under $\mathcal{H}_1$ , $\mathbf{y}\sim\mathcal{N}(\mathbf{s},\sigma^2\mathbf{I})$ , so

$p_1(\mathbf{y}) = (2\pi\sigma^2)^{-n/2} \exp\!\Bigl(-\tfrac{1}{2\sigma^2}\|\mathbf{y}-\mathbf{s}\|^2\Bigr).$

Form the log-likelihood ratio.

$\ell(\mathbf{y}) = \log\frac{p_1(\mathbf{y})}{p_0(\mathbf{y})} = -\frac{1}{2\sigma^2}\Bigl(\|\mathbf{y}-\mathbf{s}\|^2 - \|\mathbf{y}\|^2\Bigr).$ $

Expand $\|\mathbf{y}-\mathbf{s}\|^2$.

$\|\mathbf{y}-\mathbf{s}\|^2 = \|\mathbf{y}\|^2 - 2\mathbf{s}^{\mathsf{T}}\mathbf{y} + \|\mathbf{s}\|^2, KATEXPLACEHOLDER0END \ell(\mathbf{y}) = \frac{1}{\sigma^2}\,\mathbf{s}^{\mathsf{T}}\mathbf{y} - \frac{E_s}{2\sigma^2}.$ $

Reduce to a threshold test on $T(\mathbf{y})$.

Since $\ell(\mathbf{y})$ is a strictly increasing affine function of $T(\mathbf{y}) = \mathbf{s}^{\mathsf{T}}\mathbf{y}$ , the LRT $\ell(\mathbf{y}) > \log\eta$ is equivalent to $T(\mathbf{y}) > \gamma$ for a suitably chosen threshold $\gamma = \sigma^2\log\eta + E_s/2$ . By Neyman--Pearson (TNeyman-Pearson Lemma), this test is uniformly most powerful at every level $\alpha$ . $\blacksquare$

Theorem: Detection Performance in AWGN

For the correlator test $T(\mathbf{y}) > \gamma$ , the false-alarm and detection probabilities are

$P_f = Q\!\Bigl(\tfrac{\gamma}{\sigma\sqrt{E_s}}\Bigr), \qquad P_d = Q\!\Bigl(\tfrac{\gamma - E_s}{\sigma\sqrt{E_s}}\Bigr).$

Eliminating $\gamma$ yields the ROC relationship

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

and the deflection coefficient of the correlator is $d^2 = 2E_s/N_0$ .

The detection performance depends on the data only through $E_s/N_0$ . Doubling $E_s$ (more transmit energy) shifts the ROC curve up by the same amount as halving $N_0$ (lower-noise receiver).

Proof

Distribution of $T$ under $\mathcal{H}_0$.

$T = \mathbf{s}^{\mathsf{T}}\mathbf{w}$ is a linear functional of a Gaussian vector, hence $T \sim \mathcal{N}(0, \sigma^2 \|\mathbf{s}\|^2) = \mathcal{N}(0, \sigma^2 E_s)$ .

Distribution of $T$ under $\mathcal{H}_1$.

$T = \|\mathbf{s}\|^2 + \mathbf{s}^{\mathsf{T}}\mathbf{w}$ , so $T \sim \mathcal{N}(E_s, \sigma^2 E_s)$ .

Compute $P_F$ and $P_D$.

Under $\mathcal{H}_0$ the event $\{T > \gamma\}$ has probability

$P_f = \Pr[T > \gamma \mid \mathcal{H}_0] = Q\!\left(\frac{\gamma}{\sigma\sqrt{E_s}}\right).$

Under $\mathcal{H}_1$ ,

$P_d = \Pr[T > \gamma \mid \mathcal{H}_1] = Q\!\left(\frac{\gamma - E_s}{\sigma\sqrt{E_s}}\right).$

Eliminate $\gamma$.

From the first equation, $\gamma = \sigma\sqrt{E_s}\,Q^{-1}(P_f)$ . Substituting into the second and using $\sigma^2 = N_0/2$ gives

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

Read off the deflection coefficient.

$\mathbb{E}_1[T] - \mathbb{E}_0[T] = E_s$ and $\text{Var}_{0}(T) = \sigma^2 E_s$ , so

$d^2 = \frac{E_s^2}{\sigma^2 E_s} = \frac{E_s}{\sigma^2} = \frac{2E_s}{N_0}. \quad\blacksquare$

Theorem: The Matched Filter Maximises Output SNR

Let $\mathbf{h} \in \mathbb{R}^n$ be any non-zero linear filter applied to $\mathbf{y}$ , producing the output $z = \mathbf{h}^{\mathsf{T}}\mathbf{y}$ . Among all such filters, the filter $\mathbf{h}^{\star} = c\,\mathbf{s}$ (for any non-zero constant $c$ ) maximises the output signal-to-noise ratio

$\mathrm{SNR}_{\mathrm{out}}(\mathbf{h}) \;=\; \frac{|\mathbf{h}^{\mathsf{T}}\mathbf{s}|^2}{\sigma^2\|\mathbf{h}\|^2},$

and attains the maximum value $\mathrm{SNR}_{\mathrm{out}}(\mathbf{s}) = E_s/\sigma^2 = 2E_s/N_0$ .

Among all linear combinations of the data, the one weighted exactly by the signal template extracts the most signal per unit noise power. This is the discrete-time form of the matched-filter principle.

Proof

Apply the Cauchy--Schwarz inequality.

For any vectors $\mathbf{h}, \mathbf{s} \in \mathbb{R}^n$ ,

$|\mathbf{h}^{\mathsf{T}}\mathbf{s}|^2 \leq \|\mathbf{h}\|^2 \|\mathbf{s}\|^2 = \|\mathbf{h}\|^2 E_s,$

with equality iff $\mathbf{h} = c\,\mathbf{s}$ for some scalar $c$ .

Bound the output SNR.

$\mathrm{SNR}_{\mathrm{out}}(\mathbf{h}) = \frac{|\mathbf{h}^{\mathsf{T}}\mathbf{s}|^2}{\sigma^2\|\mathbf{h}\|^2} \leq \frac{\|\mathbf{h}\|^2 E_s}{\sigma^2\|\mathbf{h}\|^2} = \frac{E_s}{\sigma^2} = \frac{2E_s}{N_0}.$ $

Verify equality at $\mathbf{h}=c\mathbf{s}$.

For $\mathbf{h} = c\mathbf{s}$ , $\mathbf{h}^{\mathsf{T}}\mathbf{s} = c E_s$ and $\|\mathbf{h}\|^2 = c^2 E_s$ , so $\mathrm{SNR}_{\mathrm{out}} = c^2 E_s^2 / (\sigma^2 c^2 E_s) = E_s/\sigma^2$ . Equality is attained. $\blacksquare$

Connection to the LRT.

The maximising filter $\mathbf{h}^{\star} = \mathbf{s}$ is, up to scaling, exactly the correlator statistic derived in TLRT for Known Signal in AWGN Reduces to the Correlator. Thus the matched filter is simultaneously (i) the UMP test statistic in the Neyman--Pearson sense and (ii) the linear filter with maximum output SNR --- two independent optimality criteria that coincide in AWGN.

Example: Antipodal (BPSK) Detection

A binary antipodal modulator transmits one of two signals $\mathbf{s}_1 = +\mathbf{s}$ or $\mathbf{s}_0 = -\mathbf{s}$ with equal prior probability, where $\mathbf{s}$ has energy $E_s$ . The received vector is $\mathbf{y} = \mathbf{s}_m + \mathbf{w}$ with $\mathbf{w}\sim\mathcal{N}(\mathbf{0},\sigma^2\mathbf{I})$ . Derive the ML decision rule and the bit-error probability.

Solution

Reduce to known-signal detection.

Write $\mathbf{y} = \pm \mathbf{s} + \mathbf{w}$ , so the two hypotheses $\mathcal{H}_0: \mathbf{y}=-\mathbf{s}+\mathbf{w}$ and $\mathcal{H}_1: \mathbf{y}=+\mathbf{s}+\mathbf{w}$ are both deterministic-signal-in-AWGN hypotheses.

Apply Theorem <a href="#thm-lrt-awgn-reduces-to-correlator" class="ferkans-ref" title="Theorem: LRT for Known Signal in AWGN Reduces to the Correlator" data-ref-type="theorem"><span class="ferkans-ref-badge">T</span>LRT for Known Signal in AWGN Reduces to the Correlator</a>.

The LLR is linear in $T(\mathbf{y}) = \mathbf{s}^{\mathsf{T}}\mathbf{y}$ , with the threshold set to zero for equal priors. The ML rule is therefore $\hat{m} = \mathrm{sgn}(\mathbf{s}^{\mathsf{T}}\mathbf{y})$ .

Compute error probability.

Given $\mathcal{H}_1$ , the conditional statistic satisfies $T \sim \mathcal{N}(E_s, \sigma^2 E_s)$ . An error occurs iff $T \le 0$ :

$P_b = \Pr[T \leq 0 \mid \mathcal{H}_1] = Q\!\Bigl(\tfrac{E_s}{\sigma\sqrt{E_s}}\Bigr) = Q\!\Bigl(\sqrt{2E_s/N_0}\,\Bigr).$

This is the celebrated BPSK error formula --- a direct corollary of TDetection Performance in AWGN with an appropriate sign change.

Example: Deflection of a Sub-Optimal Linear Detector

Consider a two-sample observation $\mathbf{y} = (y_1, y_2)^{\mathsf{T}}$ with $\mathbf{s} = (1, 2)^{\mathsf{T}}$ and $\sigma^2 = 1$ . Compute the deflection coefficient of the simple averager $T_{\mathrm{avg}}(\mathbf{y}) = (y_1 + y_2)/2$ and compare it with the correlator.

Solution

Filter coefficients.

The averager is the linear filter $\mathbf{h}_{\mathrm{avg}} = (1/2, 1/2)^{\mathsf{T}}$ .

Output SNR.

$\mathbf{h}_{\mathrm{avg}}^{\mathsf{T}}\mathbf{s} = 1/2 + 1 = 3/2$ and $\|\mathbf{h}_{\mathrm{avg}}\|^2 = 1/2$ , so

$d_{\mathrm{avg}}^2 = \frac{(3/2)^2}{1 \cdot 1/2} = \frac{9/4}{1/2} = 4.5.$

Correlator output SNR.

The correlator is the matched filter, with $d_{\mathrm{MF}}^2 = E_s/\sigma^2 = (1^2 + 2^2)/1 = 5$ .

Compare.

The averager loses $10 \log_{10}(5/4.5) \approx 0.46$ dB of effective SNR. The loss is small because the two components $s_1, s_2$ have similar magnitudes; had they been very unequal (say $1$ and $10$ ), the averager would suffer a much larger penalty.

Output SNR of Matched vs. Mismatched Filters

Sweep the input SNR and compare the output SNR of the matched filter against a boxcar filter and a Gaussian filter. The matched filter upper bounds every other unit-norm choice, confirming TThe Matched Filter Maximises Output SNR.

Parameters

Min SNR (dB)-10

Max SNR (dB)20

Pulse shape

Samples per pulse128

Correlator vs. Matched Filter (Time Domain)

Observe that the running integral of the correlator and the matched filter output agree at every lag. The detection decision is taken at $t=T$ , where both produce the same number $\langle\mathbf{y},\mathbf{s}\rangle$ .

Parameters

Samples200

Noise

\sigma

0.2

Random seed7

ROC Curves for AWGN Detection

Receiver operating characteristic for the correlator at several values of $E_s/N_0$ . Drag the curves to see how the deflection coefficient $d^2 = 2E_s/N_0$ determines the achievable $(P_F, P_D)$ operating point.

Parameters

E_s/N_0

values (dB)

Common Mistake: Energy, Not Amplitude, Drives Performance

Mistake:

"Doubling the signal amplitude doubles the SNR, so $P_D$ should double at low false-alarm rates."

Correction:

Doubling the amplitude quadruples the energy, and hence quadruples $E_s/N_0$ . The argument of the $Q$ -function grows like $\sqrt{E_s/N_0}$ , so $P_D$ responds non-linearly. Performance is set by $E_s/N_0$ , not by peak amplitude. This is why energy --- not amplitude --- is the canonical quantity in the link budget.

Common Mistake: $N_0$ vs. $\sigma^2$ Conventions

Mistake:

"The deflection is $E_s/N_0$ , so the argument of $Q(\\cdot)$ should be $\\sqrt{E_s/N_0}$ ."

Correction:

The discrepancy arises from the conversion between the one-sided PSD $N_0$ and the per-sample variance $\sigma^2 = N_0/2$ . With this convention the correct argument is $\sqrt{2 E_s/N_0} = \sqrt{E_s/\sigma^2}$ . Always pin down your noise convention on the first line of any derivation.

Common Mistake: Matched Filter Is Defined Only Up to a Scalar

Mistake:

"My code normalises the filter to unit norm, so the statistic is $\\mathbf{s}^{\\mathsf{T}}\\mathbf{y} / \\|\\mathbf{s}\\|$ , and the deflection coefficient becomes $2E_s/N_0/\\|\\mathbf{s}\\|^2$ ."

Correction:

The deflection coefficient is scale-invariant: multiplying the filter by any constant multiplies the numerator of $\mathrm{SNR}_{\mathrm{out}}$ by $c^2$ and the denominator by $c^2$ . The statistic changes, and so does the threshold, but ROC performance does not.

Quick Check

The deflection coefficient for known-signal detection in AWGN is:

$E_s/N_0$

$2E_s/N_0$

$\sqrt{2E_s/N_0}$

$E_s^2/N_0$

Correction:

2E_s/N_0

Correct: $d^2 = E_s/\sigma^2 = 2E_s/N_0$ using the one-sided PSD convention.

Quick Check

Which statement about the matched filter in discrete AWGN is TRUE?

Any filter orthogonal to $\mathbf{s}$ maximises output SNR.

The matched filter is unique up to a positive scalar.

The matched filter achieves the Cramer--Rao lower bound.

The matched filter maximises $P_D$ for every fixed $P_F$ only when the noise is white.

Correction:

The matched filter is unique up to a positive scalar., The matched filter maximises

P_D

for every fixed

P_F

only when the noise is white.

Equality in Cauchy--Schwarz requires $\mathbf{h} = c\mathbf{s}$ ; scaling doesn't change the SNR.

Quick Check

For equal priors, the ML decision rule for known-signal detection in AWGN declares $\mathcal{H}_1$ when:

$\mathbf{s}^{\mathsf{T}}\mathbf{y} > E_s / 2$

$\mathbf{s}^{\mathsf{T}}\mathbf{y} > 0$

$\|\mathbf{y} - \mathbf{s}\|^2 < \|\mathbf{y}\|^2$

$\|\mathbf{y}\|^2 > E_s$

Correction:

\mathbf{s}^{\mathsf{T}}\mathbf{y} > E_s / 2

,

\|\mathbf{y} - \mathbf{s}\|^2 < \|\mathbf{y}\|^2

Setting $\eta = 1$ in the LRT gives $\gamma = \sigma^2 \cdot 0 + E_s/2 = E_s/2$ .

Historical Note: North's Matched Filter (1943)

1940s

The matched filter was introduced by Dwight O. North in a 1943 RCA technical report, "An Analysis of the Factors Which Determine Signal/Noise Discrimination in Pulsed-Carrier Systems" --- classified wartime work on radar signal processing. North proved that the filter matched to the expected pulse shape maximises the signal-to-noise ratio at the sampling instant. His report circulated widely within Allied laboratories but was not openly published until 1963 in the Proceedings of the IEEE. By then the matched filter had been independently rederived in the communications literature (by Van Vleck and Middleton, 1946) and had become the cornerstone of coherent detection.

🔧Engineering Note

Matched Filters in GPS Acquisition

A GPS receiver searching for a satellite's C/A code correlates 1 ms of received baseband with 1023 possible code-phase offsets of the satellite's known 1023-chip Gold code --- a direct application of the discrete AWGN matched filter with $n \approx 2046$ samples (two samples per chip). The receiver detects the satellite when the correlator peak exceeds a threshold set for $P_F \approx 10^{-3}$ per code--Doppler bin. At the nominal received $C/N_0 \approx 45$ dB-Hz the processing gain $10\log_{10}(1023) \approx 30$ dB lifts the buried signal out of the noise floor --- without the matched filter, acquisition would be impossible.

Practical Constraints

•
1 ms coherent integration limited by the 50 Hz navigation data bit
•
Non-coherent combining over multiple code periods for weaker signals
•
Two-dimensional search over code phase and Doppler

📋 Ref: IS-GPS-200 (Interface Specification)

⚠️Engineering Note

Matched Filtering and Pulse Compression in Radar

A radar transmits a high-bandwidth frequency-modulated pulse (LFM or "chirp") of duration $T$ and bandwidth $B$ and applies the matched filter on receive. The pulse-compression ratio $BT$ --- the time-bandwidth product --- equals the processing gain of the matched filter and typically reaches $10^3$ -- $10^5$ . This decouples range resolution ( $c/(2B)$ ) from energy-on-target ( $P_{\mathrm{peak}} T$ ), which would otherwise require an impractical peak power. The detection rule is a thresholded matched-filter output per range gate; CFAR techniques (forward reference to Chapter 4) adapt the threshold to non-stationary clutter.

Practical Constraints

•
Transmit bandwidth set by FCC/ITU allocation (e.g., 30 MHz for S-band weather radar)
•
Peak-to-average power ratio bounded by amplifier saturation
•
Ambiguity function sidelobes drive waveform design

📋 Ref: MIL-STD-461 (radar EMI), IEEE 521 standard letter bands

Matched filter

The linear filter whose impulse response is the time-reverse of the known signal, $h(t) = s(T-t)$ (continuous time) or $\mathbf{h} = c\,\mathbf{s}$ (discrete time). Its sampled output at $t=T$ is the correlator statistic $\langle y, s\rangle$ , and it maximises the output SNR for detection in AWGN.

Deflection coefficient

For a test statistic $T$ , $d^2 = (\mathbb{E}_1[T] - \mathbb{E}_0[T])^2 / \text{Var}_{0}(T)$ . Equals $2E_s/N_0$ for the known-signal correlator in AWGN.

Correlator receiver

A receiver that computes $\langle y, s\rangle = \int y(t) s(t)\,dt$ (or its discrete analogue) and compares it to a threshold. Equivalent to a matched-filter sampled at $t=T$ .

Why This Matters: From Matched Filters to Coherent Demodulation

Every coherent digital receiver --- from IEEE 802.11 through 5G NR to deep-space Mariner-class telemetry --- performs matched filtering as its first baseband operation. In 5G NR, the demodulator projects the received OFDM symbol onto known DMRS (demodulation reference signal) pilots; in DVB-S2X, onto known pilot fields; in optical coherent links, onto a local oscillator carrying a reconstructed signal phase. The theoretical backdrop is invariably Theorem TLRT for Known Signal in AWGN Reduces to the Correlator.

Detection of Known Signals in AWGN

Why Known-Signal Detection Matters

Definition: Known-Signal Detection Problem in AWGN

Definition: Correlator Statistic

Definition: Deflection Coefficient

Theorem: LRT for Known Signal in AWGN Reduces to the Correlator

Write the two Gaussian densities.

Form the log-likelihood ratio.

Expand $\|\mathbf{y}-\mathbf{s}\|^2$.

Reduce to a threshold test on $T(\mathbf{y})$.

Theorem: Detection Performance in AWGN

Distribution of $T$ under $\mathcal{H}_0$.

Distribution of $T$ under $\mathcal{H}_1$.

Compute $P_F$ and $P_D$.

Eliminate $\gamma$.

Read off the deflection coefficient.

Theorem: The Matched Filter Maximises Output SNR

Apply the Cauchy--Schwarz inequality.

Bound the output SNR.

Verify equality at $\mathbf{h}=c\mathbf{s}$.

Connection to the LRT.

Example: Antipodal (BPSK) Detection

Reduce to known-signal detection.

Apply Theorem <a href="#thm-lrt-awgn-reduces-to-correlator" class="ferkans-ref" title="Theorem: LRT for Known Signal in AWGN Reduces to the Correlator" data-ref-type="theorem"><span class="ferkans-ref-badge">T</span>LRT for Known Signal in AWGN Reduces to the Correlator</a>.

Compute error probability.

Example: Deflection of a Sub-Optimal Linear Detector

Filter coefficients.

Output SNR.

Correlator output SNR.

Compare.

Output SNR of Matched vs. Mismatched Filters

Parameters

Correlator vs. Matched Filter (Time Domain)

Parameters

ROC Curves for AWGN Detection

Parameters

Common Mistake: Energy, Not Amplitude, Drives Performance

Common Mistake: N0N_0N0​ vs. σ2\sigma^2σ2 Conventions

Common Mistake: Matched Filter Is Defined Only Up to a Scalar

Quick Check

Quick Check

Quick Check

Historical Note: North's Matched Filter (1943)

Matched Filters in GPS Acquisition

Matched Filtering and Pulse Compression in Radar

Matched filter

Deflection coefficient

Correlator receiver

Why This Matters: From Matched Filters to Coherent Demodulation

Definition:
Known-Signal Detection Problem in AWGN

Definition:
Correlator Statistic

Definition:
Deflection Coefficient

Common Mistake: $N_0$ vs. $\sigma^2$ Conventions