FFT processor which can only be used once Given an 8 point FFT processor, which can be used only once, compute the DFTs of the sequence.
$$x_1(n)=[1,8,6,7,4,2,3,1]$$
$$x_2(n)=[1,4,3,2,8,7,6,1]$$
 A: Let $y(n)=x_1(n)+jx_2(n)$.
$$\therefore y(n)=[(1+1j), (8+4j), (6+3j), (7+2j), (4+8j), (2+7j), (3+6j), (1+1j)]$$
Now we can calculate the DFT of $y(n)$ by using the single 8 point FFT processor.
$$\begin{align}
\therefore Y(k) & = \sum_{n=0}^{N-1}{y(n)e^{-{2\pi{}jnk\over N}}} \\
& =\begin{bmatrix}
(32 + 32j) \\
\left(-6 - \sqrt{2} +j \left(- 8 \sqrt{2} -10\right)\right) \\
(4 - 2j) \\
(- \sqrt{2} +j \left(- 4 \sqrt{2} -4\right)) \\
(-4 + 4j) \\
(-6 + \sqrt{2} +j \left(-10 + 8 \sqrt{2}\right)) \\
(-12 + 2j) \\
(\sqrt{2} +j \left(-4 + 4 \sqrt{2}\right))
\end{bmatrix}
\end{align}$$
Since DFTs are linear,
$$\begin{align}
Y(k) & =DFT\big(x_1(n)+jx_2(n)\big) \\ 
& =X_1(k)+jX_2(k) \tag{1}\\
\end{align}$$
where $X_1(k)$ and $X_2(k)$ are the DFTs of $x_1(n)$ and $x_2(n)$ respectively.
Taking conjugate on both sides we have,
$$Y^*(k) = X_1^*(k)-jX_2^*(k)$$
where $X^*$ denotes complex conjugation.
Also trivially,
$$Y^*(N-k) = X_1^*(N-k)-jX_2^*(N-k)$$
where $N$ is the length of the DFT, which is $8$ here.
By the property of symmetry, if $x(n)$ consists of real numbers only, then,
$$X(k)=X^*(N-k)$$
where $N$ is the length of the DFT.
$$\therefore Y^*(N-k)=X_1(k)-jX_2(k) \tag{2}$$
From equations (1) & (2), we have,
$$\begin{align}
X_1(k) & ={Y(k)+Y^*(N-k)\over 2}\\
X_2(k) & ={Y(k)-Y^*(N-k)\over 2j}\\
\text{and, } Y^*(N-k) & =\begin{bmatrix}
(32 - 32j) \\
(\sqrt{2} -j \left(-4 + 4 \sqrt{2}\right))\\
(-12 - 2j) \\
(-6 + \sqrt{2} -j \left(-10 + 8 \sqrt{2}\right)) \\
(-4 - 4j) \\
(- \sqrt{2} -j \left(- 4 \sqrt{2} -4\right)) \\
(4 + 2j) \\
\left(-6 - \sqrt{2} -j \left(- 8 \sqrt{2} -10\right)\right)
\end{bmatrix}\\
\therefore X_1(k) & = \begin{bmatrix}
32\\
-3 +j \left(- 6 \sqrt{2} -3\right)\\
-4 - 2j\\
-3 +j \left(- 6 \sqrt{2} + 3\right)\\
-4\\
-3 +j \left(-3 + 6 \sqrt{2}\right)\\
-4 + 2j\\
-3 +j \left(3 + 6 \sqrt{2}\right)
\end{bmatrix}\\
\text{and, }X_2(k) & =\begin{bmatrix}
32\\
-7 - 2 \sqrt{2} +j \left(\sqrt{2} + 3\right)\\
- 8j\\
-7 + 2 \sqrt{2} +j \left(-3 + \sqrt{2}\right)\\
4\\
-7 + 2 \sqrt{2} +j \left(- \sqrt{2} + 3\right)\\
8j\\
-7 - 2 \sqrt{2} +j \left(-3 - \sqrt{2}\right)
\end{bmatrix}
\end{align}$$
