for HWQ2 to be given on Thursday, Feb 9: Study Class Notes #4 and #5 posted on Canvas. In Lathi's text scan sections 6.1 through 6.4. Skip example 6.8 in section 6.3. Scan sections 7.1 through 7.6. We will primarily focus on the exponential version of the Fourier series. However, you need to be able to convert a trigonometric series to an exponential series and vice versa. Download CT Transforms ref.pdf posted on Canvas and bring it with you to class. You may refer to it during quizzes. Problems from Chapter 6, Lathi's book. In each case, you need do only the parts of the problem for which an answer is given. Be sure to take maximum advantage of symmetry in evaluating Dn. Note that some useful symmetry properties are given in CT Transforms ref.pdf. Some handy integrals appear on page 50-51 of the text. You may refer to these pages during the homework quiz as well. 2 n 6.3 - 1a Do 0 Dn sin for n odd n 2 Do 0 6.3 - 1d Dn j 2 n n sin( ) cos( ) n n 2 2 Hint : Consider carefully the symmetry for this waveform. 6.3 - 4 a & b Spectra sketches required, should be consistent with : Cn n (trig) |Dn| n (exp) -5 0 n/a 2/3 -3 0 n/a /2 -2 0 n/a 1 /6 0 3 0 3 0 +2 2 -/6 1 -/6 +3 1 -/2 -/2 +5 -2/3 -2/3 A low-performance amplifier has for its output the waveform shown below when its input is a \"clean\" (no higher harmonics) sinusoidal waveform. f(t) 10 2 4 6 8 10 12 14 t -10 a) Find the average value and rms value for the periodic waveform shown above. Hint: In determining the rms value, make a sketch of the squared waveform and use symmetry. Answers: Favg = 0, Frms = 7.4535 b) Find the THD for the waveform if the coefficients for the exponential Fourier series for the waveform are given by (You need not find the coefficients for this problem.): Dn j 60 n 2 2 sin n 3 for n odd; D0 0 Answer: THD = 4.64% Fourier Transform Problems: 1. Find the Fourier transform for the following time function using the defining integral relationship for the transform. f(t) = e2t[u(t) - u(t-0.5)] 1 0 0.5 t Ans: 1 ee j 0.5 1 2 j 2. Find the Fourier transform for the following time function by any method: f(t) = sin(t) [u(t) -u(t-)] 1.0 0 t 1 e j Ans: 1 2 3. Sketch the following functions (show critical points): t 20 a) rect 10 t b) sinc 5 c) sinc( t/5 )rect( t/20 ) 4. Find the Fourier transforms for: f (t ) 6e t u (t 2) 1 j 2 e Ans. 6e 2 1 j f (t ) 5 (t 2) 6e 4t u (t ) 10 6 20 ( ) 4 j 5. Problems from Chapter 7, Lathi's book: (Ignore what the book says and do these in any way you choose.) 7.1-6 part b only sin 2t sin t t 7.3-3 part a only T sin 2 2 j4 Ans. 5e j 2 6. Find the energy on a 1- basis Ef for the signal below. Ans. 40 J f(t) 2.0 +5 -5 t -2.0 7. What is the essential bandwidth for the following time function if we set our criterion at 80% of the total signal energy? f (t ) e 2t u (t ) The following integral may be helpful : x 2 1 1 x dx tan 1 2 a a a Ans. 6.16 rad/s 8. Consider the function f (t ) 100 sinc (50t ) a) Find energy on a 1- basis for the signal. Ans. 63.66 J b) Find the essential bandwidth for the signal if we set our criterion at 80% of the total energy on a 1- basis. Ans. 40 rad/s 9. What is the approx. 90% essential bandwidth for: f (t ) 40rect (1000t ) ? Ans. EBW90% 2000 rad/sec 10. A certain system has for its impulse response: h(t ) 200e 100t u (t ) . a) Which of the basic filter types best describes the system (i.e., LP, HP, BP, or BE)? Ans. LP with cutoff frequency of 100 rad/s b) Find the forced response yf(t) if the input to the filter is f (t ) 10 cos(100t 300 ) . Ans. 14.14cos(100t - 75o) 10. The input to a certain system is f(t) and the output is y(t) where: f (t ) 1000rect(1000t ) H ( j ) rect( 8000 ) Make a sketch showing Y(j). Ans. sinc function of magnitude 1.0 for -4000 < < 4000 and zero elsewhere