Answer :
The Fourier transform of this sequence does not exist
What is Fourier Transform?
The output of a Fourier transform is a function of frequency and is a mathematical transformation that breaks down functions into frequency components.
To determine the z-transform of a given sequence, we need to find the sum of the sequence's terms multiplied by z raised to the negative of the term's position. For example, given a sequence x[n], the z-transform X(z) is given by:
[tex]X(z) = sum_{n=-infinity}^{infinity} x[n] * z^{-n}[/tex]
To sketch the pole-zero plot of a z-transform, we need to plot the poles (values of z where the z-transform is infinite) and zeros (values of z where the z-transform is zero) in the complex plane. The region of convergence (ROC) is the set of values of z for which the sum defining the z-transform converges.
[tex]x[n] = a^n u[n][/tex], where a is a constant and u[n] is the unit step function[tex](u[n] = 1 for n > = 0[/tex] and [tex]u[n] = 0 for n < 0)[/tex]
The z-transform of this sequence is given by:
[tex]X(z) = sum_{n=0}^{infinity} a^n * z^{-n}= 1 / (1 - a*z^{-1})[/tex]
This z-transform has a pole at z = 1/a and a zero at z = 0. The pole-zero plot consists of a single pole at 1/a in the complex plane. The ROC is the set of values of z such that [tex]|a*z^{-1}| < 1[/tex], which is the region inside the unit circle centered at the origin. The Fourier transform of this sequence exists.
[tex]x[n] = (-1)^n u[n].[/tex]
The z-transform of this sequence is given by:
[tex]X(z) = sum_{n=0}^{infinity} (-1)^n * z^{-n}= 1 / (1 + z^{-1})[/tex]
This z-transform has a pole at z = -1 and a zero at z = 0. The pole-zero plot consists of a single pole at -1 in the complex plane. The ROC is the set of values of z such that [tex]|z^{-1}| < 1[/tex], which is the region inside the unit circle centered at the origin. The Fourier transform of this sequence exists.
[tex]x[n] = n^2 u[n].[/tex]
The z-transform of this sequence is given by:
[tex]X(z) = sum_{n=0}^{infinity} n^2 * z^{-n}= z^{-2} / (1 - z^{-1})^3[/tex]
This z-transform has poles at z = 1, z = 1, and z = 1. The pole-zero plot consists of three poles at the origin in the complex plane. The ROC is the set of values of z such that [tex]|z^{-1}| > 1[/tex], which is the region outside the unit circle centered at the origin.
Hence, The Fourier transform of this sequence does not exist.
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