Welcome to EDAboard.com

Welcome to our site! EDAboard.com is an international Electronic Discussion Forum focused on EDA software, circuits, schematics, books, theory, papers, asic, pld, 8051, DSP, Network, RF, Analog Design, PCB, Service Manuals... and a whole lot more! To participate you need to register. Registration is free. Click here to register now.

Register Log in

[SOLVED] Variance of WSS white noise process

Status
Not open for further replies.

hunter555persia

Member level 2
Joined
Jul 24, 2010
Messages
44
Helped
5
Reputation
10
Reaction score
5
Trophy points
1,288
Location
Iran
Activity points
1,540
Hi, from the theory of stochastic processes we know that for WSS zero mean processes, variance would be the autocorrelation function at zero i.e. Rx(0).
But consider the white noise process: Rx(τ) = δ(τ)No/2 and Rx(0)=∞. But we always consider No/2 as noise variance. Why is that ??

Thanks for any help.
 

zorro

Advanced Member level 4
Joined
Sep 6, 2001
Messages
1,131
Helped
356
Reputation
710
Reaction score
298
Trophy points
1,363
Location
Argentina
Activity points
8,902
Hi Hunter

No/2 is not the noise variance (whose unities would be [Watts]) but the noise spectral density (whose unities would be [Watts/Hertz]).
Regards

Z
 

hunter555persia

Member level 2
Joined
Jul 24, 2010
Messages
44
Helped
5
Reputation
10
Reaction score
5
Trophy points
1,288
Location
Iran
Activity points
1,540
What is the variance of this white process then? Is Rx(0)=∞ the variance ?? ∫S(w) also yields ∞.
 

zorro

Advanced Member level 4
Joined
Sep 6, 2001
Messages
1,131
Helped
356
Reputation
710
Reaction score
298
Trophy points
1,363
Location
Argentina
Activity points
8,902
Right.
A "real life" process can not be "really" white.
"White" means "that has uniform power spectral density over all de bandwidth of interest".
Regards
Z
 

hunter555persia

Member level 2
Joined
Jul 24, 2010
Messages
44
Helped
5
Reputation
10
Reaction score
5
Trophy points
1,288
Location
Iran
Activity points
1,540
In performance analysis we are interested in a plot of BER Vs Eb/No. Eb/No is somehow a measure of SNR which means that No/2 was considered as noise power. And noise power equals its variance. Actual noise variance would be ∫S(w)dw over the receiver bandwidth. But No/2 is always considered as noise variance without considering the receiver bandwidth in the texts I've read e.g. Proakis.
 

zorro

Advanced Member level 4
Joined
Sep 6, 2001
Messages
1,131
Helped
356
Reputation
710
Reaction score
298
Trophy points
1,363
Location
Argentina
Activity points
8,902
Eb is energy. Units: [Joule].
No is power spectral density. Units: [Watt/Hz]=[Watt*second]=[Joule]
So, Eb/No is adimensional, as it must be.

No is not noise variance or power. It is power spectral density.
Sometimes it is said that "numerically, No equals the power of the noise in the bandwidth of 1 Hz". But this is rather confusing.
 

hunter555persia

Member level 2
Joined
Jul 24, 2010
Messages
44
Helped
5
Reputation
10
Reaction score
5
Trophy points
1,288
Location
Iran
Activity points
1,540
So why do we produce the noise with variance of No/2 in simulations ?
 

zorro

Advanced Member level 4
Joined
Sep 6, 2001
Messages
1,131
Helped
356
Reputation
710
Reaction score
298
Trophy points
1,363
Location
Argentina
Activity points
8,902
Is simulations you are in discrete time. Frequency does not cover -Inf to +Inf, but -pi to +pi [radians/sample], or -fs/2 to +fs/2 [Hz] (fs is sampling frequency).
 

hunter555persia

Member level 2
Joined
Jul 24, 2010
Messages
44
Helped
5
Reputation
10
Reaction score
5
Trophy points
1,288
Location
Iran
Activity points
1,540
The simulations are discrete time. But sampling frequency is not considered. The data and noise vector are sumed and then decoded. Noise is always produced with variance No/2.
 

zorro

Advanced Member level 4
Joined
Sep 6, 2001
Messages
1,131
Helped
356
Reputation
710
Reaction score
298
Trophy points
1,363
Location
Argentina
Activity points
8,902
You are using discrete time for simulate a continuous-time system, rigt?
I don't know what are you simulating and how, but since you mention that ∫S(w) yields ∞, the simulated system is continuous-time.
So, there is a sampling frequency.
 

hunter555persia

Member level 2
Joined
Jul 24, 2010
Messages
44
Helped
5
Reputation
10
Reaction score
5
Trophy points
1,288
Location
Iran
Activity points
1,540
I mean ∫S(w)dw=∞ theoretically. In simulation S(w) is not used. Not in my simulation, but in any BER simulation I have seen before, there is no concern about the sampling frequency. I should think about the sampling frequency though. It may be considered unity I guess. Thanks for pointing out this.

But the main question is the variance of a white process. It is infinity then, right ?
 

zorro

Advanced Member level 4
Joined
Sep 6, 2001
Messages
1,131
Helped
356
Reputation
710
Reaction score
298
Trophy points
1,363
Location
Argentina
Activity points
8,902
Right.
Take into account what I said in post #4.
Regards

Z
 

David83

Advanced Member level 1
Joined
Jan 21, 2011
Messages
410
Helped
45
Reputation
92
Reaction score
45
Trophy points
1,308
Activity points
3,639
The white noise correlation is:

\[R(\tau)=\frac{N_0}{2}\delta(\tau)\]

So the power spectral density is:

\[\mathcal{S}_{nn}(f)=\int_{-\infty}^{\infty}R(\tau)e^{-j2\pi f\tau}\,d\tau=\frac{N_0}{2}\int_{-\infty}^{\infty}\delta(\tau)e^{-j2\pi f\tau}\,d\tau=\frac{N_0}{2}\]

and the autocorrelation is then:

\[R(\tau)=\int_{-\infty}^{\infty}\mathcal{S}_{nn}(f)e^{j2\pi f\tau}\,df\]

The average power is then:

\[R(0)=\sigma_n^2=\int_{-\infty}^{\infty}\mathcal{S}_{nn}(f)\,df=\int_{-\infty}^{\infty}\frac{N_0}{2}\,df\]

Now as zorro pointed out, the average power is taken over the bandwidth of interest. So:

\[\sigma_n^2=\int_{-W}^{W}\frac{N_0}{2}\,df=N_0W\]

Then the SNR will be:

\[\text{SNR}=\frac{P}{\sigma_n^2}=\frac{P}{N_0W}=\frac{PT}{N_0}=\frac{E}{N_0}\]

Hope this will help.

Regards
 

hunter555persia

Member level 2
Joined
Jul 24, 2010
Messages
44
Helped
5
Reputation
10
Reaction score
5
Trophy points
1,288
Location
Iran
Activity points
1,540
I guess this is what I was looking for: \[\text{SNR}=\frac{P}{\sigma_n^2}=\frac{P}{N_0W}=\frac{PT}{N_0}=\frac{E}{N_0}\]

Thanks guys for the help.
 

Status
Not open for further replies.
Toggle Sidebar

Part and Inventory Search

Welcome to EDABoard.com

Sponsor

Top