Let x[n] be a real causal sequence for which |x[n]| < ∞. The z-transform of x[n] is which is a Taylor’s series in the variable z^-1 and therefore converges to an analytic function everywhere outside some circular disc centered at z = 0. (The region of convergence includes the point z = ∞, and, in fact, X(∞) = x[0].) The statement that x(z) is analytic (in its region of convergence ) implies strong constraints on the function X(z). (See Churchill and Brown, 1990.) Specifically, its real and imaginaty parts each satisfy Laplace’s equation, and the real and imaginary parts are related by the Cauchy-Riemann equations, We will use these properties to determine X(z) from its real part when x[n] is a real, finite-valued, causal sequence.

Let the z-transform of such a sequence be 

X(z) = X_R(z) + j X_I(z).

where X_R(z) and X_I(z) are real-valued functions of z-suppose that X_R(z) is  

X_R(ρe^jω) = ρ + α cos ω/ρ, α real,

for z = ρe^jω. Then find X(z) (as an explicit function of z), assuming that X(z) is analytic everywhere except at z = 0, Do this using both of the following methods.

(a) Method 1, Frequency Domain. Use the fact that the real and imaginary parts of X(z) must satisfy the Cauchy-Riemann equaitons everywhere that X(z) is analytic. The Cauchy-Riemann equations are the following:

1. In Cartesian coordinates,

where z = x + jy and X(x + jy) = U(x, y) + jV(x, y).

2. In polar cooordinates,

where z = ρe^jω and X(ρe^jω) = U(ρ,ω) + jV(ρ,ω).

Since we know that U = X_R, we can integrate these equations to find V = X_I and hence X. (Be careful to treat the constant of integration properly.)

(b) Method 2, Time Domain. The sequence x[n] can be represented is x[n] = x_e[n] + x₀[n], where x_e[n] is real and even with Fourier transform X_R(e^jω) and the sequence x₀[n] is real and odd with Fourier transform j X_I(e^jω). Find x_e[n] and, using causality, find x₀[n] and hence x[n] and X(z).

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