# pulse shaping

In digital modulation, bandwidth is a function of the pulse shape g(t). Pulse shaping makes the signal fit inside a certain bandwidth. We may need to this because at high rates of modulation, there could be inter-symbol-interference from finite impulse response signals in the time domain.

If a pulse shaper \(g(t)\) is included in a basis function, then the bandwidth of the signal is exactly the bandwidth of \(g(t)\). The bandwidth is \(K/T\) where \(K\) is a constant and \(T\) is duration of the symbol.

Typically must preserve orthonormality when multiplied by a basis function

\begin{equation}
\int_0^{T} g^2(t) cos^2(2\pi f_c t) dt = 1
\end{equation}

\begin{equation}
\int_0^{T} g^2(t) cos(2\pi f_c t) sin(2\pi f_c t) dt = 1
\end{equation}