102_1_final_practice_solution

102_1_final_practice_solution - Systems and Signals EE102...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
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 different, 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 different n ? Don’t integrate! Solution: We can find the Fourier transform of cos 2 ( ω 0 t ) easily by frequency domain con- volution F h cos 2 ( ω 0 t ) i = 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 coefficients are just the Fourier transform coefficients divided by 2 π , so D ± 2 = 1 4 , and D 0 = 1 2 . 1
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
and all of the other coefficients are zero. Similarly, you found cos 3 ( ω 0 t ) to be F h cos 3 ( ω 0 t ) i = 1 2 π F h cos 2 ( ω 0 t ) i *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 )) ² = 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 coefficients 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
Background image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 02/28/2011 for the course EE 102 taught by Professor Levan during the Spring '08 term at UCLA.

Page1 / 11

102_1_final_practice_solution - Systems and Signals EE102...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online