Signals and Systems 1 Lecture 3 Dr. Ali. A. Jalali August 23, 2002

Signals and Systems 1 Lecture # 3 Introduction to Signals EE 327 fall 2002

Example 1 1. Example 1 2. Interpret and sketch the generalized function x(t) where 3. EE 327 fall 2002

Example 1:Solution: 1. To determine the meaning of x(t) we place it in an integral 2. Let  = t+4 so that 1. From the definition of the unit impulse, The integral equals e-1. Therefore EE 327 fall 2002

Example 1: Graphical solution 1. The result can also be seen graphically. The left panel shows both and  (t+4), and the right panel shows their EE 327 fall 2002 t

Unit impulse- what do we need it for? 1. The unit impulse is a valuable idealization and is used widely in science and engineering. Impulses in time are useful idealizations. 2.  Impulse of current in time delivers a unit charge instantaneously to network. 3.  Impulse of force in time delivers an instantaneous momentum to a mechanical system. EE 327 fall 2002

Unit impulse- what do we need it for? 1. Impulses in space are also useful. 2.  Impulse of mass density in space represents a point mass. 3.  Impulse of charge density in space represents a point charge. 4.  Impulse of light intensity in space represents a point of light. 5. We can imagine impulses in space and time 6.  Impulse of light intensity in space and time represents a brief flash of light at a point in space. EE 327 fall 2002

Unit Step Function 1. Integration of the unit impulse yields the unit step function which is defined as EE 327 fall 2002

Unit Step Function (Figure 1.7a text) %F1_7a Unit step function t=-2:0.01:5;% make t a vector of 701 points q=size(t); f=zeros(q(1),q(2));% set f = a vector of zeros q=size(t(201:701)); f(201:701)=ones(q(1),q(2));% set final 500 points of f to 1 plot(t,f),title('Fig.1.7a Unit step function'); axis([-2,5,-1,2]); % sets limits on axes xlabel('time, t'); ylabel(' u(t)'); grid; %F1_7a Unit step function t=-2:0.01:5;% make t a vector of 701 points q=size(t); f=zeros(q(1),q(2));% set f = a vector of zeros q=size(t(201:701)); f(201:701)=ones(q(1),q(2));% set final 500 points of f to 1 plot(t,f),title('Fig.1.7a Unit step function'); axis([-2,5,-1,2]); % sets limits on axes xlabel('time, t'); ylabel(' u(t)'); grid; Generic step function EE 327 fall 2002

%F1_7b Signal g(t) multiplied f(101:501)=2.5- cos(5*t(101:501) by a pulse functions ([u(t+1)-u(t-3)] %F1_7b Signal g(t) multiplied by a pulse functions t= -2:0.01:5; q=size(t); f=zeros(q(1),q(2)); f(101:501)=2.5-cos(5*t(101:501)); plot(t,f),title('Fig.1.7b Signal g(t) multiplied by a pulse functions'); axis([-2,5,-1,4]); xlabel('time, t'); ylabel(' g(t)[u(t+1)-u(t-3)]'); grid; %F1_7b Signal g(t) multiplied by a pulse functions t= -2:0.01:5; q=size(t); f=zeros(q(1),q(2)); f(101:501)=2.5-cos(5*t(101:501)); plot(t,f),title('Fig.1.7b Signal g(t) multiplied by a pulse functions'); axis([-2,5,-1,4]); xlabel('time, t'); ylabel(' g(t)[u(t+1)-u(t-3)]'); grid; EE 327 fall 2002

%F1_7b Signal g(t) multiplied by a pulse functions EE 327 fall 2002

Unit impulse as the derivative of the unit step 1. As an example of the method for dealing with generalized functions consider the generalized function 2. Since u(t) is discontinuous, its derivative does not exist as an ordinary function, but it does as a generalized function. To see what x(t) means, put it in an integral with a smooth testing function And apply the usual integration-by-parts theorem 5. to obtain EE 327 fall 2002

Unit impulse as the derivative of the unit step. Cont’d 1. The result is that 1. which, from the definition of the unit impulse, implies that 2. That is, the unit impulse is the derivative of the unit step in a generalized function sense. EE 327 fall 2002

Real Exponential Functions 1. Exponential signals are characterized by exponential function Where e is the Naperian constant 2.718… and a and A are real constants. f(t) Time, t

Ramp Functions 1. A shifted ram function with slop B is defined as Unit ramp function being at t=0 by making B=1 and t 0 =0 and multiplying by u(t), giving Time, t f(t) r(t)=tu(t)

Successive integration of the unit impulse 1. Successive integration of the unit impulse yields a family of functions. 2. Later we will talk about the successive derivatives of  (t), but these are too horrible to contemplate in the first lecture. EE 327 fall 2002

Sinusoidal Functions 1. A sinusoidal function frequency in hertz or cycles per second, the phase shift in radians, the radian frequency is rad/s and the period is s Exponential functions, as in Where B is the amplitude, is angular frequency in radians/second, and is the phase shift in radians. EE 327 fall 2002

Exponentialy Modulated Sinusoidal Functions 1. If s sinusoid is multiplied by a real exponential, we have an exponentially modulated sinusoid that also can arise as a sum of complex exponentials, as in EE 327 fall 2002 Example: Turned on at t = +1 by multipling shifted unit stepu(t-1)

Building-block signals can be combined to make a rich population of signals 1. Eternal complex exponentials and unit steps can be combined to produce causal and anti- causal decaying exponentials. EE 327 fall 2002

Building-block signals can be combined to make a rich population of signals 1. Unit steps and ramps can he combined to produce pulse signals. EE 327 fall 2002

Example: 1. Describe analytically the signals shown in Solution: Signal is (A/2)t at, turn on this signal at t = 0 and turn it off again at t = 2. This gives, EE 327 fall 2002 A 0 2 t f(t) See page 9 of text for more examples.

Sequences 1. Unite Sample Sequence 2. The unit sample sequence is the discrete-time version of the unit impulse in CT situations. 3. Definition of unit sample sequence: 4. Thus it is possible to represent an arbitrary sequence as the weighted sum of unit sample sequences. 1 n … … 1 n … … 0 m = n m = n-3 Plots of Unite Sample Sequence

Sequences 1. Unite Step Sequence 2. The unit step sequence is the discrete-time version of the unit step in CT situations. 3. Definition of unit step sequence: 4. The unit step sequence u(n) is related to unit sample sequence by 1 n … Plots of Unite Step Sequence U(n) Generic step sequence

Sequences 1. Ramp Sequence 2. A shifted ramp sequence with slop of B is defined by: 3. The unit ramp sequence and shifted ramp sequences 4. Example: g(t) = 2(n-10). MATLAB Code: n=-10:1:20; f=2*(n-10); stem(n,f); EE 327 fall 2002

Real Exponential Sequences 1. Real exponential sequence is defined as: Example for A = 10 and a = 0.9, as n goes to infinity the sequence approaches zero and as n goes to minus infinity the sequence approaches plus infinity. EE 327 fall 2002 Composite sequence: Multiplying point by Point by the step sequence MATLAB Code: n=-10:1:10; f =10*(.9).^n; stem(n,f); axis([ ]);

Sinusoidal Sequence 1. A sinusoidal sequence may be described as: 2. Where A is positve real number (amplitude), N is the period, and a is the phase. 3. Example: 4. A = 5, N = And 6. MATLAB Code: 7. n=-20:1:20; 8. f=5*[cos(n*pi/8+pi/4)]; 9. stem(n,f); EE 327 fall 2002

Exponentialy Modulated Sinusoidal Sequence 1. By multiplying an exponential sequence by sinusoidal sequence, we obtain an exponentially modulated sequence described by: 2. Example: 3. A = 10, N = 16, a = And 5. MATLAB Code: 6. n=-20:1:20; 7. f=10*[0.9.^n]; 8. g=[cos(2*n*pi/16+pi/4)]; 9. h=f.*g; 10. stem(n,h); 11. axis([ ]); EE 327 fall 2002

Example: 1. Use the Sequence definition, describe analytically the following sequence. 1 n A Pulse Sequence f(n) -2 Solution: This pulse sequence can be describe by f(t) = u(n) – u(n-3). The first step sequence turn on the pulse at n = 0, and second step turns it off at n = 3. EE 327 fall 2002 See page 13 of text For more examples.

A Road Map Signals and Systems Mathematical Model, usually difference or differential equations LP Filter Spring Mass Dc Motor Any phisical systems Solutions Find x(t) or x(n) or x(n+1) Ordinary DE Laplace transform Z transform Convolution Simulation Hock up Fourier transform Fourier series Transfer function State space model Signal flow graph Block diagram Unit sample response Frequency response Time Domain and Frequency Domain solutions. Continuous and Discrete-time Systems. Time Domain and Frequency Domain solutions. Continuous and Discrete-time Systems. Implementation EE 327 fall 2002

Conclusions 1. Introduction to signals and systems, 2. Signals, definitions and classifications, 3. Building block signals- eternal complex exponentials and impulse, 4. Mathematical description of signals, 5. MATLAB examples, 6. A road map. EE 327 fall 2002