In this exam, there is a total of 35 questions resulting in 53 possible points. TestResult = floor(sum(points)/4.86);

Transcription

1 .... Midterm Exam: Institut für Nachrichtentechnik und Hochfrequenztechnik In this exam, there is a total of 35 questions resulting in 53 possible points. TestResult = floor(sum(points)/4.86); The four commandments: 1. You shall sit alone at a table. - You are not allowed to go to the toilet. - You are not allowed to go out for a cigarette. 2. You shall not communicate in any form to another person. - You are not allowed to ask questions. - In case there is a problem with the exam, write it down. - You are not allowed to communicate to your neighbor. - You are not allowed to use your cell phone, Laptop, PDA, You shall state the question number next to your answer. - You can answer the questions in German and/or English. - Use legible handwriting. 4. You shall staple your exam before turning it in. - You can keep the problem statement. 1

2 Problem 1: (13 points) At the time your teaching assistant was a student, homework was usually done the following way: First, check the internet for solutions of previous years. Second, phone your friends if they have already done the homework. Third, try to understand what you have just copied. Sounds familiar? So here is the Matlab source code you have to reverse engineer (Note that the code is working flawlessly): 01 clc; clear; 02 xvalphabet = [3,3*j,-3*j,-3]; 03 xvtxsymbols = randsrc(100000,1,[1:4]); 04 xvfiltercoeff = rcosine(1, 4, fir/sqrt, 0.33, 5); 05 xvtxsamples = xvalphabet(xvtxsymbols); 06 xvtxsamplesfilt = rcosflt(xvtxsamples, 1, 4, filter, xvfiltercoeff); 07 xvrxsamplesunfilt = xvtxsamplesfilt + (randn(400040,1)+i*randn(400040,1)); 08 xvrxsamplesfilt = rcosflt(xvrxsamplesunfilt, 1, 4, filter/fs, xvfiltercoeff); 09 xvrxsamples = xvrxsamplesfilt(41:4:400040); 10 [xvnotused,xvrxsymbols] = min(abs(xvrxsamples*ones(1,4)-ones(100000,1)*xvalphabet),[],2); 11 xvresult1 = sum(xvrxsymbols~=xvtxsymbols)/(100000); 12 xvresult2 = sum(sum(abs(dec2bin(xvrxsymbols-1,2)-dec2bin(xvtxsymbols-1,2))))/200000; 1 p. 1 Calculate the average power of the symbols transmitted (xvtxsamples). 1 p. 2 Sketch the signal space diagram (=scatter plot) of the signal received (xvrxsamples). Label the axes. 1 p. 3 Sketch the envelope diagram (=IQ diagram) of the signal received (xvrxsamplesfilt). Label the axes. 1 p. 4 How many symbols are transmitted in total during one execution of the program? 1 p. 5 Calculate the expected average power of the noise that is added in line 07 to the signal received. 1 p. 6 Calculate the expected average power of the already downsampled and filtered samples received (xvrxsamples). 1 p. 7 Evaluate the following Matlab command: mean(abs(xvrxsamples).^2) 1 p. 8 Using the results from [1] and [5], show how to compute the SNR of the already filtered samples received (xvrxsamples) in db. Check: SNR= db. 2 p. 9 Calculate the expected value of xvresult1 in terms of the Q ( ) function. 2 p. 10 Calculate the expected value of xvresult2 in terms of the Q ( ) function. Hint1: Is the transmit signal Gray-Coded? Hint2: dec2bin(d,n) produces a binary representation with at least n bits. 2

3 1 p. 11 Assume that line 3 is changed to (the rest of the code is left unchanged): 03 xvtxsymbols = randsrc(100000,1,[1:3]); Calculate the new expected value of xvresult1 in terms of the Q ( ) function. Explain your findings. Problem 2: (4 points) (continuing Problem 2) Testing and comparing Matlab codes is a pain if random number generators generate different values every time the program is executed. Therefore, one is advised to reset the random number generators prior to testing a piece of code. Since you have no idea how to do so, you seek help by calling another friend of yours. He tells you that 01 of the Matlab code shown in the previous problem should be modified to: 01 clc; clear; randn( state, 0); In the Matlab help you read that the randn( state, 0); command sets randn to its default initial state. 2 p. 12 Interestingly, the modified program still deliverers slightly different results (e.g. xvresult1) every time it is executed. Explain how is this possible? 2 p. 13 You restart Matlab every time prior to executing the program. Does xvresult1 now differ between different executions? Explain why/why not. Problem 3: (15 points) Assume that you are an ML detector. When you were a child, your father has told you that the transmitter always broadcasts a 4 symbol long sequence from the following constellation, always starting with Symbol B: A 2 Im D 2 Re B C 3

4 3 p. 14 Given only the above information, name three possible modulation schemes (employing a different crest factor) the transmitter could have employed. Order them with increasing Q crest factor (= peak-to-average ratio). A D At the age of six, you receive 2 your first input signal to decode. It was transmitted over a flat Rayleigh fading AWGN channel with negligible noise added at 2 the receiver. Assume that the channel Istays constant for approx. 100 samples. You know that your brother has already root raised cosine filtered the received B C signal. After that, your sister has already synchronized the signal to tell you that the correct sampling time instances of Symbol 1 to 4 are 19, 25, 31, and 37. Real part of the already filtered signal received: Imaginary part of the already filtered signal received: sample number 1 p. 15 Assume that transmitter and receiver are using root raised cosine filters with equal oversampling ratio. What is the oversampling ratio employed? 1 p. 16 Now, knowing a lot more than in Question [14], what modulation scheme does the transmitter employ? 1 p. 17 Plot a signal space diagram (=scatter plot) of the signal received (at the correct sampling time instances). Label the axes. Only plot what is actually received. Do not sketch a general scatter plot! Hint: How many points do you have to plot? 1 p. 18 Plot an envelope diagram (=IQ diagram) of the signal received. Label the axes. Only plot what is actually received. Do not sketch a general IQ diagram! 3 p. 19 Assume that the channel is everything between the constellation transmitted and the signal received. Calculate the complex channel coefficient h in magnitude and phase. Hint: Remember what your father has told you when you were a child. 4

5 2 p. 20 ML-decode all symbols received. Clearly state all four symbols that have been transmitted by the transmitter. 1 p. 21 Assume that the sampling rate of the receiver is 10 MSamples/s, or in other words, there is a time delay of 100 ns between two received samples. Neglecting the roll-off, roughly estimate the bandwidth of the transmitted/received signal. 1 p. 22 Does this bandwidth decrease or increase when the roll-off-factor of the root raised cosine filters employed is increased. 1 p. 23 Sketch the frequency responses of a digital root-raised-cosine filter with a roll-off of 0, 0.5, and 1 in the same figure. Choose an oversampling factor of 3. Do not forget to label the axes. Problem 4: (13 points) Once upon a time, an operator named A built the GSM infrastructure to serve an urban area of square kilometers. At that time, it was the only operator in that area. The network planning engineers were assigned to guarantee to all the users a SINR of, at least, 15 db. When calculating the SINR, those engineers considered as worst case a user located at the border of the hexagonal cell he is attached to. The considered interference was caused by the first tier of interfering cells. The circle circumscribing the hexagonal area had a radius of 500 m. The pathloss exponent was assumed to be constant and equal to 4. 2 p. 24 How many cells were necessary to cover the whole area? 2 p. 25 How many cells belong to the same cluster? One day, the frequency regulator forced A to sell parts of the frequency spectrum to two competitors, named B and C. As a result, the available bandwidth is now assigned in the following way: Operator A Operator B Operator C 25.4 MHz Bandwidth 15 MHz Bandwidth 29 MHz Bandwidth The bandwidth of each full duplex channel is equal to 200 khz. The three operators decided to place different constraints on the offered service, namely: Operator A Ensure to the users a blocking probability lower than 5%. Operator B Ensure to the users a delay probability lower than 10%. Operator C Ensure to the users a blocking probability lower than 1% and a delay probability lower than 1%. 5

6 3 p. 26 Given these constraints, what is the amount of traffic, in Erlang, that each cell can support? The Erlang tables are available in the Tables 2 and 3 at the end of the problem statement. After some time, statistics about the habits of the customers have been collected. They are available in Table 1. For different time slots, they indicate in how many calls was the customer active and the duration, in minutes, of each call. 00:00-05:59 06:00-11:59 12:00-17:59 18:00-23:59 Calls Dur. [min] Calls Dur. [min] Calls Dur. [min] Calls Dur. [min] Operator A Operator B Operator C Table 1: Statistics of daily use 1 p. 27 Generate an equivalent table containing the Erlang per user. Assuming that the three operators are now working at full capacity: 2 p. 28 How many users can one cell of each operator support? 1 p. 29 How many users can be served in the whole area? 2 p. 30 Consider that the average monthly invoice of a user of operator B is equal to 30 EUR. Calculate the average cost per minute of an outgoing call. Assume that on average for two calls in Table 1 one is received. Problem 5: (8 points) The graph in Fig. 1 shows the probability that the SNR γ M at the output of a selection diversity receiver (with N branches) is lying below a given value γ = 0.01 depending on the number of branches N. Different curves have been obtained considering different values of the average SNR per branch Γ. 1 p. 31 Is the value of Γ associated to Curve A bigger, smaller or equal to the one used to draw Curve B? Justify your answer. 2 p. 32 Assuming five branches and a probability P (γ M < γ) = 10 7, calculate the value of the average SNR per branch Γ. Assume now Γ equal to 4 db, 1 p. 33 calculate the average SNR γ M when using no diversity. 6

7 2 p. 34 calculate how many branches are necessary to obtain an average SNR γ M = 8 db at the output of a selection diversity receiver. 2 p. 35 calculate how many branches are necessary to obtain an average SNR γ M = 8 db at the output of a maximum ratio combiner receiver. 7

8 N ch Blocking Probability 1% 2% 3% 4% 5% 6% 7% 8% 9% 10% Table 2: Erlang supported by a system with N ch channels and a given blocking probability 8

9 N ch Probability of a call to be queued 1% 2% 3% 4% 5% 6% 7% 8% 9% 10% Table 3: Erlang supported by a system with N ch channels and a given delay probability 9

10 A B Number of branches P(γ M < γ) Figure 1: Probability of γ M < γ 10