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greatqs

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Help! A puzzlement about noise sampling & reconstruction.
Oct 12^th, 2008, 12:57am

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!!

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HdrChopper

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Re: Help! A puzzlement about noise sampling & reconstruction.
Reply #1 - Oct 12^th, 2008, 9:07am

Hi,

The PSD of x[n] must be scaled by 1/Ts^2, not 1/Ts. The reason being that you have to consider the modulus of the square of the "transfer function" of your system. Ideal sampling can be modeled by a train of impulses equally spaced in time.
Squaring the Fourier transform of the impulses (the same way you square the reconstruction filter transfer function) gives you the sampled PSD scaled by 1/Ts^2.
Therefore the scale factor cancells out after reconstruction

Regards
Tosei

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greatqs

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Re: Help! A puzzlement about noise sampling & reconstruction.
Reply #2 - Oct 12^th, 2008, 5:12pm

HdrChopper wrote on Oct 12^th, 2008, 9:07am:

Hi,

The PSD of x[n] must be scaled by 1/Ts^2, not 1/Ts. The reason being that you have to consider the modulus of the square of the "transfer function" of your system. �Ideal sampling can be modeled by a train of impulses equally spaced in time.
Squaring the Fourier transform of the impulses (the same way you square the reconstruction filter transfer function) gives you the sampled PSD scaled by 1/Ts^2.
Therefore the scale factor cancells out after reconstruction

Regards
Tosei

A lot of thanks to your input! Tosei.
For the scaling in sampling the continuous-time noise into discrete-time noise sequence using ideal impulse sampling, you can look at �eq. 10.50, pp731, A.V Oppenheim's "Discrete-time signal processing" 2nd edition.
There is only 1/Ts scaling between the two domain's PSD not 1/Ts^2.
The reason is that discrete-time PSD can be regard as DTFT of the sampling of the autocorrelation function of continuous-time noise.

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HdrChopper

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Re: Help! A puzzlement about noise sampling & reconstruction.
Reply #3 - Oct 14^th, 2008, 6:50pm

Hi,

I'm sorry but I do not have that book at hand.

I have another reference ("Random signals" Shanmugan-Breipohl, p. 195 ) where it is clearly stated that
Sx_sx_s (f) = 1/Ts^2� ΣSxx (f-kxfs)

where Sx_sx_s is the sampled PSD, fs the sampling frequency and the summatory goes from -∞ to ∞

I agree the discrete time PSD can be seen as the DTFT of the sampling of the autocorrelation function of the continuous-time noise, but the later scaled by 1/Ts

In other words

Rx_sx_s (τ) = 1/Ts� ΣRxx (kTs)

Hpe this helps
Tosei
Sx_sx_s = 1/Ts^2� ΣSxx

However

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