For Gaussian case Matched Filter detection is optimal because it maximizes the SNR of received signal and making it apt for detection. But it is not optimal for non Gaussian case.

In general, the matched filter applies the greatest weighting to spectral components that have the greatest signal-to-noise ratio. This requires a non-flat frequency response, the associated distortion is not significant in situations such as radar and digital communications, where the original waveform is known and the objective is to detect the presence of this signal against the background noise.

But the worst part of matched filter is that signal is being detected should be known otherwise it is not worth.

MF is optimal in the sense of signal-to-noise ratio in its output. Simple like that.

To add to what Marcio said, the matched filter makes a signal which is below the noise threshold detectable.

For Gaussian case Matched Filter detection is optimal because it maximizes the SNR of received signal and making it apt for detection. But it is not optimal for non Gaussian case.

In general, the matched filter applies the greatest weighting to spectral components that have the greatest signal-to-noise ratio. This requires a non-flat frequency response, the associated distortion is not significant in situations such as radar and digital communications, where the original waveform is known and the objective is to detect the presence of this signal against the background noise.

But the worst part of matched filter is that signal is being detected should be known otherwise it is not worth.

MF is basically an integrator which extract the energy of the signal. With noise being random obviously the energy due to noise over one bit duration will be very much lower the signal energy, thus improved SNR.

MF is not just an integrator. Just in the particular case that the signal of interest is constant the MF results in an integration only. You have to multiply the time series by the interest signal before integrate.

Matched Filter linearly increase signal SNR over symbol duration. Additionally, output of the MF depends on signal energy and the noise PSD not on the shape of the signal. Finally, MF can be implemented as a Correlator because of mathematical equivalence of correlation and convolution.

Can someone shed light on mathematical equivalence of convolution and correlation in case of matched filter?

Convolution: f conv g (t) = integral -inf to +inf [ f(k) g(t - k) dk]

Crosscorrelation: f cc g (t) = integral -inf to +inf [ f*(k) g(t + k) dk]

where f* is the complex conjugate of f.

If f(t) is real only, then the two sliding products are similar, except reversed leading and trailing edges of the output signal of convolution compared with crosscorrelation.

If the function is also even, such that g(t) = g(-t), then the two operations should give identical results.

In a matched filter, f and g are the same function, but otherwise I think the rest holds.

This definition seems true for even and real signals. What if the signals are complex in nature? The reason convolution and correlation are equivalent in case of matched filter is that in correlation we just integrate product of two signals with any time-shift; whereas, in convolution ,which is operation performed by matched filter, there exist double-shift in the signal--one time-shift in signal is part of impulse response of filter and the second one is convolution operation.

Please validate my argument!

I'm confused about your last question. In matched filtering, the impulse reponse of the filter is the same as the signal you're trying to find amidst the noise.

The term "impulse response" is equivalent to "transfer function of the filter." It says that if you convolve the filter's transfer function with a delta function (impulse), the resulting signal is the transfer function of the filter. (That's true because the integral from -inf to +inf of a delta function is 1, so in the frequency domain, you are multiplying the Fourier Transform of the transfer function by 1.) However, when searching for the signal in question, it's not like you're feeding a series of impulse functions to the filter.

Maybe I misunderstood the question.

To Albert Manfredi 1st note, the 1st Eqn should read

Convolution: f conv g (t) = integral -inf to +inf [ f(k) g(k - t) dk]

To all, MF can be made an optimum detector even for non-Gaussian noise....BY using whitening filter at the head of the MF. This whitening filter task is to spread out the PSD of noise present at the channel and make it white at the MF i/p....

(see Advanced Data Transmission Systems, by A. P. Clarck).

Sorry about that. You're right. I was sloppy. I'll go back and correct what I wrote. Actually both equations change, and the final comment remains the same. Thanks, Abdulkareem.

Matched filter works even when we don't know the actual noise variance

To Abdulkarim: where such a whitening filter exists or is feasible. In channels with zeros in their spectrum, well, since first grade we've known division by zero is not good. Even if it is not an actual zero but such a deep notch your attempt to practically whiten fails under implementation failure then the same results.

To Robert Mcgwier

Yes your are right......under such extreme case some clever compensation techniques could be used.

Matched filter do not amplify any signals which are orthogonal (in any domain) to the useful signal. Whitening of the noise improves its orthogonality to the useful signal.

Main advantage of using matched filter in estimation of signal is we can perfectly estimate the signal at lower SNR

One (another) way to see match filtering is to view it in a statistical inference problem context, where you want to infer some transmitted bits from the received signal (observations). If you dick into this view, you will come along so-called sufficient statistic (http://en.wikipedia.org/wiki/Sufficient_statistic). You can only talk about sufficient statistics if you have a model of your randomness (for instance: the noise is AWGN). Sufficient statistics is in a way a transformation/projection of your observations, which does not cause any “information loss” on the inference problem, even though the raw data has been reduced (for instance a continuous signal into discrete decision values for each bit). For random Gaussian samples (observation) x_1…x_N, the sample mean and sample variance are sufficient statistics to infer the Gaussian-distribution; here N values has been reduced to two without any loss in terms of inferring the distribution.

Se the below notes:

http://old.shahed.ac.ir/references/match.pdf

http://courses.engr.illinois.edu/ece461/handouts/notes7.pdf

In vector space perspective, basis functions are used to realize the necessary signal in the signal space. Correlator type detector requires a local reference of the basis function to locate the signal, whereas in Matched filter this is not necessary and advantageous in terms of hardware complexity. Also MF adapts its impulse response according to the input signal as told by many of the above speakers and ultimately increases the signal strength and detectable against the background noise.

Matched filer or correlation circuit (in digital form) is optimal for Gaussian channels which offers MMSE. Some other detectors such as, Energy and square law detectors are other detectors but for AWGN channels, MF is optimal.

By match filtering you can the gain the signal you are looking for and suppress others as an optimal filtering. You also can detect signal emerged in noise.  Moreover in radar detection, match filtering can provide more range resolution in case of using chirp signal. 

In Gaussian channels, using matched filter, optimu

In Gaussian channels, by using matched filter in receiver, we can find optimum reception with respect to energy detection, square law or envelope detection. In digital communications, matched filter is equal to correlator detector. This type of detection is low complex but it needs to know the signal to find the filter response. Hence, for this type of detection, complete CSI is essential. 

You can use crosscorrelation. By using Convolution Therorem, time reverse the signal. Take FFT of both signals. Multiply in Frequency Domain. Take IFFT.

This is easier to implement in hardware than MF.