Ferkans — Interactive Telecom Tutor

The Simplest Estimator: Correlate and Sum

Given the forward model $\mathbf{y} = \mathbf{A}\mathbf{c} + \mathbf{w}$ , the most natural question is: what is the simplest way to produce an image? The answer is the matched filter (or backpropagation) estimator $\hat{\mathbf{c}}^{\text{BP}} = \mathbf{A}^{H} \mathbf{y}$ . It correlates each measurement with the expected response of every pixel. This operation goes by many names -- delay-and-sum in radar, backpropagation in diffraction tomography, adjoint imaging in inverse problems -- but they all compute $\mathbf{A}^{H} \mathbf{y}$ .

While the matched filter does not invert the forward model, it is the foundation on which every reconstruction algorithm in this book is built: ISTA starts from $\mathbf{A}^{H} \mathbf{y}$ , OAMP uses it at each iteration, and the MF-to-U-Net pipeline feeds it to a neural network.

Definition:
The Matched Filter (Adjoint) Estimator

Given the forward model $\mathbf{y} = \mathbf{A}\mathbf{c} + \mathbf{w}$ , the matched filter estimator is:

$\hat{\mathbf{c}}^{\text{BP}} = \mathbf{A}^{H} \mathbf{y}.$

In component form, the image value at pixel $q$ is:

$\hat{c}_{\text{MF},q} = \mathbf{A}_{q}^{H} \mathbf{y} = \sum_{m=1}^{M} A_{mq}^* y_m,$

where $\mathbf{A}_{q}$ denotes the $q$ -th column of $\mathbf{A}$ (the sensing vector for pixel $q$ ).

Interpretation: $\hat{c}_{\text{MF},q}$ is the inner product between the measurement and the "signature" of pixel $q$ . The matched filter correlates $\mathbf{y}$ with each possible scatterer location.

Theorem: SNR Optimality of the Matched Filter

For a single scatterer at pixel $q$ with reflectivity $c_q$ , the linear estimator $\hat{c}_q = \mathbf{w}_{\text{w}}^H \mathbf{y}$ that maximizes the output signal-to-noise ratio

$\text{SNR} = \frac{|c_q|^2 |\mathbf{w}_{\text{w}}^H \mathbf{A}_{q}|^2} {\mathbf{w}_{\text{w}}^H \mathbf{R}_n \mathbf{w}_{\text{w}}}$

is the matched filter $\mathbf{w}_{\text{w}} = \mathbf{A}_{q}$ (for white noise $\mathbf{R}_n = \sigma^2 \mathbf{I}$ ), achieving:

$\text{SNR}_{\max} = \frac{|c_q|^2 \|\mathbf{A}_{q}\|^2}{\sigma^2}.$

The maximum SNR grows linearly with $M$ (number of measurements) -- the processing gain of the matched filter.

The matched filter aligns the filter to the known signature of pixel $q$ . By the Cauchy-Schwarz inequality, no other linear filter can produce a higher signal-to-noise ratio.

Proof

Single-scatterer model

For a single scatterer at pixel $q$ : $\mathbf{y} = c_q \mathbf{A}_{q} + \mathbf{w}$ .

SNR expression

A linear estimator $\hat{c}_q = \mathbf{w}_{\text{w}}^H \mathbf{y}$ has:

$\text{SNR} = \frac{|c_q|^2 |\mathbf{w}_{\text{w}}^H \mathbf{A}_{q}|^2} {\sigma^2 \|\mathbf{w}_{\text{w}}\|^2}.$

Cauchy-Schwarz bound

By Cauchy-Schwarz: $|\mathbf{w}_{\text{w}}^H \mathbf{A}_{q}|^2 \leq \|\mathbf{w}_{\text{w}}\|^2 \|\mathbf{A}_{q}\|^2$ , with equality when $\mathbf{w}_{\text{w}} = \alpha \mathbf{A}_{q}$ for any $\alpha \neq 0$ .

Maximum SNR

Setting $\mathbf{w}_{\text{w}} = \mathbf{A}_{q}$ :

$\text{SNR}_{\max} = \frac{|c_q|^2 \|\mathbf{A}_{q}\|^2}{\sigma^2} = |c_q|^2 \cdot M \cdot \frac{\overline{|A_{mq}|^2}}{\sigma^2}.$

Theorem: Matched Filter and the Gram Matrix

The matched filter estimator satisfies:

$\hat{\mathbf{c}}^{\text{BP}} = \mathbf{A}^{H} \mathbf{y} = \mathbf{A}^{H} \mathbf{A} \mathbf{c} + \mathbf{A}^{H} \mathbf{w} = \mathbf{G}\mathbf{c} + \tilde{\mathbf{w}},$

where $\mathbf{G} = \mathbf{A}^{H} \mathbf{A}$ is the Gram matrix and $\tilde{\mathbf{w}} = \mathbf{A}^{H} \mathbf{w}$ is the filtered noise.

Key observations:

The MF image is the true scene convolved with $\mathbf{G}$ , which acts as the discrete point-spread function (§The Point-Spread Function (PSF)).
If $\mathbf{A}^{H} \mathbf{A} = \mathbf{I}$ (orthonormal columns), the MF is exact -- but for $M < N$ this is impossible.
The MF image equals the gradient of the least-squares cost at the origin: $\nabla_{\mathbf{c}} \frac{1}{2}\|\mathbf{y} - \mathbf{A}\mathbf{c}\|^2 \big|_{\mathbf{c}=0} = -\mathbf{A}^{H} \mathbf{y}$ .

Proof

Direct substitution

Substitute $\mathbf{y} = \mathbf{A}\mathbf{c} + \mathbf{w}$ :

$\mathbf{A}^{H} \mathbf{y} = \mathbf{A}^{H}(\mathbf{A}\mathbf{c} + \mathbf{w}) = \mathbf{A}^{H} \mathbf{A} \mathbf{c} + \mathbf{A}^{H} \mathbf{w}.$

Identification

Define $\mathbf{G} \triangleq \mathbf{A}^{H} \mathbf{A}$ and $\tilde{\mathbf{w}} \triangleq \mathbf{A}^{H} \mathbf{w}$ . The noise $\tilde{\mathbf{w}}$ has covariance $\sigma^2 \mathbf{G}$ -- it is colored, correlated across pixels, with the same structure as the PSF.

Definition:
Delay-and-Sum Beamforming for Imaging

For a scene pixel at position $\mathbf{p}$ with transmitter $i$ at $\mathbf{s}_{i}$ and receiver $j$ at $\mathbf{r}_{j}$ , the round-trip delay is:

${\tau_{i,j}}_{i,j}(\mathbf{p}) = \frac{|\mathbf{s}_{i} - \mathbf{p}| + |\mathbf{p} - \mathbf{r}_{j}|}{c}.$

The delay-and-sum (DAS) image at pixel $\mathbf{p}$ is:

$\hat{c}_{\text{DAS}}(\mathbf{p}) = \sum_{i=1}^{N_t} \sum_{j=1}^{N_r} y_{i,j}\bigl({\tau_{i,j}}_{i,j}(\mathbf{p})\bigr).$

In frequency domain (stepped frequency), this becomes:

$\hat{c}_{\text{DAS}}(\mathbf{p}) = \sum_{i,j,k} y_{i,j,k}\, e^{j2\pi f_k {\tau_{i,j}}_{i,j}(\mathbf{p})},$

which is exactly the matched filter $\hat{c}_q = \mathbf{A}_{q}^{H} \mathbf{y}$ with entries $A_{(i,j,k),q} = e^{-j2\pi f_k {\tau_{i,j}}_{i,j}(\mathbf{p}_{q})}$ .

Three Names, One Operation

The matched filter, delay-and-sum beamforming, and backpropagation are mathematically identical -- they all compute $\mathbf{A}^{H} \mathbf{y}$ . The naming depends on the community:

Matched filter: Signal processing / estimation theory.
Delay-and-sum: Radar and sonar engineering.
Backpropagation: Diffraction tomography and inverse scattering.

In the k-space (Fourier) view, the backpropagation estimator places the measured data at the corresponding k-space locations and applies an inverse Fourier transform -- equivalent to correlating $\mathbf{y}$ with the sensing vectors $\mathbf{A}_{q}$ .

Definition:
Backpropagation in the k-Space View

In the diffraction-tomography framework (Chapter 6), each measurement samples the scene's spatial spectrum at wavenumber $\kappa_{\mathbf{s},\mathbf{r}} = \kappa_{\ntn{txpos}} + \kappa_{\ntn{rxpos}}$ .

The backpropagation estimator is:

$\hat{c}_{\text{BP}}(\mathbf{p}) = \sum_{m=1}^{M} y_m \, e^{j {\kappa_{\mathbf{s},\mathbf{r}}}_{m} \cdot \mathbf{p}},$

which is an inverse discrete Fourier transform (IDFT) of the k-space data evaluated on the non-uniform grid $\{{\kappa_{\mathbf{s},\mathbf{r}}}_{m}\}_{m=1}^M$ .

When the k-space samples lie on a uniform grid, backpropagation reduces to a standard inverse FFT. For non-uniform samples, one uses a non-uniform FFT (NUFFT) or the direct summation above.

Example: Matched Filter for SAR Imaging

For stripmap SAR, the sensing matrix has Fourier structure, and the matched filter reduces to the range-Doppler algorithm:

$\hat{\mathbf{c}}^{\text{BP}} = \mathbf{F}^H \mathbf{y} = \text{IDFT}(\mathbf{y}).$

Resolution:

Range: $\Delta_r = c/(2B)$ where $B$ is the bandwidth.
Cross-range: $\Delta_{\text{cr}} = \lambda R/(2L_{\text{sa}})$ where $L_{\text{sa}}$ is the synthetic aperture length.

Sidelobes: The sinc-like PSF has sidelobes at $-13.3$ dB (uniform weighting). Windowing (Hamming, Taylor) reduces sidelobes to $-40$ dB at the cost of approximately 50% wider mainlobe.

Show that the DAS image for a monostatic radar with $N_t = 1$ , $N_r = 16$ ULA elements, and $N_f = 64$ stepped frequencies is equivalent to a 2D IDFT when the array is in the far field.

Solution

Forward model

In the far field, the round-trip delay for element $j$ and frequency $k$ to pixel $(\rho, \theta)$ is:

$\tau_j(\rho, \theta) \approx \frac{2\rho}{c} - \frac{2 j d \sin\theta}{c},$

so $A_{(j,k), (\rho,\theta)} = e^{-j 2\pi f_k \tau_j} = e^{-j 4\pi f_k \rho / c} \cdot e^{j 4\pi f_k j d \sin\theta / c}$ .

Separability

The sensing matrix factors as $\mathbf{A} = \mathbf{A}_{\text{range}} \otimes \mathbf{A}_{\text{angle}}$ , where both factors are DFT matrices. Hence $\mathbf{A}^{H} = \mathbf{A}_{\text{range}}^H \otimes \mathbf{A}_{\text{angle}}^H$ is a 2D IDFT.

Result

The DAS/matched filter image is obtained by reshaping $\mathbf{y}$ into an $N_r \times N_f$ matrix and applying a 2D IFFT -- the standard range-Doppler algorithm.

Matched Filter / Backpropagation Imaging

Demonstrates the matched filter estimator for a 2D radar scene.

Left panel: True scene -- point scatterers on a grid.

Right panel: Matched filter image $\hat{\mathbf{c}}^{\text{BP}} = \mathbf{A}^{H} \mathbf{y}$ . Observe how sidelobes blur the scatterers and how increasing SNR reduces the noise floor but does not improve resolution.

Parameters

\text{SNR}

(dB)20

Number of scatterers5

N_t

(Tx antennas)4

N_r

(Rx antennas)8

Historical Note: The Matched Filter in Radar History

1943--1960

The matched filter was introduced by D. O. North in a classified 1943 RCA Laboratories report during World War II. North showed that for a known signal in additive white Gaussian noise, the linear filter maximizing the output SNR is the time-reversed, conjugated replica of the signal -- hence "matched." The result was independently derived by J. H. Van Vleck and D. Middleton (1946) and later by G. L. Turin (1960) who generalized it to colored noise.

In imaging, the matched filter became the standard baseline with the development of synthetic aperture radar (SAR) in the 1950s at the University of Michigan, where range compression and azimuth compression were recognized as matched filter operations applied sequentially.

Quick Check

The matched filter estimator $\hat{\mathbf{c}}^{\text{BP}} = \mathbf{A}^{H} \mathbf{y}$ is:

The maximum likelihood estimator for the reflectivity

The linear filter maximizing per-pixel SNR

An unbiased estimator of the reflectivity

Correction:

The linear filter maximizing per-pixel SNR

By Cauchy-Schwarz, $\mathbf{A}_{q}^{H} \mathbf{y}$ maximizes the signal-to-noise ratio at pixel $q$ among all linear filters.

Matched filter

A linear filter whose impulse response is the time-reversed, conjugated replica of the signal of interest. In imaging, the matched filter at pixel $q$ is $\mathbf{A}_{q}^{H}$ , and the full matched filter image is $\mathbf{A}^{H} \mathbf{y}$ .

Backpropagation

The adjoint operation $\mathbf{A}^{H} \mathbf{y}$ viewed as inverse Fourier transform of k-space data. Borrowed from diffraction tomography, where the scattered field is "propagated back" through the medium to estimate the contrast function.

Delay-and-sum (DAS)

A beamforming technique that forms an image by delaying (time-shifting) received signals according to the round-trip time to each pixel and summing coherently. Mathematically equivalent to the matched filter $\mathbf{A}^{H} \mathbf{y}$ .

Processing gain

The SNR improvement achieved by the matched filter relative to a single measurement. For $M$ measurements: $G_p = M \cdot \overline{|A_{mq}|^2} / \sigma^2$ .

Gram matrix

The matrix $\mathbf{G} = \mathbf{A}^{H} \mathbf{A}$ whose $(q, q')$ entry is the inner product $\mathbf{A}_{q}^{H} \mathbf{A}_{q'}$ . The Gram matrix acts as the discrete point-spread function of the matched filter estimator.

Key Takeaway

The matched filter $\hat{\mathbf{c}}^{\text{BP}} = \mathbf{A}^{H} \mathbf{y}$ maximizes per-pixel SNR but does not deconvolve the PSF. The MF image equals the true scene convolved with the Gram matrix $\mathbf{G} = \mathbf{A}^{H} \mathbf{A}$ , and it is the negative gradient of the data-fidelity cost at the origin -- the starting point for every iterative reconstruction algorithm.

The Matched Filter / Backpropagation Estimator