# hwB17 - a 2 2 b 2 2 1 2 or its inverse as obtained in...

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ENEE 241 02* HOMEWORK ASSIGNMENT 17 Due Wed 03/12 This problem refers to the range estimation example in the textbook (p. 113). Suppose f 1 = 3 MHz, f 2 = 2 . 5 MHz and γ = 1 . 45. The vector s in the ﬁle dataB17.txt contains 101 samples of the noisy waveform s ( t ), taken at times t = 0 : 0 . 1 : 10 microseconds (i.e., the sampling rate equals 10 7 samples per second). (i) (8 pts.) Write a short piece of MATLAB code which creates the reference vectors u (1) , w (1) , u (2) and w (2) as deﬁned in the example, and then computes the coeﬃcients a 1 , b 1 , a 2 and b 2 for the least squares approximation ˆ s . Submit your code. (ii) (3 pts.) On the same graph, plot s and ˆ s (as continuous curves). Print the plot. (iii ) (3 pts.) What is the resulting estimate of the ratio R 1 /R 2 ? (iv) (6 pts.) For the same vector s , execute the commands S = fft(s); plot(abs(S)) and print the plot. By zooming in (on the graph window), or by examining the values of abs(S) , compute the ratio of the heights of two visible peaks on either half of the graph. Your result should be close in value to the value of [( a 2 1 + b 2 1
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Unformatted text preview: ) / ( a 2 2 + b 2 2 )] 1 / 2 or its inverse, as obtained in part (iii) . Solved Examples (see also Homework Problem 16) S 17.1 ( P 2.29 in textbook). Five measurements x i taken at times t i are shown below. t i 0.1 0.2 0.3 0.4 0.5 x i 2.88 3.48 4.29 5.00 6.25 It is believed that the data follow an exponential law of the form x ( t ) = ae bt or equivalently, ln x ( t ) = ln a + bt Determine the values of a and b in the least squares approximation of the vector [ s i ] = [ln x i ] by the vector [ln a + bt i ] (where i = 1 ,..., 5). S 17.2 ( P 2.30 in textbook). Five measurements x i taken at times t i are shown below. t i 10 20 30 40 50 x i 14.0 20.9 26.7 32.2 36.8 It is believed that the data obey a power law of the form x ( t ) = at r or equivalently, ln x ( t ) = ln a + r ln t Determine the values of a and r in the least squares approximation of the vector [ s i ] = [ln x i ] by the vector [ln a + r ln t i ] (where i = 1 ,..., 5)....
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