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Unformatted text preview: EE261 Raj Bhatnagar Summer 20102011 EE 261 The Fourier Transform and its Applications Problem Set 2 Due Friday 8 July 1. (15 points) Fourier Series Symmetries (a) Let f ( t ) be a periodic signal of period 1 . One says that f ( t ) has halfwave symmetry if f ( t 1 2 ) = f ( t ) . Sketch an example of a signal that has halfwave symmetry. (b) If f ( t ) has halfwave symmetry and its Fourier series is f ( t ) = X n = c n e 2 int , show that c n = 0 if n is even. (c) Let g ( t ) be a periodic signal of period 1 . One says that g ( t ) has quarterwave symmetry if g ( t 1 4 ) = ig ( t ) . If g ( t ) has quarterwave symmetry and Fourier series coe cients d n , show that d n must be zero for n 6 = 1 + 4 k where k is an integer. Solution: (a) A simple example is f ( t ) = sin(2 t ) . The graphs of sin(2 t ) and sin(2 ( t 1 2 )) are shown. Figure 1: sin2 t Figure 2: sin2 ( t 1 / 2) Algebraically, sin(2 ( t 1 2 )) = sin(2 t ) = sin2 t. (b) Apply the halfwave symmetry equation to c n : c n = Z 1 e 2 int f ( t ) dt = Z 1 e 2 int f ( t 1 2 ) dt. 1 We make a change of variable u = t 1 2 in the second integral: Z 1 e 2 int f ( t 1 2 ) dt = Z 1 / 2 1 / 2 e 2 in ( u + 1 2 ) f ( u ) du = Z 1 / 2 1 / 2 e 2 inu e 2 in 1 2 f ( u ) du = e in Z 1 / 2 1 / 2 e 2 inu f ( u ) du = e in c n , (because we can integrate over any cycle to compute c n ) . Thus c n = e in c n . If n is even then e in = 1 and we have c n = c n , hence c n = 0 . (c) We expand both sides of the quarter wave equation as a Fourier series: g ( t 1 / 4) = ig ( t ) X n = d n e 2 in ( t 1 / 4) = i X n = d n e 2 int X n = d n e 2 int in/ 2 = e i/ 2 X n = d n e 2 int X n = ( d n e in/ 2 ) e 2 int = X n = ( e i/ 2 d n ) e 2 int d n e in/ 2 = e i/ 2 d n This last equation holds automatically if n = 1 , 5 , 7 ,... . For other values of n , this equation can only be true if d n must be zero. 2. (20 points) Some practice with inner products In class we de ned the inner product between two periodic functions. In this problem, we'll extend this operation to nonperiodic functions: Let f ( t ) and g ( t ) be two nonperiodic functions and de ne their inner product as ( f,g ) = Z  f ( t ) g ( t ) dt. Also, de ne the reversed signal to be f ( t ) = f ( t ) , and the delay or shift operator by a f ( t ) = f ( t a ) . 2 (a) Show that if both f ( t ) and g ( t ) are reversed, the inner product is unchanged: ( f ,g ) = ( f,g ) . (b) Show that ( f ,g ) = ( f,g ) . (c) Show that ( a f, a g ) = ( f,g ) . (d) Show that ( a f,g ) = ( f, a g ) ....
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 Summer '09

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