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Whitened Matched Filter

Started by cyclotron March 9, 2010
I am trying to figure out why all treatments of MLSE receiver for ISI
channels use the idea of a matched-filter followed by a noise whitening
filter? Isn't the latter a simple inverse of the former? In fact if we
simply don't go into the matched-filter/whitening-filter duo, we can still
derive what Proakis refers to as the "Equivalent Discrete-time White Noise
Filter Model" for an ISI channel (and it is this model that leads into all
those sub-optimal channel equalization techniques).

To quote from Andrea Goldsmith's Wireless Communication, 

"It might seem odd at first to introduce the matched filter g*(-t) at the
receive front-end only to cancel its effect in the equalizer." 

Can any of the gurus on this forum help me _understand_ the raison d'etre
for this seeming oddity?

Thanks! 

cyclotron wrote:

> I am trying to figure out why all treatments of MLSE receiver for ISI > channels use the idea of a matched-filter followed by a noise whitening > filter?
Because this is simple and convenient concept.
> Isn't the latter a simple inverse of the former?
No, it is not. Signal path != Noise path. Vladimir Vassilevsky DSP and Mixed Signal Design Consultant http://www.abvolt.com
> > >cyclotron wrote: > >> I am trying to figure out why all treatments of MLSE receiver for ISI >> channels use the idea of a matched-filter followed by a noise whitening >> filter? > >Because this is simple and convenient concept. > >> Isn't the latter a simple inverse of the former? > >No, it is not. Signal path != Noise path. > > >Vladimir Vassilevsky >DSP and Mixed Signal Design Consultant >http://www.abvolt.com >
I am not sure if I got your second comment. Here is a block diagram of the system taken from Proakis: ______ ______ ________ smplr _____ | | | | h(t) | | / | | I(k)---| g(t) |--->| c(t) |-------> + --->| h*(-t) |---/ --->| NWF |----> |______| |______| | |________| |_____| | n(t) g(t)=transmit filter c(t)=channel impulse response h(t)=received pulse with ISI h*(-t)=filter matched to the receive pulse h(t) n(t)=White Gaussian noise NWF=Noise Whitening Filter Aren't both the noise (n(t)) and the "signal" (h(t)) going through the same path? To further elaborate my original post, if NWF simply "inverts" the matched filter h*(-t), then why do I need the last two blocks in my model? I could start out with the following: ______ ______ smplr | | | | h(t) / I(k)---| g(t) |--->| c(t) |-------> + ---/ ---> |______| |______| | | n(t) in which case the noise n(t) at the output is already white, and the T-spaced samples of the non-Nyquist pulse h(t) give me the taps of the Equivalent Discrete-time White Noise Filter model, and I am done without having to invoke a matched filter followed by a whitening filter. I am sure I'm missing something here ....
cyclotron wrote:
>> >> cyclotron wrote: >> >>> I am trying to figure out why all treatments of MLSE receiver for ISI >>> channels use the idea of a matched-filter followed by a noise whitening >>> filter? >> Because this is simple and convenient concept. >> >>> Isn't the latter a simple inverse of the former? >> No, it is not. Signal path != Noise path. >> >> >> Vladimir Vassilevsky >> DSP and Mixed Signal Design Consultant >> http://www.abvolt.com >> > > I am not sure if I got your second comment. Here is a block diagram of the > system taken from Proakis: > > > ______ ______ ________ smplr _____ > | | | | h(t) | | / | | > I(k)---| g(t) |--->| c(t) |-------> + --->| h*(-t) |---/ --->| NWF |----> > |______| |______| | |________| |_____| > | > n(t) > > g(t)=transmit filter > c(t)=channel impulse response > h(t)=received pulse with ISI > h*(-t)=filter matched to the receive pulse h(t) > n(t)=White Gaussian noise > NWF=Noise Whitening Filter > > Aren't both the noise (n(t)) and the "signal" (h(t)) going through the same > path? > > To further elaborate my original post, if NWF simply "inverts" the matched > filter h*(-t), then why do I need the last two blocks in my model? I could > start out with the following: > > > ______ ______ smplr > | | | | h(t) / > I(k)---| g(t) |--->| c(t) |-------> + ---/ ---> > |______| |______| | > | > n(t) > > > in which case the noise n(t) at the output is already white, and the > T-spaced samples of the non-Nyquist pulse h(t) give me the taps of the > Equivalent Discrete-time White Noise Filter model, and I am done without > having to invoke a matched filter followed by a whitening filter. > > I am sure I'm missing something here .... >
The signal I(k) and the noise n(t) are going through different paths. D'zat help? -- Tim Wescott Control system and signal processing consulting www.wescottdesign.com