HW8 Solutions.pdf - ESE 351-01 Fall 2017 Homework Set#8 due Oct 31 9.1 Mukai B.9.1 and B.9.2 Use key angle relationships not a calculator 9.2 Mukai

HW8 Solutions.pdf - ESE 351-01 Fall 2017 Homework Set#8 due...

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ESE 351-01, Fall 2017 Homework Set #8, due Oct. 31 9.1 Mukai B.9.1 and B.9. 2. Use key angle relationships, not a calculator. 9.2 Mukai B.9.3. Use trig identity and key angle relationships, not a calculator. 9.3 If f ( t ) is real, then the Fourier coefficients, F ( n ) have the property that F n ( ) = F n ( ) = complex conjugate . (a) Show that if the F ( n ) are even, i.e., F (- n ) = F ( n ), then they are all real. (b) Show that if the F ( n ) are odd, i.e., F (- n ) = - F ( n ), then they are all purely imaginary. (Hint: These are very easy proofs.) 9.4 In subsequent problems, you will need to evaluate the integral e at sin bt dt , for three cases: a = 0, a = - ib and cases where a 2 + b 2 0 . For the a = 0 case, the integrand is simply a sine, which is an easy integral to evaluate. Show the following for the other two cases. (Hint: use Euler’s formula to express the sine function.) e ibt sin bt dt = i t 2 1 4 b e i 2 bt + C e at sin bt dt = e at a 2 + b 2 a sin bt b cos bt ( ) + C a 2 + b 2 0 9.5 Mukai 9.9.3 as modified below. (a) As in Mukai. (b) Use a computer (I used Excel) for the sketches (over +2 periods). OK to put all on one plot. No comment required. (c) Sketch vs. index n , which represents n ω 1 radians per second. Do not comment. (Hint: the angle must be an odd function. See “convention” for negative reals.) (d) Skip part (d) entirely. 9.6 Mukai 9.9.5 as modified below. (a) As in Mukai. (b) Find the partial sums as in Mukai, but sketch only g 1 ( t ) (over +2 periods). No comment required. (c) Sketch vs. index n , which represents n ω 1 radians per second. Do not comment. (Hint: the angle must be an odd function. See “convention” for negative reals.) (d) Skip part (d) entirely. 9.7 Mukai 9.9.11 as modified below. (a) No sketch required. (b) Skip entirely. (c) As in Mukai. (d) As in Mukai. (e)&(f) Skip entirely.
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ESE 351-01, Fall 2017 Homework Set #8, due Oct. 31 9.8 Mukai 9.9.12 as modified below. (a) No sketch required. (b) Skip entirely. (c) As in Mukai. (d) As in Mukai. (e)&(f) Skip entirely. 9.9 Consider the impulse train, f t ( ) = δ t mT 1 2 T ( ) m = −∞ , sketched here. (a) Find the Exponential Fourier coefficients, F ( n ). (b) Find the partial sum, g 2 t ( ) = F n ( ) e i 2 π n T t n = 2 2 , in all real terms. (c) Sketch its Fourier spectrum. (Indicate the units of the frequency axis.) Hint: Recall the Sifting Property of the Dirac Delta Function. (See the end of Lecture 6.) Useful identities: e i 0 = 1 e i π = 1 e i π 2 = i e i π 2 = i e i π n = 1 ( ) n e i 2 π n = 1 n = integer sin θ ( ) = sin θ cos θ ( ) = cos θ sin θ ± 2 n π ( ) = sin θ ( ) cos θ ± 2 n π ( ) = cos θ ( ) sin θ ± π ( ) = sin θ ( ) cos θ ± π ( ) = cos θ ( ) sin θ + π 2 ( ) = cos θ sin θ π 2 ( ) = cos θ cos θ + π 2 ( ) = sin θ cos θ π 2 ( ) = sin θ sin 2 θ + cos 2 θ = 1 sin 2 θ ( ) = 2sin θ cos θ
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ESE 351-01, Fall 2017 Homework Set #8, due Oct. 31 Solutions 9.1 Mukai B.9.1 and B.9.2. Use key angle relationships, not a calculator. B.9.1. z = 5 + i 5 ρ = 5 2 + 5 2 = 5 2 θ = tan 1 5 5 ( ) = tan 1 1 1 ( ) = 3 π 4 radians B.9.2.
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