HW11 Solutions.pdf - ESE 351-01 Fall 2017 Homework Set#11 due Nov 21 11 Problems In problems 1-6 use known transform pairs and properties(tables results

# HW11 Solutions.pdf - ESE 351-01 Fall 2017 Homework Set#11...

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ESE 351-01, Fall 2017 Homework Set #11, due Nov. 21 11 Problems In problems 1-6, use known transform pairs and properties (tables), results from the Laplace transform, or Euler’s formula. You should not evaluate the Fourier Transform integral, or its inverse. 11.1 . Find the Fourier Transform of the function graphed below. 11.2 . Find the Fourier Transform. f t ( ) = sin 2 π t 2 ( ) t 2 11.3 . Find the Fourier Transform. (Hint: Duality.) f t ( ) = 2 α α 2 + t 2 , α > 0 11.4 . The figures below depict functions of time, f ( t ). (a) Write the Fourier Transform of the function in (a). (b) Write the Fourier Transform of the function in (b). (Hint: It is the function in (a) multiplied by t .) Write your answer so that no sine, cosine, or sinc is involved. (Euler’s formula to the rescue.)

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ESE 351-01, Fall 2017 Homework Set #11, due Nov. 21 11.5. Based on Mukai problem 11.13.7. The Fourier Transform of f ( t ) is given by. F ω ( ) = 4 25 ω 2 + i ω 6 Find the Fourier Transform, G ( ω ), of the given g ( t ). (You do not have to make the denominator all real.) a ( ) g t ( ) = f 3 t ( ) b ( ) g t ( ) = f t 10 ( ) c ( ) g t ( ) = f 3 t ( ) d ( ) g t ( ) = f t ( ) cos4 t e ( ) g t ( ) = f t ( ) sin4 t f ( ) g t ( ) = d dt f t ( ) 11.6 . Mukai 11.13.14. 11.7. Mukai 11.13.16(a). Use Parseval’s theorem to evaluate sinc 2 t ( ) dt −∞ . For problems 8-11, you will need these key results regarding Fourier transforms, linear systems (input u , output y ) and Power Spectral Density: S f ω ( ) PSD !"# = F R f τ ( ) { } , where R f τ ( ) = Autocorrelation function of f t ( ) S y ω ( ) = H i ω ( ) 2 S u ω ( ) 11.8 (a) Sketch the real part of the Fourier transform of r ( t ). (Hint: It’s all real.) r t ( ) = sin t π t 1 + cos4 t ( ) (b) Find and sketch the squared gain, | H ( i ω )| 2 , of the system below. !! y + 4 ! y + 8 y = 8 u (c) Suppose that r ( t ) from part (a) is the autocorrelation function of the input u ( t ) in part (b). Sketch the Power Spectral Density of y ( t ) in that case. (I don’t know if such an autocorrelation function is possible, but assume that it is.)