ECS 332: Principles of Communications 2012/1. HW 4 Due: Sep 7

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1 ECS 332: Principles of Communications 2012/1 HW 4 Due: Sep 7 Lecturer: Prapun Suksompong, Ph.D. Instructions (a) ONE part of a question will be graded (5 pt). Of course, you do not know which part will be selected; so you should work on all of them. (b) It is important that you try to solve all problems. (5 pt) (c) Late submission will be heavily penalized. (d) Write down all the steps that you have done to obtain your answers. You may not get full credit even when your answer is correct without showing how you get your answer. Problem 1. The spectrum of a periodic square wave can be found using: % specsquare.m plot the spectrum of a square wave close all f0=10; % "(fundamental) frequency" of the square wave EndTime=2; % Will consider from time = 0 to EndTime Ts=1/1000; % sampling interval (time interval between samples) t=0:ts:(endtime Ts); % create a time vector x=sign(cos(2 pi f0 t)); % square wave = sign of cos wave plotspec(x,ts) % call plotspec to draw spectrum The output of specsquare.m is shown in Figure 4.1. The top plot shows the first 2 seconds of a square wave with fundamental frequency f 0 = 10 cycles per second. The bottom plot shows a series of spikes that define the frequency content. In this case, the largest spike occurs at ±10 Hz, followed by smaller spikes at all the odd-integer multiples (i.e., at ±30, ±50, ±70, etc.). Modify specsquare.m to investigate the relationship between the time behavior of the square wave and its spectrum. Try square waves with different (fundamental) frequencies: f 0 = 20, 40, 100, 300 Hz. (Keep T s = 1/1000.) Describe the aliasing effect on each of cases. How do the time plots change? How do the spectra change? 4-1

2 ECS 332 HW 4 Due: Sep / Seconds 1.5 Magnitude Frequency [Hz] Figure 4.1: Plots from specsquare.m Problem 2. Determine the Nyquist sampling rate and the Nyquist sampling interval for the signals: (a) sinc(100πt) (b) sinc 2 (100πt) (c) sinc(100πt) + sinc(50πt) (d) sinc(100πt) + 3 sinc 2 (60πt) (e) sinc(50πt) sinc(100πt) Remark: Recall that in our class, sinc(x) = sin(x) x. 4-2

3 ECS 332 HW 4 Due: Sep /1 Problem 3. Consider a signal g(t) = sinc(πt). (a) Sketch the Fourier transform G(f) of g(t). (b) Find the Nyquist sampling rate. (c) Recall that the instantaneous sampled signal g δ (t) is defined by g δ (t) = g(t) δ(t nt s ) where T s is the sampling interval. n= (i) Let T s = 0.5. Sketch the Fourier transform G δ (f) of g δ (t). (ii) Let T s = 4/3. Sketch the Fourier transform G δ (f) of g δ (t). (d) The sequence of sampled values g[n] is constructed from g(t) by Recall the reconstruction equation: g r (t) = g[n] = g (t) t=nts. n= g[n] sinc(πf s (t nt s )). Note that we write g r (t) instead of g(t) to accommodate the case that the sampling rate is too low; in which case, the reconstructed signal is not the same as g(t). (i) With T s = 1, i. Find g[n] for n =..., 4, 3, 2, 1, 0, 1, 2, 3, 4,... ii. Use the reconstruction equation to find g r (t). (ii) Let s test the reconstruction equation by using MATLAB to plot g r (t). Note that the sum in the reconstruction equation extends from to +. In MATLAB, we can not add that many terms. So, we need to stop at some n. In this part, use T s = 0.5. i. Use MATLAB to plot g r (t) when only n = 0 term is included. ii. Use MATLAB to plot g r (t) and all of its sinc components. Include only n = 1, 0, 1. iii. Use MATLAB to plot g r (t) and all of its sinc components. Include only n = 5, 4,..., 1, 0, 1,..., 4, 5. iv. Use MATLAB to plot g r (t) and all of its sinc components. Include only n = 10, 9,..., 1, 0, 1,..., 9, 10. In all these plots, consider t from -4 to 4. Also include the plot of sinc(πt) for comparison. 4-3

4 Magnitude Magnitude ECS 332: Solution for Problem Set 4 Problem 1: Aliasing and periodic square wave In the time domain, the switching between the values -1 and 1 should be faster as we increase f 0. In the frequency domain, we should get impulses (spikes) at all the odd-integer multiples of f 0 Hz. The center spikes (at f 0 ) should be the largest among them. All the plots below are adjusted so that they show 10 periods of the original signal in the time domain. From the plots, as we increase f 0 from 10 to 20 Hz, the locations of spikes changes from all the odd-integer multiples of 10 Hz to all the odd-integer multiples of 20 Hz. In particular, we see the spikes at 20, 60, 100, 140, 180, 220, 260, 300, 340, 380, 420, 460. Note that plotspec only plots from [-f s /2,f s /2). So, we see a spike at -500 but not 500. Of course, the Fourier transform of the sampled waveform is periodic and hence when we replicate the spectrum every f s, we will have a spike at 500. Note that in reality, we should also see spikes at 540, 580, 620, 660, and so on. However, because the sampling rate is 1000, these high frequency spikes will suffer from aliasing and fold back into our viewing window [-f s /2,f s /2). However, they fall back to the frequencies that already have spikes (for example, 540 will fold back to 460, and 580 will fold back to 420) and therefore the aliasing effect is not easily noticeable in the frequency domain. When f 0 = 40, we start to see the aliasing effect in the frequency domain. Instead of seeing spikes only at 40, 120, 200, 280, 360, 440, the spikes at higher frequencies (such as 520, 600, and so on) fold back to lower frequencies (such as 480, 400, and so on). The plot still looks OK in the time domain Seconds Seconds Frequency [Hz] Frequency [Hz]

5 Magnitude Magnitude Magnitude Magnitude At high fundamental frequency f 0 = 100, we see stronger effect of aliasing. In the time domain, the waveform does not look quite rectangular. In the frequency domain, we only see the spikes at 100, 300, and 500. These are at the correct locations. However, there are too few of them to reconstruct a square waveform. The rest of the spikes are beyond our viewing window. We can t see them directly because they fold back to the frequencies that are already occupied by the lower frequencies Seconds Seconds Frequency [Hz] Frequency [Hz] Our problem can be mitigated by reducing the sampling interval to T S = 1/1e4 instead of T S = 1/1e3. Now, the spikes show up again as shown by the plot on the right above. Finally, at the highest frequency f 0 = 300, if we still use T = 1/1e3, the waveform will be heavily distorted in the time domain. This is shown in the left plot below. We have large spikes at 300 as expected. However, the next pair which should occur at 900 is out of the viewing window and therefore fold back to 100. Again, the aliasing effect can be mitigated by reducing the sampling time to T = 1/1e4 instead of T = 1/1e3. Now, more spikes show up at their expected places. Note that we can still see a lot of small spikes scattered across the frequency domain. These are again the spikes from higher frequency which fold back to our viewing window Seconds Seconds Frequency [Hz] Frequency [Hz]

6 Q2 Nyquist sampling rate and Nyquist sampling interval Sunday, July 17, :09 PM ECS332 HW4 Sol Page 1

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8 Q3 Sinc Reconstruction of Sinc Thursday, August 30, :51 PM ECS332 HW4 Sol Page 3

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11 ECS 332: Principles of Communications 2012/1 HW 5 Due: Sep 26 Lecturer: Prapun Suksompong, Ph.D. Instructions (a) ONE part of a question will be graded (5 pt). Of course, you do not know which part will be selected; so you should work on all of them. (b) It is important that you try to solve all problems. (5 pt) (c) Late submission will be heavily penalized. (d) Write down all the steps that you have done to obtain your answers. You may not get full credit even when your answer is correct without showing how you get your answer. Problem 1. Consider a signal g(t) = sinc (3(t 5)). (a) Is g(t) time-limited? (b) Is g(t) band-limited? (c) Carefully sketch g(t) (d) Carefully sketch the magnitude G(f) of the Fourier transform G(f). Problem 2. State the reconstruction formula. Hint: You should be able to do this by recalling the reconstruction process. Problem 3. State the Nyquist s (first) criterion for zero ISI (a) In the time domain. (b) In the frequency domain. Problem 4. In each part below, a pulse P (f) is defined in the frequency domain from f = 0 to f = 1. Outside of [0, 1], you task is to assign value(s) to P (f) so that it becomes a Nyquist pulse. Of course, you will also need to specify the symbol interval T as well. Hint: To avoid dealing with complex-valued P (f), you may assume that p(t) is real-valued and even; in which case P (f) is also real-valued and even. (a) Find a Nyquist pulse P (f) whose P (f) = 0.5 on [0, 1]. 5-1

12 ECS 332 HW 5 Due: Sep /1 (b) Find a Nyquist pulse P (f) whose P (f) = 0.25 on [0, 1]. (c) Find a Nyquist pulse P (f) whose P (f) = { 0.5, 0 f < , 0.5 f 1 (d) Find a Nyquist pulse P (f) whose { 0.5, f [0, 0.25) [0.5, 0.75) P (f) = 0.25, f [0.25, 0.5) [0.75, 1] Problem 5. Consider a raised cosine pulse p(t) and its Fourier transform P (f). Assume the rolloff factor α = 0.3 and the symbol duration T = 1. (a) Carefully sketch P (f). (b) Find p(2). (c) Find P (0.5). (d) Find P (0.3). (e) *Find P (0.4). Remark: You should be able to solve this problem without referring to the ugly formula. Problem 6. Consider a raised cosine pulse p(t) with rolloff factor α and symbol duration T. (a) Find p(t/2) as a function of α. (b) Use MATLAB to plot p(t/2) as a function of α. 5-2

13 ECS332 HW 5 Sol Page 1 Q1 Sinc Review Monday, September 17, :09 PM

14 ECS332 HW 5 Sol Page 2 Q2 Reconstruction Formula Monday, September 17, :34 PM

15 Q3 Nyquist's Criterion Monday, September 17, :47 PM ECS332 HW 5 Sol Page 3

16 Q4 Nyquist Pulses Monday, September 17, :20 PM ECS332 HW 5 Sol Page 4

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19 ECS332 HW 5 Sol Page 7 Q5 Raised Cosine Pulse Monday, September 17, :51 PM

20 ECS332 HW 5 Sol Page 8 Q6 Raised Cosine Pulse Monday, September 17, :38 PM

21 ECS 332: Principles of Communications 2012/1 HW 6 Due: Oct 5 Lecturer: Prapun Suksompong, Ph.D. Instructions (a) ONE part of a question will be graded (5 pt). Of course, you do not know which part will be selected; so you should work on all of them. (b) It is important that you try to solve all problems. (5 pt) (c) Late submission will be heavily penalized. (d) Write down all the steps that you have done to obtain your answers. You may not get full credit even when your answer is correct without showing how you get your answer. Problem 1. Consider the code {0, 01} (a) Is it nonsingular? (b) Is it uniquely decodable? (c) Is it prefix-free? Problem 2. Consider the random variable X whose support S X contains seven values: S X = {x 1, x 2,..., x 7 }. Their corresponding probabilities are given by (a) Find the entropy H(X). x x 1 x 2 x 3 x 4 x 5 x 6 x 7 p X (x) (b) Find a binary Huffman code for X. (c) Find the expected codelength for the encoding in part (b). Problem 3. Find the entropy and the binary Huffman code for the random variable X with pmf p X (x) = x for x = 1, 2,..., Also calculate E [l(x)] when Huffman code is used. 6-1

22 ECS 332 HW 6 Due: Oct /1 Problem 4. Construct a random variable X (and its pmf) whose Huffman code is {0, 10, 11}. Problem 5. These codes cannot be Huffman codes. Why? (a) {00, 01, 10, 110} (b) {01, 10} Hint: Huffman code is optimal. Problem 6. A memoryless source emits two possible message Y(es) and N(o) with probability 0.9 and 0.1, respectively. (a) Determine the entropy (per source symbol) of this source. (b) Find the expected codeword length per symbol of the Huffman binary code for the third-order extensions of this source. (c) Use MATLAB to find the expected codeword length per symbol of the Huffman binary code for the fourth-order extensions of this source. (d) Use MATLAB to plot the expected codeword length per symbol of the Huffman binary code for the nth-order extensions of this source for n = 1, 2,...,

23 Q1 Classes of Codes Sunday, August 28, :33 PM ECS332 HW 6 Sol Page 1

24 Q2 Huffman Code Sunday, August 28, :38 PM ECS332 HW 6 Sol Page 2

25 Q3 Huffman code Sunday, August 28, :13 PM ECS332 HW 6 Sol Page 3

26 ECS332 HW 6 Sol Page 4 Q4 Inverse Huffman Problem Sunday, August 28, :07 PM

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28 Q5 Non-Huffman Codes Thursday, September 27, :08 PM ECS332 HW 6 Sol Page 6

29 Q6 Source Extension Monday, August 29, :35 AM ECS332 HW 6 Sol Page 7

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32 ECS 332: Principles of Communications 2012/1 Lecturer: Prapun Suksompong, Ph.D. HW Solution 7 Due: N/A Problem 1. Optimal code lengths that require one bit above entropy: The source coding theorem says that the Huffman code for a random variable X has an expected length strictly less than H(X) + 1. Give an example of a random variable for which the expected length of the Huffman code is very close to H(X) + 1. Problem 2. Consider the AWGN channel: where N N (0, σn 2 ). Assume that X whole question, assume a > 0. = (a) In this part, use a = 5 and σ N = 3. Y = X + N N and that X takes two values, a and a. For the The channel output Y is fed into a decision device (a comparator) which will compare the value of Y to 0. The output ˆX of the decision (thresholding) device is determined by The whole system in shown in Figure 7.1. ˆX = { a, if Y 0, a, if Y < 0. X Y Decision device X N Figure 7.1: System for Q1.a Find [ ] (i) P ˆX = a X = a [ ] (ii) P ˆX = a X = a 7-1

33 ECS 332 HW Solution 7 Due: N/A 2012/1 [ ] (iii) P ˆX = a X = a [ ] (iv) P ˆX = a X = a [ ] (v) P ˆX X Hint: By the total probability theorem, [ ] [ ] [ ] P ˆX X = P ˆX X X = a P [X = a] + P ˆX X X = a P [X = a], [ ] [ ] = P ˆX a X = a P [X = a] + P ˆX a X = a P [X = a]. (b) Continue from part (a). We can use the system from part (a) to transmit/receive binary information by adding a simple mapping device with map 0 to a and 1 to a at the transmitter. At the receiver, we also have another mapping device that map the a and a back to 0 and 1, respectively. The new system is shown in Figure 7.2. S mapping X Y Decision X device mapping Z N Figure 7.2: System for Q1.b Note that S and Z are binary and that the whole system (inside the dotted box) in Figure 7.2 can be reduced to a binary symmetric channel (BSC) with crossover probability p. Find p. (c) Observe that as σ N increases, the value of p in part (b) will also increase. Is it possible to find σ N such that p > 0.5? (d) Express p using a, σ N, and the Q function. Problem 3. Consider a transmission over the BSC with crossover probability p. The random input to the BSC is denoted by S. Assume S bernoulli(p 1 ). Let Z be the output of the BSC. 7-2

34 ECS 332 HW Solution 7 Due: N/A 2012/1 (a) Suppose, at the receiver (which observes the output of the BSC), we learned that Z = 1. For each of the following scenarios, which event is more likely, S = 1 was transmitted or S = 0 was transmitted? (Hint: Use Bayes theorem.) (i) Assume p = 0.3 and p 1 = 0.1 (ii) Assume p = 0.3 and p 1 = 0.5 (iii) Assume p = 0.3 and p 1 = 0.9 (iv) Assume p = 0.7 and p 1 = 0.5 (b) Suppose, at the receiver (which observes the output of the BSC), we learned that Z = 0. For each of the following scenarios, which event is more likely, S = 1 was transmitted or S = 0 was transmitted? (i) Assume p = 0.3 and p 1 = 0.1 (ii) Assume p = 0.3 and p 1 = 0.5 (iii) Assume p = 0.3 and p 1 = 0.9 (iv) Assume p = 0.7 and p 1 = 0.5 Remark: A MAP (maximum a posteriori) detector is a detector that takes the observed value Z and then calculate the most likely transmitted value. More specifically, Ŝ MAP (z) = arg max P [S = s Z = z] s In fact, in part (a), each of your answers is ŜMAP (1) and in part (b), each of your answers is ŜMAP (0). Problem 4. Consider a repetition code with a code rate of 1/5. Assume that the code is used over a BSC with crossover probability p = 0.4. (a) Assume that the receiver uses majority vote to decode the transmitted bit. Find the probability of error. (b) Assume that the source produces source bit S with Suppose the receiver observes P [S = 0] = 1 P [S = 1] = (i) What is the probability that 0 was transmitted? (Do not forget that this is a conditional probability. The answer is not 0.4 because we have some extra information from the observed bits at the receiver.) 7-3

35 ECS 332 HW Solution 7 Due: N/A 2012/1 (ii) What is the probability that 1 was transmitted? (iii) Given the observed 01001, which event is more likely, S = 1 was transmitted or S = 0 was transmitted? Does your answer agree with the majority voting rule for decoding? (c) Assume that the source produces source bit S with Suppose the receiver observes P [S = 0] = 1 P [S = 1] = p 0. (i) What is the probability that 0 was transmitted? (ii) What is the probability that 1 was transmitted? (iii) Given the observed 01001, which event is more likely, S = 1 was transmitted or S = 0 was transmitted? Your answer may depend on the value of p 0. Does your answer agree with the majority voting rule for decoding? Problem 5. A channel encoder map blocks of two bits to five-bit (channel) codewords. The four possible codewords are 00000, 01000, 10001, and A codeword is transmitted over the BSC with crossover probability p = 0.1. Suppose the receiver observes at the output of the BSC. (a) Assume that all four codewords are equally likely to be transmitted. Given the observed at the receiver, what is the most likely codeword that was transmitted? (b) The Hamming distance between two binary vectors is defined as the number of positions at which the corresponding bits are different. For example, the Hamming distance between and is 2. The minimum distance decoder is defined as the decoder that compare the Hamming distances between the observed bits at the receiver and each of the possible codewords. The output of this decoder is the codeword that give the minimum distance. Explain why the minimum distance decoder would give the same decoded codeword as the decoder in part (a). (c) What is the minimum (Hamming) distance d min among codewords? (d) Assume that the four codewords are not equally likely. Suppose is transmitted more frequently with probability 0.7. The other three codewords are transmitted with probability 0.1 each. Given the observed at the receiver, what is the most likely codeword that was transmitted? 7-4

36 Q1 Monday, August 29, :20 AM ECS332 HW7 Sol Page 1

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38 Q2 Tuesday, September 20, :05 AM ECS332 HW7 Sol Page 3

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41 ECS332 HW7 Sol Page 6 Q3 Tuesday, October 09, :37 AM

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43 Q4 Tuesday, October 09, :21 AM ECS332 HW7 Sol Page 8

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45 ECS332 HW7 Sol Page 10 Q5 Tuesday, October 09, :50 AM

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ECS 452: Digital Communication Systems 2015/2. HW 1 Due: Feb 5

ECS 452: Digital Communication Systems 2015/2 HW 1 Due: Feb 5 Lecturer: Asst. Prof. Dr. Prapun Suksompong Instructions (a) Must solve all non-optional problems. (5 pt) (i) Write your first name and the