On Monday, January 7, 2013 11:08:58 AM UTC-5, Dave wrote:
> On Dec 24 2012, 3:18=A0am, "manishp" <58525@dsprelated> wrote:
>=20
> > Sirs,
>=20
> >
>=20
>=20
>=20
> snip
>=20
>=20
>=20
> >
>=20
> > 2) does noise always have an effect of adding to the signal (that is,
>=20
> > signal +- noise) or they can be transformation of any other form?
>=20
> >
>=20
> > Thanks once again, manish
>=20
>=20
>=20
> Another type of noise that occurs is multiplicative. One example is
>=20
> phase noise associated with clock in running an A/D converter.
>=20
> Multiplicative noise tends to be a more advanced topic - you don't
>=20
> normally see it discussed in a university undergraduate level.
>=20
>=20
>=20
> Cheers,
>=20
> Dave

Through repeated multiplicative processes, you'll often end up with a log-n=
ormal distribution.

Clay

> does it really matter what is the shape of a noise - ramp, triangle,  sinusodal etc. 

If it makes your problem harder or easier to solve, then it matters.

> does it really matter that they very unlikely to be periodic 

If it makes your problem harder or easier to solve, then it matters.

> let us assume that the information is a pure sine wave and further 
> assume noise is just one single point of a specific amplitude in a specific 
> window. In this case, when the information gets corrupted by noise, will 
> the resulting signal appear as a high frequency change in the original 
> signal. Remember, the noise is just a single pulse ... 

The power spectrum of a pure sine wave is a spike.
If the sine wave at some point in time is mixed with an impulse, your power
spectrum will briefly show "constant" power across all frequencies.

> does noise always have an effect of adding to the signal (that is, 
> signal +- noise) or they can be transformation of any other form? 

The degree to which your (linear) noise reduction algorithm is able to remove
noise, tells you how accurate your model is (assuming that your implementation
is correct).

> Leaving the theory apart, intuitively (thinking in time and frequency 
> terms) it is clear that spectrum of a sine wave is a pulse. 
> But is is not intuitive how a spectrum of a impulse can spread across 
> frequency spectrum. It is difficult to imagine that a point can contain 
> multiple frequency terms although when we see the transform equation, it 
> does make sense ... 

If you add up a lot of sines you will eventually get a spike. Try this in 
MATLAB:

clc
close all
clear all
fs = 8000;
f = 2:4000;
k = 1;
for t = 0:1/fs:1;
y(k) = sum(sin(2*pi*f*t));
k = k + 1;
end
plot(y);

The MATLAB script shows you that an impulse can be decomposed into a signal which is
a sum of lots of sine waves. In the frequency spectrum that translates into a lot of spikes
with the same amplitude across the spectrum....or if you will...a flat line..

On Dec 24 2012, 3:18=A0am, "manishp" <58525@dsprelated> wrote:
> Sirs,
>

snip

>
> 2) does noise always have an effect of adding to the signal (that is,
> signal +- noise) or they can be transformation of any other form?
>
> Thanks once again, manish

Another type of noise that occurs is multiplicative. One example is
phase noise associated with clock in running an A/D converter.
Multiplicative noise tends to be a more advanced topic - you don't
normally see it discussed in a university undergraduate level.

Cheers,
Dave

On Monday, December 24, 2012 3:14:23 AM UTC+13, manishp wrote:
> Sirs,
> 
> 
> 
> I have few questions on the characteristic of noise when it comes to
> 
> processing signals with noise:
> 
> 
> 
> 1) does it really matter what is the shape of a noise - ramp, triangle,
> 
> sinusodal etc.
> 
> 2) does it really matter that they very unlikely to be periodic
> 
> 
> 
> Thanks, Manish

Type of noise - yes can matter when doing independent component analysis cos you need the PDF of the noise ie Laplacian for instance.

On 25.12.2012 06:43, manishp wrote:

> Leaving the theory apart, intuitively (thinking in time and frequency
> terms) it is clear that spectrum of a sine wave is a pulse.

Actually, two pulses since

sin(\omega x) = 1/2i (exp(i \omega x) - exp(-i \omega x))

so one at \omega with amplitude 1/2i and another at -\omega with 
amplitude -1/2i.

> But is is not intuitive how a spectrum of a impulse can spread across
> frequency spectrum.

Take for example this:

The spectrum of the cosine (\omega x) are two pulses at \omega and 
-\omega. Make \omega smaller and smaller, then (pointwise) cos(\omega x) 
converges to the constant, and its Fourier transform (the spectrum) to 
the pulse at the origin. Since the inverse of the Fourier transform is 
the Fourier transform itself, up to a minus sign, the same holds also 
inversely: A cosine-like spectrum gives two pulses in the time domain, 
and if the cosine frequency gets smaller and smaller, the two pulses 
converge to a single pulse at the origin.


> It is difficult to imagine that a point can contain
> multiple frequency terms although when we see the transform equation, it
> does make sense ...

Probably it's time to adjust your imagination (or intuition). The more 
you work with such topics, the more intuitive these become.

Greetings,
	Thomas

On 12/25/12 12:43 AM, manishp wrote:
>> Remember that the Fourier Transform is linear, so superposition
>> applies in both domains.   In other words, when you add noise to a
>> signal, the spectra of the two signals add, too.   So in the frequency
>> domain the spectrum of the sine wave is added to the spectrum of the
>> added noise.   The spectrum of the sine wave is a spike, and the
>> spectrum of the impulse is broad and flat.
>
> Thank you very much. I do have a question on the spectrum of sine and
> impulse.
>
> Leaving the theory apart, intuitively (thinking in time and frequency
> terms) it is clear that spectrum of a sine wave is a pulse.
> But is is not intuitive how a spectrum of a impulse can spread across
> frequency spectrum. It is difficult to imagine that a point can contain
> multiple frequency terms although when we see the transform equation, it
> does make sense ...

well, we've always sorta known that sharp edges require high frequencies.

the only way i know how to prove this to you is indirect.

we know that the fourier transform of a dirac impulse is a constant 
function in frequency.  that's an easy integral.

so that means that the inverse fourier transform of the broad-banded 
constant function in frequency is a dirac impulse.


-- 

r b-j                  rbj@audioimagination.com

"Imagination is more important than knowledge."

On 12/24/12 1:02 PM, Eric Jacobsen wrote:
> On Mon, 24 Dec 2012 02:18:47 -0600, "manishp"<58525@dsprelated>
> wrote:
>
>> Sirs,
>>
>> Thanks for your response. Let me try and elaborate ...
>>
>> Since generally noise can be random at best (both in terms of when&  how it
>> occurs), I am trying to understand how they can be defined ...
>>
>> Few specific questions below:
>> 1) let us assume that the information is a pure sine wave and further
>> assume noise is just one single point of a specific amplitude in a specific
>> window. In this case, when the information gets corrupted by noise, will
>> the resulting signal appear as a high frequency change in the original
>> signal. Remember, the noise is just a single pulse ...
>
> Remember that the Fourier Transform is linear, so superposition
> applies in both domains.   In other words, when you add noise to a
> signal, the spectra of the two signals add, too.   So in the frequency
> domain the spectrum of the sine wave is added to the spectrum of the
> added noise.   The spectrum of the sine wave is a spike, and the
> spectrum of the impulse is broad and flat.
>
>> 2) does noise always have an effect of adding to the signal (that is,
>> signal +- noise) or they can be transformation of any other form?
>
> Most noise is additive, but never assume that.  A common noise model
> is AWGN, or Additive White Gaussian Noise.   The first word "additive"
> is important because it clarifies that it is, in fact, an additive
> model.   Usually noise is additive, but it can be important to clarify
> this to be sure in a particular context.


contrast this with quantization noise.  it actually is not strictly 
additive, but is a function of the signal getting quantized.  but we 
model it as "additive" simply by defining it as the difference between 
the quantized signal and the pre-quantized signal.  that makes it 
"additive".  but it's still not independently added.  if the signal 
swing is large and unrelated to the sampling frequency, this additive 
quantization noise starts appearing (or sounding) more independent and 
then it might be modeled as additive white (up to Nyquist) and uniform 
p.d.f. noise.  but i don't think they call that "AWUN" or "AWRN" 
("rectangular" p.d.f.).

with the right kind of dither, it gets more uncorrelated and looks a 
little more gaussian.  if you add triangular p.d.f. dither of 2 LSBs 
width, the quantization noise will be white and uncorrelated (first two 
moments) and the p.d.f. will be that from convolving the rectangular 
p.d.f. with itself twice.  a piece-wise quadratic function that looks a 
little bit like gaussian. but still not precisely AWGN.  but we might be 
tempted to start calling it that, just to make our lives easier.

just another $0.02

-- 

r b-j                  rbj@audioimagination.com

"Imagination is more important than knowledge."

>Remember that the Fourier Transform is linear, so superposition
>applies in both domains.   In other words, when you add noise to a
>signal, the spectra of the two signals add, too.   So in the frequency
>domain the spectrum of the sine wave is added to the spectrum of the
>added noise.   The spectrum of the sine wave is a spike, and the
>spectrum of the impulse is broad and flat.

Thank you very much. I do have a question on the spectrum of sine and
impulse.

Leaving the theory apart, intuitively (thinking in time and frequency
terms) it is clear that spectrum of a sine wave is a pulse.
But is is not intuitive how a spectrum of a impulse can spread across
frequency spectrum. It is difficult to imagine that a point can contain
multiple frequency terms although when we see the transform equation, it
does make sense ...

Thanks, Manish

On Mon, 24 Dec 2012 02:18:47 -0600, "manishp" <58525@dsprelated>
wrote:

>Sirs,
>
>Thanks for your response. Let me try and elaborate ...
>
>Since generally noise can be random at best (both in terms of when & how it
>occurs), I am trying to understand how they can be defined ...
>
>Few specific questions below:
>1) let us assume that the information is a pure sine wave and further
>assume noise is just one single point of a specific amplitude in a specific
>window. In this case, when the information gets corrupted by noise, will
>the resulting signal appear as a high frequency change in the original
>signal. Remember, the noise is just a single pulse ...

Remember that the Fourier Transform is linear, so superposition
applies in both domains.   In other words, when you add noise to a
signal, the spectra of the two signals add, too.   So in the frequency
domain the spectrum of the sine wave is added to the spectrum of the
added noise.   The spectrum of the sine wave is a spike, and the
spectrum of the impulse is broad and flat.

>2) does noise always have an effect of adding to the signal (that is,
>signal +- noise) or they can be transformation of any other form?

Most noise is additive, but never assume that.  A common noise model
is AWGN, or Additive White Gaussian Noise.   The first word "additive"
is important because it clarifies that it is, in fact, an additive
model.   Usually noise is additive, but it can be important to clarify
this to be sure in a particular context.


>Thanks once again, manish

Eric Jacobsen
Anchor Hill Communications
http://www.anchorhill.com

On 12/24/2012 3:18 AM, manishp wrote:
> Sirs,
>
> Thanks for your response. Let me try and elaborate ...
>
> Since generally noise can be random at best (both in terms of when&  how it
> occurs), I am trying to understand how they can be defined ...
>
> Few specific questions below:
> 1) let us assume that the information is a pure sine wave and further
> assume noise is just one single point of a specific amplitude in a specific
> window. In this case, when the information gets corrupted by noise, will
> the resulting signal appear as a high frequency change in the original
> signal. Remember, the noise is just a single pulse ...
>
> 2) does noise always have an effect of adding to the signal (that is,
> signal +- noise) or they can be transformation of any other form?
>
> Thanks once again, manish

If I understand what you are describing, this would be an impulse which 
has energy throughout the spectrum, not just at the high end.  But I'm 
not sure what you mean by "in a specific window".

Rick