The Z-transform of a discrete signal $$x[n]$$ is

$$X(z) = {{4z} \over {(z - {1 \over 5})(z - {2 \over 3})(z - 3)}}$$ with $$ROC = R$$.

Which one of the following statements is true?

If the Z-transform of a finite-duration discrete-time signal $x[n]$ is $X(z)$, then the Z-transform of the signal $y[n] = x[2n]$ is

The discrete-time Fourier transform of a signal $$x[n]$$ is $$X(\Omega ) = (1 + \cos \Omega ){e^{ - j\Omega }}$$. Consider that $${x_p}[n]$$ is a periodic signal of period N = 5 such that

$${x_p}[n] = x[n]$$, for $$n = 0,1,2$$

= 0, for $$n = 3,4$$

Note that $${x_p}[n] = \sum\nolimits\limits_{k = 0}^{n - 1} {{a_k}{e^{j{{2\pi } \over N}kn}}} $$. The magnitude of the Fourier series coeffiient $$a_3$$ is __________ (Round off to 3 decimal places).