Design of multiplier-free state-space digital filters

In this paper, a novel approach is presented for designing multiplier-free state-space digital filters. The multiplier-free design is obtained by finding power-of-2 coefficients and also quantizing the state variables to power-of-2 numbers. Expressions for the noise variance are derived for the quantized state vector and the output of the filter. A "structure-transformation matrix" is incorporated in these expressions. It is shown that quantization effects can be minimized by properly designing the structure-transformation matrix. Simulation results are very promising and illustrate the design algorithm.


I. INTRODUCTION
Multiplier-free filters have received widespread attention in the signal processing community due to their high computational speed and low implementation costs. Different methods have been proposed for the design of FIR multiplier-free filters [1][2][3][4][5] but the development for IIR multiplier-free filters has been limited. It has been shown [6], [7] that a multiplier-free IIR filter can be obtained by deriving a special class of digital filter transfer functions from the elliptic minimal Q-factors. In [8] multiplierless IIR filter has been obtained by appropriate extension of the approach given in [7]. In [9] and [10], the authors have used periodically shift-varying (PSV) filters for designing multiplierless IIR filters in the state-space (SS) domain.
In this paper, we provide a novel approach for designing multiplier-free IIR SS filters. A SS structure with power-of-2 coefficients and quantized state variables is employed to design the proposed multiplier-free filter. This makes the filter fast and low cost in VLSI implementation. Expressions for the noise variance are formulated for the quantized state vector and the output of the filter. A structure transformation matrix is incorporated in these expressions. It is shown in the paper that the power-of- 2 effects can be minimized by properly designing the structure transformation matrix.
The remainder of the paper is organized as follows. Section II introduces the notation and describes the filter design problem. Noise analysis is done in Section III. The procedure to obtain the structure transformation matrix is presented in Section IV. Simulation results are given in Section V and Section VI is the conclusion..

A. Multiplier-Free SS Structure
An IIR digital filter can be described by the SS equations where x(n) is the input, y(n) is the output, λ(n) is an M dimensional state vector, and A, B, C, and D are, respectively, M × M , M × 1, 1 × M , and 1 × 1 constant matrices. In fact, there is an infinite number of equivalent realizations obtained by using a similarity transformation matrix T as In this paper, we assume that D is power-of-2. This is easily done by scaling the output in (1), i.e., 1 where α is a scaling constant. Based on the transformed SS structure {A , B , C , D }, the multiplier-free implementation is given by where Q 2 [·] represents quantization to the closest powerof-2 and D = D is power-of-2. By properly designing T , B = T −1 B can also be made a power-of-2 matrix. This design procedure is described in the following sections. Each multiplication in (3) now has a power-of-2 operand, requiring only shifters in their hardware realization.
The architecture of the full precision SS filter in (1) and the multiplier free version in (5) are both illustrated in Fig. 1.

C. The Filter Design Problem
The problem of designing the multiplier-free SS filter 1) The desired output y q (n) of (5) in response to the same white noise input x(n) is as close as possible, in some sense, to the output y(n) of (1).
2) B = T −1 B is a power-of-2 matrix. The mean square error E[e 2 (n)] = E[|y q (n) − y(n)| 2 ] is used as a cost function. The output y q (n) in (5) can be separated into two components. One is the response to input x(n) and other is the response to e λ (n). Since {A, B, C, D} and {T −1 AT, T −1 B, CT, D} represent the same filter, the output noise e(n) = y q (n) − y(n) is the response to e λ (n) only. Hence the first requirement mentioned above can be equivalently stated as designing T to minimize E[e 2 (n)].

III. NOISE ANALYSIS
In this section, we formulate the expressions for the variance of the power-of-2 state noise e λ (n) and the output noise e(n).

A. Power-of-2 Quantization Noise
For power-of-2 quantization, we have is the quantized output, and Δx(n) is the quantization noise. If we assume that x(n) is zero mean white noise that is properly scaled between the upper and lower boundary, then Δx(n) is also zero mean and has a variance proportional to the variance of x(n). This relation can be formulated as where κ = 0.038 is a constant obtained through simulations. It will be seen that the optimal matrix T is independent of κ, and therefore exact value of κ is unimportant.

B. Power-of-2 State Noise
Assuming no cross correlation and using (6), we get, Defining λ(z) = [λ 1 (z), · · · , λ M (z)] T , and considering (5) without e λ (n), the input state relation of the Using (7) and Parseval's theorem, the covariance matrix of the state vector can be expressed as where dω is a Hermitian matrix and σ 2 x is the variance of the white noise input. From (7) and (8), we have where G and T −1 GT −T can be interpreted as the covariance matrices of power-of-2 state noise for the original and transformed filters, respectively.

C. Output Noise
Output noise e(n) is the response of the multiplier-free filter to input e λ (n), which is a white noise vector with covariance matrix given by (9). If we define e λ (z) = [e λ1 (z), · · · , e λM (z)] T , the state-output relation of the filter in (5) can be described by the equations After some algebraic manipulations e(z) in (10) can be expressed as Define Using (11) and (12) we get e(n) = h 1 (n) * e λ1 (n) + · · · + h M (n) * e λM (n), and E[e 2 (n)] Using Parseval's theorem on (13) and using (12)

IV. DESIGNING TRANSFORMATION MATRIX T
We now design the real non-singular matrix T to minimize the cost function given in (15), and achieve power-of-2 B = T −1 B. However, a closed form solution of T , for this problem, is very difficult to establish. We provide an iterative algorithm to find T in a scaled orthogonal subspace. That is, we design T which has the form where P is an orthogonal matrix (P T P = I) and S is a diagonal scaling matrix with nonzero diagonal elements (s ii = 0, i = 1, · · · , M). The design procedure is divided into two parts: 1) Designing the orthogonal matrix P , and 2) Determining the scaling matrix S. The design methodology will utilize the following theorem.

A. Designing the Scaling Matrix S
Theorem 1 claims that once the orthogonal matrix P is fixed, the cost function f (P S) is unaffected by the scaling matrix S. The degree of freedom in choosing S provides the opportunity to achieve power-of-2 B = T −1 B. Assuming P = [p 1 , · · · , p M ] is given, then from (16) we have where B is the desired power-of-2 matrix (b 1 , · · · , b M can be chosen arbitrarily), and Therefore, once the orthogonal matrix P is fixed, and the power-of-2 matrix B is chosen, the scaling matrix S given by can be used without affecting the cost function.

B. Designing the Orthogonal Matrix P
The problem of finding the orthogonal matrix P can be formulated as follows.
where subject to P T · P = I, G and W are given Hermitian matrices.
An iterative algorithm is derived to solve this problem At each iteration, the algorithm uses the steepest gradient update procedure and an orthogonalization procedure in order to satisfy the orthogonality constraint.
In the steepest gradient update procedure, we first assume P = [p 1 , · · · , p M ] is orthogonal and then compute the gradient of f (P ) with respect to P as Then an update of P in the steepest-descent direction is performed as where μ 1 is the iteration step and k 1 is the iteration index.
Equations (19) and (20) are repeated several times to reduce the cost function f (P ). Now we satisfy the orthogonality constraint, P T P = I. We use the following cost function where · F is the Frobenius norm. P is orthogonalized when R(P ) is reduced to zero. A steepest descent method shown in the following equations is used to minimize (21).
where μ 2 is the iteration step and k 2 is the iteration index. The algorithm to find the orthogonal matrix P is summarized in Table I. The complete procedure for designing the multiplierfree SS filter is summarized in Table II.

V. SIMULATION RESULTS
The simulation procedure is illustrated in Fig. 3 with three different filters. The full precision SS filter described by {A, B, C, D} is the desired filter. The transformed multiplier-free filter is the filter we designed. The semi multiplier-free SS filter has the same powerof-2 quantization structure as the transformed multiplierfree SS filter, except that it does not use the structure transformation matrix T . Since B and D are not powerof-2 matrices, the shifters can only be used in AQ 2 [λ(n)] and CQ 2 [λ(n)]. Hence, the name semi multiplier-free. In Input: Hermitian symmetric matrices G and W , No. of steepest descent iterations (N # ), μ 1 and μ 2 step size of steepest-descent itr., ε minimum cost for R(P ). Initialization: Randomly initialize P . Procedure: (1) Steepest-descent update of P .
for (1) Scale the output to achieve power-of-2 D.   Using (2) we obtain the following transformed SS structure. The difference between the theoretical and actual value is due to the ignored cross correlation terms in (9). The power spectrum of the filters are shown in Fig. 2.

VI. CONCLUSION
In this paper, a new approach is presented for designing multiplier-free SS digital filters. Expressions for noise variance are formulated for the quantized state vector and the output noise. A steepest-descent type method is developed for the design algorithm. The simulation results are very promising and yield low noise multiplierfree filters.