midterm_2_2004

midterm_2_2004 - BALAKRISHNAN M AR 25, 2004 ECE 301 Midterm...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: BALAKRISHNAN M AR 25, 2004 ECE 301 Midterm Examination #2 1. Enter your name, student ID number, e-mail address, and signature in the space provided on this page, NOW! Also circle your section. 2. This exam has two parts. Part I consists of questions for which no justification is required and no partial credit will be provided. Enter the answers to Part I on Page 2, which is provided to you separately for your convenience. When you return your exam, simply place Page 2 in between Pages 1 and 3 of your exam. Make sure you enter your name, student ID number and e-mail address in the space provided on Page 2, NOW! Part II consists of three problems. Unless otherwise instructed, justify your answers to these problems completely. Please note that answers provided without justification to those problems requiring a full justification will be given zero credit. 3. This exam is worth 100 points. You have one hour to complete it. 4. There are a total of 10 pages in the exam booklet (including the answer page for Part I). Use the back of each page for rough work. 5. No calculators are allowed. 6. You are allowed the use of one 8.5-inch 11-inch crib sheet (both sides). 7. You might want to read through all of the problems first, to get a feel for how long each one might take, but don’t worry—several of the questions are easier than they might appear when you just scan them. Good luck! IMPORTANT! Name: Student ID #: Whenever a certain space is provided for the final answer, be sure to enter your answer there. E-mail address: Signature: 1 Enter your answers to Part I here. 1. (35 points) (a) (b) (c) (d) (e) ¤ £ ¤ £ ¤ ¤ £ ¤ ¤ ¤ ¤ ¤ £ £ £ £ £ x1 x2 x3 x4 x5 0 50 0 50 0 50 0 50 0 50 x1 0 25 x1 0 x2 0 50 x3 0 50 x4 0 50 x5 0 50 ¤ £ ¤ £ ¤ £ ¤ £ x1 0 25 x1 0 50 2. (10 points) (a) T/F 3. (15 points) ¡ (a) (b) The average power of x n over one time period is Name: Student ID #: E-mail address: Signature: 2 §¦ §¦ § ¦ x 1 x1 ¤ x5 0 ¤ x4 0 ¤ x3 0 ¤ x2 0 (b) T/F (c) T/F (d) T/F ¡ ¥ ¡ ¢ ¡ ¢ ¡ ¢ ¡ ¢ ¡ ¢ (e) T/F Questions for Part I Do not justify your answer. Partial credit is NOT available. Enter your answers in the spaces provided, otherwise you will get zero credit. 1. (35 points) Let ak denote the kth Fourier series coefficient of a periodic continuous-time signal x t , with fundamental period 2. The values of x t for some time instances are given below: x 1 x 0 50 x 0 25 x 0 x 0 25 x 0 50 1 j 1 0 1 j (a) (5 points) Let x1 t be continuous-time signal with fundamental period 2, whose kth Fourier series coefficient bk satisfies bk 3ak . Find x1 0 50 , x1 0 25 , x1 0 , x1 0 25 , and x1 0 50 , and enter them on Page 2. Each entry is worth one point. ¤ ¤ £ ¡ ¢ ¤ £ ¡ ¢ (b) (6 points) Let x2 t be continuous-time signal with fundamental period 2, whose kth Fourier series coefficient ck satisfies ck e jkπak . Find x2 0 50 , x2 0 , and x2 0 50 , and enter them on Page 2. Each entry is worth two points. ¤ £ ¤ ¤ £ ¡ ¥ ¡ (c) (6 points) Let x3 t be continuous-time signal with fundamental period 2, whose kth Fourier series coefficient dk satisfies dk a k . Find x3 0 50 , x3 0 , and x3 0 50 , and enter them on Page 2. Each entry is worth two points. ¤ £ ¤ ¤ £ ¡ ¥ ¡ (d) (6 points) Let x4 t be continuous-time signal with fundamental period 2, whose kth 1 Fourier series coefficient ek satisfies ek 0 50 , x4 0 , and 2 ak a k . Find x4 x4 0 50 , and enter them on Page 2. Each entry is worth two points. ¤ ¤ £ ¡ ¢ ¤ ¡ ¢ (e) (12 points) Let x5 t be continuous-time signal with fundamental period 2, whose kth Fourier series coefficient f k satisfies fk ℜak , i.e., fk equals the real-part of ak . Find x5 0 50 , x5 0 , and x5 0 50 , and enter them on Page 2. Each entry is worth four points. 3 ¤ £ ¡ ¤ £ ¤ ¤ ¤ £ ¡ ¡ ¢ ¤ ¤ £ £ ¡ ¢ ¤ ¤ ¤ ¤ ¤ ¤ £ ¤ ¤ ¡ ¢ ¤ ¤ ¤ £ £ £ ¤ ¡ ¥ 2. (10 points) State whether each of the statements is true or false, following these instructions: Do not justify your answer. Enter your answers on page 2. Use abbreviation ”T” to denote ”true” and ”F” to denote false”. ¤ ¤ (a) (2 points) x t and y t are periodic signals, each with a fundamental period of T . Then xt y t has a fundamental period of T . ¤ ¢ (b) (2 points) If the Fourier series coefficients of x t are real, then x t is real. ¤ ¤ (c) (2 points) The Fourier series coefficients of x 2n are the same as that of x n . §¦ § ¦ 4 ¡ (e) (2 points) If x t is real and odd, then ak ¤ a k. ¡ (d) (2 points) If x t is purely imaginary, then ak ¤ a k . ¡ ¡ ¡ ¡ ¡ ¤ 3. (15 points) x n is a discrete-time periodic signal, with fundamental period 4. Its Fourier series coefficients are given below: ¡ (a) (10 points) Find x 1 and x 1 , and enter them on Page 2. Each entry is worth five points. Your answer must be of the form “real part” + j “imaginary part.” (b) (5 points) Find the average power of x n over one time period. §¦ §¦ § ¦ ¡ ¡ §¦ a 1 1 a0 0 a1 1 a2 0 5 Questions for Part II Justify your answer completely. No credit will be given for answers without justification. Partial credit is available. 4. (15 points) x t , a continuous-time periodic signal with fundamental period 6, is specified as follows: 0 3t 1 xt 1 1t1 01t3 As usual, let ak denote the kth Fourier series coefficient of x t . (a) (5 points) Let x1 t be a continuous-time periodic signal with fundamental period 6 specified as follows: 1 3t 1 1 1t1 x1 t 11t3 Let bk denote the kth Fourier series coefficient of x1 t . Express bk in terms of ak . Note: You will be given no credit for calculating the numerical values of b k directly. You will only be given credit for expressing bk in terms of ak . ¤ ¤ ¤ 6 ¤ ¡ £ ¤ ¦ ¦ ¤ ¡ ¦ £ ¥ ¥ ¦ ¦ ¥ ¦ ¡ ¡ ¥ ¥ ¥ ¤ ¤ ¤ ¡ ¡ ¡ ¡ ¤ ¤ ¤ ¢ £¡ ¢ £¡ ¤ ¤ ¤ ¤ (b) (5 points) Let x2 t be a continuous-time periodic signal with fundamental period 6 specified as follows: 0 3t 2 1 2t2 x2 t 02t3 Let ck denote the kth Fourier series coefficient of x2 t . Express ck in terms of ak . Note: You will be given no credit for calculating the numerical values of c k directly. You will only be given credit for expressing ck in terms of ak . ¤ ¤ ¤ (c) (5 points) Let x3 t be a continuous-time periodic signal with fundamental period 6 specified over the time-period 3 3 as δ t 1 δ t 1 . Let dk denote the kth Fourier series coefficient of x3 t . Express dk in terms of ak . Note: You will be given no credit for calculating the numerical values of d k directly. You will only be given credit for expressing dk in terms of ak . ¤ ¡ ¡ ¤ ¢ 7 ¡ £ ¦ ¦ ¦ ¥ ¥ ¥ ¡ ¡ ¤ ¤ ¤ § ¢ £¡ ¤ ¡ ¤ ¦ ¤ ¤ ¤ 5. (10 points) x n is a discrete-time periodic signal, with fundamental period 3. You are given the following data: x n is real and odd. The average power in x n over one time period is 3. The imaginary part of a1 , the first Fourier series coefficient of x n , is positive. 1 , x 0 , and x 1 . §¦ §¦ § §¦ ¡ ¦ Find x §¦ §¦ §¦ 8 6. (15 points) x n is a discrete-time periodic signal, with fundamental period N , given over one time-period as: 1n xn n 01 N1 2 ¤ 9 £££ ¢¢¢¤ ¤ (a) (10 points) Find the Fourier series coefficients ak for k answer as A ak B Ce jkΘ where A, B, C and Θ are real constants. ¤ ¢ 01 N ¡ £ ¡ ¤ £££ ¢¢¢¤ ¤ ¤ ¢¡ §¦ §¦ 1. Express your (b) (5 points) Suppose that N is even. Which of the harmonics represented by the Fourier series coefficients ak , k 0 1 N 1, has the most average power over one timeperiod, and which one has the least? ¡ ¤ £££ ¢¢¢¤ ¤ 10 ...
View Full Document

This note was uploaded on 02/18/2010 for the course ECE 301 taught by Professor V."ragu"balakrishnan during the Fall '06 term at Purdue.

Ask a homework question - tutors are online