z-transform of cosh t sinh t

The z-transform of a sampled signal x(k) is X(z) = Σ_{k=0}^{∞} x(k) z⁻ᵏ. Here the signal is sampled at t = kT; take T = 1 (for a general T, replace 2 by 2T).

Step 1: simplify the product.

cosh t sinh t = ½ sinh 2t = ¼ (e²ᵗ − e⁻²ᵗ)

Step 2: use Z[aᵏ] = z/(z − a). With t = k and T = 1:

Z[cosh t sinh t] = ¼ [ z/(z − e²) − z/(z − e⁻²) ]
                 = ¼ · z (e² − e⁻²) / (z² − z(e² + e⁻²) + 1)
                 = ½ · z sinh 2 / (z² − 2z cosh 2 + 1)

Answer: X(z) = z sinh 2 / [2 (z² − 2z cosh 2 + 1)], for |z| > e².

For general sampling period T: X(z) = z sinh 2T / [2 (z² − 2z cosh 2T + 1)].

Numerically, sinh 2 = 3.6269 and cosh 2 = 3.7622, so X(z) = 1.8134 z/(z² − 7.5244 z + 1). (Checked numerically at z = 20: both sides give 0.144778.)