# What are the differences between estimation and detection optimal filters (matched and Wiener)?

What is the differences between the two of them? I know that the matched filter is a detection filter because it detect what is the peak of the signal and return the time delay of reflected and transmitted signal. Whereas the Wiener filter is an estimation filter because it estimates the mean-square-error estimation of the degraded signal. But I'm not too sure on what is the definition of the detection filter itself, and the estimation filter definition. Any help would be appreciated.

• If you want to maximize the signal-to-noise ratio using a linear filter in the presence of additive random noise, then it can be proved that matched filter is the optimal filter. It is usually performed for detection of a known waveform.
• If you want to minimize the mean square error of estimation of a desired signal corrupted by additive random noise by a linear filter, then it can be proved that wiener filter is the optimal filter. It is usually deployed to estimate a known stationary target signal.

About what detection and estimation is, consider the radar example. Detection is to find out if there is a plane/target. If it turned out there was actually a target and it was detected, then we would like to estimate its speed, altitude, etc.

A more relevant example for detection is digital communication. Assume we use on-off keying, that is, we send a waveform if we want to represent a bit $1$, and don't send anything if the bit was $0$. During the time interval of communication of each bit (timeslot), a liner filter which is matched to the waveform of bit $1$ can be used to detect if there is a bit $1$. If no waveform was detected, then it is assumed that $0$ is sent.

An example for wiener filtering is signal compression. Assume you want to compress a stationary process $x(t)$. One approach is to estimate it using a wiener filter whose input is just white noise. Given the target signal $x(t)$, the wiener filter can form the spectrum of the white noise such that its output has the minimum mean square error compared to $x(t)$. Now the coefficients of the wiener filter (plus a source of white noise) would be enough to reproduce $x(t)$.

• thank you very much for the answer now i got the rough idea on what is match, wiener, estimation and detection filter. Though i would like to ask more about the match filter, why is that in match filter the output peak is always the same with the signal duration ? For example in a signal that i made in matlab that have a duration of 3.9 s then the output signal of match filter will have a peak exactly 3.9 s, why is that happening ? my guess is that, its related to the cross correlation but i don't know what's the specific theory from cross correlation Sep 26 '16 at 16:26
• @user3646742 You are raising a different question in this comment. You can find the answer here on dsp.SE Sep 26 '16 at 17:12
• @user3646742 Assume signal $s(t)$ has duration $T$. Consider the output of matched filter ($h(t)=s(T-t)$):$$y(t)=\int_{0}^{t}s(\tau)s(T-t+\tau)d\tau$$ At $t=T$ the lag becomes zero. It is just like autocorrelation at lag zero.
– msm
Sep 26 '16 at 22:22

For my understanding, detection is to judge whether a known signal presents in the unknown signal, and the matched filter is indeed the reversal version of known signal itself, thus the matched convolution filtering is indeed the cross correlation between a known signal and a unknown signal. Estimation is to estimate the unknown signal with noise, and there is only rough knowledge about the signal, such as the rough spectrum range, etc.