The Designer's Guide Community Forum https://designers-guide.org/forum/YaBB.pl Design >> Analog Design >> Help! A puzzlement about noise sampling & reconstruction. https://designers-guide.org/forum/YaBB.pl?num=1223798268 Message started by greatqs on Oct 12th, 2008, 12:57am

Title: Help! A puzzlement about noise sampling & reconstruction. Post by greatqs on Oct 12th, 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!!

Title: Re: Help! A puzzlement about noise sampling & reconstruction. Post by thechopper on Oct 12th, 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

Title: Re: Help! A puzzlement about noise sampling & reconstruction. Post by greatqs on Oct 12th, 2008, 5:12pm HdrChopper wrote on Oct 12th, 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.

Title: Re: Help! A puzzlement about noise sampling & reconstruction. Post by thechopper on Oct 14th, 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 Sxsxs (f) = 1/Ts^2× ΣSxx (f-kxfs) where Sxsxs 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 Rxsxs (τ) = 1/Ts× ΣRxx (kTs) Hpe this helps Tosei Sxsxs = 1/Ts^2× ΣSxx However

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