Pulse shaping

Pulse shaping is particularly important in RF communication for fitting the signal within a certain frequency band and is typically applied after line coding and modulation.

Transmitting a signal at high modulation rate through a band-limited channel can create intersymbol interference.

As a practical tool to determine ISI, one uses the Eye pattern, that visualizes typical effects of the channel and the synchronization/frequency stability.

The signal's spectrum is determined by the modulation scheme and data rate used by the transmitter, but can be modified with a pulse shaping filter.

Its Fourier transform is of the form sin(x)/x, and has significant signal power at frequencies higher than symbol rate.

To understand this completely, one needs the Hilbert transform, which induces a direction by the convolution with the Cauchy Kernel.

It is also problematic from a synchronisation point of view as any phase error results in steeply increasing intersymbol interference.

Raised-cosine is similar to sinc, with the tradeoff of smaller sidelobes for a slightly larger spectral width.

They have a configurable excess bandwidth, so communication systems can choose a trade off between a simpler filter and spectral efficiency.