Matched Filter Output SNR Calculator

A matched filter is a linear filter whose impulse response is the time-reversed, conjugated copy of a known signal, designed to maximize output signal-to-noise ratio at the moment the signal is fully captured, when the input contains that signal plus additive white Gaussian noise. It is the theoretically optimal detector for a known pulse shape in Gaussian noise, used throughout radar, sonar, and digital communications receivers.

SNR_out = 2E / N0, where E is the total energy of the transmitted pulse and N0 is the one-sided noise power spectral density. This result depends only on total pulse energy, not on the specific pulse shape, amplitude modulation, or duration individually, only their product through E.

For a rectangular pulse of constant amplitude A over duration T, E = A^2 x T. Non-rectangular pulses (Gaussian, raised-cosine, LFM chirp) use the same output SNR formula, 2E/N0, but their energy is computed by integrating the squared envelope over time instead of the simple A^2*T product.

The matched filter's gain is defined precisely so that it correlates the received signal against a copy of the transmitted pulse, and by the Cauchy-Schwarz inequality this correlation is maximized (for a fixed noise level) purely by the pulse's total energy, regardless of how that energy is distributed in time or frequency. A short, high-amplitude pulse and a long, low-amplitude pulse with the same total energy produce identical output SNR.

Input SNR is the instantaneous signal-to-noise ratio at a single sample before filtering, while output SNR is the signal-to-noise ratio at the matched filter's output at the optimal sampling instant, after coherently combining energy across the whole pulse duration. The matched filter's processing gain is exactly the ratio of these two, proportional to the pulse's time-bandwidth product.

N0 = k x T_sys, where k is Boltzmann's constant (1.38 x 10^-23 J/K) and T_sys is the system noise temperature in Kelvin, giving N0 in watts per Hz (or volts-squared per Hz for a matched impedance). A lower system noise temperature (better receiver front end) directly lowers N0 and raises the matched filter's output SNR for the same pulse energy.

The factor of 2 arises from the standard convention of defining N0 as the one-sided (positive-frequency-only) noise power spectral density while integrating signal energy over both positive and negative frequency components in the matched filter derivation. Some references instead define a two-sided PSD of N0/2, which absorbs the factor of 2 into the noise definition and yields the same numerical output SNR.

Required output SNR for a given detection probability and false alarm rate sets the minimum pulse energy 2E/N0 a radar must return from a target, which through the radar range equation directly limits maximum detection range, since received energy falls off with the fourth power of range. Increasing transmit pulse duration (and hence energy) is one of the main ways radars trade detection range against range resolution, addressed by pulse compression.

Higher output SNR directly increases the probability of detecting a target for a fixed false alarm rate, following curves derived from the noise's statistical distribution (typically Gaussian for the matched filter output amplitude). A common radar design rule of thumb requires roughly 13 dB of output SNR for a reasonable single-pulse detection probability against a Swerling target model, though this varies with the specific detection criteria used.

Yes. A digital matched filter implemented as an FIR correlator against a sampled reference pulse achieves the same 2E/N0 output SNR in the limit of adequate sampling (at or above the Nyquist rate of the pulse's bandwidth), with E computed as the discrete sum of squared sample values times the sample period instead of a continuous integral. Undersampling the pulse loses some of its energy and correspondingly reduces the achievable output SNR below this ideal figure.