# Help! A puzzlement about noise sampling & reconstruction.

Started by October 12, 2008
```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
```
```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
```
```On Oct 13, 12:12&#2013266080;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.
>
> Rune

Hi 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
```
```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.
>
> Rune

Jerry
--
Engineering is the art of making what you want from things you can get.
&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;
** Posted from http://www.teranews.com **
```
```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
```