Signals and Systems 1 Lecture 2 Dr. Ali. A. Jalali August 21, 2002

Signals and Systems 1 Lecture 2 Dr. Ali. A. Jalali August 21, 2002

Signals and Systems 1 Lecture # 2 Introduction to Signals EE 327 fall 2002

Signals: • Classification of Signals • Deterministic and Stochastic signals • Periodic and Aperiodic signals • Continuous time (CT) and Discrete time (DT) • Causal and anti-causal signals • Right and left sided signals • Bounded and unbounded signals • Even and odd signals EE 327 fall 2002

Classification of Signals • Deterministic - Predictive (An example is sin wave, square wave) • Stochastic – Non predictive (An example is noise signal or human voice) EE 327 fall 2002

Classification of Signals • Periodic and Aperiodic Signals Periodic signals have the property that x(t+T)=x(t) for all t. The smallest value of T satisfies the definition is called the period. Shown below are aperiodic signal (left) and a periodic signal (right). EE 327 fall 2002

Classification of Signals • Periodic Signals Simple Complex EE 327 fall 2002

Classification of Signals • Aperiodic Impulse Noise EE 327 fall 2002

Classification of Signals • For both exponential CT () and DT() signals, x is a complex quantity. To plot x, we can choose to plot either its magnitude and angle or its real and imaginary parts- whichever is more convenient for analysis. • For example, suppose s= j/8 • and , • then the real parts are: EE 327 fall 2002

Classification of Signals • Continuous-time (CT) and discrete time (DT) signals • CT signals take on real or complex values as a function of an independent variable that ranges over the real numbers and are denoted as x(t). • DT signals take on real or complex values as a function of an independent variable that ranges over the integers and are denoted as x[n]. EE 327 fall 2002

Classification of Signals • For example, consider the image shown on the left and its DT representation shown on the right. The image on the left consists of 302  435 picture elements (pixels) each of which is represented by a triplet of numbers {R,G,B} that encode the color. Thus, the signal is represented by c[n,m] where m and n are the independent variables that specify pixel location and c is a color vector specified by a triplet of hues {R,G,B} (red, green, and blue). EE 327 fall 2002

Classification of Signals • Real and complex signals • Signals can be real, imaginary, or complex. • An important class of signals are the complex exponentials: •  the CT signal where s is a complex number, •  the DT signal where z is a complex number. • Q. Why do we deal with complex signals? • A. They are often analytically simpler to deal with than real signals. EE 327 fall 2002

Classification of Signals • Causal and anti-causal signals • A causal signal is zero for t < 0 and • anti- causal signal is zero for t > 0. EE 327 fall 2002

Classification of Signals • Right- and left-sided signal • A right-sided signal is zero for t < T and a left-sided signal is zero for t > T where T can be positive or negative. EE 327 fall 2002

Classification of Signals • Bounded and unbounded signals EE 327 fall 2002

Classification of Signals • Even and odd signals • Even signals xe(t) and odd signals xo(t) are defined as xe(t) = xe(-t) and xo(t) = - xo(-t) EE 327 fall 2002

Classification of Signals • Any signal is a sum of unique odd and even signal. Using • x(t) = xe(t) + xo(t) and x(-t) = xe(t) - xo(t) Yields and EE 327 fall 2002

Measuring Signals • Measure Amplitude peak to peak amplitude mean amplitude (r.m.s.) • Measure Period and Repetition Frequency duration of 1 cycle (s) number of cycles per second (Hz) • How to measure shape? EE 327 fall 2002

Signal Analysis • Simple Periodic Signals called sinusoidal signals only need to know 3 things: period (or frequency), amplitude & phase EE 327 fall 2002

Representation of signalsBuilding-block signals; • We will represent signal as sums of building-block signals. • Important families of building-block signals are the eternal (everlasting ), complex exponentials and the unit impulse functions. EE 327 fall 2002

Building-block signals; • Eternal, complex exponentials • These signal have the form • for all t and for all n, where X, s, and z are complex numbers. • In general s is complex and can be written as s =  + j, • where  and  are the real and imaginary parts of s. EE 327 fall 2002

Building-block signals; • Eternal, complex exponentials- real s • If s =  is real and X is real then • , • And we get the family of real exponential functions. • Eternal, complex exponentials- imaginary s • If s = j is imaginary and X is real then • and we get the family of sinusoidal functions. EE 327 fall 2002

Building-block signals; • Eternal, complex exponentials- complex s • If s =  + j is complex and X is real then • And we get the family of damped sinusoidal functions. EE 327 fall 2002

Building-block signals; • For is plotted for different values of s superimposed on the complex s-plane. EE 327 fall 2002

Eternal complex exponentials- why are they important? • Almost any signal of practical interest can be represented as a superposition (sum) of eternal complex exponentials. • The output of a linear, time-invariant (LTI) system (to be defined next time) is simple to compute if the input is a sum of eternal complex exponentials. • Eternal complex exponentials are the eigenfuntions of characteristic (unforced, homogeneous) responses of LTI systems. EE 327 fall 2002

Building –block signals Unit impulse – definition • The unit impulse (t), is an important signal of CT systems. The Dirac delta function, is not a function in the ordinary sense. It is defined by the integral relation And is called a generalized function. • The unit impulse is not defined in terms of its values, but is defined by how it acts inside an integral when multiplied by a smooth function f(t). To see that the area of the unit impulse is 1, choose f(t) = 1 in the definition. We represent the unit impulse schematically as shown below; the number next to the impulse is its area. EE 327 fall 2002

Unit impulse- narrow pulse approximation • To obtain an intuitive feeling for the unit impulse, it is often helpful to imagine a set of rectangular pulses where each pulse has width  and height 1/ so that its area is 1. The unit impulse is the quintessential tall and narrow pulse! EE 327 fall 2002

Unit impulse- intuiting the definition • To obtain some intuition about the meaning of the integral definition of the impulse, we will use a tall rectangular pulse of unit area as an approximation to the unit impulse. • As the rectangular pulse gets taller and narrower, EE 327 fall 2002

Unit impulse- the shape does not matter • There is nothing special about the rectangular pulse approximation to the unit impulse. A triangular pulse approximation is just as good. As far as out definition is concerned both the rectangular and triangular pulse are equally good approximations. Both act as impulses. EE 327 fall 2002

Unit impulse- the values do not matter • The values of the approximation functions do not matter either. The function on the left has unit area and takes on the arbitrary value A for t=0. The function on the right, which we shall encounter frequently in later lectures, has the property that it has non-zero values at most of its values, all but a countably infinite number of points, nut still acts as a unit impulse What all these approximations have in common is that as gets small the area of each function occupies an increasingly narrow time interval centered on t=0. EE 327 fall 2002

Signals and Systems 1 Lecture 2 Dr. Ali. A. Jalali August 21, 2002