Wavelet Filter Banks Using Allpass Filters

Allpass filter is a computationally efficient versatile signal processing building block. The interconnection of allpass filters has found numerous applications in digital filtering and wavelets. In this chapter, we discuss several classes of wavelet filter banks by using allpass filters. Firstly, we describe two classes of orthogonal wavelet filter banks composed of two real allpass filters or a complex allpass filter, and then consider design of orthogonal filter banks without or with symmetry, respectively. Next, we present two classes of filter banks by using allpass filters in lifting scheme. One class is causal stable biorthogonal wavelet filter bank and another class is orthogonal wavelet filter bank, all with approximately linear phase response. We also give several design examples to demonstrate the effectiveness of the proposed method

Design of nonuniform near allpass complementary FIR filters via a semi-infinite programming technique

In this paper, we consider the problem of designing a set of nonuniform near allpass complementary FIR filters. This problem can be formulated as a quadratic semi-infinite programming problem, where the objective is to minimize the sum of the ripple energy for the individual filters, subject to the passband and stopband specifications as well as to the allpass complementary specification. The dual parameterization method is used for solving the linear quadratic semi-infinite programming problem

Design of near allpass strictly stable minimal phase real valued rational IIR filters

In this brief, a near-allpass strictly stable minimal-phase real-valued rational infinite-impulse response filter is designed so that the maximum absolute phase error is minimized subject to a specification on the maximum absolute allpass error. This problem is actually a minimax nonsmooth optimization problem subject to both linear and quadratic functional inequality constraints. To solve this problem, the nonsmooth cost function is first approximated by a smooth function, and then our previous proposed method is employed for solving the problem. Computer numerical simulation result shows that the designed filter satisfies all functional inequality constraints and achieves a small maximum absolute phase error

Design of doubly-complementary IIR digital filters using a single complex allpass filter, with multirate applications

It is shown that a large class of real-coefficient doubly-complementary IIR transfer function pairs can be implemented by means of a single complex allpass filter. For a real input sequence, the real part of the output sequence corresponds to the output of one of the transfer functions G(z) (for example, lowpass), whereas the imaginary part of the output sequence corresponds to its "complementary" filter H(z)(for example, highpass). The resulting implementation is structurally lossless, and hence the implementations of G(z) and H(z) have very low passband sensitivity. Numerical design examples are included, and a typical numerical example shows that the new implementation with 4 bits per multiplier is considerably better than a direct form implementation with 9 bits per multiplier. Multirate filter bank applications (quadrature mirror filtering) are outlined

High Input Impedance Voltage-Mode Universal Biquadratic Filters With Three Inputs Using Three CCs and Grounding Capacitors

Two current conveyors (CCs) based high input impedance voltage-mode universal biquadratic filters each with three input terminals and one output terminal are presented. The first circuit is composed of three differential voltage current conveyors (DVCCs), two grounded capacitors and four resistors. The second circuit is composed of two DVCCs, one differential difference current conveyor (DDCC), two grounded capacitors and four grounded resistors. The proposed circuits can realize all the standard filter functions, namely, lowpass, bandpass, highpass, notch and allpass filters by the selections of different input voltage terminals. The proposed circuits offer the features of high input impedance, using only grounded capacitors and low active and passive sensitivities. Moreover, the x ports of the DVCCs (or DDCC) in the proposed circuits are connected directly to resistors. This design offers the feature of a direct incorporation of the parasitic resistance at the x terminal of the DVCC (DDCC), Rx, as a part of the main resistance

On arbitrary-level IIR and FIR filters

A recently published method for designing IIR (infinite-impulse-response) digital filters with multilevel magnitude responses is reinterpreted from a different viewpoint. On the basis of this interpretation, techniques for extending these results to the case of finite-impulse-response (FIR) filters are developed. An advantage of the authors' method is that, when the arbitrary-level filter is implemented, its power-complementary filter, which may be required in specific applications, is obtained simultaneously. Also, by means of a tuning factor (a parameter of the scaling matrix), it is possible to generate a whole family of arbitrary-level filters

Cyclic LTI systems in digital signal processing

Cyclic signal processing refers to situations where all the time indices are interpreted modulo some integer L. In such cases, the frequency domain is defined as a uniform discrete grid (as in L-point DFT). This offers more freedom in theoretical as well as design aspects. While circular convolution has been the centerpiece of many algorithms in signal processing for decades, such freedom, especially from the viewpoint of linear system theory, has not been studied in the past. In this paper, we introduce the fundamentals of cyclic multirate systems and filter banks, presenting several important differences between the cyclic and noncyclic cases. Cyclic systems with allpass and paraunitary properties are studied. The paraunitary interpolation problem is introduced, and it is shown that the interpolation does not always succeed. State-space descriptions of cyclic LTI systems are introduced, and the notions of reachability and observability of state equations are revisited. It is shown that unlike in traditional linear systems, these two notions are not related to the system minimality in a simple way. Throughout the paper, a number of open problems are pointed out from the perspective of the signal processor as well as the system theorist

Digital filter design using root moments for sum-of-all-pass structures from complete and partial specifications

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Passive cascaded-lattice structures for low-sensitivity FIR filter design, with applications to filter banks

A class of nonrecursive cascaded-lattice structures is derived, for the implementation of finite-impulse response (FIR) digital filters. The building blocks are lossless and the transfer function can be implemented as a sequence of planar rotations. The structures can be used for the synthesis of any scalar FIR transfer function H(z) with no restriction on the location of zeros; at the same time, all the lattice coefficients have magnitude bounded above by unity. The structures have excellent passband sensitivity because of inherent passivity, and are automatically internally scaled, in an L_2 sense. The ideas are also extended for the realization of a bank of MFIR transfer functions as a cascaded lattice. Applications of these structures in subband coding and in multirate signal processing are outlined. Numerical design examples are included

Theory and design of uniform DFT, parallel, quadrature mirror filter banks

In this paper, the theory of uniform DFT, parallel, quadrature mirror filter (QMF) banks is developed. The QMF equations, i.e., equations that need to be satisfied for exact reconstruction of the input signal, are derived. The concept of decimated filters is introduced, and structures for both analysis and synthesis banks are derived using this concept. The QMF equations, as well as closed-form expressions for the synthesis filters needed for exact reconstruction of the input signalx(n), are also derived using this concept. In general, the reconstructed. signalhat{x}(n)suffers from three errors: aliasing, amplitude distortion, and phase distortion. Conditions for exact reconstruction (i.e., all three distortions are zero, andhat{x}(n)is equal to a delayed version ofx(n))of the input signal are derived in terms of the decimated filters. Aliasing distortion can always be completely canceled. Once aliasing is canceled, it is possible to completely eliminate amplitude distortion (if suitable IIR filters are employed) and completely eliminate phase distortion (if suitable FIR filters are employed). However, complete elimination of all three errors is possible only with some simple, pathalogical stable filter transfer functions. In general, once aliasing is canceled, the other distortions can be minimized rather than completely eliminated. Algorithms for this are presented. The properties of FIR filter banks are then investigated. Several aspects of IIR filter banks are also studied using the same framework