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102_1_final_practice_solution

# 102_1_final_practice_solution - Systems and Signals EE102...

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Systems and Signals Lee, Spring 2009-10 EE102 Final Practice Solutions Problem 1. Fourier Series A linear amplifier has an output y that is proportional to the input x , y = a 1 x where a 1 is a constant. In practice an amplifier will have a more complex characteristic y = a 0 + a 1 x + a 2 x 2 + a 3 x 3 + · · · If we apply an input x ( t ) = cos( ω 0 t ) ideally we would get an output spectrum that looks like ω 0 - ω 0 - 2 ω 0 - 3 ω 0 3 ω 0 2 ω 0 0 ω 1 2 where we’ve assumed a 1 = 1 for simplicity. In practice we get something di ff erent, and this tells us something about the amplifier char- acteristic. For each of the following amplifier characteristics, determine what the output Fourier series spectrum looks like when the inputs is x ( t ) = cos( ω 0 t ). Hint: What does the spectra of cos n ( t ) look like for di ff erent n ? Don’t integrate! Solution: We can find the Fourier transform of cos 2 ( ω 0 t ) easily by frequency domain con- volution F cos 2 ( ω 0 t ) = 1 2 π ( π ( δ ( ω - ω 0 ) + δ ( ω + ω 0 )) * ( π ( δ ( ω - ω 0 ) + δ ( ω + ω 0 )) = π 2 ( δ ( ω - 2 ω 0 ) + 2 δ ( ω ) + δ ( ω + 2 ω 0 )) The Fourier series coe ffi cients are just the Fourier transform coe ffi cients divided by 2 π , so D ± 2 = 1 4 , and D 0 = 1 2 . 1

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and all of the other coe ffi cients are zero. Similarly, you found cos 3 ( ω 0 t ) to be F cos 3 ( ω 0 t ) = 1 2 π F cos 2 ( ω 0 t ) * F [cos( ω 0 t )] = 1 2 π π 2 ( δ ( ω - 2 ω 0 ) + 2 δ ( ω ) + δ ( ω + 2 ω 0 )) * ( π ( δ ( ω - ω 0 ) + δ ( ω + ω 0 )) = π 4 ( δ ( ω - 3 ω 0 ) + 3 δ ( ω - ω 0 ) + 3 δ ( ω + ω 0 ) + δ ( ω + 3 ω 0 )) The Fourier series coe ffi cients are then D ± 3 = 1 8 , and D ± 1 = 3 8 a) Find the output Fourier series spectrum when the amplifier characteristic is y = x + 0 . 1 x 3 . Solution: y ( t ) = cos( ω 0 t ) + 0 . 1 cos 3 ( ω 0 t ) The Fourier series spectrum of y ( t ) will be the sum of the spectrum for cos( ω 0 t ) and the spectrum 0 . 1 cos 3 ( ω 0 t ). The Fourier series spectrum of cos( ω 0 t ) is D ± 1 = 1 2 The Fourier series spectrum of 0 . 1 cos 3 ( ω 0 t ) is D ± 3 = (0 . 1) 1 8 = 1 80 D ± 1 = (0 . 1) 3 8 = 3 80 The sum of these is then D ± 3 = 1 80 D ± 1 = 1 2 + 3 80 = 43 80 This is plotted below.
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102_1_final_practice_solution - Systems and Signals EE102...

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