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20085ee102_1_F-08-102-HW-5

20085ee102_1_F-08-102-HW-5 - FALL 2008 Put First Letter of...

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FALL 2008: Put First Letter of LAST Name in the corner →→ ( Otherwise Your HW May Be LOST! ) PRINT: (LAST , Middle, First):——————————————————– EE102: SYSTEMS & SIGNALS HW: # 5 A LATE HW CANNOT BE A HW! Posted: W, November 5 Hand In To Me: M, November 17 Attach This Sheet To Your HW (Otherwise It May Be Lost!) Notations: Let f ( t ) , g ( t ) , t [ a, b ] , be a real signals: f ( t ) , g ( t ) := inner product or scalar product of f ( · ) and g ( · ) f ( t ) , f ( t ) := || f ( t ) || 2 , || f ( t ) || = f ( t ) , f ( t ) := norm of f ( · ) 1. Consider the following signals: f 1 ( t ) = 1 2 , f 2 ( t ) = 3 2 t, f 3 ( t ) = 45 8 ( t 2 - 1 3 ) , t [ - 1 , 1] Define f ( t ) , g ( t ) = 1 - 1 f ( t ) g ( t ) dt (i) Verify whether the signals { f i ( t ) , t = 1 , 2 , 3 } are (mutually) orthonor- mal, that is f i , f j = 0 , i = j, and f i , f j = 1 , i = j, for i, j = 1 , 2 , 3 (ii) Given f ( t ) = e - t , t [ - 1 , 1] Calculate: f ( t ) , f n ( t ) , n = 1 , 2 1
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Sketch f ( t ) and the signal f ( t ) defined by: f ( t ) := 2 n =1 f ( t ) , f n ( t ) f n ( t ) Do they look “kind of alike”? (iii) Calculate the number (called Least (or Mean) Square Error) E 2 := 0 | e - t - f ( t ) | 2 dt If your calculations yield a negative number, you have made a mistake! Please
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