Computational Intelligence and Application of Frame Theory in Communication Systems

: In this study, we have to discuss the reduction of noise due to peak amplitude and power ratio in multi carrier modulation scheme like Orthogonal Frequency Division Multiplexing (OFDM) process using Frame theory. The frame operator of the frame which is positive, self adjoint, invertible and it commutes with synthesis operator. If X is dual frame in H with frame operator S and analysis operator T , then T is quasi normal operator. If Y is dual frame for X , T is quasi unitary operator. If T and Q are pseudo inverse and sum of its inverse with its ad joint multiplication is frame operator, then X is Bessel’s sequence and dual frame.


Introduction
Frames were formally defined in Hilbert spaces by Duffin and Schaffer (1952) to deal with non harmonic Fourier series. After a couple of years, frames were brought to life Daubechies et al. (1986), in the context of Painless nonorthogonal expansions and Peter G. Casazza and their frame theory research centre discussed Casazza, 2000;Obeidat et al., 2009). Frames are generalizations of orthonormal basis. The linear independence property for a basis which allows each element in the space to be written as a linear combination and this is very restrictive for practical problems. A frames allows each element in the space to be written as a linear combination of the elements in the frames, here linear independence between the frames element is not required. This fact plays important role in signal processing, image processing, coding theory and sampling theory. Before going to definition of Stable and unstable, let us define bounded and unbounded signal or frames. If the signal is bounded, then its magnitude is always be finite i.e., |f n |≤m n , otherwise unbounded. A system is said to be unstable if the output of the system is unbounded for bounded input. A system is called Stable if the output of system is bounded for every bounded input or BIBO stable.

Preliminaries and Notations
We begin with frame definitions. Let H be separable Hilbert space with the inner product 〈⋅,⋅〉 linear in the first entry and all index sets are assumed to be countable.

Definition 3.3
Let H be Hilbert space, then: Let H be separable Hilbert space and the frame operator Hence the proof.

Proposition 4.3
Let H be separable Hilbert space and S q is quasi normal frame operator which is bounded linear, then T(TT*) and (T*T)T have the same non zero Eigen value.

Proof
Since S q is quasi normal frame operator.

Theorem 4.4
Let H be separable Hilbert space and T∈H is reductive quasi similar to quasi normal operator and S is

Theorem 4.6
Let H be separable Hilbert space. If T and Q are pseudo inverses and (T −1 + Q −1 )*(T −1 + Q −1 ) is frame operator, then { } n n i g ∞ = is Bessel's sequence and dual frame in Hilbert space H.

Proof
Let T and Q be pseudo inverse in H (Ding, 2003): Proposition 4.7 Let H be Hilbert space and T 1 and T 2 be self ad joint operators in H, then: • S = T 2 T 1 ≥0 is self adjoin operator • If T 1 and T 2 are shift invariant operators in H • If the sequence (a n )→a∈H, then the frame operator S is stable Since T 1 and T 2 are self ad joint operators in H. We get S = T 2 T 1 which is non negative and self ad joint in H.
To prove S is shift invariant.
Since T 1 and T 2 are shift invariant operators in H: Since the sequence (a n )→a in the Hilbert space which is bounded (BIBO).
This is reconstruction of original information (Arefijamaal and Zekaee, 2013).

Conclusion
We conclude that the main problem of communication systems is noise, which is eliminated by frame theory operator in the modes of linear, Shift invariant and orthogonal. The orthonormal Frames in Hilbert space used for reduce noise to received original data. Frames play an important role not only the theoretic but also many applications in Engineering and Technology.