Question

Which of the following is not correct with respect to Z-transforms?

• A. Z-transform converts the difference equation of discrete time system into linear algebraic equation
• B. Frequency domain response is achieved and plotted
• C. Convolution in time domain is converted into multiplication in z-domain
• D. Z-transform exist for most of the signals for which discrete time Fourier transform does not exist

Hint:

Z-transform is a powerful tool for analyzing discrete-time systems.
Key properties include linearity, time shifting, convolution, and its relation to the DTFT (\(X(e^{j\omega}) = X(z)|_{z=e^{j\omega}}\)).
The ROC is crucial for the Z-transform and its relation to the DTFT.

Answer 1

The Correct Option is B

(a) TRUE. Z-transform converts linear constant-coefficient difference equations into algebraic equations in \(z\). (c) TRUE. Convolution property: \(x[n]*h[n] \longleftrightarrow X(z)H(z)\). (d) TRUE. The Z-transform can converge for signals for which the DTFT does not converge (i.e., when the ROC of \(X(z)\) does not include the unit circle). (b) "Frequency domain response is achieved and plotted". The Z-transform \(X(z)\) is a function in the complex z-plane. The frequency response is obtained by evaluating \(X(z)\) on the unit circle, i.e., \(X(e^{j\omega})\), provided the unit circle is in the ROC. So, the Z-transform itself is not directly "the frequency response that is plotted"; rather, it is used to derive it. The DTFT is the direct frequency domain representation. This statement is the least precise or potentially misleading compared to the others, making it "not correct" in a strict sense. \[ \boxed{\text{Frequency domain response is achieved and plotted}} \]