By L. Loomis, S. Sternberg

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These will be discussed in detail in chapter 19. 10 t t−1 u(τ ) dτ. Show that the system is real. Show that the system is stable. Show that the system is linear time-invariant. Calculate the response to the input u(t) = cos ωt. Calculate the response to the input u(t) = sin ωt. Calculate the amplitude response of the system. Calculate the frequency response of the system. For a discrete-time system the response y[n] to an input u[n] is given by y[n] = u[n − 1] − 2u[n] + u[n + 1]. 11 Cascade system Show that the system is linear time-invariant.

Next we equate coefficients of corresponding powers of z. For the coefficient of z 3 , z 2 and z it subsequently follows that 0 = A + C, 0 = −A + B − 2C + D, 1 = A + C − 2D. The solution to this system of equations is A = 0, B = 1/2, C = 0, D = −1/2 and so the real partial fraction expansion looks like this: z (z − 1)2 (z 2 + 1) = 1/2 1 1/2 1 = . − 2 − 2 2 2 (z − 1) z +1 2(z − 1) 2(z + 1) We finish with an example where a quadratic factor occurs twice in the denominator of F(z). 9 Let the function F(z) be given by F(z) = z 2 + 3z + 3 .

2 for the case n = 5. Note that the roots display a symmetry with respect to the real axis. This is a consequence of the fact that the polynomial z n − 1 has real coefficients and that thus the complex conjugate of a zero is a zero as well. The method described for solving the equation z n = 1 can be extended to equations of the type z n = a, where a is an arbitrary complex number. We will illustrate this using the following example. 2 The solutions of z 5 = 1. 3 We determine the roots of the equation z 3 = 8i.

### Advanced Calculus by L. Loomis, S. Sternberg

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