Abstract – A new synthesis procedure for generating orthogonal structures to implement arbitary transfer functions is described. As compared to earlier techniques associated with orthogonal filter synthesis, which used a cascade of sections or a parallel combinations of sections, this method uses feedback to simplify the design process, and results in filter structures that have the low pass band sensitivity properties associated with orthogonal structures. The advantage of the new structure is that its stability monitoring is very easy; thus, it is extremely well suited for adaptive applications.

Index Terms – Error Feedback, Orthogonality constraints, state space realization.

INTODUCTION

Signal processing is a ubiquitous aspect of most systems today, typically with the systems being so complex that not just one, but several signal processing tasks are required to be performed by the system. For instance, some possible tasks are filtering, equalization, parameter estimation, etc. These problems have typically been dealt with separately in the past, and though several solutions have been proposed to each of these problems, no attempt has been made to develop a unified solution. The merit of a unified solution is that it would represent a ‘universal- building block’ (UBB), that could be used to solve a host of problems.

In a Multi band signal processing system, the total bandwidth is divided in to sub bands and at any time signal will be present in any sub bands. The signal model in this case is assumed to have been produced by a generative model, whose form is assumed to be known. A local copy of this generative model then attempts to reproduce the input signal, and the error between the estimate produced by the local copy and the input signal is then used to correct either the parameters of the model, or its states, with the objective of minimizing the estimate error.

By using a state space representation the number of input variables, output variables and state variables are expressed as vectors and the differential and algebraic equations of a filter are written in matrix form. At a particular instant of time the state space representation of a system is given by

X(n+1) = A X(n) + B U(n) ; state equation

Y(n) = CX(n) + D U(n); output equation

In this report, a unifying approach to spectrum estimation problems and the concept of the universal signal-processing building-block, a feedback based orthogonal filter structure is presented.

DEVELOPMENT OF FILTER STRUCTURE

The general state space description of the filter structure is given by

X(n+1) = A_cl X(n) + B_cl U(n); state equation

Y(n) = C^T_cl X(n); output equation.

The subscript ‘cl’ represent closed loop.

The constraint of orthogonality can now be expressed in terms of the state transition matrix R as follows 1

Then orthogonality implies that

R*R = RR* = I_N+1 ---- (1)

Where * indicates complex conjugation of transpose and I_N+1 is an (N+1) x (N+1) identity matrix

Let us consider a feedback based orthogonal filter structure as shown

Theorem: The structure in above figure is orthogonal in the sense if and only if

A^*_ol A_ol = I_N

B^*_ol B_ol = I

C_ol = B^*_ol A_{ol .}

The subscript ‘ol’ represent open loop

Proof: The state space matrices of the closed loop filter can be expressed in terms of the open loop parameters as

A_cl = A_ol – B_ol C^T_ol

Bcl _{= B}ol

C_cl = C_ol

Substituting in equation (1) we get

R^*R = I_N+1

B^*_cl B_cl = 1

A^*_cl B_cl = 0

A^*_cl A_cl + C_cl C_cl = I_N

B^*_ol B_ol = 1

A^*_ol B_ol - C_ol B^*_ol B_ol = 0

A^*_ol A_ol – C_ol B^*_ol A_ol – A^*_ol B_ol C^T_ol + C_ol B^*_ol B_ol C^T_ol + C_ol C^T_ol = I_N

At this stage, a feedback-based orthogonalfilter structure to implement an allpass transfer function has been developed. In order to implement an arbitrary transfer function, we first extract an allpass transfer function from the desired transfer function. The allpass is forced to have the same poles as the desired transfer function, and also no delay-free forward path. The coefficients of the feedback structure are now designed to implement the allpass transfer function between U(n) and Y(n), under the constraints of orthogonality.

Y( z ) / E( z ) = H_lg (z) = C^T_ol ( z I – A_ol ) ^-1 B_ol

H_a( z ) = Y( z ) / U( z ) = H_lg( z ) / [ 1 + H_lg( z ) ] and

H( z ) = V( z ) / U( z ) = [ W^T ( z I – A_ol ) ^-1] B_ol

Where W^T = [ w₁, w₂, ....., w_N ] is a vector denoting weights, and H_lg( z ), H_a( z ), H( z ) to denote the transfer functions.

To simplify these expressions, the matrix A_ol can be expressed as

A_ol = E^* Q E

Where Q is diagonal matrix containing the eigen values of A_ol

And E is a unitary matrix containing the eigen vectors of A_ol. Based on choice of E, A_ol can be made circulant or skew circulant, in which case the computation simply becomes a circular convolution.

REFERENCES

[1] Mukund Padmanabhan, ‘FEEDBACK BASED ORTHOGONAL DIGITAL FILTERS Theory, Applications, and Implementation’

[2] A. Varga , ‘Computational Techniques Based on the Block Diagonal Form for solving large systems modeling problems’

[3] B. D. O. Anderson and J. B. Moore. Linear optimal control. Printice Hall, 1971.

[4] S. Haykin. Adaptive Filter Theory. Prentice-Hall Inc., Englewood Cliffs, New Jersey, 1986.

[5] L. B. Jackson, A. G. Lindgren, and Y. Kim. Optimal Synthesis of Second-order State Space Structures for digital filters. IEEE Transactions on circuits and systems, 26:149-153, Mar 1979