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The Detection Problem

Suppose a known signal $s(t)$ is buried in additive white Gaussian noise (AWGN). We want to design a linear filter that, when we sample its output at the right moment, gives us the best possible chance of detecting the signal. "Best" here means maximizing the signal-to-noise ratio at the sampling instant. The answer — the matched filter — is one of the most important results in all of signal processing and communications. Its optimality proof is a beautiful application of the Cauchy-Schwarz inequality, and the result has a clean physical interpretation: the optimal filter is the time-reversed, delayed replica of the signal itself.

Definition:
Signal Detection in AWGN

Consider the observation $Y(t) = s(t) + N(t),$ where $s(t)$ is a known, deterministic, finite-energy signal with energy $E_s = \int_{-\infty}^{\infty} |s(t)|^2\, dt$ , and $N(t)$ is white Gaussian noise with PSD $P_N(f) = N_0/2$ . We pass $Y(t)$ through an LTI filter with impulse response $h(t)$ and sample the output at time $t = t_0$ . The output signal-to-noise ratio (SNR) at $t_0$ is $\text{SNR}(t_0) = \frac{|g(t_0)|^2}{\mathbb{E}[|N_o(t_0)|^2]},$ where $g(t_0) = \int h(\tau) s(t_0 - \tau)\, d\tau$ is the signal component and $N_o(t)$ is the filtered noise.

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Definition:
Matched Filter

The matched filter for a signal $s(t)$ sampled at time $t_0$ is the LTI filter with impulse response $h_{\text{mf}}(t) = s^*(t_0 - t).$ Equivalently, in the frequency domain: $\check{h}_{\text{mf}}(f) = \check{s}^*(f)\, e^{-j2\pi f t_0},$ where $\check{s}(f)$ is the Fourier transform of $s(t)$ .

The matched filter is the time-reversed, conjugated, and delayed version of the signal. For real-valued signals, it is simply $h(t) = s(t_0 - t)$ : a mirror image of $s(t)$ shifted to peak at $t_0$ .

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Theorem: Matched Filter Theorem

Among all LTI filters, the matched filter $h_{\text{mf}}(t) = s^*(t_0 - t)$ maximizes the output SNR at time $t_0$ . The maximum SNR is $\text{SNR}_{\max} = \frac{2 E_s}{N_0},$ where $E_s = \int |s(t)|^2\, dt$ is the signal energy and $N_0/2$ is the two-sided noise PSD.

The matched filter correlates the received signal with a stored copy of the expected waveform. At the sampling instant, the signal contributions from all time instants add coherently (in phase), while the noise contributions add incoherently (random phases). The maximum SNR depends only on the signal energy and the noise spectral density — not on the signal's shape. A long, weak signal achieves the same SNR as a short, strong pulse of equal energy.

Proof

Signal component at $t_0$

The output signal at $t_0$ is $g(t_0) = \int_{-\infty}^{\infty} h(\tau) s(t_0 - \tau)\, d\tau.$ In the frequency domain, $\check{g}(f) = \check{h}(f) \check{s}(f)$ , so by Parseval: $g(t_0) = \int \check{h}(f) \check{s}(f) e^{j2\pi f t_0}\, df.$

Noise power at output

The output noise has PSD $P_{N_o}(f) = |\check{h}(f)|^2 \cdot N_0/2$ , so the output noise power is $\mathbb{E}[|N_o(t_0)|^2] = \frac{N_0}{2} \int |\check{h}(f)|^2\, df.$

Form the SNR

$\text{SNR}(t_0) = \frac{\left|\int \check{h}(f) \check{s}(f) e^{j2\pi f t_0}\, df\right|^2}{\frac{N_0}{2} \int |\check{h}(f)|^2\, df}.$ $

Apply Cauchy-Schwarz

By the Cauchy-Schwarz inequality for integrals, $\left|\int \check{h}(f) \check{s}(f) e^{j2\pi f t_0}\, df\right|^2 \leq \int |\check{h}(f)|^2\, df \cdot \int |\check{s}(f)|^2\, df.$ The second integral is $E_s$ by Parseval's theorem. Therefore $\text{SNR}(t_0) \leq \frac{E_s}{N_0/2} = \frac{2 E_s}{N_0}.$

Equality condition

Cauchy-Schwarz holds with equality iff $\check{h}(f) = c \cdot \check{s}^*(f) e^{-j2\pi f t_0}$ for some constant $c \neq 0$ . Choosing $c = 1$ gives the matched filter $h_{\text{mf}}(t) = s^*(t_0 - t)$ . Therefore the matched filter achieves $\text{SNR}_{\max} = 2E_s / N_0$ .

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Example: Matched Filter for a Rectangular Pulse

A rectangular pulse $s(t) = A$ for $0 \leq t \leq T$ and $s(t) = 0$ otherwise is transmitted through AWGN with PSD $N_0/2$ . Find the matched filter, the output waveform (signal component), and the maximum SNR. Sample at $t_0 = T$ .

Solution

Matched filter

$h_{\text{mf}}(t) = s(T - t) = A$ for $0 \leq t \leq T$ , zero otherwise. This is just a rectangular window of the same duration — an integrate-and-dump filter.

Signal output

$g(t) = \int_0^T A \cdot A\, d\tau = A^2 \min(t, T)$ $for$ 0 \leq t \leq T $. At$ t_0 = T $:$ g(T) = A^2 T$.

Signal energy

$E_s = A^2 T$ .

Maximum SNR

$\text{SNR}_{\max} = \frac{2 E_s}{N_0} = \frac{2 A^2 T}{N_0}.$ $ The matched filter for a rectangular pulse is the simplest possible receiver: just integrate for the pulse duration and sample.

Example: Matched Filter for a Bandlimited Pulse

A sinc pulse $s(t) = \text{sinc}(Wt)$ with bandwidth $W$ Hz is transmitted in AWGN. Show that the matched filter output at $t_0 = 0$ yields $\text{SNR}_{\max} = 2E_s/N_0$ and describe the output waveform.

Solution

Spectrum of sinc

$\check{s}(f) = \frac{1}{W} \text{rect}(f/W)$ , so $E_s = \int |\check{s}(f)|^2\, df = 1/W$ .

Matched filter

$\check{h}_{\text{mf}}(f) = \check{s}^*(f) = \frac{1}{W} \text{rect}(f/W)$ . The matched filter for a sinc is an ideal lowpass filter of bandwidth $W$ — a satisfying result.

Output at $t_0 = 0$

$g(0) = \int |\check{s}(f)|^2\, df = E_s = 1/W$ . The noise power is $(N_0/2) \int |\check{h}|^2\, df = (N_0/2)(1/W) = N_0/(2W)$ . So $\text{SNR} = E_s^2 / (N_0/(2W)) = (1/W)^2 / (N_0/(2W)) = 2/(WN_0) = 2E_s/N_0$ .

Matched Filter vs. Mismatched Filter: Output SNR Comparison

Compare the output SNR of the matched filter to that of suboptimal filters (e.g., integrator of wrong duration, Gaussian pulse filter). The matched filter always achieves $2E_s/N_0$ ; all others fall short.

Parameters

Signal waveform

T

(signal duration)2

N_0

1

Quick Check

Two signals have the same energy $E_s$ but different shapes: one is a rectangular pulse of duration $T$ , the other is a Gaussian pulse of the same energy. In AWGN with PSD $N_0/2$ , how do their matched filter SNRs compare?

They are equal: both achieve $2E_s/N_0$

The rectangular pulse achieves higher SNR because it has more bandwidth

The Gaussian pulse achieves higher SNR because it is smoother

Correction:

They are equal: both achieve

2E_s/N_0

The matched filter SNR depends only on energy and noise PSD, not on signal shape.

Common Mistake: The Matched Filter Is Not Always the Best Detector

Mistake:

Assuming the matched filter is optimal for all detection problems, including those with colored (non-white) noise.

Correction:

The matched filter maximizes SNR only when the noise is white. For colored noise with PSD $P_N(f)$ , the optimal "generalized matched filter" is $\check{h}_{\text{opt}}(f) = \frac{\check{s}^*(f) e^{-j2\pi f t_0}}{P_N(f)}$ , which pre-whitens the noise before correlating. The resulting SNR is $\text{SNR}_{\max} = 2\int \frac{|\check{s}(f)|^2}{P_N(f)}\, df$ .

Common Mistake: Correlator vs. Matched Filter

Mistake:

Treating the correlator receiver and the matched filter as fundamentally different structures.

Correction:

The correlator (multiply by $s(t)$ and integrate from $0$ to $T$ ) and the matched filter (convolve with $s(T - t)$ and sample at $T$ ) produce identical output at $t = T$ . They are two implementations of the same mathematical operation. The correlator is often preferred in digital implementations; the matched filter viewpoint is more natural for analog or spectral analysis.

Common Mistake: $\text{SNR}_{\max}$ Depends on Energy, Not Power

Mistake:

Computing matched filter SNR using signal power $P_s$ instead of signal energy $E_s$ .

Correction:

The matched filter SNR is $2E_s/N_0$ , where $E_s$ is the total energy $\int |s(t)|^2\, dt$ , not the average power. A longer signal with the same amplitude has more energy and thus higher matched filter SNR. This is the fundamental principle behind pulse compression in radar: spread the energy over a long, coded waveform and collect it coherently with the matched filter.

The Matched Filter as a Correlation Detector

The matched filter output at $t_0$ is $g(t_0) = \int h(\tau) y(t_0 - \tau)\, d\tau = \int s^*(t_0 - t_0 + \tau) y(t_0 - \tau)\, d\tau = \int s^*(\tau) y(t_0 - \tau)\, d\tau$ . This is exactly the cross-correlation of $y$ with $s$ evaluated at lag $t_0$ . The matched filter is measuring how much the received waveform "looks like" the expected signal.

🔧Engineering Note

Pulse Compression in Radar

Radar systems exploit the matched filter principle through pulse compression. A long, chirp-modulated pulse is transmitted (high energy, low peak power), and the receiver applies the matched filter, which compresses the energy into a short, high-amplitude peak. The processing gain is the time-bandwidth product $BT$ of the chirp. Modern radar systems achieve processing gains of $10^3$ to $10^6$ , enabling detection of targets at extreme ranges without requiring dangerously high peak transmit power.

Practical Constraints

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Range sidelobes require additional windowing (Hamming, Taylor)
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Doppler shift of moving targets causes SNR loss if not compensated

Matched Filter: From Noisy Input to SNR Peak

Animation showing a known signal buried in noise, the matched filter impulse response, and the output building up to a peak at the sampling instant.

The matched filter output peaks at

t = t_0

with

\text{SNR}_{\max} = 2E_s/N_0

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Matched Filter

The LTI filter $h(t) = s^*(t_0 - t)$ that maximizes the output SNR at time $t_0$ when a known signal $s(t)$ is observed in white noise. The maximum SNR is $2E_s/N_0$ .

Correlator Receiver

A receiver that computes $\int_0^T y(t) s^*(t)\, dt$ to detect a known signal $s(t)$ . Produces the same output as the matched filter sampled at $t_0 = T$ .

Related: Matched Filter

Key Takeaway

The matched filter $h(t) = s^*(t_0 - t)$ maximizes the output SNR for detecting a known signal in white noise. The maximum SNR is $\text{SNR}_{\max} = 2E_s/N_0$ , depending only on the signal energy and the noise spectral density — not on the signal shape. This is the theoretical foundation of coherent detection in communications and radar.

The Matched Filter

The Detection Problem

Definition: Signal Detection in AWGN

Definition: Matched Filter

Theorem: Matched Filter Theorem

Signal component at $t_0$

Noise power at output

Form the SNR

Apply Cauchy-Schwarz

Equality condition

Example: Matched Filter for a Rectangular Pulse

Matched filter

Signal output

Signal energy

Maximum SNR

Example: Matched Filter for a Bandlimited Pulse

Spectrum of sinc

Matched filter

Output at $t_0 = 0$

Matched Filter vs. Mismatched Filter: Output SNR Comparison

Parameters

Quick Check

Common Mistake: The Matched Filter Is Not Always the Best Detector

Common Mistake: Correlator vs. Matched Filter

Common Mistake: SNRmax⁡\text{SNR}_{\max}SNRmax​ Depends on Energy, Not Power

The Matched Filter as a Correlation Detector

Pulse Compression in Radar

Matched Filter: From Noisy Input to SNR Peak

Matched Filter

Correlator Receiver

Key Takeaway

Definition:
Signal Detection in AWGN

Definition:
Matched Filter

Common Mistake: $\text{SNR}_{\max}$ Depends on Energy, Not Power