Digital Signal Processing

Properties of Z Transform

Question 1
Marks : +2 | -2
Pass Ratio : 100%
What is the z-transform of the signal x(n)=[3(2n)-4(3n)]u(n)?
\\(\\frac{3}{1-2z^{-1}}-\\frac{4}{1-3z^{-1}}\\)
\\(\\frac{3}{1-2z^{-1}}-\\frac{4}{1+3z^{-1}}\\)
\\(\\frac{3}{1-2z}-\\frac{4}{1-3z}\\)
None of the mentioned
Explanation:
Let us divide the given x(n) into x1(n)=3(2n)u(n) and x2(n)= 4(3n)u(n)
Question 2
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Pass Ratio : 100%
What is the z-transform of the signal x(n)=δ(n-n0)?
zn0
z-n0
zn-n0
zn+n0
Explanation:
From the definition of z-transform,
Question 3
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If Z{x1(n)}=X1(z) and Z{x2(n)}=X2(z) then Z{x1(n)*x2(n)}=?
X1(z).X2(z)
X1(z)+X2(z)
X1(z)*X2(z)
None of the mentioned
Explanation:
According to the convolution property of z-transform, the z-transform of convolution of two sequences is the product of their respective z-transforms.
Question 4
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If X(z) is the z-transform of the signal x(n), then what is the z-transform of x*(n)?
X(z*)
X*(z)
X*(-z)
X*(z*)
Explanation:
According to the conjugation property of z-transform, we have
Question 5
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If X(z) is the z-transform of the signal x(n) then what is the z-transform of anx(n)?
X(az)
X(az-1)
X(a-1z)
None of the mentioned
Explanation:
We know that from the definition of z-transform
Question 6
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X(z) is the z-transform of the signal x(n), then what is the z-transform of the signal nx(n)?
\\(-z\\frac{dX(z)}{dz}\\)
\\(z\\frac{dX(z)}{dz}\\)
\\(-z^{-1}\\frac{dX(z)}{dz}\\)
\\(z^{-1}\\frac{dX(z)}{dz}\\)
Explanation:
Question 7
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Which of the following justifies the linearity property of z-transform?[x(n)↔X(z)].
x(n)+y(n) ↔ X(z)Y(z)
x(n)+y(n) ↔ X(z)+Y(z)
x(n)y(n) ↔ X(z)+Y(z)
x(n)y(n) ↔ X(z)Y(z)
Explanation:
According to the linearity property of z-transform, if X(z) and Y(z) are the z-transforms of x(n) and y(n) respectively then, the z-transform of x(n)+y(n) is X(z)+Y(z).
Question 8
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Pass Ratio : 100%
What is the z-transform of the signal x(n)=sin(jω0n)u(n)?
\\(\\frac{z^{-1} sin\\omega_0}{1+2z^{-1} cos\\omega_0+z^{-2}}\\)
\\(\\frac{z^{-1} sin\\omega_0}{1-2z^{-1} cos\\omega_0-z^{-2}}\\)
\\(\\frac{z^{-1} cos\\omega_0}{1-2z^{-1} cos\\omega_0+z^{-2}}\\)
\\(\\frac{z^{-1} sin\\omega_0}{1-2z^{-1} cos\\omega_0+z^{-2}}\\)
Explanation:
By Euler’s identity, the given signal x(n) can be written as
Question 9
Marks : +2 | -2
Pass Ratio : 100%
If Z{x1(n)}=X1(z) and Z{x2(n)}=X2(z) then what is the z-transform of correlation between the two signals?
X1(z).X2(z-1)
X1(z).X2(z-1)
X1(z).X2(z)
X1(z).X2(-z)
Explanation:
We know that rx1x2(l)=x1(l)*x2(-l)
Question 10
Marks : +2 | -2
Pass Ratio : 100%
If x(n) is an imaginary sequence, then the z-transform of the real part of the sequence is?
\\(\\frac{1}{2}\\)[X(z)+X*(z*)]
\\(\\frac{1}{2}\\)[X(z)-X*(z*)]
\\(\\frac{1}{2}\\)[X(-z)-X*(z*)]
\\(\\frac{1}{2}\\)[X(-z)+X*(z*)]
Explanation:
If x(N) is an imaginary sequence, then the real part of x(n) is given as