Digital Signal Processing

Properties of Z Transform

Question 1
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 2
Marks : +2 | -2
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 3
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 4
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 5
Marks : +2 | -2
Pass Ratio : 100%
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 6
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 7
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 8
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,
Question 9
Marks : +2 | -2
Pass Ratio : 100%
If x(n) is causal, then \\(\\lim_{z\\rightarrow\\infty}\\) X(z)=?
x(-1)
x(1)
x(0)
Cannot be determined
Explanation:
According to the initial value theorem, X(z)=x(0)+x(1)z-1+x(2)z-2+….
Question 10
Marks : +2 | -2
Pass Ratio : 100%
If X(z) is the z-transform of the signal x(n), then what is the z-transform of the signal x(-n)?
X(-z)
X(z-1)
X-1(z)
None of the mentioned
Explanation:
From the definition of z-transform, we have