Test on Z-Transforms and Transfer Functions in DSP

Question 1: The Z-transform symbolism Z{X(z)} is used to denote the Z-transform of a sequence x[n].

A. Ano

B. Ne

Explanation: The study materials show the symbolism for the Z-transform of a sequence x[n] as Z x[n] → X(z) or X(z) = Z{x[n]}. The notation Z{X(z)} is not used for this purpose.

Question 2: According to the provided study materials, which statements accurately describe the linearity property of the Z-transform?

A. The Z-transform of a sequence multiplied by a constant is the constant multiplied by the Z-transform of the sequence.

B. The Z-transform of the sum of two sequences is the sum of their individual Z-transforms.

C. The linearity property ensures that only non-linear difference equations can be transformed into algebraic equations.

D. The linearity property applies exclusively to continuous-time systems, not discrete-time systems.

Explanation: The study materials state that the Z-transform exhibits linearity, specifically noting that 'Z[a ⋅ x[n]] → a ⋅ X(z)' for multiplication by a constant, and 'Z[x1[n] + x2[n]] → X1(z) + X2(z)' for addition. Option 0 and 1 correctly reflect these properties. Option 2 is incorrect because the material states that 'linear difference equations can be transformed to algebraic equations' using the Z-transform, not non-linear ones. Option 3 is incorrect as the Z-transform is introduced for the 'Discrete time world' and its properties are discussed within that context.

Question 3: The system transfer function G(z) in the operator domain is defined as the Z-transform of the unit impulse response g[n].

A. Ano

B. Ne

Explanation: The study materials state that the System transfer function G(z) is equal to Z{g[n]}, where g[n] is the unit impulse response.

Question 4: The `filter` or `lfilter` function in MATLAB®/Python is suitable for real-time processing of a system's response based on its transfer function coefficients.

A. Ano

B. Ne

Explanation: The study materials state that the `filter` or `lfilter` function requires knowing the whole input signal x(n), which makes it not suitable for real-time processing.

Question 5: Based on the provided study materials, what is the primary purpose of Direct Form I in the context of LTI systems?

A. To directly calculate the output response using MATLAB®/Python's filter function.

B. To convert a transfer function into a signal flow diagram.

C. To implement real-time processing of signals without needing the whole input signal x(n).

D. To express the transfer function as a rational polynomial, simplifying the operator domain representation.

Explanation: The study materials state: 'We can convert the transfer function to a signal flow diagram. More variants exist, “Direct form I” and “Direct form II” are the most common.' This directly describes the purpose of Direct Form I.