Final answer:
To demonstrate that 3[x*(n)] equals X*(z*), one uses the properties of the z-transform and complex conjugates to account for linear multiplication within the z-domain.
Step-by-step explanation:
The student has asked to show that 3[x*(n)] = X*(z*), where x*(n) denotes the conjugate of the time-domain signal x(n) and X*(z*) denotes the conjugate of the z-transform of x(n). The z-transform is a complex frequency domain representation for discrete-time signals. To show the equality, one can use properties of the complex conjugate and linearity of the z-transform.
Working through the question involves taking the complex conjugate of the sequence, multiplying by three, and then taking the z-transform. Since the z-transform is a linear operation, it preserves scalar multiplication. Therefore, if you multiply the time-domain signal by three, after taking the z-transform, you can pull the scalar out and multiply the result of the z-transform by three.
The process might involve utilizing properties related to exponentials and complex conjugates, such as the complex conjugate of an exponential function, where any complex component i is replaced with -i when taking the conjugate. For example, in the context of evaluating cube of exponentials, if you have an exponential term e^(jθ), taking its conjugate would result in e^(-jθ), and cubing it would involve raising the magnitude to the third power while multiplying the angle by three: (e^(jθ))^3 = e^(3jθ).