final exam jameel communication systems fall 2013 updated

8/13/2019 FINAL Exam Jameel Communication Systems FALL 2013 Updated

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University of Management & Technology

Department of Electrical Engineering, SEN

Student ID ______________________________ Section _____________

Student Name ______________________________ Semester Fall 2013Student Signature ______________________________ EXAM: Final

Course Code EE410 Academic Year 2013

Course Title Communication Systems Time Allowed 120 min.

Exam Date Jan 22, 2014 Total Marks 100

Resource Person(s) M Ilyas Khan, Dr M. Tahir Mushtaq, Jameel Ahmad

DO NOT OPEN THIS EXAM UNTIL TOLD TO DO SO

The instructions below must be followed strictly.

You must

• Keep your eyes on your own paper.

• Switch off your mobile phones completely.

⇒ Specific instructions

Calculator Allowed, Closed Book / Closed Notes. No extra sheet will be given.

Provide your final answers in space provided.

You may use back blank sheets at the end of the paper for rough work.

Use a blue or black ball point or pen. Please do not use lead pencils.

Certificate to be filled at the time of exam

I have counted the pages and no page is missing.

Student’s Signature: ___________

Question Q1 Q2 Q3 Q4 Total

Total Marks 25 25 25 25 100

Marks Obtained



Student ID:________________ Exam Room No:__________ Section:_______

Question No.1

PART-AUsing the message signal, ( ) = 2 | |

, determine and sketch the modulated waves for the

following methods of modulation:

(a) Amplitude modulation with 50 percent modulation index. (8 points)

(b) Double sideband-suppressed carrier (DSB-SC) modulation. (8 points)




Question No.1 PART-B

A carrier wave of frequency 100 MHz is frequency modulated by a sinusoidal wave of amplitude 20 volts

and frequency 100 KHz. The frequency sensitivity k f of the modulator is 25 kHz per volt.

a) Determine the approximate bandwidth of the FM signal.(3 points)

b) Determine the bandwidth if the amplitude of the message signal is doubled. (3 points)

c) Determine the bandwidth if the frequency of the message signal is doubled. (3 points)




Question No.2a) The sinusoidal signal, ( ) = 8sin(2 ) volts, is transmitted using a 4 bit binary

PCM system. The quantizer is of midrise type, with step size of 1 volt. Sketch the

resulting PCM wave for one complete cycle of the input. Assume a sample rate of

four samples per second, with samples taken at = ± ,± ,± ,….., .

(15 points)

b) Calculate the signal to quantization noise ratio (SQNR) for the above PCM signal. (5 points)




c) In the following an NRZ waveform is shown. Draw Manchester and bipolar RZ signals for the

same. (2.5+2.5 points)

Manchester

___________________________________________________________

Bipolar RZ

________________________________________________________________________




Question-No-3 (3+2+15+5)

a) Write the general expression for the impulse response of a matched filter ( 3points)

b) Write one most important property of Matched filter. ( 2points)

c) Consider the following BPSK transmit-receive system ( 13points)

1. Derive the theoretical expression for Probability of Error in case of Baseband Transmission of

Binary wave using Polar NRZ (non-return-to-zero) signaling i.e. BPSK. In this form of signaling, symbols

1 and 0 are represented by positive and negative rectangular pulses of equal amplitude and equal duration.

The bit duration is Tb and A is the transmitted pulse amplitude. The baseband signal is corrupted by Additive

White Gaussian Noise (AWGN) ( ) as shown in the above diagram. Noise is assumed to have zero mean

and power spectral density . Show all important steps and any assumptions. The received signal is

( ) = ( ) + ( ).BPSK demodulator can be either a correlator or a Matched Filter type detector. is

bit energy. is the source bit stream of data and is the detected bit stream.

Your final expression should look like the following using either Q( ) function or complementary errorfunction erfc ( ).

= = ( )

2. Calculate for Eb=2x10-19 joules and No=10-20

Watts/Hz ( 2points)




d) Describe the Nyquist criterion for zero ISI (Inter symbol interference) (5points)

Solution

Question-No-4 (do Part-A or Part-B)

Part-Aa) Block diagram of FSK generation is shown below. Write the corresponding output equation

( ) for this scheme.




Solution: (2.5points)

(2.5 points) Describe the following FSK receiver as coherent receiver or non-coherent receiver and Why?

Solution:

b) An FSK system transmits binary data at the rate of 2.5x106 bits per second. During the course of

transmission, white Gaussian Noise of zero mean and power spectral density No=10-20 Watt/Hz is

added to the signal. In the absence of noise, the amplitude, A, of the received sinusoidal wave fordigit 1 or 0 is 1 milli Volt. Determine the average probability of error for FSK.

• Coherent Binary FSK has BER = ( )

Solution (5points)




c) What are the 3 similarities and 3 differences in MSK and GMSK? (3points)

Similarities Differences

MSK

GMSK

d) Draw the block diagram of MSK OR GMSK transmitter and receiver. Write a general time-domain

mathematical expression for a Gaussian filter impulse response used in GMSK. (1.5+1.5 points)




e) Why constant envelope is desirable in Passband Communication. (3points)

f) MSK and GMSK are continuous phase modulation schemes while PSK has phase discontinuity.What is the effect of phase discontinuity on spectrum? Why continuous phase is preferred?

(4points)




g) Explain which modulation scheme has better BER performance by observing the following BER

curves for GMSK, BPSK or MSK.(2 points)

GMSK

MSKBPSK

Part-Ba) Define the following (5points)

1) CDF (2) PDF ( 3) variance (4) Bayesian Theorem (5) Figure of merit




b) (5points)

In a random experiment, a trial of four successive tosses of a coin. If we define an RV X as the number

of heads appearing in a trial, determine Px(x) and FX(x)

c) (5points)




Determine the autocorrelation ( )and the power of a low-pass random process with a noise

PSD of Sx( )= /2 as shown in figure below

d) Explain the thermal and shot noise (5points)

e) Explain the behavior of AM signal in the presence of AWGN noise (5points)




Appendix-A



Q – Function:

An alternate form of error function. It basically defines the area under the

Standardized Gaussian tail. For a standardized Gaussian random variable X of zero

mean and unit variance, the Q-function is defined by

dx x

vQv

2exp

2

1)(

2

π-------- ( A6.6)

Relations between Q-function and erfc function:

(i)

22

1

)(

v

erfcvQ ------- ( A6.7a)

(ii) )2(2 uQuerfc ------ ( A6.7b)



E&CE 411, Spring 2009, Table of Q Function 1

Table 1: Values of Q(x) for 0 ≤ x ≤ 9x Q(x) x Q(x) x Q(x) x Q(x)

0.00 0.5 2.30 0.010724 4.55 2.6823×10−6 6.80 5.231×10−12

0.05 0.48006 2.35 0.0093867 4.60 2.1125×10−6 6.85 3.6925×10−12

0.10 0.46017 2.40 0.0081975 4.65 1.6597×10−6 6.90 2.6001×10−12

0.15 0.44038 2.45 0.0071428 4.70 1.3008×10−6 6.95 1.8264×10−12

0.20 0.42074 2.50 0.0062097 4.75 1.0171×10−6 7.00 1.2798×10−12

0.25 0.40129 2.55 0.0053861 4.80 7.9333×10−7 7.05 8.9459×10−13

0.30 0.38209 2.60 0.0046612 4.85 6.1731×10−7 7.10 6.2378×10−13

0.35 0.36317 2.65 0.0040246 4.90 4.7918×10−7 7.15 4.3389×10−13

0.40 0.34458 2.70 0.003467 4.95 3.7107×10−7 7.20 3.0106×10−13

0.45 0.32636 2.75 0.0029798 5.00 2.8665×10−7 7.25 2.0839×10−13

0.50 0.30854 2.80 0.0025551 5.05 2.2091×10−7 7.30 1.4388×10−13

0.55 0.29116 2.85 0.002186 5.10 1.6983×10−7 7.35 9.9103×10−14

0.60 0.27425 2.90 0.0018658 5.15 1.3024×10−7 7.40 6.8092×10−14

0.65 0.25785 2.95 0.0015889 5.20 9.9644×10−

8 7.45 4.667×10−

14

0.70 0.24196 3.00 0.0013499 5.25 7.605×10−8 7.50 3.1909×10−14

0.75 0.22663 3.05 0.0011442 5.30 5.7901×10−8 7.55 2.1763×10−14

0.80 0.21186 3.10 0.0009676 5.35 4.3977×10−8 7.60 1.4807×10−14

0.85 0.19766 3.15 0.00081635 5.40 3.332×10−8 7.65 1.0049×10−14

0.90 0.18406 3.20 0.00068714 5.45 2.5185×10−8 7.70 6.8033×10−15

0.95 0.17106 3.25 0.00057703 5.50 1.899×10−8 7.75 4.5946×10−15

1.00 0.15866 3.30 0.00048342 5.55 1.4283×10−8 7.80 3.0954×10−15

1.05 0.14686 3.35 0.00040406 5.60 1.0718×10−8 7.85 2.0802×10−15

1.10 0.13567 3.40 0.00033693 5.65 8.0224×10−9 7.90 1.3945×10−15

1.15 0.12507 3.45 0.00028029 5.70 5.9904×10−9 7.95 9.3256×10−16

1.20 0.11507 3.50 0.00023263 5.75 4.4622×10−9

8.00 6.221×10−16

1.25 0.10565 3.55 0.00019262 5.80 3.3157×10−9 8.05 4.1397×10−16

1.30 0.0968 3.60 0.00015911 5.85 2.4579×10−9 8.10 2.748×10−16

1.35 0.088508 3.65 0.00013112 5.90 1.8175×10−9 8.15 1.8196×10−16

1.40 0.080757 3.70 0.0001078 5.95 1.3407×10−9 8.20 1.2019×10−16

1.45 0.073529 3.75 8.8417×10−5 6.00 9.8659×10−10 8.25 7.9197×10−17

1.50 0.066807 3.80 7.2348×10−5 6.05 7.2423×10−10 8.30 5.2056×10−17

1.55 0.060571 3.85 5.9059×10−5 6.10 5.3034×10−10 8.35 3.4131×10−17

1.60 0.054799 3.90 4.8096×10−5 6.15 3.8741×10−10 8.40 2.2324×10−17

1.65 0.049471 3.95 3.9076×10−5 6.20 2.8232×10−10 8.45 1.4565×10−17

1.70 0.044565 4.00 3.1671×10−5 6.25 2.0523×10−10 8.50 9.4795×10−18

1.75 0.040059 4.05 2.5609×10−5 6.30 1.4882×10−10 8.55 6.1544×10−18

1.80 0.03593 4.10 2.0658×10−5 6.35 1.0766×10−10 8.60 3.9858×10−18

1.85 0.032157 4.15 1.6624×10−5 6.40 7.7688×10−11 8.65 2.575×10−18

1.90 0.028717 4.20 1.3346×10−5 6.45 5.5925×10−11 8.70 1.6594×10−18

1.95 0.025588 4.25 1.0689×10−5 6.50 4.016×10−11 8.75 1.0668×10−18

2.00 0.02275 4.30 8.5399×10−6 6.55 2.8769×10−11 8.80 6.8408×10−19

2.05 0.020182 4.35 6.8069×10−6 6.60 2.0558×10−11 8.85 4.376×10−19

2.10 0.017864 4.40 5.4125×10−6 6.65 1.4655×10−11 8.90 2.7923×10−19

2.15 0.015778 4.45 4.2935×10−6 6.70 1.0421×10−11 8.95 1.7774×10−19

2.20 0.013903 4.50 3.3977×10−6 6.75 7.3923×10−12 9.00 1.1286×10−19

2.25 0.012224



Complementary Error Function Table

x erfc(x) x erfc(x) x erfc(x) x erfc(x) x erfc(x) x erfc(x) x erfc(x)

0 1.000000 0.5 0.479500 1 0.157299 1.5 0.033895 2 0.004678 2.5 0.000407 3 0.00002209

0 .0 1 0 .9 887 17 0 .5 1 0 .4 70 75 6 1 .0 1 0 .1 53 19 0 1 .5 1 0 .0 32 72 3 2 .0 1 0 .00 44 75 2 .5 1 0 .0 00 38 6 3 .0 1 0.00 00 20 74

0 .0 2 0 .9 774 35 0 .5 2 0 .4 62 10 1 1 .0 2 0 .1 49 16 2 1 .5 2 0 .0 31 58 7 2 .0 2 0 .00 42 81 2 .5 2 0 .0 00 36 5 3 .0 2 0.00 00 19 47

0 .0 3 0 .9 661 59 0 .5 3 0 .4 53 53 6 1 .0 3 0 .1 45 21 6 1 .5 3 0 .0 30 48 4 2 .0 3 0 .00 40 94 2 .5 3 0 .0 00 34 6 3 .0 3 0.00 00 18 27

0 .0 4 0 .9 548 89 0 .5 4 0 .4 45 06 1 1 .0 4 0 .1 41 35 0 1 .5 4 0 .0 29 41 4 2 .0 4 0 .00 39 14 2 .5 4 0 .0 00 32 8 3 .0 4 0.00 00 17 14

0 .0 5 0 .9 436 28 0 .5 5 0 .4 36 67 7 1 .0 5 0 .1 37 56 4 1 .5 5 0 .0 28 37 7 2 .0 5 0 .00 37 42 2 .5 5 0 .0 00 31 1 3 .0 5 0.00 00 16 08

0 .0 6 0 .9 323 78 0 .5 6 0 .4 28 38 4 1 .0 6 0 .1 33 85 6 1 .5 6 0 .0 27 37 2 2 .0 6 0 .00 35 77 2 .5 6 0 .0 00 29 4 3 .0 6 0.00 00 15 08

0 .0 7 0 .9 211 42 0 .5 7 0 .4 20 18 4 1 .0 7 0 .1 30 22 7 1 .5 7 0 .0 26 39 7 2 .0 7 0 .00 34 18 2 .5 7 0 .0 00 27 8 3 .0 7 0.00 00 14 14

0 .0 8 0 .9 099 22 0 .5 8 0 .4 12 07 7 1 .0 8 0 .1 26 67 4 1 .5 8 0 .0 25 45 3 2 .0 8 0 .00 32 66 2 .5 8 0 .0 00 26 4 3 .0 8 0.00 00 13 26

0 .0 9 0 .8 987 19 0 .5 9 0 .4 04 06 4 1 .0 9 0 .1 23 19 7 1 .5 9 0 .0 24 53 8 2 .0 9 0 .00 31 20 2 .5 9 0 .0 00 24 9 3 .0 9 0.00 00 12 43

0.1 0.887537 0.6 0.396144 1.1 0.119795 1.6 0.023652 2.1 0.002979 2.6 0.000236 3.1 0.00001165

0 .1 1 0 .8 763 77 0 .6 1 0 .3 88 31 9 1 .1 1 0 .1 16 46 7 1 .6 1 0 .0 22 79 3 2 .1 1 0 .00 28 45 2 .6 1 0 .0 00 22 3 3 .1 1 0.00 00 10 92

0 .1 2 0 .8 652 42 0 .6 2 0 .3 80 58 9 1 .1 2 0 .1 13 21 2 1 .6 2 0 .0 21 96 2 2 .1 2 0 .00 27 16 2 .6 2 0 .0 00 21 1 3 .1 2 0.00 00 10 23

0 .1 3 0 .8 541 33 0 .6 3 0 .3 72 95 4 1 .1 3 0 .1 10 02 9 1 .6 3 0 .0 21 15 7 2 .1 3 0 .00 25 93 2 .6 3 0 .0 00 20 0 3 .1 3 0.00 00 09 58

0 .1 4 0 .8 430 53 0 .6 4 0 .3 65 41 4 1 .1 4 0 .1 06 91 8 1 .6 4 0 .0 20 37 8 2 .1 4 0 .00 24 75 2 .6 4 0 .0 00 18 9 3 .1 4 0.00 00 08 97

0 .1 5 0 .8 320 04 0 .6 5 0 .3 57 97 1 1 .1 5 0 .1 03 87 6 1 .6 5 0 .0 19 62 4 2 .1 5 0 .00 23 61 2 .6 5 0 .0 00 17 8 3 .1 5 0.00 00 08 40

0 .1 6 0 .8 209 88 0 .6 6 0 .3 50 62 3 1 .1 6 0 .1 00 90 4 1 .6 6 0 .0 18 89 5 2 .1 6 0 .00 22 53 2 .6 6 0 .0 00 16 9 3 .1 6 0.00 00 07 860 .1 7 0 .8 100 08 0 .6 7 0 .3 43 37 2 1 .1 7 0 .0 98 00 0 1 .6 7 0 .0 18 19 0 2 .1 7 0 .00 21 49 2 .6 7 0 .0 00 15 9 3 .1 7 0.00 00 07 36

0 .1 8 0 .7 990 64 0 .6 8 0 .3 36 21 8 1 .1 8 0 .0 95 16 3 1 .6 8 0 .0 17 50 7 2 .1 8 0 .00 20 49 2 .6 8 0 .0 00 15 1 3 .1 8 0.00 00 06 89

0 .1 9 0 .7 881 60 0 .6 9 0 .3 29 16 0 1 .1 9 0 .0 92 39 2 1 .6 9 0 .0 16 84 7 2 .1 9 0 .00 19 54 2 .6 9 0 .0 00 14 2 3 .1 9 0.00 00 06 44

0.2 0.777297 0.7 0.322199 1.2 0.089686 1.7 0.016210 2.2 0.001863 2.7 0.000134 3.2 0.00000603

0 .2 1 0 .7 664 78 0 .7 1 0 .3 15 33 5 1 .2 1 0 .0 87 04 5 1 .7 1 0 .0 15 59 3 2 .2 1 0 .00 17 76 2 .7 1 0 .0 00 12 7 3 .2 1 0.00 00 05 64

0 .2 2 0 .7 557 04 0 .7 2 0 .3 08 56 7 1 .2 2 0 .0 84 46 6 1 .7 2 0 .0 14 99 7 2 .2 2 0 .00 16 92 2 .7 2 0 .0 00 12 0 3 .2 2 0.00 00 05 27

0 .2 3 0 .7 449 77 0 .7 3 0 .3 01 89 6 1 .2 3 0 .0 81 95 0 1 .7 3 0 .0 14 42 2 2 .2 3 0 .00 16 12 2 .7 3 0 .0 00 11 3 3 .2 3 0.00 00 04 93

0 .2 4 0 .7 343 00 0 .7 4 0 .2 95 32 2 1 .2 4 0 .0 79 49 5 1 .7 4 0 .0 13 86 5 2 .2 4 0 .00 15 36 2 .7 4 0 .0 00 10 7 3 .2 4 0.00 00 04 60

0 .2 5 0 .7 236 74 0 .7 5 0 .2 88 84 5 1 .2 5 0 .0 77 10 0 1 .7 5 0 .0 13 32 8 2 .2 5 0 .00 14 63 2 .7 5 0 .0 00 10 1 3 .2 5 0.00 00 04 30

0 .2 6 0 .7 131 00 0 .7 6 0 .2 82 46 3 1 .2 6 0 .0 74 76 4 1 .7 6 0 .0 12 81 0 2 .2 6 0 .00 13 93 2 .7 6 0 .0 00 09 5 3 .2 6 0.00 00 04 02

0 .2 7 0 .7 025 82 0 .7 7 0 .2 76 17 9 1 .2 7 0 .0 72 48 6 1 .7 7 0 .0 12 30 9 2 .2 7 0 .00 13 26 2 .7 7 0 .0 00 09 0 3 .2 7 0.00 00 03 76

0 .2 8 0 .6 921 20 0 .7 8 0 .2 69 99 0 1 .2 8 0 .0 70 26 6 1 .7 8 0 .0 11 82 6 2 .2 8 0 .00 12 62 2 .7 8 0 .0 00 08 4 3 .2 8 0.00 00 03 51

0 .2 9 0 .6 817 17 0 .7 9 0 .2 63 89 7 1 .2 9 0 .0 68 10 1 1 .7 9 0 .0 11 35 9 2 .2 9 0 .00 12 01 2 .7 9 0 .0 00 08 0 3 .2 9 0.00 00 03 28

0.3 0.671373 0.8 0.257899 1.3 0.065992 1.8 0.010909 2.3 0.001143 2.8 0.000075 3.3 0.00000306

0 .3 1 0 .6 610 92 0 .8 1 0 .2 51 99 7 1 .3 1 0 .0 63 93 7 1 .8 1 0 .0 10 47 5 2 .3 1 0 .00 10 88 2 .8 1 0 .0 00 07 1 3 .3 1 0.00 00 02 85

0 .3 2 0 .6 508 74 0 .8 2 0 .2 46 18 9 1 .3 2 0 .0 61 93 5 1 .8 2 0 .0 10 05 7 2 .3 2 0 .00 10 34 2 .8 2 0 .0 00 06 7 3 .3 2 0.00 00 02 66

0 .3 3 0 .6 407 21 0 .8 3 0 .2 40 47 6 1 .3 3 0 .0 59 98 5 1 .8 3 0 .0 09 65 3 2 .3 3 0 .00 09 84 2 .8 3 0 .0 00 06 3 3 .3 3 0.00 00 02 49

0 .3 4 0 .6 306 35 0 .8 4 0 .2 34 85 7 1 .3 4 0 .0 58 08 6 1 .8 4 0 .0 09 26 4 2 .3 4 0 .00 09 35 2 .8 4 0 .0 00 05 9 3 .3 4 0.00 00 02 32

0 .3 5 0 .6 206 18 0 .8 5 0 .2 29 33 2 1 .3 5 0 .0 56 23 8 1 .8 5 0 .0 08 88 9 2 .3 5 0 .00 08 89 2 .8 5 0 .0 00 05 6 3 .3 5 0.00 00 02 16

0 .3 6 0 .6 106 70 0 .8 6 0 .2 23 90 0 1 .3 6 0 .0 54 43 9 1 .8 6 0 .0 08 52 8 2 .3 6 0 .00 08 45 2 .8 6 0 .0 00 05 2 3 .3 6 0.00 00 02 02

0 .3 7 0 .6 007 94 0 .8 7 0 .2 18 56 0 1 .3 7 0 .0 52 68 8 1 .8 7 0 .0 08 17 9 2 .3 7 0 .00 08 03 2 .8 7 0 .0 00 04 9 3 .3 7 0.00 00 01 88

0 .3 8 0 .5 909 91 0 .8 8 0 .2 13 31 3 1 .3 8 0 .0 50 98 4 1 .8 8 0 .0 07 84 4 2 .3 8 0 .00 07 63 2 .8 8 0 .0 00 04 6 3 .3 8 0.00 00 01 75

0 .3 9 0 .5 812 61 0 .8 9 0 .2 08 15 7 1 .3 9 0 .0 49 32 7 1 .8 9 0 .0 07 52 1 2 .3 9 0 .00 07 25 2 .8 9 0 .0 00 04 4 3 .3 9 0.00 00 01 63

0.4 0.571608 0.9 0.203092 1.4 0.047715 1.9 0.007210 2.4 0.000689 2.9 0.000041 3.4 0.00000152

0 .4 1 0 .5 620 31 0 .9 1 0 .1 98 11 7 1 .4 1 0 .0 46 14 8 1 .9 1 0 .0 06 91 0 2 .4 1 0 .00 06 54 2 .9 1 0 .0 00 03 9 3 .4 1 0.00 00 01 42

0 .4 2 0 .5 525 32 0 .9 2 0 .1 93 23 2 1 .4 2 0 .0 44 62 4 1 .9 2 0 .0 06 62 2 2 .4 2 0 .00 06 21 2 .9 2 0 .0 00 03 6 3 .4 2 0.00 00 01 32

0 .4 3 0 .5 431 13 0 .9 3 0 .1 88 43 7 1 .4 3 0 .0 43 14 3 1 .9 3 0 .0 06 34 4 2 .4 3 0 .00 05 89 2 .9 3 0 .0 00 03 4 3 .4 3 0.00 00 01 23

0 .4 4 0 .5 337 75 0 .9 4 0 .1 83 72 9 1 .4 4 0 .0 41 70 3 1 .9 4 0 .0 06 07 7 2 .4 4 0 .00 05 59 2 .9 4 0 .0 00 03 2 3 .4 4 0.00 00 01 15

0 .4 5 0 .5 245 18 0 .9 5 0 .1 79 10 9 1 .4 5 0 .0 40 30 5 1 .9 5 0 .0 05 82 1 2 .4 5 0 .00 05 31 2 .9 5 0 .0 00 03 0 3 .4 5 0.00 00 01 07

0 .4 6 0 .5 153 45 0 .9 6 0 .1 74 57 6 1 .4 6 0 .0 38 94 6 1 .9 6 0 .0 05 57 4 2 .4 6 0 .00 05 03 2 .9 6 0 .0 00 02 8 3 .4 6 0.00 00 00 99

0 .4 7 0 .5 062 55 0 .9 7 0 .1 70 13 0 1 .4 7 0 .0 37 62 7 1 .9 7 0 .0 05 33 6 2 .4 7 0 .00 04 77 2 .9 7 0 .0 00 02 7 3 .4 7 0.00 00 00 92

0 .4 8 0 .4 972 50 0 .9 8 0 .1 65 76 9 1 .4 8 0 .0 36 34 6 1 .9 8 0 .0 05 10 8 2 .4 8 0 .00 04 53 2 .9 8 0 .0 00 02 5 3 .4 8 0.00 00 00 86

0 .4 9 0 .4 883 32 0 .9 9 0 .1 61 49 2 1 .4 9 0 .0 35 10 2 1 .9 9 0 .0 04 88 9 2 .4 9 0 .00 04 29 2 .9 9 0 .0 00 02 4 3 .4 9 0.00 00 00 80

final exam jameel communication systems fall 2013 updated

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