HW_9_sol - HW 9 ECE 2704 Due 8.1.4 Consider the amplitude...

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DKL:11/13/06 HW 8 1/5 HW 9 ECE 2704 Due 11-13-06 8.1.4 Consider the amplitude and phase spectra shown in Figure P8.1.4. (a) What is the fundamental period of this signal? (b) Write the corresponding Fourier series in exponential form. (c) Write the corresponding Fourier series in cosine form. (d) Write the corresponding Fourier series in trigonometric form. See Table 7.6.1 Figure P8.1.4 Solution This problem illustrates the relationship between the amplitude and phase spectra, Definition 8.1.2, and the corresponding signal. (a) The fundamental frequency of this signal is ! 0 = 2 rad/sec . Therefore, the fundamental period is T 0 = 2 ! " 0 = 2 ! 2 = ! (b) The Fourier series in exponential form is x ( t ) = 0.8 e j 3 ( ) e ! j 4 t + 0.3 e j 2.5 ( ) e ! j 2 t + 0.1 + 0.3 e ! j 2.5 ( ) e j 2 t + 0.8 e ! j 3 ( ) e j 4 t (c) The Fourier series in cosine form is x ( t ) = 0.1 + 0.6cos(2 t ! 2.5) + 1.6cos(4 t ! 3) Using
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DKL:11/13/06 HW 8 2/5 A m = 2 X m , ! m = " X m (d) The Fourier series is trigonometric form (see Table 7.6.1) is x ( t ) = 0.1 + 0.6 cos( ! 2.5) ( ) cos(2 t ) + 1.6 cos( ! 3) ( ) cos(4 t ) ! 0.6 sin( ! 2.5) ( ) sin(2 t ) ! 1.6 sin( ! 3) ( ) sin(4 t ) = 0.1 ! 0.4807cos(2 t ) !
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