We have $$X(k)=4[u(k-2)-u(k)* d(k-3)]$$
I need to find the Fourier transform,$Z$ transform,as well as dhe magnitude and phase spectra.
First of all I think that I need to convert the $u(k)$ and $u(k-2)$ to $d(k)$.Then find $Z$ transform.Then study the region of convergence of the $Z$ transform,to determine if the fourier transform exists.How do I do all these steps?By the way,I dont have any idea about the magnitude/phase
Note :$u(k)* d(k-3)]$ is the convolution of u(k) and d(k-3),not the multiplication of them.
I assume that by $d(k)$ you mean the discrete delta impulse $\delta(k)$. By noting that
$$f(k)*\delta(k-l)=f(k-l)$$
for any sequence $f(k)$, the signal $x(k)$ can be written as
$$x(k)=4[u(k-2)-u(k-3)]=4\delta(k-2)$$
The $\mathcal{Z}$-transform of $X(k)$ is then simply
$$X(z)=4z^{-2}$$
which converges anywhere except for $z=0$, i.e. its region of convergence is $|z|>0$. Since the region of convergence includes the unit circle, the Fourier transform of $x(k)$ exists and is simply given by $X(z)$ with $z=e^{j\omega}$:
$$X(e^{j\omega})=4e^{-2j\omega}$$
The magnitude is $|X(e^{j\omega})|=4$, and the phase is $\arg\{X(e^{j\omega})\}=-2\omega$.