RE: [mathcad] Modelling White Noise

Search email archive for  

RE: [mathcad] Modelling White Noise
Author: Philip Oakley    Posted: Fri, 4 Apr 2008 20:00:12 +0100
Chris,

There are a couple of subtle 'mistakes' in the text.
The central limit theorem has certain assumptions that need to be true for
the result to become Gaussian - as an example, a uniform distribution never
becomes Gaussian just because a large number of samples have been taken. But
in this case we are sampling from an initial Gaussian anyway!

Also you say the variance goes to zero. We need to clarify that it is the
density that goes toward zero.

We also have to be careful about the distinction between regular samples,
random samples and the 'continuous' function values. (it is possible to have
"alias free sampling of random noise"! search for the papers...)

A regular sampling scheme has the spectral distribution Sin(x)/x (with
implied aliasing). Other sampling schemes have other implied distributions
for the natural filtering effect of that sampling.

The finite data set (regular samples) does constrain the discrete Fourier
transform to be a fixed frequencies, so original signals, not at those
frequencies, are 'forced' to be partitioned to those fixed values.

Amplitude and phase are orthogonal and form a dual space with the complex
representation, which is why they (Fourier techniques) are used so much - we
can swap between representations without a care, until someone mixes them up
then asks an awkward question ;-)

In FM we have the problem that we have two frequencies (the carrier and the
modulator) and three phases (two in and one out!) so we have a lot of scope
for confusion.

A worksheet may be called for so we can name and distinguish all these
terms!

[Back in the early '70s in UK there was the 'winter of discontent' when, to
save power, the electricity voltages were low and frequencies slow
(otherwise the transformers saturate). During that period, direct drive
clocks could loose 10 minutes during the day, then overnight they would
speed up the generators to get all the clocks back in phase - a very
practical Phase modulation system!]

Philip

-----Original Message-----
From: Chris Whitford /> Sent: 04 April 2008 10:39 AM
To: /> Subject: Re: [mathcad] Modeling White Noise


This line of argument seems to lead to the wrong conclusions. If one takes
a finite set of data, sampled independently from some distribution (i.e.
white noise), and takes the discrete Fourier transform, the sine and cosine
terms will be independent and from the central limit theorem will have a
Gaussian distribution. Converting from the components to amplitude and
phase, the phase is random with a uniform distribution and the amplitude
has a distribution f(v)dv = exp(-v^2/2*a^2)*v*dv/a^2, where a is the
variance. As more samples are taken, the samples are more closely spaced in
frequency, but have the same distribution. Smoothing the spectrum changes
the distribution and in the limit, as the number of points approaches
infinity, the spectrum becomes flat, with zero variance. Smoothing in the
frequency domain is equivalent to multiplying by a window function in the
time domain. If the smoothing function is sin(f)/f the window function is a
box-car and the effect is to truncate the data set. The snag in this is
that computing the amplitude is a non-linear function (v = sqrt(x^2 + y^2))
so smoothing the amplitude is not the same as smoothing the components. The
amplitude and phase are not Cartesian, they form a set of curvilinear
co-ordinates. The frequency components are orthogonal, amplitude and phase
are only so locally.

I think what one should do, is generate a set of random data with a
Gaussian distribution, take the Fourier transform, apply a filter to this
with the required frequency response, and transform back to the time
domain. The sample rate should be high enough to avoid aliasing (that is,
the filter amplitude is small at the Nyquist frequency).

In FM, if the noise is small, the in-phase component modulates the carrier
amplitude and this is rejected by the demodulator. The quadrature component
modulates the carrier phase. As frequency is the derivative of the phase,
the high frequency noise components have larger derivatives and add more FM
noise (which is why pre-emphasis is used in FM, to attenuate the HF noise
after demodulation).

Chris

At 20:33 03/04/2008 -0500, you wrote:
>Eden,
>
>In your first e-mail on this subject when you used the term random numbers
>you are actually speaking of complex random numbers. Is this correct? If
>it is, I can understand that.
>
>In the second e-mail, I think what you are trying to say is that for a
>good signal to noise ratio into an FM receiver (above limiting threshold)
>the phase noise in the white noise is insignificant. But for a poor
>signal to noise ratio the phase noise in the white noise is quite
>significant and the output signal to noise ratio from the FM demodulator
>is poor. Is this correct?
>
>I don't understand why you say white noise is orthogonal to an FM signal
>when random phase variations are what make white noise in the first place.
>
>Bill
>
>Eden Mei wrote:
>>Adder:
>>
>>The white noise is orthogonal to PM phase noise, since the random noise
>>generally does not affect the phase of the FM signal, unless the SNR is
>>REALLY bad. The phase of the noise has nothing to do with the phase
>>noise of the FM signal.
>>
>>TTFN,
>>Eden
>>
>>---
>>The Mathcad List - Discussion, Support & News
>>Contributions: /> >>Hosted by: Adept Scientific http://www.adeptscience.com
>>List Archive: http://lists.adeptscience.co.uk/
>>---
>>Would you like this to come to a different email address?
>>---
>>Simply leave the mailing list (see below) and re-join by
>>sending a blank email from the new address to:
or
>>---
>>You are currently subscribed to mathcad as: /> >>To unsubscribe send a blank email to
/> >
>
>---
>The Mathcad List - Discussion, Support & News
>Contributions: /> > Hosted by: Adept Scientific http://www.adeptscience.com
> List Archive: http://lists.adeptscience.co.uk/
>---
>Would you like this to come to a different email address?
>---
>Simply leave the mailing list (see below) and re-join by
>sending a blank email from the new address to:
or
>---
>You are currently subscribed to mathcad as: /> >To unsubscribe send a blank email to
/>

+------------------+
+ Chris Whitford
+ Research Fellow, University of Leicester, Space Research Centre,
+ Physics and Astronomy Department, University Road, LEICESTER LE1 7RH, UK
+ Tel: +44 116 252 3496, Fax: +44 116 252 2464
+ email: http://www.star.le.ac.uk/
+ ------------------+


---
The Mathcad List - Discussion, Support & News
Contributions: /> Hosted by: Adept Scientific http://www.adeptscience.com
List Archive: http://lists.adeptscience.co.uk/
---
Would you like this to come to a different email address?
---
Simply leave the mailing list (see below) and re-join by
sending a blank email from the new address to:
or
---
You are currently subscribed to mathcad as: /> To unsubscribe send a blank email to
/>

---
The Mathcad List - Discussion, Support & News
Contributions: /> Hosted by: Adept Scientific http://www.adeptscience.com
List Archive: http://lists.adeptscience.co.uk/
---
Would you like this to come to a different email address?
---
Simply leave the mailing list (see below) and re-join by
sending a blank email from the new address to:
or
---
You are currently subscribed to mathcad as: /> To unsubscribe send a blank email to />

Previous by date: Re: [mathcad] Modeling White Noise, Chris Whitford
Next by date: RE: [mathcad] Modeling White Noise,  Philip Oakley
Previous thread: [mathcad] 3d graphs for random functions, =?iso-8859-2?Q?Damian_Ga=B3=EAziowski?=
Next thread: [mathcad] Modeling White Noise,  Bill E Dumke

For the time being we are unable to offer the following product ranges although we are currently working hard to increase the number of products we can offer in the future. Please contact us to talk about alternative products that we may be able to offer you.