Matched Filters are frequently used in communication systems. The distinguishing characteristic of a Matched Filter is that the step response approximates a ramp, and the impulse response approximates a pulse. The purpose of the Matched Filter is to maximize the signal to noise (S/N) ratio to minimize the probability of undetected errors received from a signal.
The function of a Matched Filter is to optimize the S/N ratio at the sampling point of a bit stream. This will occur if the filter applied to the bit stream has an impulse response that is the time-inverse of the pulse shape being sampled.
If the pulse is rectangular, the filter impulse response must also be rectangular; and the step response is a ramp. FilterSolutions and Filter Quick allow one to define the Matched Filter by setting the rise time of the ramp. The proper use of the Matched Filter is to set the rise-time equal to the pulse width of the pulses in the bit stream.
Since ideal continuous and IIR matched filter solutions are not realizable, they must be approximated. FilterSolutions uses an approximation solution that optimizes the time response of the filter with the constraint that the transfer function zeros remain on the jω 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 jω zeros constraint is to allow the filter to be realized with passive elements.
Shown here are examples of a Matched Filter’s step, impulse, and frequency responses. Following the frequency response is the Matched Filter’s square wave response, wherin the rise time of the filter is set to match the pulse width of the square wave.