exercises-2011

exercises-2011 - Data Analysis Modelling and Simulation...

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Unformatted text preview: Data Analysis Modelling and Simulation MATH1711 Epiphany Term: Problems Weekly assignments will be selected from the following list of problems. Assign- ments will be set weekly on Thursdays during lecture. Assignments should be submitted Thursdays of the following week by 12:30pm (in the folders on the door of room CM110). Your assignments will contribute to your overall mark for this module, so it is essential to return assignments on time. 1. Calculate the linear least-squares fit to the following data x i 0.3 0.6 0.9 1.2 y i- 1.466- 0.062 0.492 0.822 1.086 What is the error at each point? What effect does increasing the weight in the least squares at x = 0 from 1 to 2 have on the errors in the new approximation? 2. Calculate the quadratic least-squares fit to x i- 1- 0.9- 0.8- 0.7- 0.6 y i- 1.466- 0.062 0.492 0.822 1.086 3. Derive a system of equations, similar to the normal equations, to fit a curve of the form y = a + b e- x to a set of data ( x i , y i ) , i = 0 , ··· , n . 4. By taking logarithms, or otherwise, find a best fit of the form y = a e bx to the data x i- 1- 0.5 0.0 0.5 1.0 y i 0.7716 1.3 1.9767 3.3013 5.5028 5. Find a general formula for y n which satisfies the difference equation y n +1 = x n + ay n , y = 0 with (a) x n = (- 1) n (b) x n = δ n . 6. Find the z-transform of the sequence (a) { n ( n + 1) } ∞ n =0 (b) { s n } ∞ n =0 (c) { (- 1) n ( n + 1)- 1 } ∞ n =0 (d) { sin( ωn ) } ∞ n =0 Hint for part (d): Im (e iωn ) = sin ωn . 7. Find the z-transform of the sequences (a) { x n } ∞ n =0 = { 1 , 2 , 3 , , , ···} (b) { x n } ∞ n =0 = { 1 / 2 , 1 , 1 / 2 , 1 , ···} For part (b), which linear combination of elementary sequences give rise to this sequence? 8. Calculate the z-transform of the periodic sequence { x n } ∞ n =0 = { , 1 2 , 1 , 1 2 , , 1 2 , 1 , 1 2 , , . . . } and hence deduce what elementary sequences make up x . 1 9. Find the z-transform of the sequence y n = x n- 1 2 x n- 1 + 1 3 x n- 2- 1 4 x n- 3 + ··· where x is the sequence in exercise 7(b). 10. Find a , b and possibly c or ω which satisfy (a) 1 1 + 1 2 z- 1 2 z 2 = 1 (1 + z )(1- 1 2 z ) = a 1 1 + z + b 1 1- 1 2 z ; (b) 1 + 2 z 1- 2 z + z 2 = a 1 1- z + b z (1- z ) 2 ; (c) 1 1- z + z 2 = a 1- z cos ω 1- 2 z cos ω + z 2 + b z sin ω 1- 2 z cos ω + z 2 ; (d) 5 z 2- 4 z + 1 (1- z )(1- 2 z ) 2 = a 1 1- z + b 1 1- 2 z + c 2 z (1- 2 z ) 2 . 11. Find the complete sequence { y n } satisfying y n +2 = x n + y n , y = 0 , y 1 = 0 where x is the sequence { 1 ,- 2 , 1 , 1 ,- 2 , 1 , 1 ,- 2 , 1 , ···} ∞ n =0 12. Find the complete sequence { y n } of y n +2- 2 y n +1 + 10 y n = 9 , y = 2 , y 1 = 3 . 13. If y n +2- 2 μy n +1 + μy n = c , n = 0 , 1 , 2 , ··· , with < μ < 1 show that y n → c/ (1- μ ) as n → ∞ ....
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This note was uploaded on 02/04/2011 for the course MATH 1711 taught by Professor Dru.picchini during the Spring '11 term at Durham.

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exercises-2011 - Data Analysis Modelling and Simulation...

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