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Istanbul Sehir University School of Engineering and Natural Sciences BM 504–Systems Theory for Biomedical Science Homework Assignment 3 Due to: Mar 18, 2019, 14:00 March 11, 2019

Spring Term

Instructor: Ahmet Ademoglu, PhD

1. Determine the Fourier Transform of each of the following signals (α is positive). (a) x(t) =

4t 1+t2

(b) Λ(2t + 3) + Λ(3t − 2) (c) sinc3 (t) (d) te−αt cos(βt) 2. Determine the Fourier transform of the signals shown below;

3. (a) Determine the continuous-time signal corresponding to the following transform; X(ω) = 2sin[3(ω−2π) (w−2π) (b) Let X(ω) denote the Fourier transform of the signal x(t) depicted as

Evaluate

R∞ −∞

2 2 sin(ω) e2ω dω without explicitly computing X(ω). ω

4. Find the impulse response of a system with the frequency response H(ω) =

sin2 (3ω) cos(ω) ω2

5. Evaluate the integrals (α is positive) using Fourier Transform properties; (a) (b)

R∞ −∞ R∞

sinc5 (t)dt

e−αt cos(βt)dt

0

6. The input and the output of a stable and causal LTI system are related by the differential equation d2 d y(t) + 6 y(t) + 8y(t) = 2x(t) 2 dt dt (a) Find the impulse response of this system. (b) What is the response of this system if x(t) = te−2t u(t)? 7. Determine the Fourier series expansion of the following signals where Λ(t) is the triangular pulse given as ( t + 1 −1 ≤ t < 0, 0 ≤ t < 1, Λ(t) = −t + 1 0 otherwise (a) x1 = (b) x2 =

∞ P n=−∞ ∞ P

Λ(t − 2n) Λ(t − n)u(t − n)

n=−∞ ∞ P

(c) x3 (t) =

0

δ (t − nT )

n=−∞

(d) x9 = cos(ω0 t) + |cos(ω0 t)| 8. Determine the Fourier series expansion of each of the periodic signals shown below;

9. Using Parseval’s relation, show that 1+

1 1 1 π4 + + . . . + = 34 54 (2n + 1)4 96

10. By computing the the Fourier series coefficients for the periodic signal

∞ P n=−∞

show that

∞ X

δ(t − nT ) =

n=−∞

∞ 2πt 1 X ejn T T n=−∞

Using this result, prove the following identity ∞ X n=−∞

δ(t − nT ) =

∞ n 2πt 1 X ejn T X Ts n=−∞ T

δ(t − nT )