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Unformatted text preview: EE 261 The Fourier Transform and its Applications Fall 2009 Problem Set One Due Wednesday, September 30 1. Some practice with geometric sums and complex exponentials (5 points each) Well make much use of formulas for the sum of a geometric series, especially in combination with complex exponentials. (a) If w is a real or complex number, w 6 = 1, and p and q are any integers, show that q X n = p w n = w p- w q +1 1- w . (Of course if w = 1 then the sum is q n = p 1 = q + 1- p .) Discuss the cases when p =- or q = . What about p =- and q = + ? (b) Find the sum N- 1 X n =0 e 2 in/N and explain your answer geometrically. (c) Derive the formula N X k =- N e 2 ikt = sin(2 t ( N + 1 / 2)) sin( t ) 2. Some practice combining simple signals. (5 points each) The triangle function with a parameter a > 0 is a ( t ) = ( t/a ) = ( 1- 1 a | t | , | t | a , | t | > a The graph is ! a a 1 1 The parameter a specifies the width, namely 2 a . Alternately, a determines the slopes of the sides: the left side has slope 1 /a and the right side has slope- 1 /a . We can modify a by scaling the height and shifting horizontally, forming b a ( t- c ). The slopes of the sides of the scaled function are then b/a . The graph is: c ! a c c+a b bLa(t ! c) Express each of the following as a sum of two shifted, scaled triangle functions b 1 a 1 ( t- c 1 )+ b 2 a 2 ( t- c 2 ). Think of the sum as left-triangle plus a right-triangle (right meaning to the)....
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