noise

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 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 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 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

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

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

Clay