(I have picked up the thread from http://groups.google.com/group/comp.dsp/browse_frm/thread/5151c9a999783895?hl=en. You may reply to either) OK, here is a rephrase: We now know for sure that "pure white noise" is a hypothetical entity, used for mathematical simplicity rather than any practical use. By definition, such a source should have an infinite PSD and unit dirac autocorr (rxx). Since the PSD (W/Hz) is infinitely flat, the total power consumed by the noise generator is infinite, which is impossible. So let us assume a "colored" noise generator with an extremely large, but finite, bandwidth, much larger than the most prolific measuring instrument available to us. Since the bandwidth of the measured noise is finitely large, the autocorr, rxx of the noise process is finitely small, though not delta. Let us assume for simplicity sake that rxx in our case is (2k+1) samples wide - k samples to either side of the current sample r(n). Hence the correlation occurs from r(n-k) to r(n+k). In simpler terms, noise sample r(n) for any n >= 0 is correlated to different extents with all samples in the range r(n-k) and r(n+k), but is not correlated with samples r(n-k-1) and r(n+k+1) and so on. So assuming an infinite "colored" noise sequence, the (equally infinite) subset of every kth sample are all mutually uncorrelated!!! So let us down sample the noise sequence to obtain this so called "mutually uncorrelated noise"? What my confusion is that the down sampling process LPFs the spectrum of the output noise sequence. So would that affect the premise that the down sampling should have created a sequence of uncorrelated noise samples in the first place? Is there some gap in the assumptions or thought process, or am I missing something here. Ofcourse, all this is based on the assumption that the auto corr is a finite sequence for "colored" noise instead of being infinitely (exponentially) decaying. Is this assumption wrong? Something is amiss here!!! Any clarification would be welcome Partho Randy Yates wrote:> Ben wrote: > > point taken...... > > > > but even if there were an hypothetical point in space enemating pure > > white noise (as defined by completely independent and random samples, > > with 0 rxx and a completely flat PSD from -inf to +inf, and thereby > > giving Einstein's Special Theory of Relativity a complete toss), the > > point is that given the current state and future probable direction of > > solid state physics, the very act of measuring "pure white noise" to > > determine its "(pure) whiteness" will render it non-white. > > Ahem. Let me get this straight. You're saying, even if we break > a few laws of physics and discover there is a white noise source, > then, by the laws of physics, measuring it would make it non-white? > > YYYYYeahhh. Rrrrrrigggght. > > --RY
Confusion re. nature of output when "white noise" is filtered/down sampled.....
Started by ●June 28, 2006
Reply by ●June 28, 20062006-06-28
Ben wrote: (top posting fixed)> > Randy Yates wrote: > >>Ben wrote: >> >>>point taken...... >>> >>>but even if there were an hypothetical point in space enemating pure >>>white noise (as defined by completely independent and random samples, >>>with 0 rxx and a completely flat PSD from -inf to +inf, and thereby >>>giving Einstein's Special Theory of Relativity a complete toss), the >>>point is that given the current state and future probable direction of >>>solid state physics, the very act of measuring "pure white noise" to >>>determine its "(pure) whiteness" will render it non-white. >> >>Ahem. Let me get this straight. You're saying, even if we break >>a few laws of physics and discover there is a white noise source, >>then, by the laws of physics, measuring it would make it non-white? >> >>YYYYYeahhh. Rrrrrrigggght. >> >>--RY > >> (I have picked up the thread from > http://groups.google.com/group/comp.dsp/browse_frm/thread/5151c9a999783895?hl=en. > You may reply to either) > > > OK, here is a rephrase: > > We now know for sure that "pure white noise" is a hypothetical entity, > used for mathematical simplicity rather than any practical use. By > definition, such a source should have an infinite PSD and unit dirac > autocorr (rxx). > > Since the PSD (W/Hz) is infinitely flat, the total power consumed by > the noise generator is infinite, which is impossible. So let us assume > a "colored" noise generator with an extremely large, but finite, > bandwidth, much larger than the most prolific measuring instrument > available to us. Since the bandwidth of the measured noise is finitely > large, the autocorr, rxx of the noise process is finitely small, > though not delta. > > Let us assume for simplicity sake that rxx in our case is (2k+1) > samples wide - k samples to either side of the current sample r(n). > Hence the correlation occurs from r(n-k) to r(n+k). > > In simpler terms, noise sample r(n) for any n >= 0 is correlated to > different extents with all samples in the range r(n-k) and r(n+k), but > is not correlated with samples r(n-k-1) and r(n+k+1) and so on. > > So assuming an infinite "colored" noise sequence, the (equally > infinite) subset of every kth sample are all mutually uncorrelated!!! > > So let us down sample the noise sequence to obtain this so called > "mutually uncorrelated noise"? What my confusion is that the down > sampling process LPFs the spectrum of the output noise sequence. So > would that affect the premise that the down sampling should have > created a sequence of uncorrelated noise samples in the first place? > Is there some gap in the assumptions or thought process, or am I > missing something here. > > Ofcourse, all this is based on the assumption that the auto corr is a > finite sequence for "colored" noise instead of being infinitely > (exponentially) decaying. Is this assumption wrong? Something is amiss > here!!! > > Any clarification would be welcome > > Partho > Downsampling doesn't low-pass filter anything. It _does_ mix frequency information through aliasing, so if you downsample sufficiently the result will be white, but only in the sampled-time sense. If you take your white sampled noise and reconstruct it to a continuous-time signal then your resulting signal will be random, but not stationary. -- Tim Wescott Wescott Design Services http://www.wescottdesign.com Posting from Google? See http://cfaj.freeshell.org/google/ "Applied Control Theory for Embedded Systems" came out in April. See details at http://www.wescottdesign.com/actfes/actfes.html
Reply by ●June 28, 20062006-06-28
Oops sorry....muh mistake!!!! Downsampling will only create periodic bandpass spectrum which may aliase if the sampling rate is too low......but then, the lower the sampling rate, the more likely the correlation of the individual samples is zero. What I dont understand is your statement that if the downsampling rate is low enough (which ensures that no "correlated" samples were sampled in the first place), the resultant waveform would be white "in the time series sense". I do understand that given a colored noise sequence input (with a large but finite BW) stretching from 0 to +INF in the time domain, downsampling of these infinite number of samples by a factor of k would again produce a "white - completely uncorrelated" noise sequence which stretches from 0 to +INF/k=+INF, which should have a flat spectrum from -INF to +INF.....(Here, as in the last post, the width of the correlation is 2k+1) Where am I messing up, since obv, the last statement is sheer nonsense!!!> > OK, here is a rephrase: > > > > We now know for sure that "pure white noise" is a hypothetical entity, > > used for mathematical simplicity rather than any practical use. By > > definition, such a source should have an infinite PSD and unit dirac > > autocorr (rxx). > > > > Since the PSD (W/Hz) is infinitely flat, the total power consumed by > > the noise generator is infinite, which is impossible. So let us assume > > a "colored" noise generator with an extremely large, but finite, > > bandwidth, much larger than the most prolific measuring instrument > > available to us. Since the bandwidth of the measured noise is finitely > > large, the autocorr, rxx of the noise process is finitely small, > > though not delta. > > > > Let us assume for simplicity sake that rxx in our case is (2k+1) > > samples wide - k samples to either side of the current sample r(n). > > Hence the correlation occurs from r(n-k) to r(n+k). > > > > In simpler terms, noise sample r(n) for any n >= 0 is correlated to > > different extents with all samples in the range r(n-k) and r(n+k), but > > is not correlated with samples r(n-k-1) and r(n+k+1) and so on. > > > > So assuming an infinite "colored" noise sequence, the (equally > > infinite) subset of every kth sample are all mutually uncorrelated!!! > > > > So let us down sample the noise sequence to obtain this so called > > "mutually uncorrelated noise"? What my confusion is that the down > > sampling process LPFs the spectrum of the output noise sequence. So > > would that affect the premise that the down sampling should have > > created a sequence of uncorrelated noise samples in the first place? > > Is there some gap in the assumptions or thought process, or am I > > missing something here. > > > > Ofcourse, all this is based on the assumption that the auto corr is a > > finite sequence for "colored" noise instead of being infinitely > > (exponentially) decaying. Is this assumption wrong? Something is amiss > > here!!! > > > > Any clarification would be welcome > > > > Partho > > > Downsampling doesn't low-pass filter anything. It _does_ mix frequency > information through aliasing, so if you downsample sufficiently the > result will be white, but only in the sampled-time sense. > > If you take your white sampled noise and reconstruct it to a > continuous-time signal then your resulting signal will be random, but > not stationary. >
Reply by ●June 29, 20062006-06-29
Ben wrote: (top posting fixed)> >> > OK, here is a rephrase: >> > >> > We now know for sure that "pure white noise" is a hypothetical entity, >> > used for mathematical simplicity rather than any practical use. By >> > definition, such a source should have an infinite PSD and unit dirac >> > autocorr (rxx). >> > >> > Since the PSD (W/Hz) is infinitely flat, the total power consumed by >> > the noise generator is infinite, which is impossible. So let us assume >> > a "colored" noise generator with an extremely large, but finite, >> > bandwidth, much larger than the most prolific measuring instrument >> > available to us. Since the bandwidth of the measured noise is finitely >> > large, the autocorr, rxx of the noise process is finitely small, >> > though not delta. >> > >> > Let us assume for simplicity sake that rxx in our case is (2k+1) >> > samples wide - k samples to either side of the current sample r(n). >> > Hence the correlation occurs from r(n-k) to r(n+k). >> > >> > In simpler terms, noise sample r(n) for any n >= 0 is correlated to >> > different extents with all samples in the range r(n-k) and r(n+k), but >> > is not correlated with samples r(n-k-1) and r(n+k+1) and so on. >> > >> > So assuming an infinite "colored" noise sequence, the (equally >> > infinite) subset of every kth sample are all mutually uncorrelated!!! >> > >> > So let us down sample the noise sequence to obtain this so called >> > "mutually uncorrelated noise"? What my confusion is that the down >> > sampling process LPFs the spectrum of the output noise sequence. So >> > would that affect the premise that the down sampling should have >> > created a sequence of uncorrelated noise samples in the first place? >> > Is there some gap in the assumptions or thought process, or am I >> > missing something here. >> > >> > Ofcourse, all this is based on the assumption that the auto corr is a >> > finite sequence for "colored" noise instead of being infinitely >> > (exponentially) decaying. Is this assumption wrong? Something is amiss >> > here!!! >> > >> > Any clarification would be welcome >> > >> > Partho >> > >>Downsampling doesn't low-pass filter anything. It _does_ mix frequency >>information through aliasing, so if you downsample sufficiently the >>result will be white, but only in the sampled-time sense. >> >>If you take your white sampled noise and reconstruct it to a >>continuous-time signal then your resulting signal will be random, but >>not stationary. >> > >> Oops sorry....muh mistake!!!! > > Downsampling will only create periodic bandpass spectrum which may > alias if the sampling rate is too low......but then, the lower the > sampling rate, the more likely the correlation of the individual > samples is zero. > > What I dont understand is your statement that if the downsampling rate > is low enough (which ensures that no "correlated" samples were sampled > in the first place), the resultant waveform would be white "in the > time series sense". Each sample is independent of all other samples. "White noise" is a concept that makes sense for stationary signals, which a sampled-time system isn't. > > I do understand that given a colored noise sequence input (with a > large but finite BW) stretching from 0 to +INF in the time domain, > downsampling of these infinite number of samples by a factor of k > would again produce a "white - completely uncorrelated" noise sequence > which stretches from 0 to +INF/k=+INF, which should have a flat > spectrum from -INF to +INF.....(Here, as in the last post, the width > of the correlation is 2k+1) > > Where am I messing up, since obv, the last statement is sheer > nonsense!!! Yes and no. If you model your sampling process the way it usually is done, i.e. as a multiplication by a train of impulses, then yes, you have a signal with a spectrum that goes from -INF to +INF, which requires infinite power and is therefore senseless in the real world. Of course, if you model your sampling process that way then just a DC signal has a spectrum with impulses going from -INF to +INF, and it requires infinite power and is therefore senseless in the real world. This is a case of a handy mathematical abstraction (the multiply-by-impulses model) makes the arithmetic easier and ties the math together nicely, but causes all sorts of paradoxes when you try to apply it in the real world. In the real world your sampled signal _doesn't_ exist as a train of impulses. Usually it exists as a set of digital numbers to which the concept of power _really_ doesn't apply. So the most sensible thing you can do is look at what happens after it is run out a DAC and (possibly) through a reconstruction filter. Then you will find that the spectrum goes to zero as the frequency goes up, and the power is finite. -- Tim Wescott Wescott Design Services http://www.wescottdesign.com Posting from Google? See http://cfaj.freeshell.org/google/ "Applied Control Theory for Embedded Systems" came out in April. See details at http://www.wescottdesign.com/actfes/actfes.html
