Hello forum, a real-valued power signal x(t) has a Fourier transform X(w) that involves both negative and positive frequencies w. The PSD corresponds to the S(w)=|X(w)^2|= |X(w) X*(w)|=|X(w)X(-w)| because of Hermitian symmetry. I have read that the function S(w) represents the power due to two frequencies whose sum equal to zero: w1+w2=0 where w1=w and w2=-w.... What does that really mean? What power does the integral INT S(w)dw between a frequency w and -w represent? The power due to all frequencies between 0 and w in the real signal? thanks, fisico32
PSD interpretation....
Started by ●May 4, 2010
Reply by ●May 4, 20102010-05-04
fisico32 <marcoscipioni1@n_o_s_p_a_m.gmail.com> wrote:>I have read that the function S(w) represents the power due to two >frequencies whose sum equal to zero: w1+w2=0 where w1=w and w2=-w.... >What does that really mean?Not much. sin(-wt) = -sin(wt) so unless you're analysing the signal as a complex signal there is no difference between positive and negative frequency components. (The question I find more interesting is what does it mean when the PSD evaluates to a negative value at a positivie frequency....but this is unrelated.) Steve
Reply by ●May 4, 20102010-05-04
On May 5, 5:27�am, "fisico32" <marcoscipioni1@n_o_s_p_a_m.gmail.com> wrote:> Hello forum, > > a real-valued power signal x(t) has a Fourier transform X(w) that involves > both negative and positive frequencies w. > > The PSD corresponds to the S(w)=|X(w)^2|= |X(w) X*(w)|=|X(w)X(-w)| because > of Hermitian symmetry. > > I have read that the function S(w) represents the power due to two > frequencies whose sum equal to zero: w1+w2=0 where w1=w and w2=-w.... > What does that really mean? > > What power does the integral INT S(w)dw �between a frequency w and -w > represent? The power due to all frequencies between 0 and w in the real > signal? > > thanks, > fisico32The integral is the variance (assuming zero dc) or total average power in the signal.
Reply by ●May 5, 20102010-05-05
"Steve Pope" <spope33@speedymail.org> wrote in message news:hrprb1$j81$5@blue.rahul.net...> (The question I find more interesting is what does it mean > when the PSD evaluates to a negative value at a positivie > frequency....but this is unrelated.)How can it do that?
Reply by ●May 5, 20102010-05-05
"fisico32" <marcoscipioni1@n_o_s_p_a_m.gmail.com> writes:> Hello forum, > > a real-valued power signal x(t) has a Fourier transform X(w) that involves > both negative and positive frequencies w. > > The PSD corresponds to the S(w)=|X(w)^2|If by X(w) you mean the FT of x(t), then this is not formally correct. The PSD is defined to be S(w) = FT(R(tau)), where R(tau) is the autocorrelation function of x(t). However, we commonly _estimate_ S(w) by |X(2)^2|.> = |X(w) X*(w)|=|X(w)X(-w)| because > of Hermitian symmetry. > > I have read that the function S(w) represents the power due to two > frequencies whose sum equal to zero: w1+w2=0 where w1=w and w2=-w.... > What does that really mean?I think it depends on how the author defines the power spectrum (one-sided or two-sided); the power in some positive bandwidth in a one-sided power spectrum (valid for a real input signal) includes the positive and negative frequencies (and don't get us started...).> What power does the integral INT S(w)dw between a frequency w and -w > represent? The power due to all frequencies between 0 and w in the real > signal?The way I view the world is that ALL signals (real or otherwise) have a two-sided PSD, in which case you would be correct. -- Randy Yates % "My Shangri-la has gone away, fading like Digital Signal Labs % the Beatles on 'Hey Jude'" mailto://yates@ieee.org % http://www.digitalsignallabs.com % 'Shangri-La', *A New World Record*, ELO
Reply by ●May 5, 20102010-05-05
Alfred Bovin <alfred@bovin.invalid> wrote:>"Steve Pope" <spope33@speedymail.org> wrote in message>> (The question I find more interesting is what does it mean >> when the PSD evaluates to a negative value at a positive >> frequency....but this is unrelated.)>How can it do that?For some signals and some windows functions it can sometimes happen. Beyond that, I am not certain what conditions lead to this. It is usually rare, and the negative values can usually be ignored. Steve
Reply by ●May 5, 20102010-05-05
On May 5, 8:56�am, spop...@speedymail.org (Steve Pope) wrote:> Alfred Bovin <alf...@bovin.invalid> wrote: > >"Steve Pope" <spop...@speedymail.org> wrote in message > >> (The question I find more interesting is what does it mean > >> when the PSD evaluates to a negative value at a positive > >> frequency....but this is unrelated.) > >How can it do that? > > For some signals and some windows functions it can sometimes > happen. �Beyond that, I am not certain what conditions lead > to this. � It is usually rare, and the negative values can > usually be ignored. > > SteveSteve, were you having a senior moment? :) X(f)*X*(f) is always non- negative. --Randy
Reply by ●May 5, 20102010-05-05
On 2010-05-05 16:16:33 -0300, Randy Yates <yates@ieee.org> said:> On May 5, 8:56�am, spop...@speedymail.org (Steve Pope) wrote: >> Alfred Bovin <alf...@bovin.invalid> wrote: >>> "Steve Pope" <spop...@speedymail.org> wrote in message >>>> (The question I find more interesting is what does it mean >>>> when the PSD evaluates to a negative value at a positive >>>> frequency....but this is unrelated.) >>> How can it do that? >> >> For some signals and some windows functions it can sometimes >> happen. �Beyond that, I am not certain what conditions lead >> to this. � It is usually rare, and the negative values can >> usually be ignored. >> >> Steve > > Steve, were you having a senior moment? :) X(f)*X*(f) is always non- > negative. > > --RandyIf the window concerned was being applied in the lag domain it is quite possible. In fact positive definite lag windows are not that common compared to lag windows which have lower variance and are unbiased but have the technical bother of permitting negative estimates of a parameter which is known to be positive.
Reply by ●May 5, 20102010-05-05
Randy Yates <yates@ieee.org> wrote:>On May 5, 8:56�am, spop...@speedymail.org (Steve Pope) wrote:>> For some signals and some windows functions it can sometimes >> happen. �Beyond that, I am not certain what conditions lead >> to this. � It is usually rare, and the negative values can >> usually be ignored.>Steve, were you having a senior moment? :) X(f)*X*(f) is always non- >negative.Well you have to drill down a bit more. X(t) -> window -> sum(X(t)-(X(t-u)) -> window -> cosine transform -> some non-positive-values. Depending on signal and window. Odd stuff it is. Steve
Reply by ●May 5, 20102010-05-05
On May 5, 1:11�pm, spop...@speedymail.org (Steve Pope) wrote:> Randy Yates �<ya...@ieee.org> wrote: > > >On May 5, 8:56�am, spop...@speedymail.org (Steve Pope) wrote: > >> For some signals and some windows functions it can sometimes > >> happen. �Beyond that, I am not certain what conditions lead > >> to this. � It is usually rare, and the negative values can > >> usually be ignored. > >Steve, were you having a senior moment? :) X(f)*X*(f) is always non- > >negative. > > Well you have to drill down a bit more. > > X(t) -> window -> sum(X(t)-(X(t-u)) -> window -> cosine transform -> > some non-positive-values. �Depending on signal and window. > > Odd stuff it is. > > SteveLet's see now... Someone takes a signal, windows, samples, applies a known high variance estimator [1 sections 2,3] in a processing chain, discovers that this does not produce results equivalent to the (infinite/ continuous) theoretical model, becomes surprised, reports results and surprise and has the report met with disbelief. What is unusual about this? On comp.dsp, nothing. Dale B. Dalrymple [1] Marple, S.L., Jr., �A tutorial overview of modern spectral estimation�, Acoustics, Speech, and Signal Processing, volume 4, 23-26, Page(s):2152 - 2157, May 1989 available at: http://www.cactus.org/~benjamin/X/marple.pdf
