# Approximation in orthogonal functions with pulse shaping

1. Dec 14, 2012

### EmilyRuck

Hello!
I have tried for a whole afternoon to solve this problem but I didn't succeed.
Let $\cos(2 \pi (f_0 + i/T_N) t + \phi_i)$ and $\cos(2 \pi (f_0 + j/T_N) t + \phi_j)$ be two quasi-orthogonal functions:

$\int_{0}^{T_N} \cos(2 \pi (f_0 + i/T_N) t + \phi_i) \cos(2 \pi (f_0 + j/T_N) t + \phi_j) dt \simeq \displaystyle \frac{T_N}{2} \delta (i - j)$

where $\delta()$ is the Kroneker Delta, and the approximation is good if $f_0 T_N \gg 1$. $i,j = 0, 1, \ldots, N - 1$.

This is the theory of OFDM (Orthogonal Frequency Division Multiplexing) and it is partially explained in Andrea Goldsmith, Wireless Communications (Cambridge University Press) with these simbols and this notation.

$\int_{0}^{T_N} g(t) \cos(2 \pi (f_0 + i/T_N) t + \phi_i) g(t) \cos(2 \pi (f_0 + j/T_N) t + \phi_j) dt \simeq \displaystyle \frac{T_N}{2} \delta (i - j)$???

$g(t)$ is called the pulse shaping function and it could have a raised-cosine form in $[0, T_N]$ (a sort of $sin(x)/x$). I know only that the highest frequency component of $g(t)$ is about $B_N = 1/T_N$, with $B_N \ll f_0$.

2. Relevant equations
To demonstrate the first approximation, the book uses the Werner formulas and writes:

$\int_{0}^{T_N} \cos(2 \pi (f_0 + i/T_N) t + \phi_i) \cos(2 \pi (f_0 + j/T_N) t + \phi_j) dt =$
$=\displaystyle \frac{1}{2} \int_{0}^{T_N} \cos(2 \pi (2 f_0 + (i + j)/T_N) t + \phi_i + \phi_j) + \displaystyle \frac{1}{2} \int_{0}^{T_N} \cos(2 \pi ((i - j)/T_N) t + \phi_i - \phi_j) dt \simeq$
$\simeq \displaystyle \frac{1}{2} \int_{0}^{T_N} \cos(2 \pi ((i - j)/T_N) t + \phi_i - \phi_j) dt = \frac{T_N}{2} \delta (i - j)$

and I obviously agree fot $f_0 T_N \gg 1$. But the book says nothing about the insertion of the $g(t)$ function.

3. The attempt at a solution

I tried to follow the same procedure:

$\int_{0}^{T_N} g^2 (t) \cos(2 \pi (f_0 + i/T_N) t + \phi_i) \cos(2 \pi (f_0 + j/T_N) t + \phi_j) dt =$
$=\displaystyle \frac{1}{2} \int_{0}^{T_N} g^2 (t) \cos(2 \pi (2 f_0 + (i + j)/T_N) t + \phi_i + \phi_j) + \displaystyle \frac{1}{2} \int_{0}^{T_N} g^2 (t) \cos(2 \pi ((i - j)/T_N) t + \phi_i - \phi_j) dt$

For $i = j$ I have no troubles. But for $i \neq j$:
the first integral is almost zero again because $B_N \ll 2 f_0 + (i + j)/T_N$ and so $g^2(t)$ is almost constant over each period of the integrating function; but what about the second integral? This simplifying hypothesis is not true anymore!
In the second integral the cosine has a very small frequency and the $g^2(t)$ is variable over the period $T_N/(i-j)$.
I hope this was the right section to post.