 # 6. Consider a discrete time linear time-invariant system. The input...

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6. Consider a discrete time linear time-invariant system. The input to the system X [n] is given by n x (n)=(2)nun) = and the output of the system is ly[n] is given by n - where u[n] is the unit step function. a. Determine the discrete time Fourier transform x(ejw) of the input x[n] b. Determine the discrete time Fourier transform y(ejw) of the output y[n]. C. Determine a linear constant coefficient difference equation that describes this system. 2. Consider the following signal X (t) X (t) = 2 cos(6nt + 0) + 3 sin (8mt) where 0 is a constant. i. Determine the fundamental period of x(t). ii. Express x (t) in the Fourier series representation of the form 8 x(>) E. = akelkwot k=-00 where wo is the fundamental frequency, i.e., determine the Fourier series coefficients ak for all k. 3. Consider a discrete time periodic signal x[n] with period N = 4. 1, n = 0, X [n] = 1, n = 1, - 1, n = 2, -1, n = 3. Determine the discrete time Fourier series coefficients for x[n]. Simplify your answer for full score.

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