Introduction

This chapter deals with the design methods in which a desired frequency response is approximated by a transfer function consisting of a ratio of polynomials. In general, this type of transfer function yields an impulse response of infinite duration. Therefore, the systems approximated in this chapter are commonly referred to as IIR filters.

In general, IIR filters are able to approximate a prescribed frequency response with fewer multiplications than FIR filters. For that matter, IIR filters can be more suitable for some practical applications, especially those involving real-time signal processing.

In Section 6.2 we study the classical methods of analog filter approximation, namely the Butterworth, Chebyshev, and elliptic approximations. These methods are the most widely used for approximations meeting prescribed magnitude specifications. They originated in the continuous-time domain and their use in the discrete-time domain requires an appropriate transformation.

We then address, in Section 6.3, two approaches that transform a continuous-time transfer function into a discrete-time transfer function, namely the impulse-invariance and bilinear transformation methods.

Section 6.4 deals with frequency transformation methods in the discrete-time domain. These methods allow the mapping of a given filter type to another; for example, the transformation of a given lowpass filter into a desired bandpass filter.

In applications where magnitude and phase specifications are imposed, we can approximate the desired magnitude specifications by one of the classical transfer functions and design a phase equalizer to meet the phase specifications.

Introduction

In many applications of digital signal processing, it is necessary for different sampling rates to coexist within a given system. One common example is when two subsystems working at different sampling rates have to communicate and the sampling rates must be made compatible. Another case is when a wideband digital signal is decomposed into several nonoverlapping narrowband channels in order to be transmitted. In such a case, each narrowband channel may have its sampling rate decreased until its Nyquist limit is reached, thereby saving transmission bandwidth.

Here, we describe such systems which are generally referred to as multirate systems. Multirate systems are used in several applications, ranging from digital filter design to signal coding and compression, and have been increasingly present in modern digital systems.

First, we study the basic operations of decimation and interpolation, and show how arbitrary rational sampling-rate changes can be implemented with them. Then, we describe properties pertaining to the multirate systems, namely their valid inverse operations and the noble identities. With these properties introduced, the next step is to present the polyphase decompositions and the commutator models, which are key tools in multirate systems. The design of decimation and interpolation filters is also addressed. A step further is to deal with filter design techniques which use decimation and interpolation in order to achieve a prescribed set of filter specifications.

Introduction

In Chapter 8 we dealt with multirate systems in general; that is, systems in which more than one sampling rate coexist. Operations of decimation, interpolation, and sampling-rate changes were studied, as well as some filter design techniques using multirate concepts.

In a number of applications, it is necessary to split a digital signal into several frequency bands. After such decomposition, the signal is represented by more samples than in the original stage. However, we can attempt to decimate each band, ending up with a digital signal decomposed into several frequency bands without increasing the overall number of samples. The question is whether it is possible to recover the original signal exactly from the decimated bands. Systems which decompose and reassemble the signals are generally called filter banks.

In this chapter, we deal with filter banks, showing several ways in which a signal can be decomposed into critically decimated frequency bands, and recovered from them with minimum error. We start with an analysis of M-band filter banks, giving conditions for perfect reconstruction. Then we perform both frequency- and time-domain analyses of filter banks, followed by a discussion on orthogonality. We also treat two-band perfect reconstruction filter banks, and present the special designs for quadrature mirror filters (QMFs) and conjugate quadrature filters (CQFs). In addition, we shift to M-band filter banks, analyzing block transforms, cosine-modulated filter banks, and lapped transforms.

Introduction

In previous chapters we were introduced to some design techniques for FIR and IIR digital filters. Some of these techniques can also be used in other applications related to the general field of digital signal processing. In the present chapter we consider the very practical problem of estimating the power spectral density (PSD) of a given discrete-time signal y(n). This problem appears in several applications, such as radar/sonar systems, music transcription, speech modeling, and so on. In general, the problem is often solved by first estimating the autocorrelation function associated with the data at hand, followed by a Fourier transform to obtain the desired spectral description of the process, as suggested by the Wiener–Khinchin theorem to be described in this chapter.

There are several algorithms for performing spectral estimation. Each one has different characteristics with respect to computational complexity, precision, frequency resolution, or other statistical aspects. We may classify all algorithms as nonparametric or parametric methods. Nonparametric methods do not assume any particular structure behind the available data, whereas parametric schemes consider that the process follows some pattern characterized by a specific set of parameters pertaining to a given model. In general, parametric approaches tend to be simpler and more accurate, but they depend on some a priori information regarding the problem at hand.

Introduction

In Chapter 9 we dealt with filter banks, which are important in several applications. In this chapter, wavelet transforms are considered. They come from the area of functional analysis and generate great interest in the signal processing community, because of their ability to represent and analyze signals with varying time and frequency resolutions. Their digital implementation can be regarded as a special case of critically decimated filter banks. Multiresolution decompositions are then presented as an application of wavelet transforms. The concepts of regularity and number of vanishing moments of a wavelet transform are then explored. Two-dimensional wavelet transforms are introduced, with emphasis on image processing. Wavelet transforms of finite-length signals are also dealt with. We wrap up the chapter with a Do-it-yourself section followed by a brief description of functions from the Matlab Wavelet Toolbox which are useful for wavelets implementation.

Wavelet transforms

Wavelet transforms are a relatively recent development in functional analysis that have attracted a great deal of attention from the signal processing community (Daubechies, 1991). The wavelet transform of a function belonging to ℒ2{ℝ}, the space of the square integrable functions, is its decomposition in a base formed by expansions, compressions, and translations of a single mother function ψ(t), called a wavelet.

The applications of wavelet transforms range from quantum physics to signal coding. It can be shown that for digital signals the wavelet transform is a special case of critically decimated filter banks (Vetterli & Herley, 1992).

Introduction

This chapter starts by addressing some implementation methods for digital filtering algorithms and structures. The implementation of any building block of digital signal processing can be performed using a software routine on a simple personal computer. In this case, the designer's main concern becomes the description of the desired filter as an efficient algorithm that can be easily converted into a piece of software. In such cases, the hardware concerns tend to be noncritical, except for some details such as memory size, processing speed, and data input/output.

Another implementation strategy is based on specific hardware, especially suitable for the application at hand. In such cases, the system architecture must be designed within the speed constraints at a minimal cost. This form of implementation is mainly justified in applications that require high processing speed or in large-scale production. The four main forms of appropriate hardware for implementing a given system are:

• The development of a specific architecture using basic commercial electronic components and integrated circuits (Jackson et al., 1968; Peled & Liu, 1974, 1985; Freeny, 1975; Rabiner & Gold, 1975;Wanhammar, 1981).

• The use of programmable logic devices (PLDs), such as field-programmable gate arrays (FPGAs), which represent an intermediate integrated stage between discrete hardware and full-custom integrated circuits or digital signal processors (DSPs) (Skahill, 1996).

• The design of a dedicated integrated circuit for the application at hand using computer-automated tools for a very large scale integration (VLSI) design.

• […]

Introduction

The most widely used realizations for IIR filters are the cascade and parallel forms of second-order, and, sometimes, first-order, sections. The main advantages of these realizations come from their inherent modularity, which leads to efficient VLSI implementations, to simplified noise and sensitivity analyses, and to simple limit-cycle control. This chapter presents high-performance second-order structures, which are used as building blocks in high-order realizations. The concept of section ordering for the cascade form, which can reduce roundoff noise in the filter output, is introduced. Then we present a technique to reduce the output roundoff-noise effect known as error spectrum shaping. This is followed by consideration of some closed-form equations for the scaling coefficients of second-order sections for the design of parallel-form filters.

We also deal with other interesting realizations, such as the doubly complementary filters, made from allpass blocks, and IIR lattice structures, whose synthesis method is presented. Arelated class of realizations is the wave digital filters, which have very low sensitivity and also allow the elimination of zero-input and overflow limit cycles. The wave digital filters are derived from analog filter prototypes, employing the concepts of incident and reflected waves. The detailed design of these structures is presented in this chapter.

IIR parallel and cascade filters

The Nth-order direct forms seen in Chapter 4, Figures 4.11–4.13, have roundoff-noise transfer functions Gi(z) (see Figure 11.16) and scaling transfer functions Fi(z) (see Figure 11.20) whose L2 or L∞ norms assume significantly high values.

Introduction

In this chapter, alternative realizations to those introduced in Chapter 5 for FIR filters are discussed.

We first present the lattice realization, highlighting its application to the design of linear-phase perfect reconstruction filter banks. Then, the polyphase structure is revisited, discussing its application in parallel processing. We also present an FFT-based realization for implementing the FIR filtering operation in the frequency domain. Such a form can be very efficient in terms of computational complexity, and is particularly suitable for offline processing, although widely used in real-time implementations. Next, the so-called recursive running sum is described as a special recursive structure for a very particular FIR filter, which has applications in the design of FIR filters with low arithmetic complexity.

In the case of FIR filters, the main concern is to examine methods which aim at reducing the number of arithmetic operations. These methods lead to more economical realizations with reduced quantization effects. In this chapter, we also present the prefilter, the interpolation, and the frequency-response masking approaches for designing lowpass and highpass FIR filters with reduced arithmetic complexity. The frequency-response masking method can be seen as a generalization of the other two schemes, allowing the design of passbands with general widths. For bandpass and bandstop filters, the quadrature approach is also introduced.

Lattice form

Figure 12.1 shows the block diagram of a nonrecursive lattice filter of order M, which is formed by concatenating basic blocks of the form shown in Figure 12.2.

When we hear the word “signal” we may first think of a phenomenon that occurs continuously over time that carries some information. Most phenomena observed in nature are continuous in time, such as for instance our speech or heart beating, the temperature of the city where we live, the car speed during a given trip, the altitude of the airplane we are traveling in – these are typical continuous-time signals. Engineers are always devising ways to design systems, which are in principle continuous time, for measuring and interfering with these and other phenomena.

One should note that, although continuous-time signals pervade our daily lives, there are also several signals which are originally discrete time; for example, the stock-market weekly financial indicators, the maximum and minimum daily temperatures in our cities, and the average lap speed of a racing car.

If an electrical or computer engineer has the task of designing systems to interact with natural phenomena, their first impulse is to convert some quantities from nature into electric signals through a transducer. Electric signals, which are represented by voltages or currents, have a continuous-time representation. Since digital technology constitutes an extremely powerful tool for information processing, it is natural to think of processing the electric signals generated using it. However, continuous-time signals cannot be processed using computer technology (digital machines), which are especially suited to deal with sequential computation involving numbers.

Introduction

In this chapter we will study the approximation schemes for digital filters with FIR and we will present the methods for determining the multiplier coefficients and the filter order, in such a way that the resulting frequency response satisfies a set of prescribed specifications.

In some cases, FIR filters are considered inefficient, in the sense that they require a high-order transfer function to satisfy the system requirements when compared with the order required by IIR digital filters. However, FIR digital filters do possess a few implementation advantages, such as a possible exact linear-phase characteristic and intrinsically stable implementations, when using nonrecursive realizations. In addition, the computational complexity of FIR digital filters can be reduced if they are implemented using fast numerical algorithms such as the FFT.

We start by discussing the ideal frequency response characteristics of commonly used FIR filters, as well as their corresponding impulse responses. We include lowpass, highpass, bandpass, and bandstop filters in the discussion, and also treat two other important filters, namely differentiators and Hilbert transformers.

We go on to discuss the frequency sampling and the window methods for approximating FIR digital filters, focusing on the rectangular, triangular, Bartlett, Hamming, Blackman, Kaiser, and Dolph–Chebyshev windows. In addition, the design of maximally flat FIR filters is addressed.

Following this, numerical methods for designing FIR filters are discussed. A unified framework for the general approximation problem is provided.