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### HW8_solu_S06

Course: EE 422, Fall 2009
School: Iowa State
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Word Count: 572

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of Solution Homework # 8, Course EE422 (S'06) Q1 (modified 15.1-2) A picture contains 300 picture elements. Each of the elements assume 2 brightness levels with probabilities (0.6, 0.4). Find the information content of a picture. Solution: Information per element is 0.6 log2 (1/0.6) + 0.4 log2 (1/0.4) = 0.971 bit. Information per picture is 300 0.971 = 291.29 bits. Q2 (modified 15.2-6) A source emits three...

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of Solution Homework # 8, Course EE422 (S'06) Q1 (modified 15.1-2) A picture contains 300 picture elements. Each of the elements assume 2 brightness levels with probabilities (0.6, 0.4). Find the information content of a picture. Solution: Information per element is 0.6 log2 (1/0.6) + 0.4 log2 (1/0.4) = 0.971 bit. Information per picture is 300 0.971 = 291.29 bits. Q2 (modified 15.2-6) A source emits three equiprobable messages (m1 , m2 , m3 ) randomly and independently. 1. Find the source entropy H(m). Do Huffman coding for a binary code. Find the average code length L, the code efficiency and redundancy . 2. If the probabilities of (m1 , m2 , m3 ) are now (0.5, 0.25, 0.25). Do the above in step 1 again. Solution: 1. P1 = P (m1 ) = 1/3. P2 = P (m2 ) = 1/3. P3 = P (m3 ) = 1/3. P = P1 = P2 = P3 = 1/3. H(m) = 3 (P log2 1 P) = 3 (1/3 log 2 3) = log2 3 = 1.585 bits. Original Source Messages Probabilities Code Reduced Source S1 2/3 1/3 0 1 m1 m2 m3 Huffman coded output is m1 1 1/3 1/3 1/3 m2 00 m3 01 1 00 01 We see that the code lengths for all symbols are L1 = 1, L2 = 2, and L3 = 2. The average code length is L = P1 L1 + P2 L2 + P3 L3 = 1/3 (1 + 2 + 2) = 5/3 1.67. The code efficiency is = H(m)/L = 1.585/1.67 = 94.9%. The redundancy is = 1 - = 5.1%. 2. P1 = P (m1 ) = 0.5. P2 = P (m2 ) = 0.25. P3 = P (m3 ) = 0.25. H(m) = P1 log2 = 0.5 log 2 1 0.5 1 P1 + P2 log2 1 0.25 1 P2 + 0.25 log 2 + 1 P3 1 0.25 log 2 0.25 + P3 log2 = 0.5 + 0.5 + 0.5 = 1.5 bits. Original Source Messages Probabilities Code Reduced Source S1 0 .5 0 .5 0 1 m1 m2 m3 m1 1 0 .5 0 . 25 0 . 25 m2 00 m3 01 1 00 01 Huffman coded output is We see that the code lengths for all symbols are L1 = 1, L2 = 2, and L3 = 2. The average code length is L = P1 L1 + P2 L2 + P3 L3 = 0.5 1 + 0.25 2 + 0.25 2 = 1.5. The code efficiency is = H(m)/L = 1.5/1.5 = 100%. The redundancy is = 1 - = 0. Q3 (modified ICQ5) We have four messages m1 , m2 , m3 , m4 , with probabilities in shown the table m1 m2 m3 m4 0.2 0.3 0.4 0.1 1. Find the Huffman code for these symbols, 2. Determine H(m), , and . Solution: P1 = P (m1 ) = 0.2, P2 = P (m2 ) = 0.3, P3 = P (m3 ) = 0.4, P4 = P (m4 ) = 0.1. H(m) = P1 log2 = 1 P1 1 0.2 log2 0.2 + P2 log2 + 1 1 1 P2 + P3 log 2 P3 + P4 log2 P4 1 1 1 0.3 log 2 0.3 + 0.4 log2 0.4 + 0.1 log 2 0.1 = 1.846 bits. Original Source Messages Probabilities Code Reduced Source S1 0 .4 0 .3 0 .3 1 00 01 S2 0 .6 0 .4 0 1 m3 m2 m1 m4 Huffman coded output is m1 0 .4 0 .3 0 .2 0 .1 m2 m3 1 01 000 001 m4 000 01 1 001 We see that the code lengths for all symbols are L1 = 3, L2 = 2, L3 = 1, and L4 = 3. The average code length is L = P1 L1 +P2 L2 +P3 L3 +P4 L4 = 0.23+0.32+0.41+0.13 = 1.9. The code efficiency is = H(m)/L = 1.846/1.9 = 97.2%. The redundancy is = 1 - = 2.8%. Q4 (modified 15.4-1) A binary channel matrix is given. P (y1 |x1 ) = 2/3, P (y2 |x1 ) = 1/3, P (y1 |x2 ) = 1/10, and P (y2 |x2 ) = 9/10. Also, P (x1 ) = 1/3 and P (x2 ) = 2/3. Determine H(x), H(x|y), H(y), H(y|x) and I(x; y). Solution: H(x) = 2 k=1 p(xk ) log 2 1/p(xk ) = 1/3 log2 (3) + 2/3 log2 (3/2) = 0.9183 bit. P (y1 ) = P (y1 |x1 )P (x1 ) + P (...

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Iowa State - EE - 422
Hint for HW # 7, Course EE422 (S'05)14.3-2 ) A source emits eight equiprobable messages, which are assigned signals s1 , s2 , . . . , s8 , as shown in Fig. P 14.3-2. (a) Find the optimum receiver for an AWGN channel. Solution: The optimum receiver u
Iowa State - EE - 422
Hint for HW # 8, Course EE422 (S'06)Q1 A picture contains 300 picture elements. Each of the elements assume 10 brightness levels with equal probabilities. Find the information content of a picture. Solution: P = 1/10. Information per element is 10
Iowa State - EE - 422
Project #1 for Course EE422 (S'06)Due on Thursday, April 13, 2006 Name:Programming: out of 8University ID:(scores include validity of Matlab codes, clearness to read, and efficiency.) Report writing: out of 8(scores include, for example, clar
Iowa State - EE - 422
EE 422 Communication Systems II Instructor: Dr. Yao Ma, Coover 3107, Phone: 515-294-8382, Email: mayao@iastate.edu. URL: http:/clue.eng.iastate.edu/mayao/EE422/EE422.html Class Time and Location: TU/TR 4:10-5:30 pm, at Sweeney 1126. Instructor Off
Iowa State - EE - 422
Homework #for Course EE523Full name:University ID:1
Iowa State - EE - 422
Homework # 5 for Course EE422 (S04)(Due on 4pm Thursday, Mar. 25, 2004)Full name:University ID:Q1.) The signal p(t) = A sin(o t + 1 ) (0 t T0 /2), where T0 = 2/0 . p(t) is transmitted for data 1, and q(t) = B sin(o t + 2 ) (0 t T0 /2) is t
Iowa State - EE - 422
Due on 4pm, Tuesday, Mar. 30, 2004Note: 1. This project carry 7.5 points to your final score. 2. You are required to do the programming using the Matlab software, and send the .m file (source code) to the instructor via email. Deduction of score or
Missouri S&T - EE - 265
Missouri S&T - EE - 265
Missouri S&T - EE - 265
Missouri S&T - EE - 265
Missouri S&T - EE - 265
Missouri S&T - EE - 265
Find the a's and b's of the Fourier Series for the signals shown below. Show your work for full credit. 1) xt= 5 2) xt= -5 3) xt=5 cos(2100t) 4) xt=5 cos(2200t) 5) xt=5 sin(2100t) 6) xt=5 sin(2200t) 7) xt=5t
Missouri S&T - EE - 265
Homework #6 Due Mon Mar. 2 Equations for the Fourier Series are shown below (copied out of textbook for class). The first ones we will call the trig Fourier Seriesand the next set we will call the complex exponential Fourier SeriesUsing these equ
Missouri S&T - EE - 265
Missouri S&T - EE - 265
1) In the circuit shown below, x(t) is the input, and y(t) is the outputDo all of the following: a) Write a mathematical expression for the frequency response, H(f). b) Sketch the straight-line approximation to the Bode plot for |H(f)| in dB. c) Sk
Missouri S&T - EE - 265
Missouri S&T - EE - 265
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Missouri S&T - EE - 265
Find the Fourier Transform, X(f), for the signals listed below. You may use the fundamental equation for Fourier Transforms, Fourier Transform Tables, and/or Fourier Transform Theorems. Show your work, or explain your answers, for full credit
Missouri S&T - EE - 265
0 1) In homework 8, you found the Fourier Transform of 5 0 3 73 7 , probably by using FT tables and theorems (at least that's the way I did it). Now do it using the basic equation. (If you used the basic equation in homework 8, then do it us
Missouri S&T - EE - 265
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Missouri S&T - EE - 265
1) The input to a linear, t ime invariant, system is: xt=5cos2300t+sinc(t) The frequency response, H(f) (which is the Fourier Transform of the impulse response, h(t), for this system is: Hf= 10 What is the output of this system, y(t)?2) Repeat prob
Missouri S&T - EE - 265
Laboratory #1 Due no later than January 30, 2009 Write a Matlab script that will generate a sine wave of amplitude 500 mV, at 1 kHz, and a phase of 45 degrees. Make the signal 1 second long. Do all of the following: 1. Plot the sine wave over a 1 sec
Missouri S&T - EE - 265
1) In the circuit shown below, x(t) is the input, and y(t) is the outputDo all of the following: a) Write a mathematical expression for the transfer function, H(s). b) Sketch the pole/zero plot c) Estimate the frequency response, |H(f)|, from the p
Missouri S&T - EE - 265
Missouri S&T - EE - 265
Laboratory #2 Due no later than February 6, 2009 When you press a number button on the telephone, it generates the sum of two sine waves. The frequencies of the sine waves are different depending on which key you press. This method of dialing the pho
Missouri S&T - EE - 265
Laboratory #3 Due no later than February 13, 2009 Find, or create, a .wav audio file between 1 and 10 seconds long. Read the file into Matlab, and do all of the following: 1. Plot it, labeling both axis, put a title on it, and grid lines. 2. Plot bot
Missouri S&T - EE - 265
Laboratory #4 Due no later than February 20, 2009 1. Signal x(t) is a square pulse, 2 volts in amplitude, from 0 to 200 msec. Signal y(t) is a square pulse with amplitude -3, and runs from 0 sec to 300 msec. Generate z(t), the convolution of x(t) and
Missouri S&T - EE - 265
1) Evaluate the following integrals: a) -cos2100ttdt b) -sin2100ttdt c) -11t tdt d) 67t tdt e) -11t t-5+cos2100tt-cos2100tt+5dt f) 0106t-57t+5dt g) 0106t-5+7t+5dt h) -1010-tdt i) 010-t-5dt j) -101056tdt 2) Evaluate the following convolutions. In all
Missouri S&T - EE - 265
Missouri S&T - EE - 265
Missouri S&T - EE - 265
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LSU - MATH - 7330
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LSU - MATH - 7330
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