Qassim University College of Engineering Electrical Engineering Department

1. Qassim UniversityCollege of Engineering Electrical Engineering Department Course: EE301: Signals and Systems AnalysisPrerequisite: EE202 Text Books: * B.P. Lathi, “Modern Digital and Analog Communication Systems”, 3th edition, Oxford University Press, Inc., 1998 * Sanjay Sharma; “Signals and Systems”, S.K. Kataria & Sons, 6th edition, 2008. Ref Books: * Oppenheim, A. Willsky and W. Nawab; “Signals and Systems”, Prentice- Hall, 1997 Instructor: Associate Prof. Dr. Ahmed Abdelwahab

2. INTRODUCTION Nowadays, electrical systems can play very vital roles with diverse applications such as circuit design, communications, biomedical engineering, machines, energy generation and distribution, etc. For example, wireless communications can transmit information signals over much longer distances (even to distant planets and galaxies) and at the speed of light. Moreover, important discussions now mostly communicated face to face in meetings or conferences, often requiring travel, are increasingly using teleconferencing. Similarly, teleshopping and telebanking provide reliable and economic services by electronic communication, and newspapers are now replaced by electronic news services.

3. Signals and Systems Signals are processed by systems • A signal, as the term implies, is a set of information or data. Examples include a telephone or a television signal, monthly sales of a corporation, or the daily closing prices of a stock market (e.g., the Dow Jonesaverages). However, in electrical sense, the signal (voltage or current) is defined as a function of one or more independent variables such as time and/or space. • A system may be made up of physical components connected together to produce an output signal in response to an input signal, as in electrical, mechanical, or hydraulic systems (hardware realization), or it may be an algorithm that computes an output from an input signal (software realization). The response (output) of the system depends upon the transfer function of the system.

4. CLASSIFICATION OF SIGNALS A signal that is specified for every value of time t is a continuous-time signal, g(t). A signal that is specified only at discrete values of t is a discrete-time signal, g(nT). Where T is the time duration between two successive samples (data values). Simply it can be written as g(n), where n is any integer number to represent the time independent discrete variable. A signal whose amplitude can take on any value in a continuous finite range is an analog signal. This means that an analog signal amplitude can take on an infinite number of values within a finite range. A digital signal, on the other hand, is one whose amplitude can take on only a finite number of values. A digital signal whose amplitudes can take on M values is called an M-ary signal of which binary (M = 2) is a special case. The terms continuous time and discrete time qualify the nature of a signal along the time (horizontal) axis. The terms analog and digital, on the other hand, qualify the nature of the signal amplitude (vertical axis). A signalis said to be periodic when g(t) = g (t ± To) for all t, where To is the signal period. Similarly, g(n) = g (n ± N), where N is the period of the signal. A signalis said to be aperiodic if it is not periodic. The second important property of a periodic signal g(t) is that g(t) can be generated by periodic extension of any segment of g(t) of duration To (the period).

6. The sum of two or more sinusoids may be periodic if the ratio of their periods is a rational number. A signal whose physical description is known completely, in either a mathematical form or a graphical form, is a deterministic signal. If a signal is known only in terms of its probabilistic description, such as mean value, mean squared value, and so on, rather than its complete mathematical or graphical description, is a random signal. Most of the noise signals encountered in practice are random signals. All message signals are random signals because, as will be shown later, a signal, to convey information, must have some uncertainty (randomness) about it.

7. Even (Symmetric) and Odd (Asymmetric) Signals • For Even (symmetrical) signals: g(t) = g(-t) or g(n) = g(-n) • For Odd (asymmetrical) signals: g(t) = - g(-t) or g(n) = - g(-n) Any continuous (or discrete) time signal g(t) can be expressed as the summation of an even signal ge(t) and an odd signal go(t) : g(t) =ge(t) + go(t) then, g(-t) =ge(-t) + go(-t) = ge(t) - go(-t) Therefore, ge(t) = ½ [g(t) +g(-t)]. Similarly, go(t) = ½ [g(t) - g(-t)].

8. Multichannel and Multidimensional Signals • Multichannel signal is generated by multiple sources (or sensors). Thee resultant signal is the vector sum of signals from all channels such as ECG signal where different leads are connected to the body of the patient and each lead is acting as an individual channel. • Multidimensional signal is a function of more than one independent variable. For example, speech signal is one-dimensional signal while, an image is two-dimensional signal and the moving picture is three-dimensional signal.

9. Size of A Signal The area under a signal g(t) may be considered as a possible measure of its size, because it takes account of not only the amplitude, but also the duration. However, this will be a defective measure because g(t) could be a large signal, yet its positive and negative areas could cancel each other, indicating a signal of small size. This difficulty can be corrected by defining the signal size as the area under g2(t), which is always positive. This measure is called the signal energy Egdefined (for a general complex signals) as

10. The signal energy must be finite for it to be a meaningful measure of the signal size. A necessary condition for the energy to be finite is that Otherwise the integral in the previous equation will not converge. If the amplitude of g(t) does not go to zero as |t| goes to ∞, the signal energy is infinite. A more meaningful measure of the signal size in such a case would be the time average of the energy (if it exists), which is the average power Pgdefined (for a general complex signal) by

11. Asignal with finite and nonzero energy is an energysignalwith zero power, while a signal with finite and nonzero power is a power signal with infinite energy. Every signal that can be generated in a lab bas a finite energy. In other words, every signal observed in real life is an energy signal. A power signal, on the other hand, must necessarily have an infinite duration with infinite energy. Otherwise, its power which is its average energy (averaged over an infinitely large interval) will not approach a nonzero limit. However, practical periodic signals are considered power signals. Note that periodic signals for which the area under |g(t)|2 over one period is finite are power signals, however, not all power signals are periodic. Aperiodic signals are energy signals with zero power.

13. Transformation of the independent variable in the analysis of signals and systems, we need the transformation of the independent variable. In case of continuous-time signal, the independent variable is time t, whereas in case of discrete-time signals, the independent variable is time n. Some manipulations or transformations of the independent variable t are (1) Time shifting. (2) Time-scaling (3) Time-Inversion or folding.

14. Time shifting

15. Time-scaling The expansion or compression of a signal in time is called as time-scaling. g(2t) is the signal g(t) compressed in time by a factor 2. This means that whatever happens in g(t) at some instant t, also happens in g(2t) at the instant t/2 as shown in (b) Similarly, g(t/2) is the signal g(t) expanded in time by a factor of 2as shown in (c) To summarize, we note that to time-scale a signal by a factor a, we should replace (t) with (at). if a > 1, the scaling is compression and if a < I, the scaling will be expansion.

16. Example of time compression(Fast Forward) (a) (b) The signal x(3t) shown in Figure (b) is the signal x(t) shown in figure (a)compressed in time by a factor of 3 This indicates that the values of x(t)at t = 6,12,15 and 24 occur in x(3t)at the instantst = 2,4,5 and 8 respectively.

17. Example of time expansion(Slow Forward) The signal x(t/2)is the signal x(t)expanded in time by a factor of 2. This means that the values of x(t) at t = 1, - 1 and - 3occur in x(t/2)at instants 2,- 2-ahd – 6 respectively.

18. Time-Inversion or folding Time inversion is achieved by replacing the independent variable t by - t. This results in folding of the signal about the origin, i.e., t = 0. Time-inversion or folding may be considered as a special case of time-scaling with a = - 1.

19. Some Standard Signals

21. A basic sine wave

24. δ(n) = u(n) – u(n-1) g(n) δ(n) = g(0) g(n) δ(n-no) = g(no) Delta or Unit Impulse Function δ(t) g(t) δ(t) = g(0) δ(t) g(t) δ(t-T) = g(T) δ(t-T) g(t) * δ(t) = g(t)

25. Real Exponential Signals and damped Sinusoidal x(t) = e-at x(t) = eαt

26. Sinc Function (Pulse)