The SNR Maximization Problem
Consider a received signal $r(t) = s(t) + n(t)$ on $t \in [0, T]$, where $s(t)$ is the known deterministic signal of energy $E = \int_0^T s^2(t)\,dt$, and $n(t)$ is zero-mean white Gaussian noise with one-sided PSD $N_0/2$ (i.e., $\mathbb{E}[n(t)n(\tau)] = (N_0/2)\delta(t-\tau)$).
We filter with a linear time-invariant filter $h(t)$ and sample the output at time $t_0$:
\[y(t_0) = \int_{-\infty}^\infty h(t_0 - \tau) r(\tau)\, d\tau\]The output SNR at $t_0$ is:
\[\text{SNR} = \frac{y_s^2(t_0)}{\mathbb{E}[y_n^2(t_0)]}\]where $y_s(t_0) = \int h(t_0-\tau)s(\tau)d\tau$ is the signal component and $y_n(t_0)$ is the noise component. By Parseval’s theorem and the Cauchy-Schwarz inequality:
\[y_s^2(t_0) = \left|\int H(f)S(f)e^{j2\pi f t_0}\,df\right|^2 \leq \int |H(f)|^2\,df \cdot \int |S(f)|^2\,df\]| The noise power at the output is $\sigma_n^2 = (N_0/2)\int | H(f) | ^2\,df$. Therefore: |
The Matched Filter Solution
Equality in the Cauchy-Schwarz inequality holds when $H(f) \propto S^*(f)e^{-j2\pi f t_0}$. The matched filter has transfer function:
\[H_{\text{opt}}(f) = S^*(f)\, e^{-j2\pi f t_0}\]and equivalently the impulse response:
\[h_{\text{opt}}(t) = s(t_0 - t)\]The matched filter is the time-reversed, delayed replica of the signal. The output SNR is the maximum achievable value:
\[\text{SNR}_{\text{max}} = \frac{2E}{N_0}\]This is a profound result: the maximum SNR depends only on the ratio of signal energy to noise spectral density, not on the waveform shape. A 1 ms Gaussian pulse and a 1 ms chirp with the same energy achieve identical maximum SNR, though their time-bandwidth products differ.
Matched Filter as a Correlator
The matched filter output $y(t) = r(t) * h_{\text{opt}}(t) = r(t) * s(-t)$ is the cross-correlation of the received signal with the reference:
\[y(t_0) = \int_{-\infty}^\infty r(\tau)\, s(\tau - (t_0 - t_0))\, d\tau = \int_{-\infty}^\infty r(\tau)\, s(\tau)\, d\tau\]At the sampling instant $t_0$, the matched filter computes the inner product of the received waveform with the transmitted waveform — the projection of the received signal onto the signal space. This correlation receiver interpretation is central to optimal demodulation in digital communications.
The output waveform of the matched filter applied to a clean signal $s(t)$ is the autocorrelation function of $s$:
\[y_s(t) = \int s(\tau) s(\tau - (t_0 - t))\, d\tau = R_{ss}(t - t_0)\]This has a peak at $t = t_0$ of value $E$, and rolls off according to the autocorrelation shape of $s$.
Detection Theory and the Neyman-Pearson Framework
The detection problem is a binary hypothesis test:
\(H_0: r(t) = n(t) \qquad \text{(noise only)}\) \(H_1: r(t) = s(t) + n(t) \qquad \text{(signal present)}\)
The sufficient statistic is the matched filter output $y_0 = \int_0^T r(t)s(t)\,dt$. Under $H_0$, $y_0 \sim \mathcal{N}(0, N_0 E/2)$; under $H_1$, $y_0 \sim \mathcal{N}(E, N_0 E/2)$.
The Neyman-Pearson (NP) detector sets a threshold $\eta$ and decides $H_1$ if $y_0 > \eta$:
-
Probability of false alarm: $P_{\text{FA}} = P(y_0 > \eta H_0) = Q!\left(\frac{\eta}{\sqrt{N_0 E/2}}\right)$ -
Probability of detection: $P_D = P(y_0 > \eta H_1) = Q!\left(\frac{\eta - E}{\sqrt{N_0 E/2}}\right)$
where $Q(x) = \frac{1}{\sqrt{2\pi}}\int_x^\infty e^{-t^2/2}\,dt$. Eliminating $\eta$:
\[P_D = Q\!\left(Q^{-1}(P_{\text{FA}}) - \sqrt{\frac{2E}{N_0}}\right)\]The quantity $d’ = \sqrt{2E/N_0}$ is the deflection or detectability index. This formula directly yields the receiver operating characteristic (ROC) curve — the trade-off between $P_D$ and $P_{\text{FA}}$ as the threshold $\eta$ varies.
| $E/N_0$ (dB) | $P_{\text{FA}} = 10^{-6}$ | $P_{\text{FA}} = 10^{-3}$ |
|---|---|---|
| 10 dB | $P_D \approx 0.21$ | $P_D \approx 0.60$ |
| 13 dB | $P_D \approx 0.63$ | $P_D \approx 0.92$ |
| 16 dB | $P_D \approx 0.93$ | $P_D \approx 0.998$ |
Pulse Compression in Radar
A long pulse has high energy but poor range resolution ($\Delta R = cT/2$ for pulse duration $T$). A short pulse has good range resolution but low energy. Pulse compression combines both advantages using a wideband modulated waveform followed by matched filtering.
The linear frequency modulated (LFM) chirp:
\[s(t) = A\,\text{rect}(t/T)\, e^{j\pi \beta t^2/T}\]sweeps frequency from $-\beta/2$ to $+\beta/2$ over duration $T$, giving bandwidth $B = \beta$. The time-bandwidth product $BT$ is the compression ratio. After matched filtering, the range resolution is:
\[\Delta R = \frac{c}{2B}\]with a peak-to-sidelobe ratio of $-13.2$ dB for the rectangular envelope. The pulse compression gain is the ratio of output to input SNR improvement: $G = BT$, which for a 100 $\mu$s chirp with 10 MHz bandwidth gives $G = 1000$ (30 dB).
Tapering (windowing) the matched filter reduces sidelobes at the cost of widening the main lobe and losing $\sim 1$–$2$ dB of SNR. A Hamming-weighted matched filter achieves $-43$ dB sidelobes with $\sim 40\%$ resolution broadening.
The Ambiguity Function
The ambiguity function characterizes matched filter performance in the joint delay-Doppler plane:
\[|\chi(\tau, \nu)|^2 = \left|\int_{-\infty}^\infty s(t)\, s^*(t-\tau)\, e^{j2\pi\nu t}\, dt\right|^2\]where $\tau$ is the time delay (range) and $\nu$ is the Doppler shift (velocity). The ambiguity function satisfies:
-
$ \chi(0,0) ^2 = E^2$ (maximum at origin) -
$\int!!\int \chi(\tau,\nu) ^2\, d\tau\, d\nu = E^2$ (volume invariant) -
$ \chi(-\tau,-\nu) = \chi(\tau,\nu) $ (symmetry)
The volume invariance is fundamental: total ambiguity is conserved. A thumbtack ambiguity function (narrow main lobe everywhere) is impossible — sidelobes must exist somewhere. Waveform design trades off range-Doppler coupling, sidelobe levels, and Doppler tolerance.
The LFM chirp has a ridge-shaped ambiguity function — good Doppler tolerance (the matched filter output remains high even for moderate Doppler shifts, just displaced in range). Phase-coded waveforms (Barker codes, m-sequences) have thumbtack-like ambiguity functions — accurate simultaneous range and velocity estimation but sensitive to Doppler.
| Waveform | Range resolution | Doppler tolerance | Sidelobe level |
|---|---|---|---|
| CW pulse | $c/2B$ | Excellent | N/A |
| LFM chirp | $c/2B$ | Good | $-13$ dB |
| Barker code (13-bit) | $c/2B$ | Poor | $-22$ dB |
| Polyphase codes | $c/2B$ | Fair | $< -30$ dB |