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

# SNR of multiple sinusoids

Started by ●February 21, 2006

Reply by ●February 21, 20062006-02-21

"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 analyzethe> signal at various bandwidths - say by using FFTs of various resolutionsand> doing spectral estimates. > > What I don't know right off hand, and would like to confirm, is whathappens> 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. Isit?> > 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 sinusoidsaren'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 signaland> the highest noise. > So, only when the highest signal and the highest noise are in the sameband> 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

Reply by ●February 21, 20062006-02-21

"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. > > TamBut you know that superposition doesn't work with (uncorrelated) noise.... Fred

Reply by ●February 22, 20062006-02-22

>>>What I don't know right off hand, and would like to confirm, is what happensto 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

Reply by ●February 22, 20062006-02-22

"Mark" <makolber@yahoo.com> wrote in message news:1140621069.327006.91000@g14g2000cwa.googlegroups.com...>>>>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