Module 4 Laplace transforms, ROC, rational systems, Z transform, properties of LT and ZT, rational functions, system properties from ROC, inverse transforms Related web links and videos Sl no Web link Topic covered. http://www.youtube.com/watch?v=rp-mxc_kojo FT and ZT 2. http://www.youtube.com/watch?v=ncn_ofrmw3 Introduction to LT 3. https://www.khanacademy.org/math/differential-equations/laplacetransform/laplace-transform-tutorial/v/laplace-transform- 4. http://ocw.mit.edu/courses/electrical-engineering-and-computerscience/6-3-signals-and-systems-fall-2/lecture-videos-andslides/lecture-6-laplace-transform/ Introduction to LT Laplace Transform 5. http://freevideolectures.com/course/377/signals-and-systems/29 Properties of Z transform 6. http://www.wikihow.com/calculate-the-laplace-transform-of-a- Function Different ways of calculating Laplace transform 7. http://lpsa.swarthmore.edu/laplaceztable/laplacezfunctable.html LT and ZT table 8. http://dspcan.homestead.com/files/ztran/zlap.htm LT and ZT definitions with very good graphics 9. http://math.stackexchange.com/questions/5863/relationshipbetween-the-z-transform-and-the-laplace-transform. http://dea.brunel.ac.uk/cmsp/home_saeed_vaseghi/chapter4-z- Transform.pdf Relationship between LT and ZT Z transform. http://en.wikibooks.org/wiki/control_systems/z_transform_mappings Z transform mapping Page of 8
2. http://sertoz.bilkent.edu.tr/courses/math26/24/transformations.pdf Lecture notes on ZT and LT. Which of the following Integrals converge? a. b. +t +t 2 dt 3 dt c. cos( e 2 t )e t dt d. e t dt e. cos(t 2 ) dt Also give their value if they converge. ASSIGNMENT I OBJECTIVE TYPE QUESTIONS 2. ROC cannot contain poles because the very definition of ROC is that it must contain s = σ + jω, such that Transform will. 3. If x[n] is a finite-duration sequence, then the ROC is the entire z-plane, except possibly a. z = b. z =. c. z < d. z > 4. If the line Re{s} = σ is in the ROC, then all values of s for which Re{s} σ will also be in the ROC. This statement is true for a. Right sided signals b. Left sided signals c. Finite length signals d. All signals 5. If x[n] is summable, i.e. its z transform exists with a pole at z =.999 then x[n] is a a. A finite duration signal b. A left sided signal c. A right sided signal d. A two sided signal 6. If X(s) = (s+2)(s+3) a. A finite duration signal b. A left sided signal, and it is known that ROC exists, then x(t) must be a Page 2 of 8
c. A right sided signal d. A two sided signal 7. Polynomials of s can be expressed as a Laplace transformed sum of k th derivatives of x(t). Where k n, n being the order of the polynomial. Then x(t) is equal to, a. u(t) b. δ(t) c. t d. e at, where a is a constant If the ROC of transform of x (t), x 2 (t), x 3 (t) are R, R2 and R3 respectively then the ROC of the following expressions will be, 8. d dt [x (t) + {x 2 (t) x 3 (t)}] a. R R2 R3 b. RUR2UR3 c. R (R2UR3) d. RU(R2 R3) 9. n{ax [ n] bx 2 [n]} a. R R2 b. RUR2 c. R - R2 d. {R R2} -. The inverse Laplace transform of a function X(s) is given by Here, the path C of integration is any line a) Parallel to σ axis b) Parallel to jω axis and on the left half of the complex plane c) Parallel to jω axis and on the right half of the complex plane d) Parallel to jω axis and in the ROC.. Given L F(s), then which of the following statements are true? s3 a) f 3 (t)dt t v w b) f(u)du dw dv t t t c) f(u) du 3 d) d3 dt 3 f(t) Page 3 of 8
2. Known that the ROC for Laplace transform of x(t)is a region to the right side of a constant σ line where σ >. What would you expect the ROC to be, if the z transform of the sampled version of x(t) is found? 3. A rational system function has a pole at origin, and at s = 5, 7 ± j3.42. The system is known to be causal. Which of the following ROCs are possibilities? a) σ <.5 b) σ > c) σ < 6 d) σ < 4. A second order system (2 poles) has a negative damping. Then one or both of the poles lie in Region R in the s plane. R= 5. A Discrete time system is known to be causal and BIBO stable. What can be said about the ROC of the system? a) z > α, α > b) z > α, α < c) z < α, α > d) z < α, α < Note: Indicate whether the following statements (Q6 to Q9) are true or false 6. If there is a pole on the unit circle, then the signal is not absolutely summable. 7. For stability all the poles must be located at finite locations on the s plane. 8. The following plots represent all pass systems. Hint: An all-pass system is a stable system for which the magnitude of the frequency response is a constant, independent of frequency. 9. If the poles and zeros at z = are counted then the rational transfer function will always have the same number of poles and zeros. Page 4 of 8
2. Can the method of Laplace transform be used to find the solution of Y + Y = sec (t)? Explain. 2. For a system with a rational system function, causality of the system is equivalent to a) ROC being right half of s plane b) ROC being left half of the leftmost pole c) ROC being the right half the rightmost pole. d) ROC being the left half of s plane. 22. Which of the following systems are stable? a) b) c) d) 2 (s+)(s+2)(s+3) 2(s+) (s )(s 2 +2s+2) (s 2 +4) 2 (s+) s 4 +3s 3 +s 2 +s If the system turns out to unstable plot the poles and zeros, and explain graphically why the system is unstable. 23. The system H(z) = z3 +2Z 2 +Z is Causal. (True/False/Can t say) 2Z 2 +2z+ 24. Consider a rational system function H(z) = 4z2 3z with ROC z > 2. Plot the pole zero 2z 2 5z+2 locations and determine stability and causality of the system. The system turns out to be a) Stable and Causal b) Stable but not Causal c) Not Stable but Causal d) Not Stable and not Causal 25. A second order system with complex poles is given by the equation H(z) = (2rcosθ)z +r 2 z 3 Assume the system is causal. Now, for what values of r and z will this system be stable? (a) z > r, r < (b) z < r, r > (c) z > r, r > (d) z < r, r < Page 5 of 8
26. Given X(z) =.5z ( z )( 3z ). Then x[] = 27. We know that a constant σ line on s-plane is equivalent to a circle with constant radius e σ on the z-plane. The ROC of the Laplace transform of a signal x(t) is the region to the right of σ< constant line. If the z-transform of the sampled signal x[n] is analyzed, x[n] will be found to bea) Unstable and non-causal b) Only causal c) Only stable d) Causal and stable. 28. The voltage and current in an Inductor are related by the equation V = L di. To represent the inductor as a resistor, analysis is carried out in frequency domain using Laplace transform. In the new domain, the Voltage and current are related as: a. V = s I L b. V I = L s c. V I = sl d. V I = sl dt 29. The Laplace transform of 5 sin(3t) is equal to 5 a. b. c. s 2 4 3 s 2 25 5 s 2 9 5 d. s 2 4 e. None of these t < 2 3. Which of the following is the Laplace transform of f(t) = t 2 4t + 4 t 2 a. 2e 2s s 3 b. 2e 2s s 3 c. e 2s s d. 2 2e 2s s 3 + e 2s s + 2 2e 2s s 3 3. Suppose the function y(t) satisfies the differential equation y 2y y =, with initial values y() =, y () =. Find the Laplace transform. Page 6 of 8
a. b. c. d. s 2 2s+ s(s 2 2s+) + s+ s(s 2 2s+) s 2 2s+ + s+3 s(s 2 2s+) s 2 2s+ t 32. The Laplace transform of f(t) = sinh(α) cosh(t α) dα a. F(s) = s2 (s 2 +) 2 b. F(s) = s (s 2 ) 2 c. F(s) = (s 2 +) 2 d. F(s) = s (s 4 ) t 33. Find the Laplace transform of the function f(t) = cos(t τ) e τ sin(τ) dτ a. b. c. d. s (s 2 +)((s ) 2 +) s (s 2 +)((s ) 2 +) s ((s ) 2 +) 2 s (s 2 +) 2 34. z-transform of the signal y[n] = {,,3,2,,, }, y[] =, is a. + 2 z + 3 z 2 b. + 3 z + 2 z 2 c. + 3 z d. + 2 z 2 35. The ROC of Question 34 is: a. whole of z-plane b. whole of z-plane except z = c. whole of z-plane except of z = d. none of these 36. If x[n] X[z], then which of the following are valid a. x[ n] X[z] b. x[ n] zx[z] c. x[ n] X[z] z d. x[ n] X[ z ] is Page 7 of 8
37. If X(z) = a., b.,.5 c., d.,.5z 2 (z )(z 2.95z+.45), then the Initial and final values of x[n] are 38. If x[n] = for n < and x[n] = 3 n for n >, the z transform is a. b. z 3 (z 3) 2 z c. z 3 d. z 3 39. If a time domain sequence is shifted by n samples, then poles are introduced at a. Zero b. Infinity c. Infinity and Zero d. Infinity or Zero 4. The multiplication of a sequence by a complex exponential will result in a. Rotation of the real axis by angle ω b. Rotation of the Z plane by angle ω c. Rotation of complex plane d. Rotation of imaginary plane 4. The pole zero plot of a system is shown below. The time domain response of the system is a: a. Falling Exponential b. Rising Exponential c. Cosine d. Damped Cosine Page 8 of 8
42. Given x(t) = e t, y(t) = e 2t. Then x(t) y(t) is equal to a. e 3t b. e t + e 2t c. e t e 2t d. e t 43. Which one of the following transfer functions represents a critically damped system? a. b. c. d. s 2 +4s+4 s 2 +3s +4 s 2 +2s+4 s 2 +s+4 44. Two linear time-invariant discrete time systems s and s2 are cascaded. Each system is modeled by a second order difference equation. The difference equation of the overall cascaded system can be of the order of a.,, 2, 3, or 4 b. either 2 or 4 c. 2 d. 4 45. Match the system function in column with the impulse response in column 2 and select the appropriate choice. (A) e s s + (B) s 2 + s + Page 9 of 8
(C) (s + ) 2 (D) s 2 + s A B C D (a) 3 4 2 (b) 5 2 3 4 (c) 3 2 4 (d) 5 4 3 2 46. Match the time domain functions in column with its Laplace transform in column2. (A) n u[n] z ( z ROC: z > α ) 2 (B) n u[ n ] ROC: z < α z (C) n n u[ n ] ROC: z < α z (D) nα n u[n] ( z ROC: z < α ) 2 A B C D (a) 2 4 3 Page of 8
(b) 3 4 2 (c) 4 3 2 (d) 2 3 4 47) If Ris the region of convergence of x (n) and R2 is the region of convergence of y(n), then the region of convergence of x (n) convoluted y (n) is a) R + R2 b)r R2. c) R R2. d) R R2. 48) The Laplace transform of u(t) is A(s) and the Fourier transform of u(t) is B(jω) Then a) B(jω)=A(s) s=jω b) A(s) =/s but B(jω) /jω c) A(s) /s but B(jω) = /jω d) A(s) /s but B(jω) /jω 49) Region of convergence of a causal LTI system a) Is the entire s-plane. b) Is the right-half of s-plane c) Is the left-half of s-plane c) Does not exist. 5) The region of convergence of a causal finite duration discrete time signal is a) The entire z plane except z = b) The entire z plane except z = c) The entire z plane d) A circle in z plane ASSIGNMENT - II. Find the Laplace transform of the following signals (If it exists) and also determine ROC, show the pole, zero locations also. a. e 2t cos(65πt) u(t) b. e 2t u( t) e t u(t) c. sin(t) e t u(t) + cos(t) e t u(t) d. Unipolar Square wave Page of 8
e. Bipolar square wave f. {Unit Ramp(t) with period π}{u(t) u(t )} g. Rect(t) h. 2. Find the LT and ROC of the following signals. (If it exists) a. 3e 2t u(t) 2e 6t u(t) b. 2e 6 t 3. Find the Z transform and ROC (if it exists) a. x[n] = 4 n u[n] 2 n u[ n ] b. x[n] = 3 n u[n] 2 n u[n ] c. x[n] = k n Once Laplace Transform has been obtained, calculate the DTFT (If possible) and represent graphically. 4. Consider the signal x[n] = ( 2 )n u[n] + α n u[ n n ] + 3 n u[n]. Find the range of α such that the ROC for the given signal is < z < 2. 5. Determine whether each of the following statements is true or false, Justify: a. The Laplace transform of t 2 u(t) does not converge anywhere on the s plane. Page 2 of 8
b. The Laplace transform of e t2 u(t) does not converge anywhere on the s plane. c. The Laplace transform of t does not converge anywhere on the s plane. 6. Using the time-shifting property, find the Laplace transform of the following signals: a. x(t) = u(t) u(t a), a > b. x(t) = 3e (3t 6) u(t 2) c. x(t) = π cos π(t ) u(t ) 7. Using the time scaling property, find the Laplace transform of the following signals: a. x(t) = δ(5t) b. x(t) = u(5t 5) + u(5t ) + u(5t 5) u(5t 2) u(5t 25) u(5t 3) 8. Using the time differentiation property, find the Laplace transform of the following signals: a. x(t) = d dt u(t) b. x(t) = d dt e 2t u(t) c. x(t) = d dt (Asin(ωt)u(t)) 9. Evaluate the following convolutions using Laplace transform a. e t u(t) u(t) b. 2 sin(πt) u(t) {u(t) u(t )}. Using the property for periodic functions determine the Laplace transform of the following signal.. Determine the z transform for each of the following sequences. Sketch the ROC and indicate whether the Fourier transform exists. a. δ[n ] b. δ[n + ] c. (.5) n u[n + ] d. 4 n u[n] 2 n u[ n ] Page 3 of 8
2. Find the Inverse Laplace transform of the following signals: a) X(s) = (s+)(s+2) b) X(s) = 2(s+2) c) X(s) = s 2 +7s+2 s (s+2)(s+3)(s 2 +s+) Re{s} > Re{s} > 3 d) X(s) = s2 s+ (s+) 2 Re{s} > e) X(s) = s+ (s+) 2 +9 Re{s} < f) s sin s 3. Use Heaviside s expansion formula to find (a) L 2s { (b) L { } (s+2)(s 3) 9s+37 )} (s 2)(s+)(s+3) 4. Find Inverse z transform of H(z) = H (z)h 2 (z) where H (z) = + 4 z 8 z 2 H 2 (z) = 7 4 z z 2 2 5. A continuous time signal is sampled for discrete processing. The sampled signal x(t) is x(t) = e nt δ(t nt) (a) Find X(s) and its ROC. (b) Sketch the pole zero plot and determine whether the system is stable, causal. (c) Geometrically interpret that X(jω) is periodic in nature. n= 6. Let h(t) be the impulse response of a causal and stable system with a rational system function. (a)is the system with impulse response dh(t) guaranteed to be stable and causal? (b) Is the system with impulse response dt t h(τ)dτ guaranteed to be stable and causal? 7. Let an LTI system H(s) have an inverse H (s). It is evident that when H(s) and H (s) are cascaded the overall transfer function will be unity. (a) Determine the general algebraic relation between H(s) and H (s). (b)shown below is the pole zero plots of G(s), a stable and causal system. Determine the pole zero plot for the associated inverse system. Page 4 of 8
8. We are given the five following facts about a DT signal x[n] with z transform X[z].. x[n] is real and right sided 2. X[z] has exactly two poles. 3. X[z] has two zeroes at the origin. 4. X[z] has a pole at z =.5e jπ/3 5. X[]=8/3 Determine X[z] and specify its ROC. 9. A transformer with a primary and secondary has a mutual inductance M, and individual self inductances L and L 2 respectively. The primary is excited by a voltage E. The currents in the primary and secondary circuits are I and I 2 respectively. The circuit is shown below. If the currents I and I 2 in the circuits are zero at time t =, show that at time t > they are given by I = EL eαt 2 e α2t L L 2 M 2 + ER 2 eαt eα2t + E α α 2 α α 2 α α 2 R EM eαt e α2t I 2 = L L 2 M 2 α 2 α Where, α_ and α 2 are the roots of the equation (L L 2 M 2 )α 2 + (L R 2 + L 2 R ) + R R 2 =. What happens if L L 2 = M 2? 2. A particle moving on a straight line is acted upon by a force of repulsion which is proportional to its instantaneous distance from a fixed point O on the line. If the particle is placed at a distance Page 5 of 8
a from O, is given a velocity towards O of magnitude V, find the distance of closest approach to O. 2. Suppose we are given the following information about an LTI system: (a) If the input to the system is x [n] = 6 n u[n], then the output is: y [n] = a 2 n + 3 3 u[n], Where a is real number. (b) If x 2 [n] = ( ) n, then the output is y 2 [n] = 7 ( )n 4 Determine a and the system function H(z). Also find the difference equation and represent the system in time domain using a block diagram.. Laplace transform is a unique transform. True or False 2. For to converge, Re(s) <. 3. The real part of s always determines the ROC of the LT. 4. The LT of a signal cannot exist, if the FT does not exist. 5. The ROC does not contain poles of the system. 6. The ROC does not contain zeros of the system. 7. No two signals can have the same LT. 8. The ROC consists of strips parallel to the real axis of the s-plane. 9. The ROC consists of strips parallel to the imaginary axis of the s-plane.. The ROC in the s-plane is connected.. The ROC of sx(s) is > ROC of X(s). 2. If the ROC of the LT does not contain the imaginary axis, the FT does not exist. 3. FT is a special case of LT. 4. LT can be applied even to unstable signals. 5. Systems with rational functions are realizable. 6. ROC of the LT of tx(t) is contained within the ROC of LT of x(t). 7. The solution for the time domain signal is not possible without the knowledge of the ROC. 8. In a stable system, the largest pole of H(s) should lie on the RHS of the s-plane. 9. The z-transform exists for every signal. 2. There is no relation between FT,LT and ZT. Page 6 of 8
Review Questions. What is a transform domain? 2. What is natural domain? 3. When is e st an eigen function of the LSI system? 4. The real part of s has to be greater than zero, for ROC to exist. Justify. 5. What is a double sided Laplace Transform? 6. What is the ROC of the function h(t) = e t u(t)? 7. Does the ROC indicate the stability of the system? 8. If the ROCs are not specified, two signals would be indistinguishable if their LTs are same. Give an example. 9. If, it is not Laplace Transformable. Why?. What is the nature of ROC of the ZT of a discrete two sided signal?. The ROC of a signal which is sum of two signals, can be greater than the intersection of the two ROCs. Give an example. 2. The ROC of the ZT of a time shifted signal, is same as the original signal, with either or deleted. Justify. 3. How is the LT of a signal related to its FT? 4. LT is a general transform, and FT is a special case of LT. Justify. 5. Give an example of an unstable signal which has the LT. 6. What is the LT of tx(t)? 7. If X(s) =/(s-α) what would be the nature of the ROC? 8. If X(z)= /(-βz - ) is x(t) unique? 9. What is the h(t) is H(s)=A+Bs -? 2. How can the stability of the system be inferred from the poles of the system function H(s)? 2. What is the condition on ROC for a rational system to be causal? 22. Is the condition Re(s) =, be a part of ROC, necessary, sufficient or both? 23. Two sequences x (n) and x2 (n) are related by x2 (n) = x (- n). In the z- domain, how are their ROC s related? FAQs. Why is it necessary at times to operate with a transformed domain? 2. What is the ROC for LT? 3. Is it possible to define other transforms for signals which do not have FT? 4. If a signal is time limited and its LT exists, what is the ROC? 5. What is the ROC of a two-sided signal? 6. X(t)=X(t)+X2(t). The ROCs of the two LTs do not intersect. Is X(t) Laplace Transformable? 7. What is the LT of x(at)? 8. What is the concept of time expansion in discrete time signals? 9. What is the LT of the function e at x(t)?. Why do we study extensively systems with rational functions? Page 7 of 8
. What are poles and zeros of a system function? 2. In general what is the ROC of a right sided signal? 3. What is a rational system? 4. What is the condition on the poles of the system function H(z) in a stable system? 5. The region of convergence of a discrete time system is given by z>2. Is the system stable? 6. The region of convergence of a discrete time system is given by z<2. Is the system stable? 7. The region of convergence of a discrete time system is given by z>2. Is the system causal? 8. The region of convergence of a discrete time system is given by z<2. Is the system causal? 9. What is the necessary and sufficient condition for a rational CT system to be causal and stable? 2. What is the necessary and sufficient condition for a rational DT system to be causal and stable? Page 8 of 8