HW_6_R1_sol[1]

# HW_6_R1_sol[1] - HW 6 ECE 2704 Due 7.3.7 Consider the...

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DKL:10/30/06 HW 6 Solutions 1/3 HW 6 ECE 2704 Due 10-27-06 7.3.7 Consider the periodic signal shown in Figure P7.3.7. Find the Fourier series of this signal. 1 2 3 -1 -2 -3 1 ... ... t x ( t ) Figure P7.3.7 (a) (5) Find the fundamental frequency. (b) (10) Set up the computation(s) for the coefficients of the Fourier series. (c) (15) Integrate to find an expression for the coefficients. (d) (10) Write out the Fourier series. Solution This problem illustrates the calculation of the exponential Fourier series . (a) The fundamental period and the fundamental frequency are T 0 = 2 ! " 0 = 2 # 2 = # (b) The coefficients are given by X m = 1 T 0 x ( t ) e ! jm " 0 t dt T 0 # = 1 2 te ! jm \$ t dt 0 1 # + (0) e ! jm \$ t dt 0 1 # % & ( ) * = 1 2 te ! jm \$ t dt 0 1 # (c) The constant coefficient is X 0 = 1 2 x ( t ) dt T 0 ! = 1 2 t dt 0 1 ! = 1 2 1 2 " # \$ % & = 1 4

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DKL:10/30/06 HW 6 Solutions 2/3 The coefficients are given by X m = 1 2 te ! jm " t dt 0 1 # = e ! jm " t 2 t ! jm " ! 1 ! jm " ( ) 2 \$ % & ( ) \$ % & & ( ) ) t = 0 1 = e ! jm " 2 1 ! jm " ! 1 ! jm " ( ) 2 \$ % & ( ) ! 1 2 ! 1 ( ! jm " ) 2 \$ % & ( ) = e ! jm " 2 1 m " ( ) 2 ! 1 jm " \$ % & ( ) ! 1 2 1 ( m " ) 2 \$ % & ( ) (d) The Fourier series is given by x ( t ) = 1 4 + e ! jm " 2 1 m " ( ) 2 !
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