# SNR of multiple sinusoids

Started by February 21, 2006
```Here's a question that came up recently in a project I'm working on:

Assume that you have a signal made up of two (or more) equal-amplitude
sinusoids. Add white noise.

Now, I understand what happens to the SNR of each sinusoid if I analyze the
signal at various bandwidths - say by using FFTs of various resolutions and
doing spectral estimates.

What I don't know right off hand, and would like to confirm, is what happens
to the SNR of the *composite* signal if I pass the total signal through a
filter that bandpasses the sinusoids - as I vary the bandwidth of the
passbands together?  Intuitively it seems that it would be the same.  Is it?

Does this work?

{S1 + N1*sqrt(B)} "+" {S2 + N2*sqrt(B)}  "+" {S3 + N3(sqrt(B)}

= sqrt(S1^2 + S2^2 + S3^2) + sqrt{B*N1^2 + B*N2^2 + B*N3^2}
= sqrt(3)*S + sqrt(3B)*N where Sn is the rms value of the sinusoid and Nn
or N is the rms value of the 1Hz noise.

So, the SNR of the composite is the same as the SNR of any individual
component.

Then, what can we say about the SNR of the composite if the sinusoids aren't
of equal amplitude and the noises aren't of equal density?

{S1 + N1*sqrt(B)} "+" {S2 + N2*sqrt(B)}  "+" {S3 + N3(sqrt(B)}

= sqrt(S1^2 + S2^2 + S3^2) + sqrt{{N1^2*B) + N2^2*B + N3^2*B}
= sqrt(S1^2 + S2^2 + S3^2) + sqrt(3B)*sqrt{N1^2 + N2^2 + N3^2}

In this case, the SNR would appear to be dominated by the highest signal and
the highest noise.
So, only when the highest signal and the highest noise are in the same band
are the two similar.  Otherwise, the composite could be degraded relative
the best band and could be better than the worst band?

Fred

```
```"Fred Marshall" <fmarshallx@remove_the_x.acm.org> wrote in message
news:jOqdnXhBoYq5_GbeRVn-gg@centurytel.net...
> Here's a question that came up recently in a project I'm working on:
>
> Assume that you have a signal made up of two (or more) equal-amplitude
> sinusoids. Add white noise.
>
> Now, I understand what happens to the SNR of each sinusoid if I analyze
the
> signal at various bandwidths - say by using FFTs of various resolutions
and
> doing spectral estimates.
>
> What I don't know right off hand, and would like to confirm, is what
happens
> to the SNR of the *composite* signal if I pass the total signal through a
> filter that bandpasses the sinusoids - as I vary the bandwidth of the
> passbands together?  Intuitively it seems that it would be the same.  Is
it?
>
> Does this work?
>
> {S1 + N1*sqrt(B)} "+" {S2 + N2*sqrt(B)}  "+" {S3 + N3(sqrt(B)}
>
>  = sqrt(S1^2 + S2^2 + S3^2) + sqrt{B*N1^2 + B*N2^2 + B*N3^2}
>  = sqrt(3)*S + sqrt(3B)*N where Sn is the rms value of the sinusoid and Nn
> or N is the rms value of the 1Hz noise.
>
> So, the SNR of the composite is the same as the SNR of any individual
> component.
>
> Then, what can we say about the SNR of the composite if the sinusoids
aren't
> of equal amplitude and the noises aren't of equal density?
>
> {S1 + N1*sqrt(B)} "+" {S2 + N2*sqrt(B)}  "+" {S3 + N3(sqrt(B)}
>
>  = sqrt(S1^2 + S2^2 + S3^2) + sqrt{{N1^2*B) + N2^2*B + N3^2*B}
>  = sqrt(S1^2 + S2^2 + S3^2) + sqrt(3B)*sqrt{N1^2 + N2^2 + N3^2}
>
> In this case, the SNR would appear to be dominated by the highest signal
and
> the highest noise.
> So, only when the highest signal and the highest noise are in the same
band
> are the two similar.  Otherwise, the composite could be degraded relative
> the best band and could be better than the worst band?
>
> Fred
>
>
>
By suporposition I am led to believe it must be the same.

Tam

```
```"Dougal McDougal of that Elk" <FU2@yahoo.co.zpc> wrote in message
news:4yJKf.151744\$vH5.1305196@news.xtra.co.nz...
>
> "Fred Marshall" <fmarshallx@remove_the_x.acm.org> wrote in message
> news:jOqdnXhBoYq5_GbeRVn-gg@centurytel.net...
>> Here's a question that came up recently in a project I'm working on:
>>
>> Assume that you have a signal made up of two (or more) equal-amplitude
>> sinusoids. Add white noise.
>>
>> Now, I understand what happens to the SNR of each sinusoid if I analyze
> the
>> signal at various bandwidths - say by using FFTs of various resolutions
> and
>> doing spectral estimates.
>>
>> What I don't know right off hand, and would like to confirm, is what
> happens
>> to the SNR of the *composite* signal if I pass the total signal through a
>> filter that bandpasses the sinusoids - as I vary the bandwidth of the
>> passbands together?  Intuitively it seems that it would be the same.  Is
> it?
>>
>> Does this work?
>>
>> {S1 + N1*sqrt(B)} "+" {S2 + N2*sqrt(B)}  "+" {S3 + N3(sqrt(B)}
>>
>>  = sqrt(S1^2 + S2^2 + S3^2) + sqrt{B*N1^2 + B*N2^2 + B*N3^2}
>>  = sqrt(3)*S + sqrt(3B)*N where Sn is the rms value of the sinusoid and
>> Nn
>> or N is the rms value of the 1Hz noise.
>>
>> So, the SNR of the composite is the same as the SNR of any individual
>> component.
>>
>> Then, what can we say about the SNR of the composite if the sinusoids
> aren't
>> of equal amplitude and the noises aren't of equal density?
>>
>> {S1 + N1*sqrt(B)} "+" {S2 + N2*sqrt(B)}  "+" {S3 + N3(sqrt(B)}
>>
>>  = sqrt(S1^2 + S2^2 + S3^2) + sqrt{{N1^2*B) + N2^2*B + N3^2*B}
>>  = sqrt(S1^2 + S2^2 + S3^2) + sqrt(3B)*sqrt{N1^2 + N2^2 + N3^2}
>>
>> In this case, the SNR would appear to be dominated by the highest signal
> and
>> the highest noise.
>> So, only when the highest signal and the highest noise are in the same
> band
>> are the two similar.  Otherwise, the composite could be degraded relative
>> the best band and could be better than the worst band?
>>
>> Fred
>>
>>
>>
> By suporposition I am led to believe it must be the same.
>
> Tam

But you know that superposition doesn't work with (uncorrelated) noise....

Fred

```
```>>>What I don't know right off hand, and would like to confirm, is what happens
to the SNR of the *composite* signal if I pass the total signal through
a
filter that bandpasses the sinusoids - as I vary the bandwidth of the
passbands together? <<<<

Does the filter for the composite signal consist of a single passband
that has ONE passband that has a BW wide enough to pass all the
sines..............  or...............  is it a composite filter made
up of individual passbands, onme for EACH  sine?

Mark

```
```"Mark" <makolber@yahoo.com> wrote in message
>>>>What I don't know right off hand, and would like to confirm, is what
>>>>happens
> to the SNR of the *composite* signal if I pass the total signal through
> a
> filter that bandpasses the sinusoids - as I vary the bandwidth of the
> passbands together? <<<<
>
> Does the filter for the composite signal consist of a single passband
> that has ONE passband that has a BW wide enough to pass all the
> sines..............  or...............  is it a composite filter made
> up of individual passbands, onme for EACH  sine?
>
> Mark
>

Oh, yes, one for each sine.  They are fairly well separated in frequency.

Fred

```