Basics of Signals

1 Chapter 2 Basics of What are Signals ?As mentioned in Chapter XX, a system designed to perform a particular taskoften uses measurements obtained from the environment and/or inputs from auser. These in turn may be converted into other forms. The physical variablesof interest are generally calledsignals. In an electrical system, the physicalvariables of interest might be a voltage, current, amount of charge, etc. Ina mechanical system, the variables of interest might be the position, velocity,mass, volume, etc. of various objects. Financial examples might include theprice of a stock, commodity, or option, an interest rate, or an exchange performing its tasks, the system may need to manipulate or combine varioussignals, extract information, or otherwise process the Signals . These actions arecalledsignal processingorsignal convenient abstraction is to model the value of a physical variable ofinterest by a number. We are usually interested in the physical variable notat just a single time, but rather at a set of times.

2 In this case, the signal is afunctionof time, sayf(t). For example,f(t) might denote a voltage level, orthe velocity of an object, or the price of a stock at some cases, we might be interested in measuring the quantity as a func-tion of some variable other than time. For example, suppose we are interestedin measuring the water temperature in the ocean as a function of depth. In thiscase, the signal is a function of a spatial variable, withf(x) denoting tempera-ture at signal need not be a function of just a single variable. To continue theexample above, suppose we are interested in the temperature at particular pointsin the ocean, not simply as a function of depth. In this case, we might letf(x, y, z) denote the temperature at the point (x, y, z), so the signal is a function c 1999-2002 by Sanjeev R. Kulkarni. All rights reserved. Lecture Notes for ELE201 Introduction to Electrical Signals and Systems. Thanks to Richard Radke for producing the 2. Basics OF SIGNALSof three variables.

3 Now, if we are also interested in how the temperature evolvesin time, the signalf(x, y, z, t) would be a function of four 3 2 10123secondsintensityThe word "Matlab"Figure : Someone saying the word Matlab Examples of Signals that we will encounter frequently are audio Signals ,images, and video. An audio signal is created by changes in air pressure, andtherefore can be represented by a function of timef(t) withfrepresentingthe air pressure due to the sound at timet. An example of an audio signalof someone saying Matlab is shown in Figure A black and white imagecan be represented as a functionf(x, y) of two variables. Here (x, y) denotes aparticular point on the image, and the valuef(x, y) denotes the brightness (orgray level) of the image at that example of a black and white image is shown in Figure A video canbe thought of as a sequence of images. Hence, a black and white video signal canbe represented by a functionf(x, y, t) of three variables (two spatial variablesand time).

4 In this case, for a fixedt,f( , , t) represents the still image/frame attimet, while for a fixed (x, y),f(x, y, ) denotes how the brightness at the point(x, y) changes as a function of frames of a video of a commercial are shown in Figure It turnsout that color images (or video) can be represented by a combination of threeintensity images (or video, respectively), as will be discussed later in Analog and Digital SignalsOften the domain and the range of a signalf(x) are modeled as is, the time (or spatial) coordinatexis allowed to take on arbitrary values(perhaps within some interval) and the value of the signal itself is allowed totake on arbitrary values (again within some interval). Such Signals are ANALOG AND DIGITAL SIGNALS3 Figure : A gray-scale :13 0:14 0:15 Figure : Video frames from a 2. Basics OF Signals analogsignals. A continuous model is convenient for some situations, but inother situations it is more convenient to work withdigitalsignals , signalsthat have a discrete (often finite) domain and range.

5 Two other related wordsthat are often used to describe Signals arecontinuous-timeanddiscrete-time,refer ring to Signals where the independent variable denotes time and takes oneither a continuous or discrete set of values, version of temperature graphFigure : Sampling an analog a signal that starts out as an analog signal needs to bedigitized( , converted to a digital signal ). The process of digitizing the domain is calledsampling. For example, iff(t) denotes temperature as a function of time, andwe are interested only in the temperature at 1 second intervals, we can samplefat the times of interest as shown in Figure example of sampling is shown in Figure An original imagef(x, y) is shown together with sampled versions of the image. In the sampledversions of the image, the blocks of constant intensity are calledpixels, and thegray level is constant within the pixel. The gray level value is associated withthe intensity at the center of the pixel. But rather than simply showing a smalldot in the center of the pixel, the whole pixel is colored with the same gray levelfor a more natural appearance of the image.

6 The effect of more coarse samplingcan be seen in the various images. Actually, the so-called original image inFigure is also sampled, but the sampling is fine enough that we don t noticeany process of digitizing the range is calledquantization. In quantizing asignal, the valuef(x) is only allowed to take on some discrete set of values (asopposed to the variablextaking on discrete values as in sampling).Figure shows the original temperature signalf(t) (shown previously inFigure ) as well various quantized versions off. Figure shows the imagefrom Figure and various quantized versions. In the quantized versions of theimages, the gray levels can take on only some discrete set of values. Actually, ANALOG AND DIGITAL SIGNALS5No subsampling4 x 4 blocks8 x 8 blocks16 x 16 blocksFigure : Sampling an 2. Basics OF SIGNALS0 1 0 1 Unquantized signal 0 1 0 1 32 levels 0 1 0 1 16 levels 0 1 0 1 8 levels Figure : Quantized versions of an analog so-called original image is also quantized, but because of the resolutionof the printer and limitations of the human visual system, a technique knownas halftoning (discussed in Chapter XX) can be used so that we don t noticeany artifacts due to quantization.

7 It is typical in images to let the gray leveltake on 256 integer values with 255 being the brightest gray level and 0 thedarkest. In Figures there are only 8, 4, and 2 gray levels respectively, andquantization artifacts become quite and quantization to digitize a signal seem to throw away muchinformation about a signal , and one might wonder why this is ever done. Themain reason is that digital Signals are easy to store and process with digitalcomputers. Digital Signals also have certain nice properties in terms of robust-ness to noise, as we ll discuss in Section XX. However, there are also situationsin which analog Signals are more appropriate. As a result there is often a needforanalog-to-digital conversionanddigital-to-analog conversion(also writtenA/D and D/A conversion). In digitizing Signals , one would also like to knowhow much information is lost by sampling and quantization, and how best todo these operations. The theory for sampling is clean and elegant, while thetheory for quantization is more difficult.

8 It turns out that choices for samplingrates and number of quantization levels also depend to a large extent on systemand user requirements. For example, in black-and-white images, 256 gray levelsis adequate for human viewing much more than 256 would be overkill, whilemuch less would lead to objectionable artifacts. We defer a more detailed con-sideration of sampling and quantization until Chapter XX after we have ANALOG AND DIGITAL SIGNALS7256 levels 8 levels 32 levels 16 levels 4 levels 2 levels Figure : Quantized versions of a gray-scale 2. Basics OF Signals some additional background discussing some basic operations on Signals , we describe a fairly com-mon notational convention which we will also follow. Continuous-time signalswill be denoted using parentheses, such asx(t), while discrete-time Signals willuse brackets such asx[n]. This convention also applies even if the indepen-dent variable represents something other than time. That is,y(u) denotes asignal where the domain is continuous, whiley[k] indicates a discrete domain,whether or not the independent variablesuandkrefer to time.

9 Often the lettersi, j, k, l, m, nare used to denote a discrete independent Some basic signal OperationsIn addition to the obvious operations of adding or multiplying two Signals , anddifferentiating or integrating a signal , certain other simple operations are quitecommon in signal processing . We give a brief description of some of these original signal is denoted byx(t). 8 6 4 2024680246 Original signal , f(x) 8 6 4 2024680246 Amplitude shifted signal , f(x)+ 8 6 4 2024680246 Time shifted signal , f(x+3)Figure : Amplitude- and time-shifted versions of a signala+x(t) whereais some number is just adding a constant signaltox(t) and simply shifts the range (or amplitude) of the signal by the amounta. A somewhat different operation is obtained when one shifts the domain ofthe signal . Namely, the signalx(t t0) is atime-shiftof the original signalx(t)by the amountt0. It s like a delayed version of the original signal . Figure amplitude and time-shifted versions of a SOME basic signal periodic signalsecondsFigure : A periodic some Signals , appropriate time shifts can leave the signal , a signal is said to beperiodicwith periodPifx(t P) =x(t) for allt.

10 That is, the signal simply repeats itself every P seconds. Figure shows anexample of a periodic scaling a signal to getax(t) is simply multiplyingx(t) with aconstant signala. However, a rather different operation is obtained when onescales the time domain. Namely, the signalx(at) is like the original signal ,but with the time axis compressed or stretched (depending on whethera >1 ora <1). Of course, ifa= 1 the signal is unchanged. Figure shows the effectsof amplitude and time scaling. For negative values ofa, the signal is flipped (or reflected ) about the range axis, in addition to any compression or particular, ifa= 1, the signal is reflected about the range axis, but thereis no stretching or compression. For some functions, the reflection about therange axis leaves the function unchanged, that is, the signal is symmetric aboutthe range axis. Formally, the property required for this isx( t) =x(t) for allt. Such functions are calledeven. A related notion is that of anoddfunction,for whichx( t) = x(t).