<< Chapter < Page Chapter >> Page >

The following sections will explain the three filters in detail.

  • B(ω) , the separation filter.
  • Bm+ , the normalization filter used to solve the scaling problem that arises after applying independent component analysis.
  • P(ω) , the permutation filter used to solve the frequency distortion problem that arises after applying independent component analysis.

Separation filter

The separation filter, shown in the previous section can be broken down into the following components.

Separation Filter

W(ω) is the preprocessing filter used to reduce reflections and ambient noise by using the subspace method. U(ω) is the filter obtained by applying independent component analysis after preprocessing.

I. subspace method

The spatial correlation, or autocorrelation, matrix at frequency ω is defined below.

Spatial Correlation

Given that there are M inputs and M sources, the resulting matrix will be M × M . This matrix can be rewritten in the following way.

Rewritten Correlation Matrix
The spatial correlation matrix.
Noise Cross Spectrum
The noise correlation matrix.
Q Matrix
The sources' cross-spectrum matrix.

In other words, K is the correlation matrix of n(t) and Q is the cross-spectrum matrix of the sources. However, since the source signals, s(t) , are unknown at this point, this equation cannot be directly solved. Instead, we can represent the generalized eigenvalue decomposition of the spatial correlation matrix in the following manner.

Generalized Decomposition

E refers is the eigenvector matrix, E = [e1,e2, …,eM] , and Λ = diag(λ1,λ2,…,λM) refers to the eigenvalues of R . Since K , the noise correlation matrix, cannot be directly observed apart from the source signals, we will assume that K = I , which will evenly distribute the reflections and ambient noise induced by the environment among the estimated sources. As such, the eigenvalue decomposition of R can be rewritten as the following.

Standard Decomposition

The final preprocessing filter uses the eigevector and eigenvalue matrices resulting from the standard eigenvalue decomposition of the spatial correlation matrix and is shown below.

Final Preprocessing Filter

Ii. independent component analysis

By applying the preprocessing filter obtained from the subspace method, our observed signals are now of the following form.

Applying the Subspace filter

To estimate the source signals from the preprocessed input signals, we can use independent component analysis to solve the linear system displayed below, for the filter U(ω) .

Applying the ICA Filter

Now that the separation filter at each frequency has been found, the problems that arise from the separation process must be addressed.

Scaling

The scaling problem that results from applying independent component analysis can be resolved using the filter shown below.

Scaling Filter

Bm,n+(ω) refers to the (m,n) th entry in B+(ω) , which is the pseudo-inverse of B(ω) (regular inverse also works here since we assume that the number of separated sources and the number of inputs are both M , so that the resulting separation matrix is square). Additionally, m refers to an arbitrary row or microphone in B+(ω) . By applying this matrix, each signal, i , is amplified by the component of signal i observed at microphone or input m . Essentially, this filter amplifies the frequency components of the separated source signals so that the waveforms of the resulting sources will be distinguishable and audible.

Permutation

A second critical problem that arises after applying independent component analysis is that the order in which the source signals are returned is unknown. Since independent component analysis is applied at each frequency, we must find the permutation of components that has the highest chance of being the correct permutation to reconstruct the source signals and minimize the amount of frequency distortion caused by independent component analysis.

We define the permutation matrix, P , as the following.

Pseudo Inverse of Separation Filter
Permutation Filter Applied to Pseudo Inverse of Separation Filter
Applied Pseudo Inverse Matrix

The matrix, P , exchanges the column vectors of Z(ω) to get different permutations. We define the cosine of θn between the two vectors zn(ω) and zn(ω0) , where ω0 is the reference frequency is as follows.

Angle Between Filter Vectors

The permutation is, then, determined by the following.

Permutation Filter Definition

The cost function, F(P) , used above, is defined as follows.

Cost Function for Permutation Filter

This compares the inverse vector filter at each frequency with a filter at a reference frequency, ω0 . However, if we use the same reference frequency to find the permutation at every other frequency, then the problem arises if the filter at the reference frequency has the incorrect permutation. In order to minimize this error, a new reference frequency is chosen after the permutations of filters at K frequencies are found. As such, the frequency range of a reference frequency, ω0 , is as follows.

Frequency Range

Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
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
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
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.
Bharti
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
Daniel
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
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
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
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
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.
Tarell
what is the actual application of fullerenes nowadays?
Damian
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.
Tarell
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.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
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.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Elec 301 projects fall 2014. OpenStax CNX. Jan 09, 2015 Download for free at http://legacy.cnx.org/content/col11734/1.2
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Elec 301 projects fall 2014' conversation and receive update notifications?

Ask