The Z-transform of a unit step signal is given by:

• A. 1
• B. z
• C. z / (z − 1)
• D. 1 / (z − 1)

Answer: (C)

The Z-transform of a unit step signal, defined mathematically as u[n] = 1 for n ≥ 0, is derived from the definition of the Z-transform, which is the summation of the signal multiplied by z raised to the negative power of the discrete time index. For the unit step signal, the Z-transform is calculated as follows:

U(z) = Σ (from n=0 to ∞) u[n] * z^(-n)

Substituting the value of u[n]:

U(z) = Σ (from n=0 to ∞) 1 * z^(-n)

= Σ (from n=0 to ∞) z^(-n)

This infinite series can be recognized as a geometric series with the first term a = 1 and the common ratio r = 1/z, which converges for |z| > 1. The formula for the sum of an infinite geometric series is:

Sum = a / (1 - r)

Applying this to our series:

U(z) = 1 / (1 - (1/z))

= 1 / (1 - 1/z)

= 1 / ((z - 1)/z)

= z