How to demonstrate the orthogonality of OFDM subcarriers? Assume if we have $N$ OFDM subcarriers represented  by results of the inverse FFT of $N$ data symbols $\mathbf x$. As I know, the subcarriers of OFDM should be orthogonal. It means that
$$X(n)X(n+1) = 0\quad\text{where}\quad n = 0,\ldots,N-1$$
My question is that I can't get that right when doing it in MATLAB. For example suppose that $N=4$ and
$$\mathbf x = \begin{bmatrix}
0.7+0.7i\\ 
0.7-0.7i\\ 
-0.7+0.7i\\ 
0.7+0.7i\end{bmatrix}$$
is a data symbols gotten after QAM modulation.  The OFDM $N\times 1$ subcarriers are gotten by iFFT of the data symbol as below:
X = ifft(x)*sqrt(4); 

It means that
$$X = \begin{bmatrix}
0.7+ 0.7i\\ 
1.4 + 0.0i\\
-0.7 + 0.7i\\
0.0 + 0.0i\end{bmatrix}$$
Then,
$$\begin{align}
X(1)\cdot X(2) &amp;= (0.7000 + 0.7000i)\cdot(1.4000 + 0.0000i)\\
&amp; = 0.9800 + 0.9800i
\end{align}$$
If subcarriers are orthogonal, that wil be zero.
Thus, my question, how can I demonstrate the otrhognality of OFDM subcarriers ?
Orthogonality is defined as &quot;the inner product of two vectors equals zero&quot;.
Now, in OFDM, the transmit vector for a single subcarrier is exactly one row vector $\mathbf D_k$ of the DFT Matrix $\mathbf D$, multiplied by the complex value of a symbol $c_k$, i.e. $c_K \mathbf D_K$.
Two different subcarriers $k, l, k\ne l$ hence have the inner product $\langle c_k\mathbf D_k,c_l\mathbf D_l\rangle$; inner products are linear things, hence that's
\begin{align}
\langle c_k\mathbf D_k,c_l\mathbf D_l\rangle &amp;= c_kc_l \langle \mathbf D_k,\mathbf D_l\rangle\\
&amp;= c_kc_l \begin{cases}0&amp;k\ne l\\\|\mathbf D_k\| &amp; k = l\end{cases} &amp;\text{q.e.d.},
\end{align}
because the DFT matrix is unitary.