Z Transform MCQ [Free PDF] - Objective Question Answer for Z Transform Quiz

21. If X(z) is the z-transform of the signal x(n) then what is the z-transform of anx(n)?

A. X(az)
B. X(az-1)
C. X(a-1z)
D. None of the mentioned

Answer: C

We know that from the definition of z-transform

Z{anx(n)} = \(\sum_{n = -\infty}^{\infty}a^n x(n) z^{-n} = \sum_{n = -\infty}^{\infty}x(n)(a^{-1}z)^{-n} = X(a^{-1}z)\).

22. If the ROC of X(z) is r1<|z|<r2, then what is the ROC of X(a-1z)?

A. |a|r1<|z|<|a|r2
B. |a|r1>|z|>|a|r2
C. |a|r1<|z|>|a|r2
D. |a|r1>|z|<|a|r2

Answer: A

Given ROC of X(z) is r1<|z|<r2

Then ROC of X(a-1z) will be given by r1<|a-1z |<r2 = |a|r1<|z|<|a|r2

23. What is the z-transform of the signal x(n) = an(sinω0n)u(n)?

A.\(\frac{az^{-1} sin\omega_0}{1+2 az^{-1} cos\omega_0+a^2 z^{-2}}\)

B.\(\frac{az^{-1} sin\omega_0}{1-2 az^{-1} cos\omega_0- a^2 z^{-2}}\)

C.\(\frac{(az)^{-1} cos\omega_0}{1-2 az^{-1} cos\omega_0+a^2 z^{-2}}\)

D.\(\frac{az^{-1} sin\omega_0}{1-2 az^{-1} cos\omega_0+a^2 z^{-2}}\)

Answer: D

we know that by the linearity property of z-transform of sin(ω0n)u(n) is

X(z) = \(\frac{z^{-1} sin\omega_0}{1-2z^{-1} cos\omega_0+z^{-2}}\)

Now by the scaling in the z-domain property, we have z-transform of an (sin(ω0n))u(n) as

X(az) = \(\frac{az^{-1} sin\omega_0}{1-2az^{-1} cosω_0+a^2 z^{-2}}\)

24. If X(z) is the z-transform of the signal x(n), then what is the z-transform of the signal x(-n)?

A. X(-z)
B. X(z-1)
C. X-1(z)
D. None of the mentioned

Answer: B

From the definition of z-transform, we have

Z{x(-n)} = \(\sum_{n = -\infty}^{\infty} x(-n) z^{-n} = \sum_{n = -\infty}^{\infty} x(-n) (z^{-1})^{-(-n)} = X(z^{-1})\)

so, If X(z) is the z-transform of the signal x(n), then the z-transform of the signal x(-n) is X(z-1)

25. X(z) is the z-transform of the signal x(n), then what is the z-transform of the signal nx(n)?

A. \(-z\frac{dX(z)}{dz}\)

B. \(z\frac{dX(z)}{dz}\)

C. \(-z^{-1}\frac{dX(z)}{dz}\)

D. \(z^{-1}\frac{dX(z)}{dz}\)

Answer: A

From the definition of z-transform, we have

X(z) = \(\sum_{n = -\infty}^{\infty} x(n) z^{-n}\)

On differentiating both sides, we have

\(\frac{dX(z)}{dz} = \sum_{n = -\infty}^{\infty} (-n) x(n) z^{-n-1} = -z^{-1} \sum_{n = -\infty}^{\infty}nx(n) z^{-n} = -z^{-1}Z\{nx(n)\}\)

Therefore, we get \(-z\frac{dX(z)}{dz}\) = Z{nx(n)}.

26. What is the z-transform of the signal x(n) = nanu(n)?

A. \(\frac{(az)^{-1}}{(1-(az)^{-1})^2}\)

B. \(\frac{az^{-1}}{(1-(az)^{-1})^2}\)

C. \(\frac{az^{-1}}{(1-az^{-1})^2}\)

D. \(\frac{az^{-1}}{(1+az^{-1})^2}\)

Answer: C

We know that Z{anu(n)} = \(\frac{1}{1-az^{-1}}\) = X(z)

Now the z-transform of nanu(n) = \(-z\frac{dX(z)}{dz} = \frac{az^{-1}}{(1-az^{-1})^2}\)

27. Sampling rate conversion by the rational factor I/D is accomplished by what connection of interpolator and decimator?

A. Parallel
B. Cascade
C. Convolution
D. None of the mentioned

Answer: B

A sampling rate conversion by the rational factor I/D is accomplished by cascading an interpolator with a decimator.

28. Which of the following has to be performed in sampling rate conversion by rational factor?

A. Interpolation
B. Decimation
C. Either interpolation or decimation
D. None of the mentioned

Answer: A

We emphasize that the importance of performing the interpolation first and decimation second is to preserve the desired spectral characteristics of x(n).

29. Which of the following operation is performed by the blocks given in the figure below?

A. Sampling rate conversion by a factor I
B. Sampling rate conversion by a factor D
C. Sampling rate conversion by a factor D/I
D. Sampling rate conversion by a factor I/D

Answer: D

In the diagram given, an interpolator is in a cascade with a decimator which together performs the action of sampling rate conversion by a factor I/D.

30. The Nth root of unity WN is given as __________

A. ej2πN
B. e-j2πN
C. e-j2π/N
D. ej2π/N

Answer: C

We know that the Discrete Fourier transform of a signal x(n) is given as

X(k) = \(\sum_{n = 0}^{N-1} x(n)e^{-j2πkn/N} = \sum_{n = 0}^{N-1} x(n) W_N^{kn}\)

Thus we get Nth rot of unity WN = e-j2π/N

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