Lecture 1A: Introduction to Signals and Systems and Mathematical Review PDF

LECTURE 1A: INTRODUCTION TO SIGNALS AND SYSTEMS AND MATHEMATICAL REVIEW Eric Amoateng Kwabi GHANA COMMUNICATION TECHNOLOGY UNIVERSITY [email protected] March 9, 2...

LECTURE 1A: INTRODUCTION TO SIGNALS AND SYSTEMS AND MATHEMATICAL REVIEW Eric Amoateng Kwabi GHANA COMMUNICATION TECHNOLOGY UNIVERSITY [email protected] March 9, 2022 Eric Amoateng Kwabi (GCTU) ENEE 351: SIGNALS AND SYSTEMS March 9, 2022 1 / 21 Outline 1 What is signals 2 Classification of Signals Continous Time and Discrete Time Analog and Digital Real and Complex Deterministic and Random Even and Odd Periodic and Non Periodic Energy and Power 3 Basic Continous Time Signals Unit Step Unit Impulse Eric Amoateng Kwabi (GCTU) ENEE 351: SIGNALS AND SYSTEMS March 9, 2022 2 / 21 What is a Signal A signal is a function representing a physical quantity or variable, and typically it contains information about the behavior or nature of the phenomenon. Example 1 : The patterns of variation over time in the source and capacitor voltages, Vs and vc, Example 2 : The variations over time of the applied force f and the resulting automobile velocity v are signals. Figure: An automobile responding to an applied force f from the engine and to a retarding frictional force pv proportional to the automobile’s velocity v. Figure: A simple RC circuit with source voltage Vs and capacitor voltage Vc. Eric Amoateng Kwabi (GCTU) ENEE 351: SIGNALS AND SYSTEMS March 9, 2022 3 / 21 What is Signal 2 Mathematically: A signal is a function of one or more dependent variable. Example : A speech signal can be represented mathematically by acoustic pressure as a function of time. A picture can be represented by brightness as a function of two spatial variables. Signal is represented mathematically as f (x1 , x2 , x3 ,...xn ) where f is the dependent variable and x1 , x2 , x3 ,... xn is the independent variable Signal can be classified as being ; Single variable signal and multiple variable signal. Signal variable signal is a function of only one variable. Example: f (x), g (x) Multiple variable signal is a function of more than one variable. Example: f (x1 , x2 ), g (t1 , t2 , t3 ) Eric Amoateng Kwabi (GCTU) ENEE 351: SIGNALS AND SYSTEMS March 9, 2022 4 / 21 Classification of Signals 1 The course will focus our attention on signals involving a single independent variable, in this case time(t) A. Continous Time (CT) and Discrete Time (DT) Signals CT signals, the independent variable is continous. That is the signal is defined for a continuum of values of the independent variable. On the other hand, DT signals are defined only at discrete times and consequently the independent variable takes on only a discrete set of values. Figure: The weekly Dow-Jones stock market index is a DT signal Figure: Speech signal as a function of time is a CT signal Eric Amoateng Kwabi (GCTU) ENEE 351: SIGNALS AND SYSTEMS March 9, 2022 5 / 21 Classification of Signals 2 Figure: Illustration of (a) is CT and (b) is DT signals 1 The term t enclosed in parenthesis (.) denotes CT independent variable and n enclosed in bracket [.] denotes DT independent variable. 2 CT signal is represented as x(t) whilst DT signal as x[n]. 3 The DT signal is defined only for integer values of the independent variable. 4 DT signals x[n] may be obtained by sampling a CT signal x(t) such as; x(t0 ), x(t1 ), x(t2 ),...x(tn ), we can deduce that; xn = x[n] = x(tn ) and xn′ s are called samples and the time interval between them is called the sampling interval. When the sampling intervals are equal (uniform sampling) then; x[n] = x(nTs ), where the constant Ts is the sampling interval. Eric Amoateng Kwabi (GCTU) ENEE 351: SIGNALS AND SYSTEMS March 9, 2022 6 / 21 Classifications of Signals 3 B. Analog and Digital Signal 1 Analog and Digital signals are type of signals carrying information. 2 If a CT signal can take on any value in the continuous interval (a,b) where a may be -∞ and b may be +∞, then the CT signal x(t) is called an analog signal 3 If a DT signal x[n] can take on only finite number of distinct values, then we call the signal a digital signal Figure: Analog and Digital Signals Eric Amoateng Kwabi (GCTU) ENEE 351: SIGNALS AND SYSTEMS March 9, 2022 7 / 21 Classification of Signals 4 C. Real and Complex Signals 1 A signal x(t) is a real signal if its value is a real number and the signal is a complex signal if its value is a complex number. 2 A general complex signal x(t) is a function of the form ; √ x(t) = x1 (t) + jx2 (t) where x1 (t) and x2 (t) are real signals and j = - −1 Eric Amoateng Kwabi (GCTU) ENEE 351: SIGNALS AND SYSTEMS March 9, 2022 8 / 21 Classification of Signals 5 D. Deterministic and Random Signals 1 A signal which has no uncertainty at any given instant of time is called a deterministic signal 2 Example: x(t) = A cos wt for -∞ t +∞ 3 At any instant the behaviour of the signal can be predicted. So, for a deterministic signal, the amplitude value of x(t) can be computed. 4 The pattern of deterministic signal is regular and can be characterized mathematically (ie. can be put in an equation) Figure: No uncertainty with respect to its value at any time (past, present, and future). Eric Amoateng Kwabi (GCTU) ENEE 351: SIGNALS AND SYSTEMS March 9, 2022 9 / 21 Classification of Signals 6 D. Deterministic and Random Signals 1 The behaviour of random signal can’t be predicted in advance. 2 Pattern of random signal is quite irregular. Random signal is also known as non-deterministic signal. 3 Examples of random signals are noise generated in the amplifier of a radio receiver, ECG signal (ie heart-beat-signal), speech signal. Figure: Some degree of uncertainty about the value of random signal at any given time. Eric Amoateng Kwabi (GCTU) ENEE 351: SIGNALS AND SYSTEMS March 9, 2022 10 / 21 Classification of Signals 6 E. Even and Odd Signal A signal x(t) or x[n] is referred to an even signal if; x(−t) = x(t) or x[−n] = x[n] Figure: Example of even signal A signal x(t) or x[n] is referred to an odd signal if; x(−t) = -x(t) or x[−n] = -x[n] Eric Amoateng Kwabi (GCTU) ENEE 351: SIGNALS AND SYSTEMS March 9, 2022 11 / 21 Classification of Signals 7 E. Even and Odd Signal Figure: Example of odd signal Any signal x(t) or x[n] can be expressed as a sum of two signals, one of which is even and one of which is odd. That is; x(t) = xe (t) + xo (t) x[t] = xe [t] + xo [t] where; where; xe (t) = 21 {x(t) + x(−t)} xe [t] = 12 {x[t] + x[−t]} xo (t) = 21 {x(t) - x(−t)} xo [t] = 21 {x[t] - x[−t]} Eric Amoateng Kwabi (GCTU) ENEE 351: SIGNALS AND SYSTEMS March 9, 2022 12 / 21 Classification of Signals 8 F. Periodic and Non Periodic Signal 1 A continous time signal x(t) is said to be periodic with period T, if there is a positive nonzero value of T for which; x(t + T ) = x(t) for all t 2 An example of such is given below; it follows that; x(t + mT ) = x(t) ; for all t and any integer m. The fundamental period T0 of x(t) is the smallest positive value of T for which x(t + T ) = x(t) holds Figure: Example of periodic signal Eric Amoateng Kwabi (GCTU) ENEE 351: SIGNALS AND SYSTEMS March 9, 2022 13 / 21 Classification of Signals 8 F. Periodic and Non Periodic Signal 1 A discrete time signal is defined analogously. A sequence x[n] is periodic with period N, if there is a positive integer N for which for which; x[n + mN] = x[n] for all n and any integer M 2 The fundamental period N0 of x[n] is the smallest positive integer N for which x[n + mN] = x[n] holds 3 Any sequence which is not periodic is called a non-periodic (aperiodic) signal. Eric Amoateng Kwabi (GCTU) ENEE 351: SIGNALS AND SYSTEMS March 9, 2022 14 / 21 Classification of Signals 9 G. Energy and Power Signals 1 Consider v (t) to be the voltage across a resistor R producing a current i(t). The instantaneous power p(t) per ohm is defined as; v (t)i(t) p(t) = = i 2t R 2 The total energy E and average power P on a per ohm basis are ; 0. Z −∞ E= i 2 (t) dt ∞ Z T 1 2 P = lim i 2 (t) dt T →∞ T −T 2 3 For an arbitrary CT signal x(t), the normalized energy content E of x(t) is defined as; Z −∞ E= x 2 (t) dt ∞ Eric Amoateng Kwabi (GCTU) ENEE 351: SIGNALS AND SYSTEMS March 9, 2022 15 / 21 Classification of Signals 10 The normalized average power P of x(t) is defined as; Z T 1 2 P = lim |x 2 (t)| dt T →∞ T −T 2 Similarly, for a discrete time signal x[n], the normalized energy content E of x[n] is defined as; ∞ X |x[n]2 | n=−∞ The normalized average power P of x[n] is defined as; N 1 X P = lim |x[n]2 | N→∞ 2N + 1 n=−N x(t) or x[n] is said to be an energy signal if and only if 0