Digital Signal Processing

Properties of Z Transform

Question 1
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
Question 2
Marks : +2 | -2
Pass Ratio : 100%
What is the z-transform of the signal x(n)=an(sinω0n)u(n)?
\\(\\frac{az^{-1} sin\\omega_0}{1+2 az^{-1} cos\\omega_0+a^2 z^{-2}}\\)
\\(\\frac{az^{-1} sin\\omega_0}{1-2 az^{-1} cos\\omega_0- a^2 z^{-2}}\\)
\\(\\frac{(az)^{-1} cos\\omega_0}{1-2 az^{-1} cos\\omega_0+a^2 z^{-2}}\\)
\\(\\frac{az^{-1} sin\\omega_0}{1-2 az^{-1} cos\\omega_0+a^2 z^{-2}}\\)
Explanation:
Question 3
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 4
Marks : +2 | -2
Pass Ratio : 100%
What is the z-transform of the signal defined as x(n)=u(n)-u(n-N)?
\\(\\frac{1+z^N}{1+z^{-1}}\\)
\\(\\frac{1-z^N}{1+z^{-1}}\\)
\\(\\frac{1+z^{-N}}{1+z^{-1}}\\)
\\(\\frac{1-z^{-N}}{1-z^{-1}}\\)
Explanation:
Question 5
Marks : +2 | -2
Pass Ratio : 100%
If the ROC of X(z) is r1<|z|<r2, then what is the ROC of X(a-1z)?
|a|r1<|z|<|a|r2
|a|r1>|z|>|a|r2
|a|r1<|z|>|a|r2
|a|r1>|z|<|a|r2
Explanation:
Given ROC of X(z) is r1<|z|<r2
Question 6
Marks : +2 | -2
Pass Ratio : 100%
What is the convolution x(n) of the signals x1(n)={1,-2,1} and x2(n)={1,1,1,1,1,1}?
{1,1,0,0,0,0,1,1}
{-1,-1,0,0,0,0,-1,-1}
{-1,1,0,0,0,0,1,-1}
{1,-1,0,0,0,0,-1,1}
Explanation:
Question 7
Marks : +2 | -2
Pass Ratio : 100%
What is the signal whose z-transform is given as X(z)=\\(\\frac{1}{2Ï€j} \\oint X_1 (v) X_2 (\\frac{z}{v})v^{-1} dv\\)?
x1(n)*x2(n)
x1(n)*x2(-n)
x1(n).x2(n)
x1(n)*x2*(n)
Explanation:
From the convolution property in z-domain we have,
Question 8
Marks : +2 | -2
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 9
Marks : +2 | -2
Pass Ratio : 100%
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 10
Marks : +2 | -2
Pass Ratio : 100%
If Z{x(n)}=X(z) and the poles of X(z) are all inside the unit circle, then the final value of x(n) as \\(n\\rightarrow\\infty\\) is given by i.e., \\(\\lim_{n\\rightarrow\\infty}\\)x(n)=?
\\(\\lim_{z \\rightarrow 1} [(z-1) ⁡ X(z)] \\)
\\(\\lim_{z \\rightarrow 0} [(z-1) ⁡ X(z)] \\)
\\(\\lim_{z \\rightarrow -1} [(z-1) X(z)] \\)
\\(\\lim_{z \\rightarrow 1} [(z+1) ⁡ X(z)] \\)
Explanation:
According to the Final Value theorem of z-transform we have,