07hw02 - fi’Z 2.3 Two random variables. X, and X2. are...

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Unformatted text preview: fi’Z 2.3 Two random variables. X, and X2. are produced in a digitai communication receiver. An error occurs if X1 >X2: otherwise. the receiver output is correct. The random variables X, and x2 are independent. Gaussian random variables. with means p, and fig and standard deviations o, and oz. respectively. Find the probability of error for this receiver. Express your answer in terms of the function 4’0. and then convert it to be in terms of the function 00). Hint: perhaps the best approach is to let x=x, -X2 so that X is a Gaussian random variable and PfX>UJ is the probability of error {or the communication receiver. 2.4 Suppose the random variables X1. X2. and X, are jointlg Gaussian with means pi =Elxii and covariances A” =Eiixi-pi)(Xj-pj)}. isisa and lsjss. Let Y be the random variable defined by Y=X1 «we. Express the distribution function for Y in terms of the function #170} and the parameters pi and A” . 3.15 Suppose X(t) and Yit} are zero-mean. widowsense stationarg. continuous-time random processes. ff Kit) and Yit} are independent. find the autocorrelation function for zit) in terms of the autocorrelation functions for X following cases. stationary. (ti and fit) in each of the in each case. determine if the random process zit} is wide-sense (a) Ziti = c Xft)Y(tJ i d. where c and d are deterministic constants (b) ZitJ = Xiticosimgt}+Yithinfmat) 4.5 Suppose Kit) is a zero~mean was continuous-time process with autocorrelation function int) =‘nexpi-a'l 2| }. «st-m. Let this be the input to the unear- time-invariant filter Wii'h imPuiSe WSan Ht): Wt). expected vaiue of the instantaneous power in the output process at time t. 5.5 A. time—invariant linear filter has transfer function Him) = expt-m'i‘). m sea s so. The input to this filter is a white noise process Kit} with power spectral density ' $x(m}=Nuf2, dufmcoo. and the corresponding output is the random process Yit). Find the expected value of the instantaneous power In the output process. iHint: To heip evaluate the resulting integral. notice that the transfer function has a Gaussian shape. Can gou relate it to a probability density function? How might this help solve the problem?] is. 6. l. 2 Binary data is to be transmitted over an AUGN channel (noise spectral den— sity ‘5N0} using the rectangular pulses soit) = n pT(t) and slit) = -o pTit), where pTit) = i for 0_S t < T and pT(t} = U elsewhere. The receiver forms an output I and decides "50 was sent" if 2 > O and "s1 was sent“ if 2's 0. (a) Suppose that the output 3 is given by I z =I git) fit) at o where Y(t) is the transmitted signal plus noise and g(t} is the triangular waveform 0 otherwise Find the error probabilities Pa 0 and Peal. = YflfiT); that isw it merely (b) Suppose the receiver produces the output 2 Find the error probabilities ssmples the received signal at time kT. P p . 2,0 and e.1 the type shown in Figure 6.2 The channel is an additive 5 Kit) has zero mean and Suppose a binary communications System of empIOys the signal set of Figure 6.1(b). Gaussian noise channel for which the noise proces autocorrelation function IrliT Lei $0M: Refit) anci 5 ‘A'Frlfi [1| > T Find Pe 0 and Fe 1 if the threshold is Y = 0. I 1 Threshold Device Channel Figure 8.2 A binary data transmission system model 300:) 510:) f I 0 '1‘ D ‘1‘ so“) 51(t) f—“""“'. I Z 0 T W —‘-—-—-—r—:P t E ' I 0 I : : 50(t) 51(t) 5“" r“‘1; t a T 0 : , ——i——i—-—i——-'h- t —:-----T-—l—* t 0 I § . . I l i. 1 500) 510:) 0 ’1‘ H t t 0 T .Figure 6.1 Four examples of baseband signal sets. I: - Ifi’i 6.3. Consider the binary communication system of Figure 6.3 (see pages fig through 425 for a description of this system}. The signals sgft} and sl{t} are defined by and Suppose T =2 2 and ND = 5r. li- Wfih sfeih'ui JEI'ISIf"! £5. (a) {b} {c} {d} sfl(t} = i prt} 51m = 2 pr”) Find the error probabilities Pelg and PERI far the system with threshold $ = F- ‘ What is the minimax threshold far this systEm? Find the minimax errpr probability. Add a squaring device to the system at the input to the threshald device. The decisian statistic is new 22 rather than I. Finfi expressions in terms of the threshold Y and the function Q for the_ aura: probabilities Pg’fl and Pei] for this new systea. Warning: 33 is not Gaussian. ...
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This note was uploaded on 02/13/2012 for the course ECE 544 taught by Professor Jameslehnert during the Spring '12 term at Purdue.

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07hw02 - fi’Z 2.3 Two random variables. X, and X2. are...

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