SNR Question

Hi,

Suppose you have a signal consisting of 2 stationary signals plus a
stochastic component that is WSS. We'll also suppose that one of the
deterministic signal components (say signal 1) is our "desired" signal
and the other deterministic signal component (signal 2) an
"interference" signal.

If signals 1 and 2 are correlated, how does one *correctly* represent
the SNR for signal 1?

TIA,

Matt

"MathMan" <mathimagical@netscape.net> wrote in message 
news:945d6ae0-dbee-4364-917b-bd96c41c69f1@k39g2000hsf.googlegroups.com...
> Hi,
>
> Suppose you have a signal consisting of 2 stationary signals plus a
> stochastic component that is WSS. We'll also suppose that one of the
> deterministic signal components (say signal 1) is our "desired" signal
> and the other deterministic signal component (signal 2) an
> "interference" signal.
>
> If signals 1 and 2 are correlated, how does one *correctly* represent
> the SNR for signal 1?
>
> TIA,
>
> Matt

If signal 2 is correlated with signal 1 then it isn't really "noise" is it?
There is the concept of "signal to interference ratio" which you might 
Google.  I find it's "the ratio of the power of the wanted signal to the 
total residue power of the unwanted signals."

I'm not sure that correlation has a lot to do with the definition - rather, 
the definition has more to do with how filtering is done or is successful, 
effective etc.  Correlated signals may make it harder to achieve but not 
affect the definition.

Fred 

On Jan 15, 4:04 am, "Fred Marshall" <fmarshallx@remove_the_x.acm.org>
wrote:
> "MathMan" <mathimagi...@netscape.net> wrote in message
>
> news:945d6ae0-dbee-4364-917b-bd96c41c69f1@k39g2000hsf.googlegroups.com...
>
> > Hi,
>
> > Suppose you have a signal consisting of 2 stationary signals plus a
> > stochastic component that is WSS. We'll also suppose that one of the
> > deterministic signal components (say signal 1) is our "desired" signal
> > and the other deterministic signal component (signal 2) an
> > "interference" signal.
>
> > If signals 1 and 2 are correlated, how does one *correctly* represent
> > the SNR for signal 1?
>
> > TIA,
>
> > Matt
>
> If signal 2 is correlated with signal 1 then it isn't really "noise" is it?
> There is the concept of "signal to interference ratio" which you might
> Google.  I find it's "the ratio of the power of the wanted signal to the
> total residue power of the unwanted signals."
>
> I'm not sure that correlation has a lot to do with the definition - rather,
> the definition has more to do with how filtering is done or is successful,
> effective etc.  Correlated signals may make it harder to achieve but not
> affect the definition.
>
> Fred

You are correct that it's a measure of the SINR I want: thanks for the
clarification. The basic problem however that I have still stands,
which is that the expression in this case:

SINR =  (PowerDesiredSignal)/ (PowerNoise + PowerInterference).

seems to me to be suggesting that the total signal covariance matrix
can be expressed as the sum of 3 separate covariance matrices. This
last condition, we know, however is only true generally when all 3
signal components are pairwise uncorrelated.

So my question, it seems, still remains as to how one takes into
account signal correlation in expressing the SINR? (Or does no such
definition exist under such cases?)

TA,

Matt

I think it may depend on the actual problem you're trying to solve.

Some hint, you might look at the signal in the same way the actual
receiver "sees" it. For example if an interferer falls beyond a matched
filter's frequency response, I would not consider it noise anymore,
because the receiver can reject it.

If a and b are correlated (for example a multipath reflection), then
adding b) may effectively improve the SNR in some applications (example
wireless link). In other cases it will wreck your performance (range
sensing etc).

I usually use one signal as reference, and get the normalized
crosscorrelation coefficient c.
Then a part c^2 of my signal is explained by the reference, and a part
1-c^2 is unexplained (uncorrelated). It follows that SNR=c^2/(1-c^2).

But as said, it may depend on the application and there will be several
possibilities for defining SNR.

On Jan 15, 5:33 am, "mnentwig" <mnent...@elisanet.fi> wrote:
> I think it may depend on the actual problem you're trying to solve.
>
> Some hint, you might look at the signal in the same way the actual
> receiver "sees" it. For example if an interferer falls beyond a matched
> filter's frequency response, I would not consider it noise anymore,
> because the receiver can reject it.
>
> If a and b are correlated (for example a multipath reflection), then
> adding b) may effectively improve the SNR in some applications (example
> wireless link). In other cases it will wreck your performance (range
> sensing etc).
>
> I usually use one signal as reference, and get the normalized
> crosscorrelation coefficient c.
> Then a part c^2 of my signal is explained by the reference, and a part
> 1-c^2 is unexplained (uncorrelated). It follows that SNR=c^2/(1-c^2).
>
> But as said, it may depend on the application and there will be several
> possibilities for defining SNR.

In this case, the second signal is due to multipath. I like your idea
of the using the normalized cross-correlation, as I think it is more
in line with the measure of interest to me.

Thank you,

Matt

On Jan 15, 5:04&#4294967295;am, MathMan <mathimagi...@netscape.net> wrote:
> On Jan 15, 4:04 am, "Fred Marshall" <fmarshallx@remove_the_x.acm.org>
> wrote:
>
>
>
>
>
> > "MathMan" <mathimagi...@netscape.net> wrote in message
>
> >news:945d6ae0-dbee-4364-917b-bd96c41c69f1@k39g2000hsf.googlegroups.com...
>
> > > Hi,
>
> > > Suppose you have a signal consisting of 2 stationary signals plus a
> > > stochastic component that is WSS. We'll also suppose that one of the
> > > deterministic signal components (say signal 1) is our "desired" signal
> > > and the other deterministic signal component (signal 2) an
> > > "interference" signal.
>
> > > If signals 1 and 2 are correlated, how does one *correctly* represent
> > > the SNR for signal 1?
>
> > > TIA,
>
> > > Matt
>
> > If signal 2 is correlated with signal 1 then it isn't really "noise" is it?
> > There is the concept of "signal to interference ratio" which you might
> > Google. &#4294967295;I find it's "the ratio of the power of the wanted signal to the
> > total residue power of the unwanted signals."
>
> > I'm not sure that correlation has a lot to do with the definition - rather,
> > the definition has more to do with how filtering is done or is successful,
> > effective etc. &#4294967295;Correlated signals may make it harder to achieve but not
> > affect the definition.
>
> > Fred
>
> You are correct that it's a measure of the SINR I want: thanks for the
> clarification. The basic problem however that I have still stands,
> which is that the expression in this case:
>
> SINR = &#4294967295;(PowerDesiredSignal)/ (PowerNoise + PowerInterference).
>
> seems to me to be suggesting that the total signal covariance matrix
> can be expressed as the sum of 3 separate covariance matrices. This
> last condition, we know, however is only true generally when all 3
> signal components are pairwise uncorrelated.
>
> So my question, it seems, still remains as to how one takes into
> account signal correlation in expressing the SINR? (Or does no such
> definition exist under such cases?)
>
> TA,
>
> Matt- Hide quoted text -
>
> - Show quoted text -

My thought is that it would depend upon the implementation of your
receiver, specifically if the receiver makes good use of the
"correlated interference" or not.

The example I'm thinking of is multi-path. Depending upon how the
particular receiver deals with it, the delayed multipath
"interference" signal can be interference (hurt the SNR) , it can be
ignored (no impact on SNR) , or it can be added as useful signal
(improve the SNR).

Mark

On Jan 15, 10:00 am, Mark <makol...@yahoo.com> wrote:
> On Jan 15, 5:04 am, MathMan <mathimagi...@netscape.net> wrote:
>
>
>
> > On Jan 15, 4:04 am, "Fred Marshall" <fmarshallx@remove_the_x.acm.org>
> > wrote:
>
> > > "MathMan" <mathimagi...@netscape.net> wrote in message
>
> > >news:945d6ae0-dbee-4364-917b-bd96c41c69f1@k39g2000hsf.googlegroups.com...
>
> > > > Hi,
>
> > > > Suppose you have a signal consisting of 2 stationary signals plus a
> > > > stochastic component that is WSS. We'll also suppose that one of the
> > > > deterministic signal components (say signal 1) is our "desired" signal
> > > > and the other deterministic signal component (signal 2) an
> > > > "interference" signal.
>
> > > > If signals 1 and 2 are correlated, how does one *correctly* represent
> > > > the SNR for signal 1?
>
> > > > TIA,
>
> > > > Matt
>
> > > If signal 2 is correlated with signal 1 then it isn't really "noise" is it?
> > > There is the concept of "signal to interference ratio" which you might
> > > Google.  I find it's "the ratio of the power of the wanted signal to the
> > > total residue power of the unwanted signals."
>
> > > I'm not sure that correlation has a lot to do with the definition - rather,
> > > the definition has more to do with how filtering is done or is successful,
> > > effective etc.  Correlated signals may make it harder to achieve but not
> > > affect the definition.
>
> > > Fred
>
> > You are correct that it's a measure of the SINR I want: thanks for the
> > clarification. The basic problem however that I have still stands,
> > which is that the expression in this case:
>
> > SINR =  (PowerDesiredSignal)/ (PowerNoise + PowerInterference).
>
> > seems to me to be suggesting that the total signal covariance matrix
> > can be expressed as the sum of 3 separate covariance matrices. This
> > last condition, we know, however is only true generally when all 3
> > signal components are pairwise uncorrelated.
>
> > So my question, it seems, still remains as to how one takes into
> > account signal correlation in expressing the SINR? (Or does no such
> > definition exist under such cases?)
>
> > TA,
>
> > Matt- Hide quoted text -
>
> > - Show quoted text -
>
> My thought is that it would depend upon the implementation of your
> receiver, specifically if the receiver makes good use of the
> "correlated interference" or not.
>
> The example I'm thinking of is multi-path. Depending upon how the
> particular receiver deals with it, the delayed multipath
> "interference" signal can be interference (hurt the SNR) , it can be
> ignored (no impact on SNR) , or it can be added as useful signal
> (improve the SNR).
>
> Mark

Hi Mark,

Would that I were so lucky ;>).

In my case, even when the multipath "adds" to the SNR this would be
undesirable. I don't want to get into gory details, but in an
application like GPS where you are using a delay lock loop to estimate
the carrier phase of your desired signal *any* multipath is bad,
*even* if increases the SNR. That sounds somewhat counter-intuitive
perhaps, but the presence of the multipath effectively degrades the
correlation of the signal with which you are working with your
reference signal, and this has the ultimate consequence of degrading
the position solution sought.

So, back to square 1 , although I am starting to think there is no pat
answer of the type I seek at this point.

Matt

>> but in an application like GPS 
good point... when I said "multipath" I was assuming the purpose was to
recover the information in the signal. But if it's the timing you're
interested, then it's not the same.

One thing that comes to my mind (but this is just an idea!) is to use
known timing, amplitude and phase, and subtract the shifted / rotated /
scaled reference from the received signal. What remains is by definition
all unwanted energy. Then SNR as the ratio of powers.
Whether it's meaningful is another question, because I need to know the
timing result before I can get the SNR... however in a simulation one
could make it work.

On Jan 15, 7:40&#4294967295;pm, MathMan <mathimagi...@netscape.net> wrote:
> On Jan 15, 10:00 am, Mark <makol...@yahoo.com> wrote:
>
>
>
>
>
> > On Jan 15, 5:04 am, MathMan <mathimagi...@netscape.net> wrote:
>
> > > On Jan 15, 4:04 am, "Fred Marshall" <fmarshallx@remove_the_x.acm.org>
> > > wrote:
>
> > > > "MathMan" <mathimagi...@netscape.net> wrote in message
>
> > > >news:945d6ae0-dbee-4364-917b-bd96c41c69f1@k39g2000hsf.googlegroups.com...
>
> > > > > Hi,
>
> > > > > Suppose you have a signal consisting of 2 stationary signals plus a
> > > > > stochastic component that is WSS. We'll also suppose that one of the
> > > > > deterministic signal components (say signal 1) is our "desired" signal
> > > > > and the other deterministic signal component (signal 2) an
> > > > > "interference" signal.
>
> > > > > If signals 1 and 2 are correlated, how does one *correctly* represent
> > > > > the SNR for signal 1?
>
> > > > > TIA,
>
> > > > > Matt
>
> > > > If signal 2 is correlated with signal 1 then it isn't really "noise" is it?
> > > > There is the concept of "signal to interference ratio" which you might
> > > > Google. &#4294967295;I find it's "the ratio of the power of the wanted signal to the
> > > > total residue power of the unwanted signals."
>
> > > > I'm not sure that correlation has a lot to do with the definition - rather,
> > > > the definition has more to do with how filtering is done or is successful,
> > > > effective etc. &#4294967295;Correlated signals may make it harder to achieve but not
> > > > affect the definition.
>
> > > > Fred
>
> > > You are correct that it's a measure of the SINR I want: thanks for the
> > > clarification. The basic problem however that I have still stands,
> > > which is that the expression in this case:
>
> > > SINR = &#4294967295;(PowerDesiredSignal)/ (PowerNoise + PowerInterference).
>
> > > seems to me to be suggesting that the total signal covariance matrix
> > > can be expressed as the sum of 3 separate covariance matrices. This
> > > last condition, we know, however is only true generally when all 3
> > > signal components are pairwise uncorrelated.
>
> > > So my question, it seems, still remains as to how one takes into
> > > account signal correlation in expressing the SINR? (Or does no such
> > > definition exist under such cases?)
>
> > > TA,
>
> > > Matt- Hide quoted text -
>
> > > - Show quoted text -
>
> > My thought is that it would depend upon the implementation of your
> > receiver, specifically if the receiver makes good use of the
> > "correlated interference" or not.
>
> > The example I'm thinking of is multi-path. Depending upon how the
> > particular receiver deals with it, the delayed multipath
> > "interference" signal can be interference (hurt the SNR) , it can be
> > ignored (no impact on SNR) , or it can be added as useful signal
> > (improve the SNR).
>
> > Mark
>
> Hi Mark,
>
> Would that I were so lucky ;>).
>
> In my case, even when the multipath "adds" to the SNR this would be
> undesirable. I don't want to get into gory details, but in an
> application like GPS where you are using a delay lock loop to estimate
> the carrier phase of your desired signal *any* multipath is bad,
> *even* if increases the SNR. That sounds somewhat counter-intuitive
> perhaps, but the presence of the multipath effectively degrades the
> correlation of the signal with which you are working with your
> reference signal, and this has the ultimate consequence of degrading
> the position solution sought.
>
> So, back to square 1 , although I am starting to think there is no pat
> answer of the type I seek at this point.
>
> Matt- Hide quoted text -
>
> - Show quoted text -

Hi Matt,

in that case I think it's apples and oranges and you can't mix them,
you need two seperate measures:

1) is SNR and is a measure of noise that casues RANDOM errors

2) is BIAS ERROR (or pick another name)   and  causes NON RANDOM bias
errors but are still errors.

The random errors can be averaged out, the bias errors cannot.

Mark

On Jan 16, 7:50 am, "mnentwig" <mnent...@elisanet.fi> wrote:
> >> but in an application like GPS
>
> good point... when I said "multipath" I was assuming the purpose was to
> recover the information in the signal. But if it's the timing you're
> interested, then it's not the same.
>
> One thing that comes to my mind (but this is just an idea!) is to use
> known timing, amplitude and phase, and subtract the shifted / rotated /
> scaled reference from the received signal. What remains is by definition
> all unwanted energy. Then SNR as the ratio of powers.
> Whether it's meaningful is another question, because I need to know the
> timing result before I can get the SNR... however in a simulation one
> could make it work.

I think you are correct about this. It would seem that the ratios
would make sense because the numerator would be the true desired
source's power, while the denominator would be (by construction) the
total power from the interference + noise components of the signal.
So, perhaps in this case, the correlation between the desired signal
and its multipath was a bit of a red herring, and never even enters
into the picture here. It seems so obvious to me now once you laid it
out (but doesn't everything once you get it right ;>)).

Thank you for the suggestion,

Matt