a bandlimit white noise x(t) with PSD of S0 is sampled (no aliasing) to produce x[n]. The PSD of x[n] is calculated to be S0/Ts (Ts is the sample period). Now I just reconstruct the continuous noise xr(t) by passing x[n] impulses to the ideal reconstruction filter (gain=Ts, -fs<f<fs). The output PSD is calculated to be S0/Ts*Ts^2=S0*Ts. There is an offset from the input noise PSD by a ratio of Ts! There must be some scaling error in above statement because ideal sampling and reconstructing a bandlimit white noise should produce itself. Please correct me!! Thanks!!

# Help! A puzzlement about noise sampling & reconstruction.

Started by ●October 12, 2008

Reply by ●October 12, 20082008-10-12

On 12 Okt, 12:04, "Qian.S...@gmail.com" <Qian.S...@gmail.com> wrote:> a bandlimit white noise x(t) with PSD of S0 is sampled (no aliasing) > to produce x[n]. The PSD of x[n] is calculated to be S0/Ts (Ts is the > sample period). > Now I just reconstruct the continuous noise xr(t) by passing x[n] > impulses to the ideal reconstruction filter (gain=Ts, -fs<f<fs). The > output PSD is calculated to be S0/Ts*Ts^2=S0*Ts. There is an offset > from the input noise PSD by a ratio of Ts! > There must be some scaling error in above statement because ideal > sampling and reconstructing a bandlimit white noise should produce > itself. Please correct me!! Thanks!!First, if the missing scaling factor is 1/Ts I would check out the definitions of the Fourier transforms. With the discrete- domain DFT there is a 'skewness' between the forward and inverse transforms. The scale factor is 1 in the forward transform and 1/N in the inverse tranform. So check the definitions of the FTs to see if there are more issues like that around, and what effects they might have. That said, scaling factors are usually ignored unless there are very good reasons for keeping them. Which is done only in calibrated systems. So if the missing 1/Ts factor is associated with the ADC or DAC, chances are that it is just dropped. Rune

Reply by ●October 12, 20082008-10-12

On Oct 13, 12:12�am, Rune Allnor <all...@tele.ntnu.no> wrote:> On 12 Okt, 12:04, "Qian.S...@gmail.com" <Qian.S...@gmail.com> wrote: > > > a bandlimit white noise x(t) with PSD of S0 is sampled (no aliasing) > > to produce x[n]. The PSD of x[n] is calculated to be S0/Ts (Ts is the > > sample period). > > Now I just reconstruct the continuous noise xr(t) by passing x[n] > > impulses to the ideal reconstruction filter (gain=Ts, -fs<f<fs). The > > output PSD is calculated to be S0/Ts*Ts^2=S0*Ts. There is an offset > > from the input noise PSD by a ratio of Ts! > > There must be some scaling error in above statement because ideal > > sampling and reconstructing a bandlimit white noise should produce > > itself. Please correct me!! Thanks!! > > First, if the missing scaling factor is 1/Ts I would check > out the definitions of the Fourier transforms. With the discrete- > domain DFT there is a 'skewness' between the forward and inverse > transforms. The scale factor is 1 in the forward transform and > 1/N in the inverse tranform. > > So check the definitions of the FTs to see if there are more > issues like that around, and what effects they might have. > > That said, scaling factors are usually ignored unless there > are very good reasons for keeping them. Which is done only in > calibrated systems. So if the missing 1/Ts factor is associated > with the ADC or DAC, chances are that it is just dropped. > > RuneHi Rune, only with the current literature. If you go back to the 60s then you see people have more common sense and use 1/N for the direct FFT. This makes more sense since for dc we need the average and this means dividing by N. Hardy

Reply by ●October 12, 20082008-10-12

Rune Allnor wrote:> On 12 Okt, 12:04, "Qian.S...@gmail.com" <Qian.S...@gmail.com> wrote: >> a bandlimit white noise x(t) with PSD of S0 is sampled (no aliasing) >> to produce x[n]. The PSD of x[n] is calculated to be S0/Ts (Ts is the >> sample period). >> Now I just reconstruct the continuous noise xr(t) by passing x[n] >> impulses to the ideal reconstruction filter (gain=Ts, -fs<f<fs). The >> output PSD is calculated to be S0/Ts*Ts^2=S0*Ts. There is an offset >> from the input noise PSD by a ratio of Ts! >> There must be some scaling error in above statement because ideal >> sampling and reconstructing a bandlimit white noise should produce >> itself. Please correct me!! Thanks!! > > First, if the missing scaling factor is 1/Ts I would check > out the definitions of the Fourier transforms. With the discrete- > domain DFT there is a 'skewness' between the forward and inverse > transforms. The scale factor is 1 in the forward transform and > 1/N in the inverse tranform.Or 1 in the inverse direction and 1/n forward. Or 1/sqrt(N) in each direction. Think of it as different dialects.> > So check the definitions of the FTs to see if there are more > issues like that around, and what effects they might have. > > That said, scaling factors are usually ignored unless there > are very good reasons for keeping them. Which is done only in > calibrated systems. So if the missing 1/Ts factor is associated > with the ADC or DAC, chances are that it is just dropped. > > RuneJerry -- Engineering is the art of making what you want from things you can get. ����������������������������������������������������������������������� ** Posted from http://www.teranews.com **

Reply by ●October 13, 20082008-10-13

On 12 Okt, 13:12, Rune Allnor <all...@tele.ntnu.no> wrote:> On 12 Okt, 12:04, "Qian.S...@gmail.com" <Qian.S...@gmail.com> wrote: > > > a bandlimit white noise x(t) with PSD of S0 is sampled (no aliasing) > > to produce x[n]. The PSD of x[n] is calculated to be S0/Ts (Ts is the > > sample period). > > Now I just reconstruct the continuous noise xr(t) by passing x[n] > > impulses to the ideal reconstruction filter (gain=Ts, -fs<f<fs). The > > output PSD is calculated to be S0/Ts*Ts^2=S0*Ts. There is an offset > > from the input noise PSD by a ratio of Ts! > > There must be some scaling error in above statement because ideal > > sampling and reconstructing a bandlimit white noise should produce > > itself. Please correct me!! Thanks!! > > First, if the missing scaling factor is 1/Ts I would check > out the definitions of the Fourier transforms. With the discrete- > domain DFT there is a 'skewness' between the forward and inverse > transforms. The scale factor is 1 in the forward transform and > 1/N in the inverse tranform.Actually, I think the problem is to preserve the physical dimensions through the sampling. DSP algorithms work on dimensionless data while you seem to work with physical data in the sense that dimensions and scales are preseved. The missing 1/Ts [s] factor is consistent with that you seem to expect an answer in dimension [Hz], which you don't get. So the first place to look for errors is to work through the ADC model in painstaking detail and make sure all the scaling factors etc are preserved. The result of this exercise would be a constant scaling factor, so it would have little effect on the overall algorithm. And don't be surprised if the factor turns out to be 1/Ts... This is one of thise things people tend to skip unless one works with physics simulators or DSP in calibrated systems. I have done neither, so I can't help out with the details. Rune