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Why the Matched Filter is the Starting Point

The matched filter is the optimal linear filter for detecting a known signal in additive white Gaussian noise. In the imaging context, it is the adjoint operator $\mathbf{A}^{H}$ — the simplest and fastest way to form an image. Every more sophisticated reconstruction method (LASSO, ADMM, deep unfolding) starts from or improves upon the matched-filter image.

We derive the matched filter from first principles, show its connection to pulse compression, and quantify its limitations through sidelobe analysis.

Theorem: Optimality of the Matched Filter

Consider the detection problem: observation $r(t) = \alpha s(t - \tau_0) + w(t)$ , where $s(t)$ is a known waveform, $\alpha$ is the target complex amplitude, $\tau_0$ is the delay, and $w(t)$ is white Gaussian noise with spectral density $N_0/2$ .

The linear filter $h(t)$ that maximizes the output $\text{SNR}$ at time $\tau_0$ is:

$h_{\text{MF}}(t) = s^*(\tau_0 - t),$

and the maximum output $\text{SNR}$ is:

$\text{SNR}_{\text{out}} = \frac{2E_s}{N_0},$

where $E_s = |\alpha|^2 \int |s(t)|^2\,dt$ is the received signal energy. The output $\text{SNR}$ depends only on signal energy, not on waveform shape.

The matched filter correlates the received signal with a time-reversed copy of the transmitted waveform. It accumulates energy coherently at the delay corresponding to the target, while noise — being uncorrelated with $s(t)$ — averages out.

Proof

Output at time $\tau_0$

The filter output is $(r * h)(\tau_0) = \int r(t) h(\tau_0 - t)\,dt = \int r(t) s^*(t)\,dt$ . The signal component is $\alpha \int |s(t)|^2\,dt = \alpha E_s / |\alpha|^2$ .

Noise output power

The noise output variance is $\frac{N_0}{2}\int |h(t)|^2\,dt = \frac{N_0}{2}\int |s(t)|^2\,dt$ .

Apply Cauchy-Schwarz

By the Cauchy-Schwarz inequality in $L^2$ : $|\int r(t) h^*(\tau_0 - t)\,dt|^2 \leq \int |r(t)|^2\,dt \cdot \int |h(t)|^2\,dt,$ with equality when $h(t) = c \cdot s^*(\tau_0 - t)$ for any constant $c$ . The maximum $\text{SNR} = 2|\alpha|^2 E_s / (|\alpha|^2 N_0) = 2E_s/N_0$ .

Definition:
Pulse Compression

Pulse compression is the process of transmitting a long, bandwidth-modulated waveform and applying the matched filter to achieve range resolution determined by bandwidth rather than pulse duration.

For an LFM chirp with bandwidth $W$ and duration $T$ :

Uncompressed pulse: range extent $cT/2$ (poor resolution).
After matched filtering: range resolution $\Delta R = c/(2W)$ (fine resolution).
Compression ratio: $W T$ (the processing gain in linear units, $10\log_{10}(W T)$ in dB).

The matched filter output is approximately:

$y(\tau) \approx W T \cdot \text{sinc}(W(\tau - \tau_0)) \cdot e^{j2\pi f_0 \tau_0},$

a sinc function with mainlobe width $1/W$ and peak amplitude $W T$ times the input.

Example: Pulse Compression Gain for a Weather Radar

A weather radar transmits an LFM chirp with $W = 5$ MHz and $T = 100$ $\mu$ s. The peak transmit power is 250 W. Compute (a) the pulse compression ratio, (b) the effective transmit energy, and (c) the range resolution.

Solution

Compression ratio

$W T = 5 \times 10^6 \cdot 10^{-4} = 500$ , or $10\log_{10}(500) = 27$ dB.

Effective energy

$E_p = P_t \cdot T = 250 \cdot 10^{-4} = 25$ mJ. After compression, the effective peak power is $P_{\text{eff}} = P_t \cdot W T = 125$ kW — equivalent to a short pulse radar with 125 kW peak power but only 250 W transmitter hardware.

Range resolution

$\Delta R = c/(2W) = 3 \times 10^8 / (2 \times 5 \times 10^6) = 30$ m.

Definition:
Range Sidelobes and Windowing

The matched filter output for a single point target has a sinc-like shape with sidelobes at $-13.3$ dB relative to the peak. These range sidelobes mask weak targets near strong ones.

Windowing (or weighting) applies a taper $w(f)$ to the frequency-domain matched filter:

$H_w(f) = w(f) \cdot S^*(f),$

reducing sidelobes at the cost of mainlobe broadening:

Window	Peak sidelobe	Mainlobe width	Mismatch loss
Rectangular	$-13.3$ dB	$1/W$	0 dB
Hamming	$-42.7$ dB	$1.5/W$	1.34 dB
Blackman	$-58$ dB	$1.7/W$	1.73 dB
Taylor ( $\bar{n}=5$ , SLL $-35$ )	$-35$ dB	$1.3/W$	0.85 dB

Matched Filter Output and Sidelobe Control

Observe the matched filter output for an LFM chirp with two point targets. Adjust the bandwidth to see how range resolution changes, and apply different windows to suppress range sidelobes.

Parameters

B

(MHz)50

SNR (dB)20

Window

Definition:
Stretch Processing (Dechirp-on-Receive)

For wideband LFM waveforms (large $W$ ), direct digitization requires very high ADC sampling rates. Stretch processing avoids this by mixing the received signal with a reference chirp:

$r_{\text{beat}}(t) = r(t) \cdot s^*(t - \tau_{\text{ref}}),$

where $\tau_{\text{ref}}$ is the reference delay (center of the range swath). Each target at delay $\tau_0$ produces a sinusoidal beat signal at frequency:

$f_b = \mu(\tau_0 - \tau_{\text{ref}}),$

where $\mu = W/T$ is the chirp rate. An FFT of $r_{\text{beat}}(t)$ maps beat frequencies to ranges.

The required ADC bandwidth drops from $W$ to $\mu \cdot 2\Delta R_{\text{swath}}/c$ , which is typically orders of magnitude smaller.

🔧Engineering Note

Mismatch Loss and the Sidelobe-Resolution Trade-Off

Applying a window to the matched filter creates a mismatched filter — it is no longer the SNR-optimal processor. The mismatch loss quantifies this penalty:

$L_{\text{mm}} = \frac{|\int w(f)|S(f)|^2\,df|^2}{\int |S(f)|^2\,df \cdot \int |w(f)|^2|S(f)|^2\,df}.$

For typical windows, $L_{\text{mm}} = 1$ - $2$ dB. This is usually an acceptable trade-off: losing 1 dB of SNR to gain 20+ dB of sidelobe suppression is worthwhile in most imaging scenarios, where dynamic range matters more than marginal SNR.

Common Mistake: Matched Filter in Colored Noise

Mistake:

Applying the standard matched filter $h(t) = s^*(-t)$ when the noise is not white (e.g., in the presence of clutter or interference).

Correction:

The matched filter is optimal only for white Gaussian noise. In colored noise with spectral density $P_n(f)$ , the optimal filter is the whitening-matched filter:

$H_{\text{opt}}(f) = \frac{S^*(f)}{P_n(f)}.$

This pre-whitens the noise and then applies the matched filter. In STAP (Section 9.6), this generalizes to the matrix form $\mathbf{w} = \mathbf{R}_{cn}^{-1}\mathbf{v}$ .

Pulse Compression

The technique of using a long, bandwidth-modulated waveform and matched filtering to achieve range resolution $c/(2W)$ independent of pulse duration, with processing gain $W T$ .

Key Takeaway

The matched filter $h(t) = s^*(-t)$ maximizes $\text{SNR}$ in AWGN and achieves range resolution $\Delta R = c/(2W)$ via pulse compression. Windowing trades 1-2 dB of SNR for 20+ dB sidelobe suppression. The matched filter output is the starting point for all imaging algorithms — it is the adjoint $\mathbf{A}^{H}\mathbf{y}$ .

Matched Filtering and Pulse Compression