Continuous Time Signals Basic Signals – Singularity Functions Transformations of Continuous Time Signals Signal Characteristics Common Signals

Continuous-Time Signals Assumptions: Functions, x(t), are of the one independent variable that typically represents time, t. Time t can assume all real values: -∞ < t < ∞, Function x(t) is typically a real function. 20 April 2017 Veton Këpuska

Singularity Functions 20 April 2017 Veton Këpuska

Unit Step Function Unit step function definition: 20 April 2017 Veton Këpuska

Unit Step Function Properties Scaling: Unit step function can be scaled by a real constant K (positive or negative) Multiplication: Multiplication of any function, say x(t), by a unit step function u(t) is equivalent to defining the signal x(t) for t≥0. 20 April 2017 Veton Këpuska

Unit Ramp Function Unit Ramp Function is defined as: 20 April 2017 Veton Këpuska

Unit Ramp Function Properties Scaling: Unit step function can be scaled by a real constant K (positive or negative) Integral of the unit step function is equal to the ramp function: Derivative of the unit ramp function is the unit step function. Slope of the straight line 20 April 2017 Veton Këpuska

Unit Impulse Function Unit Impulse Function, also know as Dirac delta function, is defined as: 20 April 2017 Veton Këpuska

Unit Impulse Function Properties Scaling: Unit impulse function can be scaled by a real constant K (positive or negative) Delta function can be approximated by a pulse centered at the origin 20 April 2017 Veton Këpuska

Unit Impulse Function Properties Unit impulse function is related to unit step function: Conversely: Proof: t<0 t>0 20 April 2017 Veton Këpuska

Time Transformation of Signals 20 April 2017 Veton Këpuska

Time Reversal: 20 April 2017 Veton Këpuska

Time Scaling |a| > 1 – Speed Up |a| < 1 – Slow Down 20 April 2017 Veton Këpuska

Time Shifting 20 April 2017 Veton Këpuska

Example 1 20 April 2017 Veton Këpuska

Independent Variable Transformations 20 April 2017 Veton Këpuska

Example 2 20 April 2017 Veton Këpuska

Example 3 20 April 2017 Veton Këpuska

Independent Variable Transformations Replace t with , on the original plot of the signal. Given the time transformation: Solve for Draw the transformed t-axis directly below the -axis. Plot y(t) on the t-axis. 20 April 2017 Veton Këpuska

Amplitude Transformations 20 April 2017 Veton Këpuska

Example 4 Consider signal in the figure. Suppose the signal is applied to an amplifier with the gain of 3 and introduces a bias (a DC value) of -1. That is: 20 April 2017 Veton Këpuska

Example 5 20 April 2017 Veton Këpuska

Transformations of Signals Name y(t) Time reversal x(-t) Time scaling x(at) Time shifting x(t-t0) Amplitude reversal -x(t) Amplitude scaling Ax(t) Amplitude shifting x(t)+B 20 April 2017 Veton Këpuska

Signal Characteristics

Even and Odd Signals xe(t)=xe(-t) xo(t)=-xo(-t) Even Functions Odd Functions xo(t)=-xo(-t) 20 April 2017 Veton Këpuska

Even and Odd Signals Any signal can be expressed as the sum of even part and on odd part: 20 April 2017 Veton Këpuska

Average Value Average Value of the signal x(t) over a period of time [-T, T] is defined as: The average value of a signal is contained in its even function (why?). 20 April 2017 Veton Këpuska

Properties of even and odd functions The sum of two even functions is even. The sum of two odd functions is odd. The sum of an even function and an odd function is neither even nor odd. The product of two even functions is even. The product of two odd functions is even. The product of an even function and an odd function is odd. 20 April 2017 Veton Këpuska

Periodic Signals 20 April 2017 Veton Këpuska

Periodic Signals Continuous-time signal x(t) is periodic if: T is period of the signal. A signal that is not periodic is said to be aperiodic. 20 April 2017 Veton Këpuska

Periodic Signals If constant T is a period of of a function x(t) than nT is also its period, where T>0 and n is any positive integer. The minimal value of the constant T >0 is a that satisfies the definition x(t)= x(t+ T) is called a fundamental period of a signal and it is denoted by T0. 20 April 2017 Veton Këpuska

Examples of Periodic Signals 20 April 2017 Veton Këpuska

Sinusoidal Signal Properties A – Amplitude of the signal  - is the frequency in rad/sec  - is phase in radians 20 April 2017 Veton Këpuska

Sinusoidal Function Properties Note: 20 April 2017 Veton Këpuska

Periodicity of Sinusoidal Signal 20 April 2017 Veton Këpuska

Example: Sawtooth Periodic Waveform 20 April 2017 Veton Këpuska

Period and Frequency Fundamental Period T0 – Measured in seconds. Fundamental Frequency f0 – Measured in Hz – number of periods (cycles) per second or equivalently in radian frequency rad/s. 20 April 2017 Veton Këpuska

Testing for Periodicity 20 April 2017 Veton Këpuska

Testing for Periodicity 20 April 2017 Veton Këpuska

Composite Signals Each signal can be decomposed into a sum of series of pure periodic signals (Taylor Series Expansion/Fourier Series Expansion) The sum of continuous-time periodic signals is periodic if and only if the ratios of the periods of the individual signals are ratios of integers. 20 April 2017 Veton Këpuska

Composite Signals If a sum of N periodic signals is periodic, the fundamental period can be found as follows: Convert each period ratio, To1/Toi≤ i ≤ N , to a ratio of integers, where To1 is the period of the first signal considered and Toi is the period of one of the other N-1 signals. If one or more of these ratios is not rational, the sum of signals is not periodic. Eliminate common factors from the numerator and denominator of each ratio of integers. The fundamental period of the sum of signals is To=koTo1 ; kois the least common multiple of the denominators of the individual ratios of integers. 20 April 2017 Veton Këpuska

Composite Signals If x1(t) is periodic with period T1, and Then x1(t)+x2(t) is periodic with period equal to the least common multiple (T1, T2) if the ratio of the two periods is a rational number, where k1 and k2 are integers: 20 April 2017 Veton Këpuska

Composite Signals Let T’= k1T1 = k2T2 y(t) = x1(t)+x2(t) Then y(t+T’) = x1(t+T’)+x2(t+T’)= x1(t+ k1T1)+x2(t+ k2T2)= x1(t)+x2(t) = y(t) 20 April 2017 Veton Këpuska

Example 2.7 a) Assume that v(t) is a sum of periodic signals given below. Determine if the signal is periodic and what its periodicity? 20 April 2017 Veton Këpuska

Solution Determine whether v(t) constituent signals have periods with ratios that are integers (rational numbers): 20 April 2017 Veton Këpuska

Solution Ratios of periods are rational numbers thus the composite signal v(t) is periodic. Elimination of common factors: T01/T02 = 4/7 T01/T03 = 7/21=1/3 Least common multiple of the denominator ratios: n1= 3*7=21 Fundamental period of v(t) is: T0= n1 T01 = 21*2/3.5=12 20 April 2017 Veton Këpuska

Example 2.7 b) Assume that to v(t) is added a periodic signal x4(t) given below. Determine if the signal is periodic and what its periodicity? 20 April 2017 Veton Këpuska

Solution Since ratio of the x1(t) and x4(t) periods is not a rational number the v(t) is not periodic. 20 April 2017 Veton Këpuska

Homework #1: For x(t)=Acos(t+) find What are its maximum and minimum values? What are corresponding times when they occur? What is the value of the function when it crosses vertical y- axis (ordinate) and horizontal x-axis (abscissa)? At what time instances the function becomes zero? Indicate all the above point values in a plot. 20 April 2017 Veton Këpuska

Homework #1 Use the following MATLAB script to test your calculations and plot the function: function pfunc(A, f, th1, th2) % % Periodic Sine Function % A - gain (1) % f - frequency (1) % th1 - phase of the first signal (0) % th2 - phase of the second signal (pi/6) w = 2.*pi.*f; % radial frequency fs = 0.0001*f; mint = -pi*f/2; maxt = pi*f/2; miny = -1.2*A; maxy = 1.2*A; t = mint:fs:maxt; % time axis y = A*cos(w*t+th1); plot(t, y, 'b', 'LineWidth',2); title('Periodic Signal'); grid on; hold; axis([mint maxt miny maxy]); y = A*cos(w*t+th2); plot(t, y, 'r', 'LineWidth',2); ylabel('cos(\omegat+\theta)'); xlabel('Angle x\pi [rad]'); x=-0.8; text(x,A*cos(w*x+th1),sprintf('%s+%3.2f)','\leftarrow cos(-\pit',th1),... 'HorizontalAlignment','left',... 'BackgroundColor','b'); x=-0.6; text(x,A*cos(w*x+th2),sprintf('%s+%3.2f)','\leftarrow cos(-\pit',th2),... 'BackgroundColor','r'); 20 April 2017 Veton Këpuska

Homework #1 Problems 2.1, 2.2, 2.9, 2.10, 2.13, 2.14, 2.20. 20 April 2017 Veton Këpuska

Example Consider the signal to the left and its time reversed version. The signal is decomposed into its even and odd functions: 20 April 2017 Veton Këpuska

Common Signals in Engineering 20 April 2017 Veton Këpuska

Common Signals Continuous-time physical systems are typically modeled with ordinary linear differential equations with constant coefficients. 20 April 2017 Veton Këpuska

Exponential Signals Useful Complex Exponential Relations 20 April 2017 Veton Këpuska

Euler’s Formula 20 April 2017 Veton Këpuska

Example of Exponential Functions C and a real, x(t)=Ceat a=s, s>0 - Increasing Exponential: Chemical Reactions, Uninhibited growth of bacteria, human population? a=s, s<0 - Decaying Exponential: Radioactive decay, response of an RC circuit, damped mechanical system. a=s, s=0 - Constant (DC) signal. 20 April 2017 Veton Këpuska

Time Constant of the Exponential Function The constant parameter t is called the time constant in of the exponential function presented below. To relate to the time constant the following is necessary:

Example of Exponential Functions C complex, a imaginary, x(t)=Ceat a=jw, s=0; C=Aejf – A and f are real: For C – real (f=0) x(t) is periodic: Why x(t) is periodic? 20 April 2017 Veton Këpuska

Periodicity of Complex Exponential 20 April 2017 Veton Këpuska

Example of Complex Exponentials C complex, a complex, x(t)=Ceat C=Aejf , A and f are real; a=s+jw, s, w are also real. 20 April 2017 Veton Këpuska

Time-Shifted Signals Time-Shifted Impulse Function Shift d(t) by t0: “Sifting” Property of the Impulse Function: 20 April 2017 Veton Këpuska

Examples of Impulse Functions 20 April 2017 Veton Këpuska

Properties of the Unit Impulse Function 20 April 2017 Veton Këpuska

Shifted Unit Step Function 20 April 2017 Veton Këpuska

Continuous & Piece-wise Continuous Functions Rectangular Function 20 April 2017 Veton Këpuska

Rectangular Function 20 April 2017 Veton Këpuska

Triangular Pulse Function 20 April 2017 Veton Këpuska

Straight Line Equation [x0,y0] - a point on the line [x1,y1] - a point on the line [x2,y2] - a point on the line m = (y2-y1)/(x2-x1) - slope 20 April 2017 Veton Këpuska

Composite Signal from Straight Lines 20 April 2017 Veton Këpuska

Example of Composite Signal 20 April 2017 Veton Këpuska