Widely Linear Matched Filter: A Lynchpin towards the Interpretability of Complex-valued CNNs

TL;DR

This paper introduces a general Widely Linear Matched Filter (WLMF) to illuminate the interpretability of complex-valued CNNs, extending matched-filtering to noncircular (improper) complex data without assuming a noise pdf. By employing augmented inputs and noise, it derives the WLMF solution and shows its output SNR ${\text{SNR}}_{\text{WL}}={\bf z}^H {\bf R}_q^{-1} {\bf z}$ exceeds the strictly linear MF ${\text{SNR}}_{\text{SL}}={\bf x}^H {\bf R}_v^{-1} {\bf x}$, with the mutual relation ${\bf f}_1={\bf f}_2^*$ and ${\bf R}_q$ incorporating both covariance ${\bf R}_v$ and pseudo-covariance ${\bf C}_v$. Theoretical results establish a universal SNR gain (Theorem 1), a doubling effect for proper noise (Corollary 1), and a computable lower bound via an AUT-based transformation (Theorem 2) that can grow arbitrarily large for highly improper noise. Simulations validate the theory and show that WLMF enhances CNN interpretability by aligning the convolution-activation-pooling chain with physically meaningful matched-filtering operations; in practice, WL-Net outperformed SL-Net in a two-pattern identification task, with slightly faster training due to higher SNR. Overall, the work provides a principled, interpretable framework for complex-valued CNNs grounded in matched filtering, with implications for enhanced explainability and performance in complex-domain signal processing tasks.

Abstract

A recent study on the interpretability of real-valued convolutional neural networks (CNNs) {Stankovic_Mandic_2023CNN} has revealed a direct and physically meaningful link with the task of finding features in data through matched filters. However, applying this paradigm to illuminate the interpretability of complex-valued CNNs meets a formidable obstacle: the extension of matched filtering to a general class of noncircular complex-valued data, referred to here as the widely linear matched filter (WLMF), has been only implicit in the literature. To this end, to establish the interpretability of the operation of complex-valued CNNs, we introduce a general WLMF paradigm, provide its solution and undertake analysis of its performance. For rigor, our WLMF solution is derived without imposing any assumption on the probability density of noise. The theoretical advantages of the WLMF over its standard strictly linear counterpart (SLMF) are provided in terms of their output signal-to-noise-ratios (SNRs), with WLMF consistently exhibiting enhanced SNR. Moreover, the lower bound on the SNR gain of WLMF is derived, together with condition to attain this bound. This serves to revisit the convolution-activation-pooling chain in complex-valued CNNs through the lens of matched filtering, which reveals the potential of WLMFs to provide physical interpretability and enhance explainability of general complex-valued CNNs. Simulations demonstrate the agreement between the theoretical and numerical results.

Figures (6)

• Figure 1: Block diagram of the widely linear matched filter.
• Figure 2: Normalized bias in SNR gain of WLMF over SLMF against the degree of impropriety $\rho_u$, which varies from an almost proper 0.04 to very improper 0.8.
• Figure 3: The SNR gain of an observed signal $r(n)$ generated by a deterministic input $x(n)$ of length $N=106$, corrupted by noise $v(n)$, of which the impropriety $\rho_u$ varies from an almost proper 0.04 to very improper 0.8.
• Figure 4: Interpretation of complex-valued convolution as a matched filtering operation. (a) A noisy complex-valued input signal which contains a feature to be detected; the red dots represent noise and the blue dots the feature of interest corrupted by noise. (b) The MF template corresponding to the feature of interest, depicted in blue dots. (c) The outputs of WL and SL convolutional layers, with the maxima corresponding to the position of the identified feature of interest.
• Figure 5: Block diagram of two simple complex-valued CNNs used in the simulations. The SL-Net employs standard complex input and SLC, and while the WL-Net uses augmented complex input and WLC.