Matched Filters

The Matched Filter is for use in communications. The distinguishing characteristic of a Matched Filter is the step response approximating a ramp, and the impulse response approximates a pulse. The purpose of the Matched Filter is to maximize the signal to noise ratio and to minimize the probability of undetected errors received from a signal.

The function of a Matched filter is to optimize the signal to noise ratio at the sampling point of a bit stream. This happens if the filter applied to the bit stream has an impulse response that is the time inverse of the pulse shape that is being sampled. If the pulse is rectangular, the filter impulse response must therefore also be a rectangle, and the step response is a ramp.

Filter Solutions and Filter Light allow you to define your Matched Filter by setting the rise time of the ramp. The proper use of the matched filter is to set the rise time to be equal to the pulse width of the pulses in a bit stream.

Since ideal continuous and IIR matched filter solutions are not realizable, they must be approximated. Filter Solutions uses an approximate solution that optimizes the time response of the filter with the constraint that the transfer function zeros remain on the JW axis. Specifically, the integration of the square of the error between the filter impulse response and the ideal impulse response (a square pulse) is minimized under the mentioned restraint conditions. The purpose of the JW zeros constraint is to allow the filter to be realized with passive elements.

Below are examples of Matched filters step, impulse, and frequency responses.  Below the frequency response is the Matched filter square wave response when the rise time of the filter is set to match the pulse width of the square wave.