1.1 Objectives . To investigate the significance of each coefficient in the Fourier series. To use...
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1.1 Objectives . To investigate the significance of each coefficient in the Fourier series. To use MATLAB to gain an understanding of the decomposed function. 1.2 Introduction The Fourier series is one of the most important tools in mathematical physics and engineering. It is used to represent a given periodic function in terms of sine and cosine functions. It yields a series representation of a given function in a simple fashion. The Fourier series of a periodic function of period 2L is given by nat nat f(t) = ao + n=1 (an cos + bn sin L where ao=f(t)dt an = and bn = 13 Prior investigations A periodic square wave is often employed as the input signal in circuit simulation. The periodic square wave shown in Fig. 1 can be described mathematically as f(t) cos dt, n= 1, 2, 3, ... 1.4 Lab exercise Let f(t) sindt, n = 1, 2, 3, ... f(t) = where, the function f(t) is periodic with period 7, that is, f(t +T) = f(t). T is the period of the waveform. Determine analytically the Fourier series of the periodic wave shown in Fig. 1 for an arbitrary period 7. In other words, find expressions for an and b A = So < t <0 A 0 Exercise 3 3.1 Objectives for values of N= 5, 10, 20, for the time interval 0 SIST. Plot the original periodic square wave on the same graph. Comment on the difference between the original periodic square wave and its truncated Fourier series presentation. f(t) A- 3.2 Lab exercise Matrix A and vector bare defined by 1 +0.1 +0.2 I 200 1 1 200 T 200 1 -23456 200 1 200 I 200 I 200 To use MATLAB to solve Linear Algebra problems. (c) (d) (c) +0.7 +1.3 +1.1 +0.5 +0.1 200 1 200 1 200 1 200 1 200 I 200 +0.8 +1.4 f(1) = a + (a, cos 772 + b, s sin T12 +1.1 +0.6 -T -7/2 +0.9 200 1 200 I 200 1 200 I 200 +0.3 +0.9 +1.5 +1.1 +0.7 +0.8 I 200 I 200 I 200 I 200 1 200 I 200 Figure 1 +0.4 +1.0 +1.6 0 +1.1 0.7 1 200 I 200 I 200 I 200 1 200 1 200 T/2 T +0.5 +1.1 +1.7 +1.1 HTU T12 +1.1 +0.6 1 200 T 200 1 200 I 200 1 200 I 200 +0.6 +1.2 +1.8 +1.1 +1.1 +0.5 + 500000 050000 005000 000500 00 0050 00000 5 (a) Solve the linear system of equations Ax=b using MATLAB. Find A and verify that AA-I, where I is the 6x6 identity matrix, Computex-A"b using MATLAB and check the result with that of (a). Compute the determinant of A. Take matrix A' as A'=A+A, where A' is the transpose of A. Using MATLAB, solve the linear system of equations, A's-b. Take matrix S as 1 2 3 S = 9 8 7 2.2 8.8 6.9 Compute the rank of S manually and use the MATLAB function to verify the result. 1.1 Objectives . To investigate the significance of each coefficient in the Fourier series. To use MATLAB to gain an understanding of the decomposed function. 1.2 Introduction The Fourier series is one of the most important tools in mathematical physics and engineering. It is used to represent a given periodic function in terms of sine and cosine functions. It yields a series representation of a given function in a simple fashion. The Fourier series of a periodic function of period 2L is given by nat nat f(t) = ao + n=1 (an cos + bn sin L where ao=f(t)dt an = and bn = 13 Prior investigations A periodic square wave is often employed as the input signal in circuit simulation. The periodic square wave shown in Fig. 1 can be described mathematically as f(t) cos dt, n= 1, 2, 3, ... 1.4 Lab exercise Let f(t) sindt, n = 1, 2, 3, ... f(t) = where, the function f(t) is periodic with period 7, that is, f(t +T) = f(t). T is the period of the waveform. Determine analytically the Fourier series of the periodic wave shown in Fig. 1 for an arbitrary period 7. In other words, find expressions for an and b A = So < t <0 A 0 Exercise 3 3.1 Objectives for values of N= 5, 10, 20, for the time interval 0 SIST. Plot the original periodic square wave on the same graph. Comment on the difference between the original periodic square wave and its truncated Fourier series presentation. f(t) A- 3.2 Lab exercise Matrix A and vector bare defined by 1 +0.1 +0.2 I 200 1 1 200 T 200 1 -23456 200 1 200 I 200 I 200 To use MATLAB to solve Linear Algebra problems. (c) (d) (c) +0.7 +1.3 +1.1 +0.5 +0.1 200 1 200 1 200 1 200 1 200 I 200 +0.8 +1.4 f(1) = a + (a, cos 772 + b, s sin T12 +1.1 +0.6 -T -7/2 +0.9 200 1 200 I 200 1 200 I 200 +0.3 +0.9 +1.5 +1.1 +0.7 +0.8 I 200 I 200 I 200 I 200 1 200 I 200 Figure 1 +0.4 +1.0 +1.6 0 +1.1 0.7 1 200 I 200 I 200 I 200 1 200 1 200 T/2 T +0.5 +1.1 +1.7 +1.1 HTU T12 +1.1 +0.6 1 200 T 200 1 200 I 200 1 200 I 200 +0.6 +1.2 +1.8 +1.1 +1.1 +0.5 + 500000 050000 005000 000500 00 0050 00000 5 (a) Solve the linear system of equations Ax=b using MATLAB. Find A and verify that AA-I, where I is the 6x6 identity matrix, Computex-A"b using MATLAB and check the result with that of (a). Compute the determinant of A. Take matrix A' as A'=A+A, where A' is the transpose of A. Using MATLAB, solve the linear system of equations, A's-b. Take matrix S as 1 2 3 S = 9 8 7 2.2 8.8 6.9 Compute the rank of S manually and use the MATLAB function to verify the result.
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Essentials of Business Analytics
ISBN: 978-1285187273
1st edition
Authors: Jeffrey Camm, James Cochran, Michael Fry, Jeffrey Ohlmann, David Anderson, Dennis Sweeney, Thomas Williams
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