Matched filter

In signal processing, the output of the matched filter is given by correlating a known delayed signal, or template, with an unknown signal to detect the presence of the template in the unknown signal. This is equivalent to convolving the unknown signal with a conjugated time-reversed version of the template. The matched filter is the optimal linear filter for maximizing the signal-to-noise ratio (SNR) in the presence of additive stochastic noise.

Matched filters are commonly used in radar, in which a known signal is sent out, and the reflected signal is examined for common elements of the out-going signal. Pulse compression is an example of matched filtering. It is so called because the impulse response is matched to input pulse signals. Two-dimensional matched filters are commonly used in image processing, e.g., to improve the SNR of X-ray observations. Additional applications of note are in seismology and gravitational-wave astronomy.

Matched filtering is a demodulation technique with LTI (linear time invariant) filters to maximize SNR. It was originally also known as a North filter.

The following section derives the matched filter for a discrete-time system. The derivation for a continuous-time system is similar, with summations replaced with integrals.

The matched filter is the linear filter, ${\displaystyle h}$, that maximizes the output signal-to-noise ratio.

where ${\displaystyle x[k]}$ is the input as a function of the independent variable ${\displaystyle k}$, and ${\displaystyle y[n]}$ is the filtered output. Though we most often express filters as the impulse response of convolution systems, as above (see LTI system theory), it is easiest to think of the matched filter in the context of the inner product, which we will see shortly.

We can derive the linear filter that maximizes output signal-to-noise ratio by invoking a geometric argument. The intuition behind the matched filter relies on correlating the received signal (a vector) with a filter (another vector) that is parallel with the signal, maximizing the inner product. This enhances the signal. When we consider the additive stochastic noise, we have the additional challenge of minimizing the output due to noise by choosing a filter that is orthogonal to the noise.

Let us formally define the problem. We seek a filter, ${\displaystyle h}$, such that we maximize the output signal-to-noise ratio, where the output is the inner product of the filter and the observed signal ${\displaystyle x}$.