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Unformatted text preview: BALAKRISHNAN M AR 25, 2004 ECE 301 Midterm Examination #2 1. Enter your name, student ID number, email 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 email 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 8.5inch 11inch 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. Email 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 #: Email 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 continuoustime 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 continuoustime 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 continuoustime 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 continuoustime 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 continuoustime 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 continuoustime signal with fundamental period 2, whose kth Fourier series coefﬁcient f k satisﬁes fk ℜak , i.e., fk equals the realpart 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 .
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¢ (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 discretetime 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 continuoustime 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 continuoustime 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 .
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¤ ¤ 6 ¤ ¡ £ ¤ ¦ ¦ ¤ ¡ ¦ £ ¥ ¥ ¦ ¦ ¥ ¦ ¡ ¡ ¥ ¥ ¥ ¤ ¤ ¤ ¡ ¡ ¡ ¡ ¤ ¤ ¤ ¢ £¡ ¢ £¡ ¤ ¤ ¤ ¤ (b) (5 points) Let x2 t be a continuoustime 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 .
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¤ ¤ (c) (5 points) Let x3 t be a continuoustime periodic signal with fundamental period 6 speciﬁed over the timeperiod 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 discretetime 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 discretetime periodic signal, with fundamental period N , given over one timeperiod 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.
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¢ 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?
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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.
 Fall '06
 V."Ragu"Balakrishnan

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