# 102_1_hw04 - 1 EE102 Spring 2009-10 Systems and Signals Lee...

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1 EE102 Spring 2009-10 Lee Systems and Signals Homework #4 Due: Tuesday, May 4, 2010 at 5 PM. 1. A signal f ( t ) is periodic with period T 0 . You compute the Fourier series assuming the period is T 0 = 2 T 0 and hence ω 0 = ω 0 / 2 . Find the resulting values of the Fourier series coefficients, D n,T 0 , D n,T 0 = 1 T 0 t 0 + T 0 t 0 f ( t ) e - jn ω 0 t dt Express your answer in terms of D n,T 0 , the Fourier coefficients when you assume the cor- rect fundamental period of T 0 . 2. Suppose that f ( t ) is a periodic signal with period T 0 , 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. 3. 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 T 2 T - 2 T - T 0 t ! T A The duty cycle of the switching amplifier is α , and the width of the pulses is α T . When α = 1 , the amplifier is constantly on and produces its maximum output A .

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2 (a) Reducing the duty cycle reduces the output voltage. After the lowpass filter, only the
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