Which of the following represents the gain of a system in Z-transform?

Which of the following represents the gain of a system in Z-transform?

• A. H(z) = Y(z)/X(z)
• B. H(s) = Y(s)/X(s)
• C. H(z) = X(z)/Y(z)
• D. H(s) = X(s)*Y(s)

Answer: (A)

The gain of a system in the context of the Z-transform is defined as the ratio of the output transform \( Y(z) \) to the input transform \( X(z) \). This relationship is fundamental in control theory and signal processing as it provides a way to analyze the behavior of linear time-invariant systems in the Z-domain. In the Z-transform domain, a system's transfer function is commonly denoted as \( H(z) \), which characterizes how the input signal transforms into the output signal through the system. Therefore, the correct formulation for describing the gain or transfer function \( H(z) \) is expressed as \( H(z) = \frac{Y(z)}{X(z)} \). This ratio allows one to ascertain how the input signal is modified by the system to produce the output, both in terms of amplitude (gain) and phase. The other options do not apply to the Z-transform context or the definitions typically utilized for discrete-time systems. For instance, the options using \( H(s) \) represent the gain in the Laplace transform domain, which deals with continuous-time systems instead. Also, the last option includes a multiplication of transforms, which does not correspond to the definition of gain or transfer function in either

The gain of a system in the context of the Z-transform is defined as the ratio of the output transform ( Y(z) ) to the input transform ( X(z) ). This relationship is fundamental in control theory and signal processing as it provides a way to analyze the behavior of linear time-invariant systems in the Z-domain.

In the Z-transform domain, a system's transfer function is commonly denoted as ( H(z) ), which characterizes how the input signal transforms into the output signal through the system. Therefore, the correct formulation for describing the gain or transfer function ( H(z) ) is expressed as ( H(z) = \frac{Y(z)}{X(z)} ). This ratio allows one to ascertain how the input signal is modified by the system to produce the output, both in terms of amplitude (gain) and phase.

The other options do not apply to the Z-transform context or the definitions typically utilized for discrete-time systems. For instance, the options using ( H(s) ) represent the gain in the Laplace transform domain, which deals with continuous-time systems instead. Also, the last option includes a multiplication of transforms, which does not correspond to the definition of gain or transfer function in either