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The dual frame vectors are also not unique but a set can be found such that [link] and, therefore, [link] hold (but [link] does not). A set of dual frame vectors could be found by adding a set of arbitrary but independent rows to F until it is square, inverting it, then taking the first N columns to form F ˜ whose rows will be a set of dual frame vectors. This method of construction shows the non-uniqueness of the dual framevectors. This non-uniqueness is often resolved by optimizing some other parameter of the system [link] .

If the matrix operations are implementing a frame decomposition and the rows of F are orthonormal, then F ˜ = F T and the vector set is a tight frame [link] , [link] . If the frame vectors are normalized to | | x k | | = 1 , the decomposition in [link] becomes

x = 1 A n ( x , x ˜ n ) x n

where the constant A is a measure of the redundancy of the expansion which has more expansion vectors than necessary [link] .

The matrix form is

x = 1 A F F T x

where F has more columns than rows. Examples can be found in [link] .

Sinc expansion as a tight frame

The Shannon sampling theorem [link] can be viewied as an infinite dimensional signal expansion where the sinc functions are an orthogonal basis. The sampling theorem with critical sampling, i.e. at the Nyquist rate, is the expansion:

g ( t ) = n g ( T n ) sin ( π T ( t - T n ) ) π T ( t - T n )

where the expansion coefficients are the samples and where the sinc functions are easily shown to be orthogonal.

Over sampling is an example of an infinite-dimensional tight frame [link] , [link] . If a function is over-sampled but the sinc functions remains consistentwith the upper spectral limit W , using A as the amount of over-sampling, the sampling theorem becomes:

A W = π T , for A 1

and we have

g ( t ) = 1 A n g ( T n ) sin ( π A T ( t - T n ) ) π A T ( t - T n )

where the sinc functions are no longer orthogonal. In fact, they are no longer a basis as they are not independent. They are, however, a tightframe and, therefore, have some of the characteristics of an orthogonal basis but with a “redundancy" factor A as a multiplier in the formula [link] and a generalized Parseval's theorem.Here, moving from a basis to a frame (actually from an orthogonal basis to a tight frame) is almost invisible.

Frequency response of an fir digital filter

The discrete-time Fourier transform (DTFT) of the impulse response of an FIR digital filter h ( n ) is its frequency response. The discrete Fourier transform (DFT) of h ( n ) gives samples of the frequency response [link] . This is a powerful analysis tool in digital signal processing (DSP) and suggests that an inverse (or pseudoinverse)method could be useful for design [link] .


Frames tend to be more robust than bases in tolerating errors and missing terms. They allow flexibility is designing wavelet systems [link] where frame expansions are often chosen.

In an infinite dimensional vector space, if basis vectors are chosen such that all expansions converge very rapidly, the basis is called an unconditional basis and is near optimal for a wide class of signal representation and processing problems. This is discussed by Donoho in [link] .

Still another view of a matrix operator being a change of basis can be developed using the eigenvectors of an operator asthe basis vectors. Then a signal can decomposed into its eigenvector components which are then simply multiplied by the scalar eigenvalues toaccomplish the same task as a general matrix multiplication. This is an interesting idea but will not be developed here.

Questions & Answers

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
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?
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
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
many many of nanotubes
what is the k.e before it land
what is the function of carbon nanotubes?
I'm interested in nanotube
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
preparation of nanomaterial
Victor 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, Basic vector space methods in signal and systems theory. OpenStax CNX. Dec 19, 2012 Download for free at http://cnx.org/content/col10636/1.5
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