f$)$ A DSP system samples a continuous time signal $\backslash ($ x$($t$)\backslash )$ for $1$ second generating a sequence of $\backslash (\backslash$mathrm$\{$N$\}=4096\backslash )$ samples. If a $4096-$point DFT of the sampled signal is computed,

$($i$)$ What is the frequency spacing $($resolution$)$ in Hertz between the DFT coefficients?

$(1$ Mark$)$

Suppose we are only interested in the DFT samples that correspond to frequencies in the range $\backslash (300\backslash$mathrm$\{$~Hz$\}\backslash$leq f $\backslash$leq $400\backslash$mathrm$\{$~Hz$\}\backslash ),$

$($ii$)$ How many complex multiplications are required to evaluate these values computing the DFT directly?

$(2$ Marks$)$

$($iii$)$ How many complex multiplications are required if a full $\backslash (\backslash$mathrm$\{$N$\}=4096\backslash )-$point FFT algorithm $($Radix$-2,$ DIT$)$ is used?

$(1$ Mark$)$

$($iv$)$ How many input samples would be needed for the FFT algorithm to be more efficient than evaluating the DFT directly?

$(1$ Mark$)$