# Digital Filters

Gordana Jovanovic Dolecek (Institute for Astrophysics, Optics and Electronics, INAOE, Mexico)
DOI: 10.4018/978-1-60566-014-1.ch050

## Abstract

A signal is defined as any physical quantity that varies with changes of one or more independent variables, and each can be any physical value, such as time, distance, position, temperature, or pressure (Elali, 2003; Smith, 2002). The independent variable is usually referred to as “time”. Examples of signals that we frequently encounter are speech, music, picture, and video signals. If the independent variable is continuous, the signal is called continuous-time signal or analog signal, and is mathematically denoted as x(t). For discrete-time signals, the independent variable is a discrete variable; therefore, a discrete-time signal is defined as a function of an independent variable n, where n is an integer. Consequently, x(n) represents a sequence of values, some of which can be zeros, for each value of integer n. The discrete–time signal is not defined at instants between integers, and it is incorrect to say that x(n) is zero at times between integers. The amplitude of both the continuous and discrete-time signals may be continuous or discrete. Digital signals are discrete-time signals for which the amplitude is discrete. Figure 1 illustrates the analog and the discrete-time signals. Most signals that we encounter are generated by natural means. However, a signal can also be generated synthetically or by computer simulation (Mitra, 2006). Signal carries information, and the objective of signal processing is to extract useful information carried by the signal. The method of information extraction depends on the type of signal and the nature of the information being carried by the signal. “Thus, roughly speaking, signal processing is concerned with the mathematical representation of the signal and algorithmic operation carried out on it to extract the information present,’’ (Mitra, 2006, pp. 1).
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## Introduction

A signal is defined as any physical quantity that varies with changes of one or more independent variables, and each can be any physical value, such as time, distance, position, temperature, or pressure (Elali, 2003; Smith, 2002). The independent variable is usually referred to as “time”. Examples of signals that we frequently encounter are speech, music, picture, and video signals. If the independent variable is continuous, the signal is called continuous-time signal or analog signal, and is mathematically denoted as x(t). For discrete-time signals, the independent variable is a discrete variable; therefore, a discrete-time signal is defined as a function of an independent variable n, where n is an integer. Consequently, x(n) represents a sequence of values, some of which can be zeros, for each value of integer n. The discrete–time signal is not defined at instants between integers, and it is incorrect to say that x(n) is zero at times between integers. The amplitude of both the continuous and discrete-time signals may be continuous or discrete. Digital signals are discrete-time signals for which the amplitude is discrete. Figure 1 illustrates the analog and the discrete-time signals.

Figure 1.

Examples of analog and discrete-time signals

Most signals that we encounter are generated by natural means. However, a signal can also be generated synthetically or by computer simulation (Mitra, 2006).

Signal carries information, and the objective of signal processing is to extract useful information carried by the signal. The method of information extraction depends on the type of signal and the nature of the information being carried by the signal. “Thus, roughly speaking, signal processing is concerned with the mathematical representation of the signal and algorithmic operation carried out on it to extract the information present,’’ (Mitra, 2006, pp. 1).

Analog signal processing (ASP) works with the analog signals, while digital signal processing (DSP) works with digital signals. Since most of the signals that we encounter in nature are analog, DSP consists of these three steps:

• A/D conversion (transformation of the analog signal into the digital form);

• Processing of the digital version; and

• Conversion of the processed digital signal back into an analog form (D/A).

We now mention some of the advantages of DSP over ASP (Diniz, Silva, & Netto, 2002; Ifeachor & Jervis, 2001; Mitra, 2006; Stearns, 2002; Stein, 2000):

• Less sensitivity to tolerances of component values and independence of temperature, aging, and many other parameters;

• Programmability, that is, the possibility to design one hardware configuration that can be programmed to perform a very wide variety of signal processing tasks simply by loading in different software;

• Several valuable signal processing techniques that cannot be performed by analog systems, such as for example linear phase filters;

• More efficient data compression (maximum amount of information transferred in the minimum amount of time);

• Any desirable accuracy can be achieved by simply increasing the word length;

• Applicability of digital processing to very low frequency signals, such as those occurring in seismic applications (An analog processor would be physically very large in size.); and

• Recent advances in very large scale integrated (VLSI) circuits make it possible to integrate highly-sophisticated and complex digital signal processing systems on a single chip.

## Key Terms in this Chapter

FIR Filter: A digital filter with a finite impulse response. FIR filters are always stable. FIR filters have only zeros (all poles are at the origin).

Stable Filter: A filter for which a bounded input always results in a bounded output.

Digital Signal Processing: Extracts useful information carried by the digital signals and is concerned with the mathematical representation of the digital signals and algorithmic operations carried out on the signal to extract the information.

Magnitude Response: The absolute value of the Fourier transform of the unit sample response. For a real impulse response digital filter, the magnitude response is a real even function of the frequency.

Phase Response: The phase of the Fourier transform of the unit sample response. For a real impulse response digital filter, the phase response is an odd function of the frequency.

Digital Signal: A discrete-time signal whose amplitude is also discrete. It is defined as a function of an independent, integer-valued variable n. Consequently, a digital signal represents a sequence of discrete values (some of which can be zeros), for each value of integer n.

Signal: Any physical quantity that varies with changes of one or more independent variables which can be any physical value, such as time, distance, position, temperature, and pressure.

Digital Filter: The digital system which performs digital signal processing, that is, transforms an input sequence into a desired output sequence.

Impulse Response: The time domain characteristic of a filter and represents the output of the unit sample input sequence.

IIR Filter: A digital filter with an infinite impulse response. IIR filters always have poles and are stable if all poles are inside the unit circle.

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