Initial value theorem; find x(0), x(1)

Initial value theorem

If X(z) = Z[x(k)] and the limit exists, then

x(0) = lim_{z→∞} X(z).

Further, x(1) = lim_{z→∞} z [X(z) − x(0)].

(Reason: X(z) = x(0) + x(1)z⁻¹ + x(2)z⁻² + …, and every term except x(0) tends to 0 as z → ∞.)

Finding x(0) and x(1)

X(z) = (1 − eᵀ) z⁻¹ / [ (1 − z⁻¹)(1 − e⁻ᵀ z⁻¹) ]

x(0): as z → ∞, z⁻¹ → 0, so the numerator → 0 and the denominator → 1.

x(0) = 0

x(1): multiply X(z) by z (x(0) = 0):

z X(z) = (1 − eᵀ) / [ (1 − z⁻¹)(1 − e⁻ᵀ z⁻¹) ]
lim (z → ∞) = (1 − eᵀ) / (1 · 1)

x(1) = 1 − eᵀ

Check. Expand in powers of z⁻¹: X(z) = (1 − eᵀ) z⁻¹ [1 + (1 + e⁻ᵀ) z⁻¹ + …]. The z⁻¹ coefficient is 1 − eᵀ, confirming x(1). (For T = 0.9 the series gives x(1) = −1.4596 = 1 − e^{0.9}.)

Answer: x(0) = 0, x(1) = 1 − eᵀ.