Correlated noise samples

I always had this question in my mind: The autocorrelation function of the
filtered white Gaussian noise is simply a sinc function (inverse transform
of the output of a brickwall filter with bandwidth 1/2T). Now if we sample
the autocorrelation function with rate 1/T, the discrete-time noise samples
are uncorrelated due to the sinc function passing through 0s at the
T-spaced sampling instants. However, T/2 spaced samples are correlated with
each other and the discrete-time noise is not white anymore. What are the
implications of this correlation in the subsequent receiver
stages/algorithms? 
Thanks.	 

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On 8/22/13 6:01 PM, commsignal wrote:
> I always had this question in my mind: The autocorrelation function of the
> filtered white Gaussian noise is simply a sinc function (inverse transform
> of the output of a brickwall filter with bandwidth 1/2T).

that's a premise that might be a little dubious.

-- 

r b-j                  rbj@audioimagination.com

"Imagination is more important than knowledge."

okay, i take back what i said previously.

On 8/22/13 6:01 PM, commsignal wrote:
> I always had this question in my mind: The autocorrelation function of the
> filtered white Gaussian noise is simply a sinc function (inverse transform
> of the output of a brickwall filter with bandwidth 1/2T).

a very skinny sinc() function.  this is not the discrete-time 
autocorrelation, but the continuous-time autocorrelation of what would 
come out of an ideal reconstruction of an gaussian p.d.f. random number 
generator.

> Now if we sample
> the autocorrelation function with rate 1/T, the discrete-time noise samples
> are uncorrelated due to the sinc function passing through 0s at the
> T-spaced sampling instants.

yes.  like a discrete-time impulse function (which is more like the 
discrete-time autocorrelation).

> However, T/2 spaced samples are correlated with
> each other and the discrete-time noise is not white anymore.

well, what *is* the DTFT of

    x[n]  =  sinc(n + 1/2)

?

it's gonna look *sorta* white.

> What are the
> implications of this correlation in the subsequent receiver
> stages/algorithms?

dunno.

-- 

r b-j                  rbj@audioimagination.com

"Imagination is more important than knowledge."

Hi,

one very hands-on implication is that you can suppress white (wideband)
noise by averaging nearby samples, but not if the noise samples are
correlated. 

Replace the word "averaging" with "filtering" and we've got the profound
insight that you can remove wideband noise with a filter, but not if it's
inside your signal bandwidth. 	 

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Umm, a slightly less profound interpretation is that if your noise is already lowpass filtered, adding another lowpass at or above the cutoff of the original might not do much good. This does not quite make into the realm of profundity, IMHO. 

Bob

>> This does not quite make into the realm of profundity, IMHO.

well, that was my point, more or less.

Another way to look at it: filtering a narrow-band signal with a high-rate
FIR filter is inefficient, as the taps become increasingly dependent on
each other.
	 

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