Question

Let a sequence and let denote its DFT $X=\left \{ X_{k} \right \}_{k=0}^{N-1}$ . We now define two sequences in the following way:

= {n}

$s=\left \{ s_{n} \right \}_{n=0}^{2N-1}=(x_{0},x_{1},...,x_{N-1},x_{0},x_{1},...,x_{N-1})$ ,

$y=\left \{ y_{n} \right \}_{n=0}^{2N-1}=(x_{0},x_{1},...,x_{N-1},\underbrace{0,0,...,0})$ N zeros

Let denote $S=\left \{ S_{k} \right \}_{k=0}^{2N-1}$ and $Y=\left \{ Y_{k} \right \}_{k=0}^{2N-1}$ ? their respective DFT.

1. Show that $S{_{k}}=\left\{\begin{matrix} 2X_{k/2} &if k is even \\ 0 & if k is odd \end{matrix}\right.$

2. Show that, for

$Y_{2m}=X_{_{m}}$

3. Using the fact that for

$y_{n}=s_{n}w_{n}$ where $w=({\underbrace{1,1,...,1,}\underbrace{0,0,...0})$ ?N times each of ones and zeros

show that for    $Y_{2m+1}=\frac{1}{N}\sum_{p=0}^{N-1}X_{p}W_{mod}(2(m-p)+1,2N)$

Answer 1

2N-) 2N nco ㄚ