Clifford T. Mullis

Synthesis of minimum roundoff noise fixed point digital filters

Beginning with an external specification of a digital filter, structures which minimize roundoff noise are investigated. After fixing the probability of overflow through an l_{2} scaling procedure, roundoff noise is studied via the internal structure of the filter using a state variable formulation. An output noise variance formula in terms of the internal structure is derived. Conditions for minimizing this output noise are established and realizations which meet these conditions are constructed. A new set of filter invariants called second-order modes are defined and shown to play a definitive role in minimal noise realizations. From these invariants, for example, one can calculate the minimal output noise variance of a given external specification. Numerical results are given which compare these new filter structures with the usual parallel and cascade connections of second-order filters, both theoretically and through simulations. For narrow-band filters, these new structures can be orders of magnitude better (in terms of output noise variance). One drawback of these new structures is a large increase in the number of multipliers needed to realize them. However, by applying the theory to subfilters connected in parallel and cascade, a good compromise between output noise and number of multipliers is obtained.

Roundoff noise in digital filters: Frequency transformations and invariants

The family of filters {H(F(z)):F(z) a frequency transformation} generated from a prototype filter H(z) is shown to possess certain common properties. These are coordinate-free quantities (called second-order modes) which are invariant under frequency transformation. The invariance is significant in the design of low-noise fixed-point digital filter structures since the second-order modes characterize the minimum attainable noise. Filter structures (including parallel, cascade, and ladder configurations) are studied whose output noise is essentially independent of bandwidth and center frequency. An analysis of direct form structures (whether isolated or as one section within a cascade or parallel configuration) results in an expression giving the dominant term in the output noise as a function of the parameter in the low-pass-low-pass transformation. This noise term approaches infinity as bandwidth approaches zero. Thus, for narrowband filters, a difference of several orders of magnitude in the output noise can exist between a scaled direct form (having six multiplications per two-pole section) and the optimal form (having nine multiplications per two-pole section).

Canonical coordinates and the geometry of inference, rate, and capacity

Canonical correlations measure the cosines of principal angles between random vectors. These cosines multiplicatively decompose concentration ellipses for second-order filtering and additively decompose the information rate for the Gaussian channel. More over, they establish a geometrical connection between error covariance, error rate, information rate, and principal angles. There is a limit to how small these angles can be made, and this limit determines the channel capacity.

Detection and estimation of improper complex random signals

Nonstationary complex random signals are in general improper (not circularly symmetric), which means that their complementary covariance is nonzero. Since the Karhunen-Loeve (K-L) expansion in its known form is only valid for proper processes, we derive the improper version of this expansion. It produces two sets of eigenvalues and improper observable coordinates. We then use the K-L expansion to solve the problems of detection and estimation of improper complex random signals in additive white Gaussian noise. We derive a general result comparing the performance of conventional processing, which ignores complementary covariances, with processing that takes these into account. In particular, for the detection and estimation problems considered, we find that the performance gain, as measured by deflection and mean-squared error (MSE), respectively, can be as large as a factor of 2. In a communications example, we show how this finding generalizes the result that coherent processing enjoys a 3-dB gain over noncoherent processing.

The Euclid algorithm and the fast computation of cross-covariance and autocovariance sequences

A simple linear procedure is given to compute the cross-covariance sequence associated with the outputs of two rational digital transfer functions driven by the same white noise sequence. Such a computation often appears in the study of digital filters, in Wiener filtering, in noise variance estimation, in the study of low-order approximations, and in the study of multichannel systems. A fast algorithm based on the Euclid algorithm is introduced to solve the linear system of equations involved in the computation, and a detailed analysis of the matrix is given. The special case of the autocovariance computation is reviewed, and the same study is performed. Alternate polynomial presentations are given and are shown to involve the same matrices and similar fast algorithms. >

Low roundoff noise and normal realizations of fixed point IIR digital filters

Explicit design and performance equations are given in terms of the parameters of the desired transfer function H(z) for two classes of realizations. These realizations offer the digital circuit designer an alternative to the usual designs based on direct forms. In addition to low roundoff noise, these new realizations offer other desirable properties such as low coefficient sensitivities and freedom from overflow oscillations. Numerical results are presented which detail the design process and verify the theory.

A Newton-Raphson method for moving-average spectral factorization using the Euclid algorithm

An implementation of the Newton-Raphson approach to compute the minimum phase moving-average spectral factor of a finite positive definite correlation sequence is presented. Each step in the successive approximation method involves a system of linear equations that is solved using either the Levinson algorithm backwards (the Jury stability test), or a symmetrized version of the Euclid algorithm. Various properties of the Newton-Raphson map are studied. The algorithm is generalized to other symmetries (other than with respect to the unit circle). The special case of the symmetry with respect to the imaginary axis is presented and related to the Routh-Hurwitz stability test for continuous time transfer function. >

Minimum mean-squared error transform coding and subband coding

Knowledge of the power spectrum of a stationary random sequence can be used for quantizing the signal efficiently and with minimum mean-squared error. A multichannel filter is used to transform the random sequence into an intermediate set of variables that are quantized using independent scalar quantizers, and then inverse-filtered, producing a quantized version of the original sequence. Equal word-length and optimal word-length quantization at high bit rates is considered. An analytical solution for the filter that minimizes the mean-squared quantization error is obtained in terms of its singular value decomposition. The performance is characterized by a set of invariants termed second-order modes, which are derived from the eigenvalue decomposition of the matrix-valued power spectrum. A more general rank-reduced model is used for decreasing distortion by introducing bias. The results are specialized to the case when the vector-valued time series is obtained from a scalar random sequence, which gives rise to a filter bank model for quantization. The asymptotic performance of such a subband coder is derived and shown to coincide with the asymptotic bound for transform coding. Quantization employing a single scalar pre- and postfilter, traditional transform coding using a square linear transformation, and subband coding in filter banks, arise as special cases of the structure analyzed here.

A modular and orthogonal digital filter structure for parallel processing

An orthogonal filter structure is presented, whose synthesis is based on a generalized Levinson algorithm. This structure has a ladder-like flow graph topology. It is modular and has a high degree of computational parallelism. Implementation guidelines in the framework of parallel data processing are considered. A closed form for the round off noise error variance is given, from which it appears that the structure will be particularly useful for the implementation of narrow bandwidth filters.

Least squares approximation of perfect reconstruction filter banks

Designing good causal filters for perfect reconstruction (PR) filter banks is a challenging task due to the unusual nature of the design constraints. We present a new least squares (LS) design methodology for approximating PRFBs that avoids most of these difficult constraints. The designer first selects a set of subband analysis filters from an almost unrestricted class of rational filters. Then, given some desired reconstruction delay, this design procedure produces the causal and rational synthesis filters that result in the best least squares approximation to a PRFB. This technique is built on a multi-input multi-output (MIMO) system model for filter banks derived from the filter bank polyphase representation. Using this model, we frame the LS approximation problem for PRFBs as a causal LS equalization problem for MIMO systems. We derive the causal LS solution to this design problem and present an algorithm for computing this solution. The resulting algorithm includes a MIMO spectral factorization that accounts for most of the complexity and computational cost for this design technique. Finally, we consider some design examples and evaluate their performance.