Hi,
This is something that's been bugging me for the last couple of days.
With the understanding that both are linear, we define the matched
filter as the one that maximises the output SNR, and the Wiener filter
as the one that minimises the mean square error (MSE). Superficially,
these definitions sound almost identical. However, even in the simplest
model (estimating a scalar value with independent AWGN), we clearly end
up with different filters:
MF: g = h*
Wiener: g = Sh*(Shh* + sigma^2.I)^-1
such that x_est = gy, where y = hx + v (where x is scalar data with S =
E|x|^2, v is AWGN with sigma^2.I = E|v|^2, {y,h,v} are Nx1 vectors, and
* denotes conjugate transpose).
My question is, what is the intuitive reason (i.e. not just "because the
maths says so") why these don't end up with the same result, or more
specifically, how the Wiener filter can minimise the MSE without
maximising the SNR?
[This has been brought up before
(http://groups.google.com/group/comp.dsp/msg/b3d159784bc28486?dmode=source
and subsequent thread branch), but wasn't closed out on.]
--
Oli
Comparing matched and Wiener filters
Started by ●December 24, 2008
Reply by ●December 24, 20082008-12-24
On Wed, 24 Dec 2008 11:12:45 +0000, Oliver Charlesworth wrote:> Hi, > > This is something that's been bugging me for the last couple of days. > > With the understanding that both are linear, we define the matched > filter as the one that maximises the output SNR, and the Wiener filter > as the one that minimises the mean square error (MSE). Superficially, > these definitions sound almost identical. However, even in the simplest > model (estimating a scalar value with independent AWGN), we clearly end > up with different filters: > > MF: g = h* > Wiener: g = Sh*(Shh* + sigma^2.I)^-1 > > such that x_est = gy, where y = hx + v (where x is scalar data with S = > E|x|^2, v is AWGN with sigma^2.I = E|v|^2, {y,h,v} are Nx1 vectors, and > * denotes conjugate transpose). > > My question is, what is the intuitive reason (i.e. not just "because the > maths says so") why these don't end up with the same result, or more > specifically, how the Wiener filter can minimise the MSE without > maximising the SNR? > > [This has been brought up before > (http://groups.google.com/group/comp.dsp/msg/b3d159784bc28486?dmode=source> and subsequent thread branch), but wasn't closed out on.]I don't think I could give a satisfactory answer to your question without doing more mathematics than I care to. So here's an answer to a closely related question, and some hand-waving: What this tells us is that SNR and MSE, although they sound very similar, really aren't. If you consider what the Wiener filter does to a signal, though, you can certainly see that it's hacking off parts of the signal that the matched filter doesn't. -- Tim Wescott Control systems and communications consulting http://www.wescottdesign.com Need to learn how to apply control theory in your embedded system? "Applied Control Theory for Embedded Systems" by Tim Wescott Elsevier/Newnes, http://www.wescottdesign.com/actfes/actfes.html
Reply by ●December 24, 20082008-12-24
Hi Oli, The two are designed for different applications, even if their formulations are similar. Matched filter applies to the *detection* of the presence of a known signal, whereas Wiener filter applies to the *estimation* of an unknown random signal with known power spectrum. In either case, the spectrum of the signal is assumed to be known. If you consider the outputs of the two for a binary communication signal, the output of the MF looks like triangles with peak amplitudes at the sampling times; that is, the MF maximizes the SNR only at these sampling instances. On the other hand, the output of a Wiener filter, if you really wanted to use it, would look more like the transmitted binary sequence; this could not be the last stage for a detector, though. Hope this helps, Emre
Reply by ●December 24, 20082008-12-24
>If you consider the outputs of the two for a binary communication signal, >the output of the MF looks like triangles with peak amplitudes at the >sampling times; that is, the MF maximizes the SNR only at these sampling >instances. On the other hand, the output of a Wiener filter, if youreally>wanted to use it, would look more like the transmitted binary sequence; >this could not be the last stage for a detector, though.Just to clarify this example a bit more: you do not care about the output of the matched filter except for the sampling instances (which is usually thresholded for decision), whereas you care about the output of the Wiener filter equally at all times. Emre
Reply by ●December 24, 20082008-12-24
On Dec 24, 5:12�am, Oliver Charlesworth <ca...@olifilth.co.uk> wrote:> Hi, > > This is something that's been bugging me for the last couple of days. > > With the understanding that both are linear, we define the matched > filter as the one that maximises the output SNR, and the Wiener filter > as the one that minimises the mean square error (MSE). �Superficially, > these definitions sound almost identical. However, even in the simplest > model (estimating a scalar value with independent AWGN), we clearly end > up with different filters: > > MF: � � � � �g = h* > Wiener: � � �g = Sh*(Shh* + sigma^2.I)^-1 > > such that x_est = gy, where y = hx + v (where x is scalar data with S = > E|x|^2, v is AWGN with sigma^2.I = E|v|^2, {y,h,v} are Nx1 vectors, and > * denotes conjugate transpose). > > My question is, what is the intuitive reason (i.e. not just "because the > maths says so") why these don't end up with the same result, or more > specifically, how the Wiener filter can minimise the MSE without > maximising the SNR? > > [This has been brought up before > (http://groups.google.com/group/comp.dsp/msg/b3d159784bc28486?dmode=so... > and subsequent thread branch), but wasn't closed out on.] > > -- > OliOne is SIGNAL based, and the other is MODEL based. Maurice Givens
Reply by ●December 24, 20082008-12-24
emre wrote:> Hi Oli, > > The two are designed for different applications, even if their > formulations are similar. Matched filter applies to the *detection* of the > presence of a known signal, whereas Wiener filter applies to the > *estimation* of an unknown random signal with known power spectrum. In > either case, the spectrum of the signal is assumed to be known.I'm not sure there's much of a distinction. The purpose of detection is to determine some property of the signal, either presence or absence (OOK), or some more complicated modulation scheme (or something else entirely in e.g. radar applications). Therefore, the requirement is to *estimate* some parameter behind the signal.> If you consider the outputs of the two for a binary communication signal, > the output of the MF looks like triangles with peak amplitudes at the > sampling times; that is, the MF maximizes the SNR only at these sampling > instances. On the other hand, the output of a Wiener filter, if you really > wanted to use it, would look more like the transmitted binary sequence; > this could not be the last stage for a detector, though.I see what you're saying. However, isn't this just an artifact of the "filter across all time and sample" viewpoint of matched-filtering, as opposed to the "integrate-and-dump" viewpoint, where we only calculate the output for the sampling points (i.e. correlation, equivalent to the vector formation). -- Oli
Reply by ●December 24, 20082008-12-24
On 24 Des, 17:31, maury <maury...@core.com> wrote:> On Dec 24, 5:12�am, Oliver Charlesworth <ca...@olifilth.co.uk> wrote:> > My question is, what is the intuitive reason (i.e. not just "because the > > maths says so") why these don't end up with the same result, or more > > specifically, how the Wiener filter can minimise the MSE without > > maximising the SNR?..> One is SIGNAL based, and the other is MODEL based.Agreed. The matched filter is derived from the transient signal in time domain, whereas the Wiener filter is derived from the signal and noise covariances. Since infinitely many time sequences share one autocovariance sequence, this means that phase information is lost, even if the computations take place in temporal domain. Rune
Reply by ●December 24, 20082008-12-24
Oli Charlesworth <catch@olifilth.co.uk> writes:> emre wrote: >> Hi Oli, >> >> The two are designed for different applications, even if their >> formulations are similar. Matched filter applies to the *detection* of the >> presence of a known signal, whereas Wiener filter applies to the >> *estimation* of an unknown random signal with known power spectrum. In >> either case, the spectrum of the signal is assumed to be known. > > I'm not sure there's much of a distinction. The purpose of detection is > to determine some property of the signal, either presence or absence > (OOK), or some more complicated modulation scheme (or something else > entirely in e.g. radar applications). Therefore, the requirement is to > *estimate* some parameter behind the signal. > > >> If you consider the outputs of the two for a binary communication signal, >> the output of the MF looks like triangles with peak amplitudes at the >> sampling times; that is, the MF maximizes the SNR only at these sampling >> instances. On the other hand, the output of a Wiener filter, if you really >> wanted to use it, would look more like the transmitted binary sequence; >> this could not be the last stage for a detector, though. > > I see what you're saying. However, isn't this just an artifact of the > "filter across all time and sample" viewpoint of matched-filtering, as > opposed to the "integrate-and-dump" viewpoint, where we only calculate > the output for the sampling points (i.e. correlation, equivalent to the > vector formation).Yeah. A matched filter in digital comms assumes you have timing. -- % Randy Yates % "I met someone who looks alot like you, %% Fuquay-Varina, NC % she does the things you do, %%% 919-577-9882 % but she is an IBM." %%%% <yates@ieee.org> % 'Yours Truly, 2095', *Time*, ELO http://www.digitalsignallabs.com
Reply by ●December 24, 20082008-12-24
maury wrote:> On Dec 24, 5:12 am, Oliver Charlesworth <ca...@olifilth.co.uk> wrote: >> Hi, >> >> This is something that's been bugging me for the last couple of days. >> >> With the understanding that both are linear, we define the matched >> filter as the one that maximises the output SNR, and the Wiener filter >> as the one that minimises the mean square error (MSE). Superficially, >> these definitions sound almost identical. However, even in the simplest >> model (estimating a scalar value with independent AWGN), we clearly end >> up with different filters: >> >> MF: g = h* >> Wiener: g = Sh*(Shh* + sigma^2.I)^-1 >> >> such that x_est = gy, where y = hx + v (where x is scalar data with S = >> E|x|^2, v is AWGN with sigma^2.I = E|v|^2, {y,h,v} are Nx1 vectors, and >> * denotes conjugate transpose). >> >> My question is, what is the intuitive reason (i.e. not just "because the >> maths says so") why these don't end up with the same result, or more >> specifically, how the Wiener filter can minimise the MSE without >> maximising the SNR? >> >> [This has been brought up before >> (http://groups.google.com/group/comp.dsp/msg/b3d159784bc28486?dmode=so... >> and subsequent thread branch), but wasn't closed out on.] >> >> -- >> Oli > > One is SIGNAL based, and the other is MODEL based.I appreciate that the optimisation criteria may have stemmed from different sources, but I guess what I'm having most trouble reconciling is that MSE and SNR aren't equivalent (even if we constrain S to be fixed). -- Oli
Reply by ●December 24, 20082008-12-24
On 24 Des, 23:27, Oli Charlesworth <ca...@olifilth.co.uk> wrote:> maury wrote: > > On Dec 24, 5:12 am, Oliver Charlesworth <ca...@olifilth.co.uk> wrote: > >> Hi, > > >> This is something that's been bugging me for the last couple of days. > > >> With the understanding that both are linear, we define the matched > >> filter as the one that maximises the output SNR, and the Wiener filter > >> as the one that minimises the mean square error (MSE). �Superficially, > >> these definitions sound almost identical. However, even in the simplest > >> model (estimating a scalar value with independent AWGN), we clearly end > >> up with different filters: > > >> MF: � � � � �g = h* > >> Wiener: � � �g = Sh*(Shh* + sigma^2.I)^-1 > > >> such that x_est = gy, where y = hx + v (where x is scalar data with S = > >> E|x|^2, v is AWGN with sigma^2.I = E|v|^2, {y,h,v} are Nx1 vectors, and > >> * denotes conjugate transpose). > > >> My question is, what is the intuitive reason (i.e. not just "because the > >> maths says so") why these don't end up with the same result, or more > >> specifically, how the Wiener filter can minimise the MSE without > >> maximising the SNR? > > >> [This has been brought up before > >> (http://groups.google.com/group/comp.dsp/msg/b3d159784bc28486?dmode=so... > >> and subsequent thread branch), but wasn't closed out on.] > > >> -- > >> Oli > > > One is SIGNAL based, and the other is MODEL based. > > I appreciate that the optimisation criteria may have stemmed from > different sources, but I guess what I'm having most trouble reconciling > �is that MSE and SNR aren't equivalent (even if we constrain S to be fixed).The main difference is that MSE is the deviation between the desired and actual response. SNR includes an additive noise term as well, which is missing in the MSE. Rune
