Notes to Teachers and Students:

Filtering is one of the most important things that electrical and computer engineers do. In this chapter we extend everyday understanding of filters to numerical filters. We then study weighted moving averages and exponential averages. We define the important test signals for electrical and computer engineering and show how filters respond to them. The idea that filters are characterized by their response to simple test signals is fundamental. In the numerical experiment, students explore the frequency response of a simple filter, a concept that forms the basis of circuit theory, electronics, optics and lasers, solid-state devices, communications, and control.

Introduction

A filter is any device that passes material, light, sound, current, velocity, or information according to some rule of selectivity. Material (or mechanical) filters are commonplace in your everyday life:

The first three of these examples selectively pass material according to size; the last two selectively pass material according to its mass density.

Typical filters for light are

Satellite Television. Among current filters, the tuner in a super-heterodyne receiver is, perhaps, the first example that comes to mind. But satellite TV filters are another fascinating example. A typical C-band satellite has twelve transponders (or repeaters), each of which transmits microwave radiation in a personalized 36 MHz band. (The abbreviation MHz stands for megahertz, or 106106 Hz, or 106106 cycles per second. Other common abbreviations are Hz for 1 Hz, kHz for 103103 Hz, and GHz for 109109 Hz.) Each transponder actually transmits two channels of information, one vertically polarized and one horizontally polarized. There is an 8 MHz guard band between each band, and the vertical and horizontal channels are offset by 20 MHz. The transmission scheme for the 24 channels is illustrated in Figure 1. The entire transmission band extends over 540 MHz, from 3.7×1093.7×109 Hz to 4.24×1094.24×109 Hz. The satellite receiver has two different microwave detectors, one for vertical and one for horizontal polarization, and a microwave tuner to tune into the microwave band of interest.

Figure 1: Satellite TV– V_i, Vertically Polarized Channel i;Hji;Hj, Horizontally Polarized Channel j

An Aside on Hertz and Seconds. The abbreviation Hz stands for hertz, or cycles/second. It is used to describe the frequency of a sinusoidal signal. For example, house current is 60 Hz, meaning that it has 60 cycles each second. The inverse of Hz is seconds or, more precisely, seconds/cycle, the period of 1 cycle. For example, the period of 1 cycle for house current is 1/60 second. When we are dealing with sound, electricity, and electromagnetic radiation, we need a concise language for dealing with signals and waves whose frequencies range from 0 Hz (called DC or direct current) to 10181018 Hz (visible light). Table 1 summarizes the terms and symbols used to describe the frequency and period of signals that range in frequency from 0 Hz to 10121012 Hz.

Numerical Filters. Rather amazingly, these ideas extend to the domain of numerical filters, the topic of this chapter. Numerical filters are just schemes for weighting and summing strings of numbers. Stock prices are typically averaged with numerical filters. The curves in Figure 2 illustrate the daily closing average for Kellogg's common stock and two moving averages. The 50-day moving average is obtained by passing the daily closing average through a numerical filter that averages the most current 50 days' worth of closing averages. The 200-day moving average for the stock price is obtained by passing the daily closing prices through a numerical filter that averages the most current 200 days' worth of daily closing averages. The daily closing averages show fine-grained variation but tend to conceal trends. The 50-day and 200-day averages show less fine-grained variation but give a clearer picture of trends. In fact, this is one of the key ideas in numerical filtering: by selecting our method of averaging, we can filter out fine-grained variations and pass long-term trends (or vice versa), or we can filter out periodic variations and pass nonperiodic variations (or vice versa). Figure 2 illustrates that moving averages typically lag increasing sequences of numbers and lead decreasing sequences. Can you explain why?

We will call any algorithm or procedure for transforming one set of numbers into another set of numbers a numerical filter or digital filter. Digital filters, consisting of memories and arithmetic logic units (ALUs), are implemented in VLSI circuits and used for communication, control, and instrumentation. They are also implemented in random–or semicustom–logic circuits and in programmable microcomputer systems. The inputs to a digital filter are typically electronic measurements that are produced by A/D (analog-to-digital) conversion of the output of an electrical or mechanical sensor. The outputs of the filter are “processed,” “filtered,” or “smoothed” versions of the measurements. In your more advanced courses in electrical and computer engineering you will study signal processing and system theory, assembly language programming, microprocessor system development, and computer design. In these courses you will study the design and programming of hardware that may be used for digital filtering.

Figure 2: Dow-Jones Averages (Adapted from the New York Stock Exchange, Daily Graphs, William O'Neil and Co., Inc., Los Angeles, California)