UNIVERSITY OF MAURITIUS FACULTY OF ENGINEERING PAPER NO EXAMINATION DATE Yearly 2006/2007 BEEEE/ECE/06S/8 BEng (Hons) Electrical and Electronic / Electronic and Communication Engineering Level 2 BEM/06S/18 BEng (Hons) Mechatronics Level 2 SERIES PAPER TIME May 2007 Analytical Techniques [ELEC 2001Y(3)] 9:30 – 12:30 Hours Friday 25 May 2007 This paper contains SIX (6) Questions. Candidates are required to answer ANY FIVE (5) Questions. ‘The Standard Normal Distribution Function’ and ‘Critical Values for the Normal Distribution’ is attached. Question 1 (a) The output of a half wave rectifier is shown in Figure Q1a. By making use of the properties of signals, derive the Fourier Series of the half wave rectified signal. [12 marks] 1 -T0/2 T0/2 0 -T /4 T /4 -T0 0 0 T0 t Fig Q1a: Half-wave rectified cosine wave (b) Show that convolution in the time domain y (t ) = ∫ +∞ −∞ h(t − t ′) x (t ′) dt ′ implies spectral multiplication in the frequency domain Y ( jω ) = H ( jω ) X ( jω ) where +∞ − jωt e y (t ) dt −∞ +∞ − jωt Y ( jω ) = ∫ X ( jω ) = ∫ e x(t ) dt −∞ +∞ − jωt e h(t ) dt −∞ H ( jω ) = ∫ (c) [5 marks] A signal x(t) is given by x(t ) = 0 t < −to x(t ) = a − to ≤ t ≤ t o x(t ) = 0 t > to where a is a constant. (i) Show that its Fourier transform X ( jω ) is given by 2a X ( jω ) = sin ωto ω (ii) [6 marks] If a → ∞ and to → 0 so that ato → 1 determine the limiting value of X ( jω ) and comment on the result. [2 marks] Page 1 of 7 Question 2 (a) Figures Q2a1 – Q2a3 below show s-plane diagrams for 3 different signals. The crosses mark the poles of the s-domain functions. In each case, sketch the signal response in the time domain, and write down the corresponding time domain function. Im[s] Im[s] 2 4 Re[s] Re[s] Figure Q2a1 Figure Q2a2 Im[s] 2 1 -3 -2 -1 Re[s] -1 -2 Figure Q2a3 [6 marks] (b) An electrical circuit comprises a 2H inductor, a 16 Ω resistor, a 0.025 F capacitor connected in series, and powered by a 10 V dc supply. Sketch the s-domain equivalent circuit, and hence obtain an expression for the Laplace Transform, I(s), of the circuit current, assuming that the initial current, and the initial charge on the capacitor, are both zero. Show that the Laplace Transform of the voltage across the capacitor is given by: V (s) = 200 s ( s + 8s + 20 ) 2 [8 marks] (c) Use the Final Value Theorem to determine the voltage across the capacitor in the steady state. [3 marks] (continued next page) Page 2 of 7 Question 2 (continued) (d) By taking inverse Laplace Transforms, show that the capacitor voltage in the time domain is given by: v ( t ) = a + exp ( −4t ) [b cos 2t + c sin 2t ] and give values for a, b and c. [8 marks] Laplace Transforms ( ) L e at sin ω t = ( L e at cos ω t ) ω ( s − a) = 2 +ω2 ( s − a) 2 (s − a) + ω 2 Page 3 of 7 Question 3 An electrical circuit is described by the equations 2q&&1 + 3q&&2 = 18q1 − 8q2 − q&&1 − 2q&&2 = −11q1 + 6q2 (a) Rewrite the equations in matrix form: && = AQ BQ q1 && q&&1 where Q = , Q = q2 q&&2 and define the matrices A and B. Show that the system can be simplified in the form && = CQ Q Define C in terms of B and A, and hence derive C. [6 marks] (b) Find the eigenvalues of matrix C and hence derive its corresponding eigenvectors. [6 marks] (c) The matrix C in part (a) may be diagonalised by using a matrix P to give D = P-1 C P. Write down D and P. [3 marks] (d) By making the substitution q1 = P q2 z1 z2 where z1 and z2 both depend on time, solve for z1 and z2, and hence derive q1 and q2. [10 marks] Page 4 of 7 Question 4 (a) Let S be the set of all possible outcomes of an experiment. Assume A1, A2 and A3 are mutually exclusive and exhaustive events within the sample space S. Let D be any other event. Illustrate this scenario with the aid of a Venn diagram, and thus explain in set notation (using ∩ and ∪ symbols) the meanings of the words (i) mutually exclusive and (ii) exhaustive in the context of the experiment. [3 marks] (b) With the aid of a Venn diagram explain why for any two events A and B, P(A|B) = P(A∩B)/P(B) and P(B|A) = P(B∩A)/P(A). [3 marks] (c) (i) If A and B are two events such that P( A) ≠ 0 and P( B) ≠ 0 , prove that if P ( A B) > P ( A) then P ( B A) > P ( B) . [3 marks] (ii) If A , B and C are three events such that P (C ) ≠ 0 and P( B ∩ C ) ≠ 0 , show that P ( A ∩ B ∩ C ) = P( A B ∩ C ) P( B C ) P (C ) . [2 marks] (d) For A1, A2, A3 and D in part (a), since P ( D ) = P ( D ∩ A1 ) + P( D ∩ A2 ) + P( D ∩ A3 ) , then starting with the result in part (b), prove Bayes’ Rule: P( A1 | D) = P( A1) P( D | A1) . P( D | A1 ) P( A1) + P( D | A2 ) P( A2 ) + P( D | A3 ) P( A3 ) [6 marks] (e) Let 100 bits/sec be received from transmitter A1 with error rate =1/5; let 200 bits/sec be received from transmitter A2 with error rate =1/10; let 400 bits/sec be received from transmitter A3 with error rate =1/40. Let all three bit streams be randomly multiplexed together, and let one bit be selected at random from the composite signal. Let D represent the event that this bit is in error. By using the results from part (d): (i) What is the probability (P(D)) that the selected bit is in error? [4 marks] (ii) If the selected bit is in error, what is the probability that it is from transmitter A1? [4 marks] Page 5 of 7 Question 5 (a) Let the voltage V(t) represent some parameter in a communications system. Let it also be a continuous random variable with pdf: a λ , for λ ≤ 3 pV (λ ) = . for λ > 3 0, (i) Sketch pV (λ ) , and from this explain why the value of the constant ‘a’ must equal 1/9; [2 marks] (ii) Determine the mean, µV , the variance, σ V2 , and the power ( V 2 ) of V(t); [4 marks] (iii) Calculate the probability that the voltage has a value 1 ≤ V (t ) ≤ 2 . [2 marks] (b) A shop sells rose plants in boxes, each box containing the same number of plants. The number of plants per box which produce yellow flowers has a binomial distribution with mean 11 and variance 4.95. (i) Calculate the number of plants per box. [2 marks] (ii) Calculate the probability that a box contains exactly 12 plants which produce yellow flowers. [2 marks] (iii) Another shop sells rose plants in boxes of 100. The shop’s advertisement states that the probability of any rose plant producing a pink flower is 0.3. Use a suitable approximation to calculate the probability that a box contains fewer than 35 plants which produce pink flowers. [3 marks] (c) The probability distribution function for a Poisson random variable with parameter λ is given by P( X = k ) = e (i) −λ λk k! k = 0,1,......... Under what conditions does this probability distribution function arise. [2 marks] (ii) For a Poisson random variable X with parameter λ show that µ X = λ and σ X2 = λ . [8 marks] Page 6 of 7 Question 6 (a) A packet contains six biscuits, each of which is individually wrapped. The mass of a biscuit can be taken to be normally distributed with mean 70 g and standard deviation 4 g. The mass of the individual wrapping of a biscuit is normally distributed with mean 10 g and standard deviation 1 g. The mass of the outer packaging is normally distributed with mean 30 g and standard deviation 3 g. Assuming that the masses of the biscuits, wrappings and packaging are independent, calculate the probability that the total mass of a randomly chosen packet and its contents lies between 500 g and 520 g. [6 marks] (b) An experiment carried out to compare the bond strength of modified mortar to that of unmodified mortar resulted in x = 18.12 kgf/cm2 for the modified mortar (sample size n1 = 40) and y = 16.87 kgf/cm2 for the unmodified mortar (sample size n 2 = 32). Assume that the bond strength distributions are both normal. (i) Assuming that standard deviations σ 1 = 1.6 and σ 2 = 1.4 for the modified mortar and the unmodified mortar respectively, test at level 0.01 if the bond strength for modified mortar is greater than that for the unmodified mortar. [4 marks] (ii) Calculate the probability of a type II error for the test of part (i) when mean difference between the bond strengths of modified mortar and unmodified mortar is actually µ1 − µ 2 = 1. [4 marks] (iii) How would the analysis and conclusions of part (i) change if the population standard deviations σ 1 and σ 2 were unknown, but instead the sample standard deviations s1 = 1.6 and s 2 = 1.4 were known? [4 marks] (c) /nr Each helmet in a random sample of 42 helmets was subjected to a certain impact test and 27 showed damage. (i) Calculate a 99 % Confidence Interval of the proportion of all helmets that will show damage when tested in this manner. [4 marks] (ii) What sample size would be required for the width of a 99 % Confidence Interval to be at most 0.10? [3 marks] END OF QUESTION PAPER Page 7 of 7 Page 8 of 7