20115ee102_1_hw4_soln

20115ee102_1_hw4_soln - 1 EE102 Systems and Signals Fall...

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Unformatted text preview: 1 EE102 Systems and Signals Fall Quarter 2011 Jin Hyung Lee Homework #4 Solutions 1. Suppose that f ( t ) is a periodic signal with period T , and that f ( t ) has a Fourier series. If τ is a real number, show that f ( t- τ ) can be expressed as a Fourier series identical to that for f ( t ) except for the multiplication by a complex constant, which you must find. Solution: f ( t- τ ) = ∞ X n =-∞ D n,τ e janω t where D n,τ = Z t + T t f ( t- τ ) e- jω nt dt Let t = t- τ , D n,τ = Z t + τ + T t + τ f ( t ) e- jω n ( t + τ ) dt = e- jnω τ Z t + τ + T t + τ f ( t ) e- jω nt dt = e- jnω τ D n where we have used the fact that f ( t ) is periodic in the last step. We get the same Fourier series no matter where we choose the one period to integrate over. The Fourier series is then f ( t- τ ) = ∞ X n =-∞ D n e- jnω τ e janω t 2. Switching amplifiers are a very efficient way to generate a time-varying output voltage from a fixed supply voltage. They are particularly useful in high-power applications. The basic idea is that an output voltage a is generated by rapidly switching between zero and the supply voltage A . The output is then lowpass filtered to remove the harmonics generated by the switching operation. For our purposes we can consider the lowpass filter as an integrator over many switching cycles, so the output voltage is the average value of the switching waveform.. Varying the switching rate varies the output voltage. In this problem we will only consider the case where the desired output voltage is constant. We can analyze this system with the Fourier series. If the output pulses are spaced by T , the waveform the amplifier generates immediately before the lowpass filter is 2 T 2 T...
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20115ee102_1_hw4_soln - 1 EE102 Systems and Signals Fall...

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