Suppose we get a sequence by sampling a bandlimited random signal corrupted by additive White noise with PSD N_0. Noise and signal are independent. We upsample the sequence by factor K by inserting (K-1) zeros between two consecutive samples. The upsampling will cause the spectrum of the bandlimited signal to be amplitude scaled by factor K and frequency scaled by factor (1/K). The question is what will be the impact on white noise PSD which is flat over -pi to pi? Will there be any sort of magnitude scaling for noise? Since noise PSD is falt over -pi to pi, there must not be any sort of frequency scaling on noise PSD? Can somebody make me clear about this please?
Effect of Upsampling on White Noise
Started by ●November 11, 2008
Reply by ●November 11, 20082008-11-11
RIMalhi wrote:> Suppose we get a sequence by sampling a bandlimited random signal corrupted > by additive White noise with PSD N_0. Noise and signal are independent. We > upsample the sequence by factor K by inserting (K-1) zeros between two > consecutive samples.> The upsampling will cause the spectrum of the > bandlimited signal to be amplitude scaled by factor K and frequency scaled > by factor (1/K).you divide by those factors, right?> The question is what will be the impact on white noise PSD > which is flat over -pi to pi?As far as upsampling is concerned, the noise is part of the signal. Don't make the process needlessly complicated.> Will there be any sort of magnitude scaling for noise? Since noise PSD is > falt over -pi to pi, there must not be any sort of frequency scaling on > noise PSD? > > Can somebody make me clear about this please?The only change after suitable filtering which you didn't mention is the sample rate. If the initial and final bandwidths are the same, the noise is independent of sample rate. Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●November 11, 20082008-11-11
>RIMalhi wrote: >> Suppose we get a sequence by sampling a bandlimited random signalcorrupted>> by additive White noise with PSD N_0. Noise and signal are independent.We>> upsample the sequence by factor K by inserting (K-1) zeros between two >> consecutive samples. > >> The upsampling will cause the spectrum of the >> bandlimited signal to be amplitude scaled by factor K and frequencyscaled>> by factor (1/K). > >you divide by those factors, right? > > >> The question is what will be the impact on white noise PSD >> which is flat over -pi to pi? > >As far as upsampling is concerned, the noise is part of the signal. >Don't make the process needlessly complicated. > >> Will there be any sort of magnitude scaling for noise? Since noise PSDis>> falt over -pi to pi, there must not be any sort of frequency scalingon>> noise PSD? >> >> Can somebody make me clear about this please? > >The only change after suitable filtering which you didn't mention is the>sample rate. If the initial and final bandwidths are the same, the noise>is independent of sample rate. > >Jerry >-- >Engineering is the art of making what you want from things you can get. >����������������������������������������������������������������������� >Hi Jerry, Thanks for your explanation. Still i am not satisfied because i have to use this idea somewhere and i must be clear about it before i can use it.>you divide by those factors, right?No Jerry, we multiply by those factors.The chapter 4 of Discrete-time Signal Processing book by Shafer has some explanation of upsampling and downsampling concepts. But it says nothing about the imapct on White Noise. If the Noise is bandlimited, frequency and amplitude scaling should also apply to it. But in our case, noise is not bandlimited.. RIMalhi
Reply by ●November 11, 20082008-11-11
RIMalhi wrote:>> RIMalhi wrote: >>> Suppose we get a sequence by sampling a bandlimited random signal > corrupted >>> by additive White noise with PSD N_0. Noise and signal are independent. > We >>> upsample the sequence by factor K by inserting (K-1) zeros between two >>> consecutive samples. >>> The upsampling will cause the spectrum of the >>> bandlimited signal to be amplitude scaled by factor K and frequency > scaled >>> by factor (1/K). >> you divide by those factors, right? >> >> >>> The question is what will be the impact on white noise PSD >>> which is flat over -pi to pi? >> As far as upsampling is concerned, the noise is part of the signal. >> Don't make the process needlessly complicated. >> >>> Will there be any sort of magnitude scaling for noise? Since noise PSD > is >>> falt over -pi to pi, there must not be any sort of frequency scaling > on >>> noise PSD? >>> >>> Can somebody make me clear about this please? >> The only change after suitable filtering which you didn't mention is the > >> sample rate. If the initial and final bandwidths are the same, the noise > >> is independent of sample rate. >> >> Jerry >> -- >> Engineering is the art of making what you want from things you can get. >> > > > Hi Jerry, > Thanks for your explanation. Still i am not satisfied because i have to > use this idea somewhere and i must be clear about it before i can use it. > >> you divide by those factors, right? > > No Jerry, we multiply by those factors.The chapter 4 of Discrete-time > Signal Processing book by Shafer has some explanation of upsampling and > downsampling concepts. But it says nothing about the imapct on White Noise.Interpolating zeros (and subsequent)low-pass filtering) reduces the amplitude because the zeros dilute the energy per sample. The sampling frequency is increased, as you say. Check your understanding of the reference.> If the Noise is bandlimited, frequency and amplitude scaling should also > apply to it. But in our case, noise is not bandlimited..The noise is bandlimited by virtue of being sampled. Aside from the inherent nature of a sampled signal, there's an anti-alias filter in front of the sampler, is there not? Jerry -- Engineering is the art of making what you want from things you can get.
Reply by ●November 11, 20082008-11-11
RIMalhi wrote:>> RIMalhi wrote: >>> Suppose we get a sequence by sampling a bandlimited random signal > corrupted >>> by additive White noise with PSD N_0. Noise and signal are independent. > We >>> upsample the sequence by factor K by inserting (K-1) zeros between two >>> consecutive samples. >>> The upsampling will cause the spectrum of the >>> bandlimited signal to be amplitude scaled by factor K and frequency > scaled >>> by factor (1/K). >> you divide by those factors, right? >> >> >>> The question is what will be the impact on white noise PSD >>> which is flat over -pi to pi? >> As far as upsampling is concerned, the noise is part of the signal. >> Don't make the process needlessly complicated. >> >>> Will there be any sort of magnitude scaling for noise? Since noise PSD > is >>> falt over -pi to pi, there must not be any sort of frequency scaling > on >>> noise PSD? >>> >>> Can somebody make me clear about this please? >> The only change after suitable filtering which you didn't mention is the > >> sample rate. If the initial and final bandwidths are the same, the noise > >> is independent of sample rate. >> >> Jerry >> -- >> Engineering is the art of making what you want from things you can get. >> > > > Hi Jerry, > Thanks for your explanation. Still i am not satisfied because i have to > use this idea somewhere and i must be clear about it before i can use it. > >> you divide by those factors, right? > > No Jerry, we multiply by those factors.The chapter 4 of Discrete-time > Signal Processing book by Shafer has some explanation of upsampling and > downsampling concepts. But it says nothing about the imapct on White Noise.Interpolating zeros (and subsequent)low-pass filtering) reduces the amplitude because the zeros dilute the energy per sample. The sampling frequency is increased, as you say. Check your understanding of the reference.> If the Noise is bandlimited, frequency and amplitude scaling should also > apply to it. But in our case, noise is not bandlimited..The noise is bandlimited by virtue of being sampled. Aside from the inherent nature of a sampled signal, there's an anti-alias filter in front of the sampler, is there not? Jerry -- Engineering is the art of making what you want from things you can get.
Reply by ●November 11, 20082008-11-11
RIMalhi wrote: ...> Hi Jerry, > Thanks for your explanation. Still i am not satisfied because i have to > use this idea somewhere and i must be clear about it before i can use it. > >> you divide by those factors, right? > > No Jerry, we multiply by those factors.The chapter 4 of Discrete-time > Signal Processing book by Shafer has some explanation of upsampling and > downsampling concepts. But it says nothing about the imapct on White Noise.Interpolating zeros (and subsequent)low-pass filtering) reduces the amplitude because the zeros dilute the energy per sample. The sampling frequency is increased, as you say. Check your understanding of the reference.> If the Noise is bandlimited, frequency and amplitude scaling should also > apply to it. But in our case, noise is not bandlimited..The noise is bandlimited by virtue of being sampled. Aside from the inherent nature of a sampled signal, there's an anti-alias filter in front of the sampler, is there not? Jerry -- Engineering is the art of making what you want from things you can get.
Reply by ●November 12, 20082008-11-12
On Tue, 11 Nov 2008 14:59:01 -0600, RIMalhi wrote:> Suppose we get a sequence by sampling a bandlimited random signal > corrupted by additive White noise with PSD N_0. Noise and signal are > independent. We upsample the sequence by factor K by inserting (K-1) > zeros between two consecutive samples. The upsampling will cause the > spectrum of the bandlimited signal to be amplitude scaled by factor K > and frequency scaled by factor (1/K). The question is what will be the > impact on white noise PSD which is flat over -pi to pi? > > Will there be any sort of magnitude scaling for noise? Since noise PSD > is falt over -pi to pi, there must not be any sort of frequency scaling > on noise PSD? > > Can somebody make me clear about this please?Your confusion is coming about earlier than you think, and this is what is locking you up. No real noise is truly white; "white noise" is a convenient mathematical construct that only works under certain circumstances*. Sampling white noise that has not been filtered is one of the circumstances where the white noise model leads to absurd results. Consider: if you had real, true white noise its power would be infinite (finite PSD * infinite bandwidth = infinity). Infinite power implies infinite amplitude. Sample that infinite amplitude signal, and all of a sudden you've folded up that infinite spectrum into a finite one via aliasing, your sample magnitudes are infinite, and you have no way of pulling your signal out of the noise. (Note that this notion of white noise having impossibly infinite power is, more or less, what led physicists in the late 19th century to the foundations of quantum mechanics). You need to start with an assumption of band limited noise. It doesn't have to be less than the Nyquist rate, but it does have to be band limited or your problem statement will burst into flame. * Namely, white noise is principally useful as an approximation for "noise with a bandwidth that's way bigger than my system's bandwidth". -- Tim Wescott Wescott Design Services http://www.wescottdesign.com Do you need to implement control loops in software? "Applied Control Theory for Embedded Systems" gives you just what it says. See details at http://www.wescottdesign.com/actfes/actfes.html
Reply by ●November 12, 20082008-11-12
Things are much more clear to me now. There must be some anti-aliasing filter before the sampler. Thanks Jerry and Tim... RIMalhi
Reply by ●November 12, 20082008-11-12
Tim Wescott wrote:> Consider: if you had real, true white noise its power would be infinite > (finite PSD * infinite bandwidth = infinity). �Infinite power implies > infinite amplitude. �Sample that infinite amplitude signal, and all of a > sudden you've folded up that infinite spectrum into a finite one via > aliasing, your sample magnitudes are infinite, and you have no way of > pulling your signal out of the noise.Uh, are you saying that a continuous time white noise process has infinite amplitude because it has infinite power?
Reply by ●November 12, 20082008-11-12
Tim Wescott wrote: ...> * Namely, white noise is principally useful as an approximation for > "noise with a bandwidth that's way bigger than my system's bandwidth".Why "way bigger"? wouldn't "at least as big as" do? Jerry -- Engineering is the art of making what you want from things you can get. ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
