ColorUntitled Jeffrey Schiano 2014. All rights reserved. Rec 3. EE 350Continuous-Time Linear SystemsRecitation 31 Jeffrey Schiano 2014. All rights reserved. Rec 3. Recitation 3 Topics• Solved Problems– Classification of Systems– Complex Numbers• MATLAB Programming– Complex Functions– Vector Manipulation– Built-in Housekeeping Functions2 Jeffrey Schiano 2014. All rights reserved. Rec 3. Problem 1• Consider the following circuit that contains an ideal diode1. Sketch the output y(t) as a function of the input f(t)2. Is the system zero-state linear or nonlinear?3. Is the system instantaneous (memoryless) or dynamic?3RRf(t)2Vy(t) Jeffrey Schiano 2014. All rights reserved. Rec 3. Problem 1 Solution4 Jeffrey Schiano 2014. All rights reserved. Rec 3. Problem 1 Solution5 Jeffrey Schiano 2014. All rights reserved. Rec 3. Problem 1 Solution6 Jeffrey Schiano 2014. All rights reserved. Rec 3. Problem 2• Consider the circuit below with input f(t) and output y(t)1. Is the system linear or nonlinear? 2. Is the system time-invariant or time-varying?3. Is the system instantaneous (memoryless) or dynamic?4. Is the system causal or noncausal?7Rf(t)y(t)Cv( )C Jeffrey Schiano 2014. All rights reserved. Rec 3. Problem 2 Solution8 Jeffrey Schiano 2014. All rights reserved. Rec 3. Problem 2 Solution9 Jeffrey Schiano 2014. All rights reserved. Rec 3. Problem 2 Solution10 Jeffrey Schiano 2014. All rights reserved. Rec 3. Problem 2 Solution11 Jeffrey Schiano 2014. All rights reserved. Rec 3. Problem 2 Solution12 Jeffrey Schiano 2014. All rights reserved. Rec 3. Problem 2 Solution13 Jeffrey Schiano 2014. All rights reserved. Rec 3. Problem 3• The relationship between the input f(t) and zero-state response y(t) of two systems are given below. Determine if each system is1. Zero-state linear or nonlinear2. Time-invariant or time-varying3. Causal or noncausal4. Instantaneous (memoryless) or dynamic14|| 2(a) ( ) ( 1)(b) ( ) ( )etyt e f tyt f t d Jeffrey Schiano 2014. All rights reserved. Rec 3. Problem 3 Solution15 Jeffrey Schiano 2014. All rights reserved. Rec 3. Problem 3 Solution16 Jeffrey Schiano 2014. All rights reserved. Rec 3. Problem 3 Solution17 Jeffrey Schiano 2014. All rights reserved. Rec 3. Problem 3 Solution18 Jeffrey Schiano 2014. All rights reserved. Rec 3. Problem 3 Solution19 Jeffrey Schiano 2014. All rights reserved. Rec 3. Problem 3 Solution20 Jeffrey Schiano 2014. All rights reserved. Rec 3. Problem 4• The input f(t) and zero-state response y(t) of a linear time-invariant (LTI) system is shown below1. Sketch the zero-state response y1(t) to the input f1(t)2. Sketch the input f2(t) yielding the zero-state response y2(t)21f(t)t110y(t)t11LTISystemf(t)y(t)01f(t)t2202y(t)t311 Jeffrey Schiano 2014. All rights reserved. Rec 3. Problem 4 Solution22 Jeffrey Schiano 2014. All rights reserved. Rec 3. Problem 4 Solution23 Jeffrey Schiano 2014. All rights reserved. Rec 3. Problem 4 Solution24 Jeffrey Schiano 2014. All rights reserved. Rec 3. Problem 4 Solution25 Jeffrey Schiano 2014. All rights reserved. Rec 3. Complex Number Review• A complex number is a number that can be expressed in the form a + j b, where a and b are real numbers and jsatisfies the equation j2= 1• Notation• Complex numbers extend the concept of the one-dimensional number line to the two-dimensional complex plane by using the horizontal axis for the real part and the vertical axis for the imaginary part26 Complex Number: Real part of Re , Imaginary part of Imzajbzza z zb Jeffrey Schiano 2014. All rights reserved. Rec 3. Rectangular and Polar Form• A complex number z can be represented either in rectangular or polar form27zReImabRectangular Form zajbzReImPolar Form r zr221cos( ) | |sin( ) = z = Tanar r z a bbbra Jeffrey Schiano 2014. All rights reserved. Rec 3. Polar Form and Euler’s Identity28Rectangular Form: cos( ) sin( )r cos( ) sin( )Polar Form: jzajbzr jrzjzreEuler's Identity: cos( ) sin( )jej Jeffrey Schiano 2014. All rights reserved. Rec 3. Operations on Complex Numbers• Use rectangular form for addition and subtraction• Use polar form for multiplication and division• The complex conjugate of the complex number z = a + jb is defined as z* = a – jb29121212If and , then z ( ) ( ) z ( ) ( )zajb z cjdzacjbdzacjbd 1212 12112211 2 2()12 1 2 12()11 122 2If e and e , then z e ee ejjjj jjjjzr zrzr r rrezr rezr r Jeffrey Schiano 2014. All rights reserved. Rec 3. Problem 5• Given z = a + jb = r ejshow that1. z* = r e-j2. z + z* = 2a = 2Re{z}3. z z* = 2jb = j2Im{z}4. z z* = a2+ b2= r230 Jeffrey Schiano 2014. All rights reserved. Rec 3. Problem 5 Solution31 Jeffrey Schiano 2014. All rights reserved. Rec 3. Problem 5 Solution32 Jeffrey Schiano 2014. All rights reserved. Rec 3. Entering Complex Numbersin MATLAB33• The complex number z = 2 + j3 may be entered into MATLAB as>> z = 2 + 3*i>> z = 2 + 3i>> z = 2 + 3*j>> z = 2 + 3j>> z = 2 + 3*1i Jeffrey Schiano 2014. All rights reserved. Rec 3. Complex Functions in MATLAB34Operation Functionmagnitude (r) of z abs(z)angle (q) of z in radiansangle(z)construct z from a and b complex(a,b)conjugate of z conj(z)imaginary part of z imag(z)real part of z real(z) Jeffrey Schiano 2014. All rights reserved. Rec 3. Problem 6• Use MATLAB to determine the rectangular form of• Explain the difference between the results in parts 3 and 4354121.122. 23. Execute the MATLAB command >>exp(pi/2*i)4. Execute the MATLAB command >>exp(pi/2i)jjje Jeffrey
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