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Basic Question about Additive White Gaussian Noise

Started by Your Name Heres March 28, 2005
When we talk about an AWGN signal, n(t), which parameter(s) has the Gaussian 
distribution? Is it the amplitude, power, or phase of the signal?? Also is the 
parameter linear or logarithmic under the Gaussian distribution? For 
example, are we to assume that AWGN describes a power signal with a 
mean and variance on a logarithmic scale versus time? Any help 
understanding this will be appreciated.

Thanks.

-Please respond to this forum.

pleaserespondhere@ok.com (Your Name Heres) writes:

> When we talk about an AWGN signal, n(t), which parameter(s) has the Gaussian > distribution? Is it the amplitude, power, or phase of the signal??
Amplitude.
> Also is the parameter linear or logarithmic under the Gaussian > distribution? For example, are we to assume that AWGN describes a > power signal with a mean and variance on a logarithmic scale versus > time? Any help understanding this will be appreciated.
An AWGN signal is a random process, and all random processes are power signals (i.e., infinite-energy signals and finite-power signals). What makes such a signal Gaussian is the fact that the distribution of the amplitudes of the random variable n(t0) is Gaussian. Remember that a random process is a collection of ensembles of time-domain signals, so that at any one time t0 we can talk of the "ensemble" statistics. -- Randy Yates Sony Ericsson Mobile Communications Research Triangle Park, NC, USA randy.yates@sonyericsson.com, 919-472-1124
Your Name Heres wrote:
> When we talk about an AWGN signal, n(t), which parameter(s) has the Gaussian > distribution? ...
Amplitude distribution is Gaussian. Phase distribution is uniform. Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
"Jerry Avins" <jya@ieee.org> wrote in message 
news:08ydnQbAp9mq9dXfRVn-2Q@rcn.net...
> Your Name Heres wrote: >> When we talk about an AWGN signal, n(t), which parameter(s) has the >> Gaussian distribution? ... > > Amplitude distribution is Gaussian. Phase distribution is uniform. > > 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;
If you are talking complex representation then the amplitude of the noise in In-phase is gaussian distributed and so is the amplitude of the noise in the quadrature direction , the two are uncorrelated (independent) so the phase of the noise is uniformly sistributed. In this representation the magnitude of the noise is rayleigh distributed.. Best of Luck - Mike
in article xxpy8c7zfme.fsf@usrts005.corpusers.net, Randy Yates at
randy.yates@sonyericsson.com wrote on 03/28/2005 15:20:

> pleaserespondhere@ok.com (Your Name Heres) writes: > >> When we talk about an AWGN signal, n(t), which parameter(s) has the Gaussian >> distribution? Is it the amplitude, power, or phase of the signal?? > > Amplitude. > >> Also is the parameter linear or logarithmic under the Gaussian >> distribution? For example, are we to assume that AWGN describes a >> power signal with a mean and variance on a logarithmic scale versus >> time? Any help understanding this will be appreciated. > > An AWGN signal is a random process, and all random processes are > power signals (i.e., infinite-energy signals and finite-power > signals).
one of the nasty things about white noise is that it has INfinite power (the area under the power spectrum curve is infinite). the concept exists only for this white noise to be processed with a system of finite bandwidth. then the noise in the pass band is finite power but isn't really white. it might be viewed as white within the context of that pass band.
> What makes such a signal Gaussian is the fact that the distribution > of the amplitudes of the random variable n(t0) is Gaussian. Remember > that a random process is a collection of ensembles of time-domain > signals, so that at any one time t0 we can talk of the "ensemble" > statistics.
and then there is the "ergodic" property that is sometimes appealed to that says that time averages is equal to the corresponding ensemble average. that is what allows us to predict the behavior of linear systems with random processes as input. -- r b-j rbj@audioimagination.com "Imagination is more important than knowledge."
Mike Yarwood wrote:
> "Jerry Avins" <jya@ieee.org> wrote in message > news:08ydnQbAp9mq9dXfRVn-2Q@rcn.net... > >>Your Name Heres wrote: >> >>>When we talk about an AWGN signal, n(t), which parameter(s) has the >>>Gaussian distribution? ... >> >>Amplitude distribution is Gaussian. Phase distribution is uniform. >> >>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; > > > If you are talking complex representation then the amplitude of the noise in > In-phase is gaussian distributed and so is the amplitude of the noise in the > quadrature direction , the two are uncorrelated (independent) so the phase > of the noise is uniformly sistributed. In this representation the magnitude > of the noise is rayleigh distributed.. > Best of Luck - Mike
It seems to me that you've confused two ideas there. It is indeed true that orthogonal Gaussian distributions -- examples: I and Q, horizontal and vertical position errors -- produce a Rayleigh distribution in magnitude. It is not true that a Gaussian distribution can be a Rayleigh distribution if it is squinted at just so .... 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;
Hi there,

If you consider a transmitted signal s(t), the effect of additive noise 
on the signal is represented as

r(t) = s(t) + n(t)

When you say that n(t) is AWGN, it means that the amplitude of n(t) is 
Gaussian distributed, and the frequency spread N(f) is from -inf to 
+inf, that is, it's "white".

The reason why the noise is modeled as "Gaussian" or "Normal" is that if 
you model noise from multiple sources as INDEPENDENT random variables of 
ANY distribution (not necessarily Gaussian), then the pdf of the 
resulting random variable tends to have a Gaussian distribution when the 
number of independent sources is large (Central Limit Theorem).

Now if s(t) were to be considered a complex signal made up of in-phase 
and quadrature components, then, using S(t) and N(t) to represent the 
complex baseband versions of s(t) and n(t)

S(t) = s_i(t) + j s_q(t)
N(t) = n_i(t) + j n_q(t)
R(t) = S(t) + N(t) = [s_i(t) + n_i(t)] + j [s_q(t) + n_q(t)]

there is an additive gaussian noise component on each of the in-phase 
and quadrature components separately

The received signal can be written in the amplitude-phase notation as
R(t) = A(t) exp (j phi(t)) where
A(t) = sqrt([s_i(t)+n_i(t)]^2 + [s_q(t)+n_q(t)]^2])
phi(t) = arctan ([s_q(t)+n_q(t)]/[s_i(t)+n_i(t)])

Now it can be shown that A(t) is Rayleigh distributed and phi(t) is 
uniformly distributed.

Regards,
- Ravi

Your Name Heres wrote:
> When we talk about an AWGN signal, n(t), which parameter(s) has the Gaussian > distribution? Is it the amplitude, power, or phase of the signal?? Also is the > parameter linear or logarithmic under the Gaussian distribution? For > example, are we to assume that AWGN describes a power signal with a > mean and variance on a logarithmic scale versus time? Any help > understanding this will be appreciated. > > Thanks. > > -Please respond to this forum. >
"Jerry Avins" <jya@ieee.org> wrote in message 
news:fvudnfUswdgpd9XfRVn-3g@rcn.net...
> Mike Yarwood wrote: >> "Jerry Avins" <jya@ieee.org> wrote in message >> news:08ydnQbAp9mq9dXfRVn-2Q@rcn.net... >> >>>Your Name Heres wrote: >>> >>>>When we talk about an AWGN signal, n(t), which parameter(s) has the >>>>Gaussian distribution? ... >>> >>>Amplitude distribution is Gaussian. Phase distribution is uniform. >>> >>>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; >> >> >> If you are talking complex representation then the amplitude of the noise >> in In-phase is gaussian distributed and so is the amplitude of the noise >> in the quadrature direction , the two are uncorrelated (independent) so >> the phase of the noise is uniformly sistributed. In this representation >> the magnitude of the noise is rayleigh distributed.. >> Best of Luck - Mike > > It seems to me that you've confused two ideas there. It is indeed true > that orthogonal Gaussian distributions -- examples: I and Q, horizontal > and vertical position errors -- produce a Rayleigh distribution in > magnitude. It is not true that a Gaussian distribution can be a Rayleigh > distribution if it is squinted at just so .... > > 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;
Hi Jerry! I intended to say uniform phase distribution and rayleigh magnitude is the polar representation alternative to the two orthogonal gaussian distributions (left out all the identical variance and zero mean stuff that you need to be sure you get the uniform phase distribution). I assume you were talking about a zero-mean , 2_Dimensional gaussian distribution in your earlier post so the uniform phase distribution made sense but I'm used to generating I and Q components so this representation is less confusing for me at any rate. Best of Luck - Mike
Jerry Avins <jya@ieee.org> writes:

> Your Name Heres wrote: > > When we talk about an AWGN signal, n(t), which parameter(s) has the > > Gaussian distribution? ... > > > Amplitude distribution is Gaussian. Phase distribution is uniform.
Jerry, I'll jump in as at least one other has and ask about this phase thing. If we're talking about a real signal, then what do you mean by "phase"? -- Randy Yates Sony Ericsson Mobile Communications Research Triangle Park, NC, USA randy.yates@sonyericsson.com, 919-472-1124
A few questions..

robert bristow-johnson wrote:
> one of the nasty things about white noise is that it has INfinite
power (the
> area under the power spectrum curve is infinite). the concept exists
only
> for this white noise to be processed with a system of finite
bandwidth.
> then the noise in the pass band is finite power but isn't really
white. it
> might be viewed as white within the context of that pass band. >
Is the amplitude distribution of AWGN still Gaussian after it has been band limited? Is clipped AWGN still white? Is a random signal with a uniform amplitude distribution white? Are the amplitude distribution and frequency shape of a random signal independent? Can I create a random signal of any amplitude distribution and any frequency distribution independently or are there some interdependencies? thanks Mark