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Pipelined Lattice and Wave Digital Recursive Filters uses look-ahead transformation and constrained filter design approaches. It is also shown that pipelining often reduces the roundoff noise in a digital filter. The pipelined recursive lattice and wave digital filters presented are well suited where increasing speed and reducing area or power or roundoff noise are important. Examples are wireless and cellular codec applications, where low power consumption is important, and radar and video applications, where higher speed is important.
The book presents pipelining of direct-form recursive digital filters and demonstrates the usefulness of these topologies in high-speed and low-power applications. It then discusses fundamentals of scaling in the design of lattice and wave digital filters. Approaches to designing four different types of lattice digital filters are discussed, including basic, one-multiplier, normalized, and scaled normalized structures. The roundoff noise in these lattice filters is also studied. The book then presents approaches to the design of pipelined lattice digital filters for the same four types of structures, followed by pipelining of orthogonal double-rotation digital filters, which eliminate limit cycle problems. A discussion of pipelining of lattice wave digital filters follows, showing how linear phase, narrow-band, sharp-transition recursive filters can be implemented using this structure. This example is motivated by a difficult filter design problem in a wireless codec application. Finally, pipelining of ladder wave digital filters is discussed.
Pipelined Lattice and Wave Digital Recursive Filters serves as an excellent reference and may be used as a text for advanced courses on the subject.

Inhaltsverzeichnis

Frontmatter

1. Introduction

Abstract
In order to exploit VLSI for high performance, we need to understand the characteristics of the scaled VLSI technologies. For example, VLSI offers a greater potential for complexity than speed, favors replication of one function, and imposes a high cost in performance for non-localized communication. Design costs can be minimized by composing the system as a replication of simple processing elements. These considerations favor implementations which feature arrays of identical or easily parametrized processing elements (since, these are easily given a software procedural definition) with mostly localized interconnections (for reduced communication costs). This has led to an interest in systolic- and wavefront-array implementations [1, 2].
Jin-Gyun Chung, Keshab K. Parhi

2. Pipeline Interleaving in Digital Filters

Abstract
Pipeline interleaving notion is an old idea, and has been used in general purpose computers. Pipeline interleaving approach has been advocated for programmable implementation of signal processing systems using deeply pipelined programmable digital signal processors [65], and for cyclostatic implementation of signal processing systems [66].
Jin-Gyun Chung, Keshab K. Parhi

3. Pipelining Direct Form Recursive Digital Filters

Abstract
This chapter presents approaches for pipelining direct-form recursive digital filter topologies using clustered and scattered look-ahead techniques. With clustered look-ahead, pipelined realizations require a linear complexity in the number of loop pipeline stages, and are not guaranteed to be stable. We illustrate use of scattered look-ahead approach to derive stable pipelined filters and then propose a decomposition technique to implement the non-recursive portion generated due to the scattered look-ahead process in a decomposed manner to obtain an implementation with logarithmic increase in hardware with respect to the number of loop pipeline stages. The decomposition technique is the key in obtaining area-efficient implementations, and makes pipelined realizations attractive for high speed VLSI IIR filter implementations. In addition to the clustered and the scattered look-ahead techniques, constrained filter design techniques are introduced which achieve pipelining without pole-zero cancellation. We also present fully pipelined and fully hardware efficient linear bidirectional systolic arrays for recursive filters based on scattered look-ahead.
Jin-Gyun Chung, Keshab K. Parhi

4. Roundoff Noise in Pipelined Recursive Digital Filters

Abstract
When a digital filter transfer function is implemented using a digital system, it invariably involves quantization of signals and coefficients in the system. As a result, the overall input-output behavior is not ideal. There are two basic types of quantization effects in any implementation [80, 81]. The first is due to parameter (coefficient) quantization. The result of parameter quantization is that the actual implemented transfer function is different from the ideal transfer function.
Jin-Gyun Chung, Keshab K. Parhi

5. Schur Algorithm

Abstract
The Schur algorithm was originally used to test if a power series is analytic and bounded in the unit disk [84]. If an N-th order polynomial Φ N (z) has all zeros inside the unit circle, N + 1 polynomials {Φ i (z), i = N, N − 1,⋯, 0} can be generated by the Schur algorithm. One of the most important properties of the Schur algorithm is that these N + 1 polynomials are orthonormal to each other and can be used as orthonormal basis functions to expand any N-th order polynomial. This orthonormality of the Schur algorithm has been exploited to synthesize various types of lattice filters.
Jin-Gyun Chung, Keshab K. Parhi

6. Digital Lattice Filter Structures

Abstract
The simplest form of the IIR digital filter structures is the direct-form structure, where the numerator and the denominator coefficients are directly used as multiplier coefficients in the implementation. However, this structure has very-high sensitivity. The reason for this is that the roots of a polynomial are very sensitive to the coefficients, so the poles and zeros of the given transfer function are very sensitive to the quantized multiplier coefficients [80]. With standard filters such as lowpass, highpass, and bandpass, the poles are generally crowded at angles close to the band edge. Sensitivity of the structure becomes worse as the number of crowded poles increases. This sensitivity problem can be avoided by implementing the transfer function as a sum or product of first and second-order sections, i.e., parallel or cascade form structures. However, for complex conjugate poles with small angles (e.g., narrow-band sharp-transition filters), we still have high sensitivity problems even with second order sections.
Jin-Gyun Chung, Keshab K. Parhi

7. Pipelining of Lattice IIR Digital Filters

Abstract
In this chapter, we address pipelining of lattice digital filters. Although the lattice filters can be pipelined by the cutset localization procedure, the maximum sample rate of these pipelined filters is limited by the feedback loop computations. For example, in normalized lattice filters, the cutset localization procedure can be applied to transfer one half of each delay on the right directed edges to the left directed edges as shown in Fig. 7.1. The half delays can be implemented by time rescaling. For example, using one clock cycle to represent a half delay, we can input one sample every two clock cycles and generate the output samples once every two clock cycles. With this transformation, the clock speed can be increased, but the sample rate cannot be increased, since multiple clock cycles are needed to process one sample. The maximum sample rate of this structure is limited by the feedback loop computation which involves two multiplications and two additions.
Jin-Gyun Chung, Keshab K. Parhi

8. Pipelining of Orthogonal Double-Rotation Digital Lattice Filters

Abstract
The ODR (Orthogonal Double-Rotation) digital lattice filters were developed for the realization of any stable, passive digital rational transfer function in a cascaded interconnection of orthogonal sections. Therefore, these filters possess desirable properties for VLSI implementation such as local connection, regularity and pipelinability. Each section of these filters is realized involving only Givens rotations and storage elements, which enables the translation of the sensitivity arguments for analog lossless ladder realizations to the ODR digital lattice filters. The simulations in [28] show that the ODR lattice filters have very low sensitivities with respect to perturbations of its parameters. Also, limit cycle oscillations can always be eliminated in this filter structure.
Jin-Gyun Chung, Keshab K. Parhi

9. Pipelined Lattice WDF Design for Wideband Digital Filters

Abstract
When a digital filter transfer function is implemented using a digital system, it invariably involves quantization of signals and coefficients in the system. The effects of quantization in most IIR digital filters can be improved by realizing the transfer function in terms of a cascade or parallel arrangement of second-order filter sections instead of a direct-form realization. However, for narrow/wideband sharp-transition filters, high sensitivity problems occur even with second-order sections since most of the poles are very close to the unit circle. In this case, the WDF, which is based upon the digital simulation of classical lossless filters, can be a good solution. In some applications, however, the required word-length is still quite high even with WDFs.
Jin-Gyun Chung, Keshab K. Parhi

10. Synthesis and Pipelining of Ladder WDFs in Digital Domain

Abstract
Classical doubly-terminated lossless networks designed to meet maximum available power bounds are known to have good passband sensitivity, as explained by Orchard’s principle [106]. Several of the good properties of these networks are a direct consequence of their passivity and losslessness. WDFs are a family of filter structures derived by imitating these classical doubly-terminated lossless networks. A typical WDF design starts with a prototype LC network, and each element in the LC network is transformed into an equivalent digital element. The transformation is carried out using the wave quantities known from the classical circuits [107]. Due to the inherent passivity of the LC prototype, the WDF is also passive in a certain sense [31], and consequently the WDF exhibits very low passband sensitivity, leading to reduced accuracy requirements for the multiplier coefficients and good dynamic range.
Jin-Gyun Chung, Keshab K. Parhi

Backmatter

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