The relationship between s-domain and z-domain is given by:

• A. z = sT
• B. s = ln(z)
• C. z = e^(sT)
• D. s = z − 1

Answer: (C)

The relationship between the s-domain and z-domain can be understood through the process of converting continuous-time systems into discrete-time systems. This conversion is often accomplished using the bilinear transform or the relationship defined by the exponential mapping.

The correct option, which states that ( z = e^{sT} ), illustrates this relationship effectively. Here, ( s ) represents the complex frequency variable in the Laplace transform (s-domain), and ( z ) represents the complex frequency variable in the Z-transform (z-domain). The term ( T ) represents the sampling period or the time interval associated with the discrete representation of a continuous signal.

When you express a continuous-time signal in the s-domain, such as in the context of control systems or signal processing, converting it to the z-domain for discrete-time analysis involves exponentiating the s-domain variable. The exponential function reflects how signals sampled at intervals ( T ) relate to their continuous representations over time. This relationship is foundational for digital signal processing, allowing for the analysis and design of digital systems based on their continuous-time counterparts.

Understanding this equation is crucial, as it helps grasp the dynamics between continuous and discrete systems, enabling one to analyze and design filters, controllers, and other electronic systems effectively.