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Enter the Wiener filter, an adaptive filter. Derivations and applications.

The wiener filter is an adaptive filter. It tailors itself to be the “best possible filter” for a given dataset. Below is a simplistic version of the derivation for the wiener formula.

Consider our standard equation to model a signal with noise:

y n x n n n

We want to pass this y[n] through a filter ‘h’ such that we get back something that very closely resembles our original signal x, i.e. x.

So basically we want to design a filter that minimizes the difference between x and x. Lets start out by minimizing the least mean square error between x and x.

2 x x

But we know x is h*y (h convolved with y), so we have:

2 x x * y

We can expand this expression known algebraic rules. Then we can take the Fourier transform of the expression to find the power spectra.

j X j H j Y j 2
j X j H j X j N j 2

We minimize this expression over H and in the end after all the simplification we get the following formula for H, our filter optimized to minimize the difference between x and x.

H f X f 2 X f 2 N f 2

Where X(f) is the power of the signal and N(f) is the power of the noise.

Just by inspection we can see that this filter will take care of the basics. For example when there is no noise, the filter response goes to just 1. When the signal is 0 the filter response goes to 0.

Notice to use this filter we will need to know the power spectrum of the actual signal. We also need to estimate the power spectrum of the noise. In real life this is done in numerous ways. Oftentimes out of convenience engineers will approximate the noise to Gaussian. This works well with varying results.

Building the filter

One way to approximate the noise is “by inspection” or what is more commonly known as “guess and check”. In this method, you take a close look at your received signal and your pure signal.

Pure signal spectrum.
Received signal spectrum.

Then we try to figure out what the noise could look like based on this information. For this data set, we have traced out our “best guess” for the noise spectrum below

Noise trace.

Just by looking at the noise spectrum we want, we look for patterns we could fit a mathematical formula to. In this example you can see that the function decays like a polynomial on one side and exponentially on the other.

So our general noise toggling function becomes:

And so we fix our parameters alpha, beta and gamma until we get something that resembles the function we drew in red earlier.

This function is particular to this data set. For other types of noise, you would have to fit a new appropriate noise function.

So now using the values we calculated for X(f) and N(f) we create our wiener filter.

Magnitude Plot of the Wiener filter.

So lets try the filter on the data shall we?

Unfiltered data.

We see that the wiener filter does its job pretty well. It even wins over the bandpass filter in the simpler example since it even removes the excess amplitude added by the noise.

Lets take a look at the image of all the time series vectors.

A single unfiltered time series vector.
Image of unfiltered time vectors.
Image of filtered time vectors.

We can see that the unfiltered image is pretty blurry with not much distinguishable from the background. On the other hand the picture of the filtered version comes out pretty well and we can see distinct values for every time series.

So that concludes the section on filtering. Hopefully you have a better understanding on how to design adaptive filters, the wiener filter in particular. Now its time to move the data to the next stage of the process: Imaging.

Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
characteristics of micro business
for teaching engĺish at school how nano technology help us
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
what is the actual application of fullerenes nowadays?
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Seismic imaging. OpenStax CNX. Dec 16, 2004 Download for free at http://cnx.org/content/col10251/1.1
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