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Circulant and Skew-Circulant Matrices As New Normal-Form Realization Of IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS, VOL. 35, NO. 6, JUNE 1988 625 Cir culan t and Skew-Circulant Ma trices as New Normal-Form Realization of IIR Digital Filters Abstract -Normal-form fixed-point state-space realizations of IIR filters In this paper, we shall present a new family of normal- are known to be free from both overflow oscillations and roundoff limit form state-space structures. ‘The method used allows us to cycles, provided magnitude truncation type of arithmetic is used together with two’s complement overflow features. The eigenvalues of the state synthesize in normal form, most IIR transfer functions. transition matrix have low sensitivity. In this paper two new normal-form We shall refer to the structures in this paper as circulant realizations are presented which utilize circulant and skew-circulant and skew-circulant forms, since the state transition matrices matrices as their state transition matrices. The advantage of these realiza- involved are either circulant or skew-circulant matrices. tions is that the A -matrix has only N (rather than N2) distinct elements, The well known second-order coupled form [4] is a special and is amenable to efficient memory-oriented implementation. The prob- lem of scaling the internal signals in these structures is addressed and an case of the skew-circulant form. approximate solution can be obtained through a numerical optimization In Section 11, the low sensitivity and limit-cycle free method. Several numerical examples are included. property of normal-form structures in general are re- viewed. In addition we introduce a new measure for the eigenvalue sensitivity of a state transition matrix. In Sec- I. INTRODUCTION tion 111, the method by which circulant and skew-circulant N THE fixed-point implementation of IIR filters, forms are derived is presentled. We then address the prob- I several undesirable effects occur due to the finite lem of scaling these structures in Section IV, which also wordlength of the implementation. State-space approach includes a study of roundoff noise in these filters. has been used in the past to minimize such effects. In Although the circulant and skew-circulant realizations particular, normal-form digital filters have been shown to have in general N2 nonzero entries in the state transition possess several good qualities with regard to finite word- matrix, yet there are only N distinct elements and the length effects such as the absence of limit cycles, and low circulant structure of the matrix makes these filters amena- coefficient sensitivity [I], [2], [7]. The condition for the ble to efficient memory-oriented hardware implementa- absence of overflow oscillations had been derived by Barnes tions. It has been recognized in the past that circular and Fam [l],which is to restrict the I, norm of the state convolution is a key operation in a number of signal transition matrix to be less than unity, and normal-form processing algorithms, including finite and infinite length structures satisfy this requirement. It had been pointed out linear convolutions [lo], [17]. The use of circular convolu- further by Jackson [6] that the same condition would also tion in recursive linear filtering is particularly noticeable in lead to the suppression of quantization limit cycles when block realization of IIR filters [18]-[20]. If we have a the quantization is done by magnitude truncation. So with signal processing hardware which primarily implements the use of magnitude truncation type of quantization arith- circular convolutions, then lit offers a wide range of appli- metic together with two’s complement overflow features, cations. The result of this paper adds one more application both types of limit cycles may be eliminated from normal- to ths list, namely, a limit-cycle free, low eigenvalue form filters. Most of the normal-form structures that have sensitivity implementation of IIR filters. In this connec- been presented in the past made use of second order tion, it is worth noting that block realization of digital normal sections as building blocks. They consist of either filters in the state-space context has been studied by cascaded or parallel connections of second-order sections Barnes er al. [21]; it is easily verified that the “block PI, [31. version” of a circulant realization retains the circulant nature, since the state transition matrix A is merely re- Manuscript received August 29, 1985; revised May 5, 1986 and July 22, placed by A where L is the block length. 1987. This work was supported in part by the National Science Founda- Purely from the viewpoint of computational complexity, tion under Grant ECS 84-04245 and in part by CalTech’s programs in advanced technology sponsored by Aerojet General, General Motors, (i.e., the number of multiplications and additions per sam- GTE, and TRW. V. C. Liu was under a Schlumberger Fellowshp for the ple), the structures introduced here are not necessarily period during which ths work was performed. This paper was recom- mended by Associate Editor A. N. Venetsanopoulos. more efficient than the second-order block normal form in The authors are with the Department of Electrical Engineering, Cali- [l].However, as mentioned above, there are at least two fornia Institute of Technology, Pasadena, CA 91125. IEEE Log Number 8820490. contexts where the results of this paper can be useful: 0098-4094/88/0600-0625$01.00 01988 IEEE 626 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS, VOL. 35, NO. 6. JUNE 1988 memory oriented implementations, and implementations [16] and can be used to explain the absence of limit cycles based on convolution building blocks. in several well-known structures, such as the wave-digital filters, orthogonal filters, cascaded lattice structures, nor- 11. REVIEW AND A NEWEIGENVALUE mal-form structures and second-order minimum-noise SENSITIVITYMEASURE structures. If the quantization scheme is of the magnitude A state-space structure for an IIR filter with input U( n) truncation type, along with 2's complement overflow char- and output y(n)is characterized by the equations [5] acteristics, then both quantization limit cycles and over- flow oscillations can be suppressed if the A matrix satisfies x(n+l) =Ax(n)+Bu(n) (14 the conditions given in [16]. Such an arithmetic scheme involves the use of two's complement representation for y(n)=Cx(n)+du(n). Ob) negative numbers, and the quantization is done in two In this paper, we shall deal with single-input-single-output different ways depending on the sign of the number to be systems where A = [akr]is an N X N matrix with 0 < quantized. If the number is positive, a straightforward k,l<N-l. B is Nxl, and C is IXN. Let hi be an magnitude truncation is performed; if the number is nega- eigenvalue of A. We shall denote the column eigenvector tive, then it is also truncated, but at the end 2-' is added associated with Xi as ai,while' q! will denote the corre- to the truncated number b being the wordlength of the sponding row eigenvector, i.e., implementation. The norm of A, denoted by IIAl12, is defined to be A@,= A,@., and = A,+:. (2) X~A~AX 11A11: = max -. (5) Throughout this paper we shall assume that 'k, and @, are x+o xtx scaled such that \k;t@, =l.The kth components of 0,and The norm IIA1I2 is at least as large as the magnitude of the 'k, are denoted by +,(k) and +f(k), respectively. The dominant eigenvalue (i.e., the spectral radius) of A : transfer function H(z) of the IIR filter is related to the state-space parameters by llAll2 max IW)I. (6) A normal matrix [8] has its norm strictly equal to its H(Z)= d + c(~I-A)-'B. (3) spectral radius, therefore. llAl12 <1 as long as the eigenval- It is well known that the following similarity transforma- ues of A are inside the unit circle. So provided the filter is tion: stable, a normal-form realization will always satisfy the conditions given in [7], [16]. Hence it is possible to sup- A' = TA T-' B' = TB cf= CT-~ (4) press limit cycles in these structures. The purpose of ths paper is to introduce new normal-form filters and to leaves H( z) unchanged. The eigenvalues of A correspond explore their advantages. to the poles of the transfer function H(z).A normal form A normal matrix is any square matrix that satisfies the realization is one in which the state transition matrix A is condition AtA = AAt. This condition in turn holds if and a normal matrix. We shall assume that the poles H(z)are only if A has a complete set of N orthogonal eigenvectors all distinct, because a minimal system with non-distinct [8]. This is the same as saying A is diagonalizable by a poles does not have a normal-form realization. This can be unitary matrix. If A is normal, { @, } is a complete set of seen from the fact that if the N X N state transition matrix orthogonal eigenvectors. Then it can be shown that @, = is diagonalizable and has less than N distinct eigenvalues, for each i. Conversely, if for each i @, = 'k, within a scalar then such a realization cannot be minimal. Since a normal factor, then A is necessarily normal. Typical examples of matrix is always diagonalizable [8], any normal-form real- normal matrices are Hermitian matrices, and unitary ization having less than N distinct eigenvalues is neces- matrices [8]. sarily non-minimal. In the study of eigenvalue sensitivity, a commonly used In state-space implementation, it is possible to avoid global sensitivity measure [2] is defined to be both overflow oscillations and quantization limit cycles by requiring an upper bound of unity on the norm of the state transition matrix A [l].Barnes and Fam have introduced (7) the novel minimum-norm structures [l]by constraining A to be a normal matrix.
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