Pulse shaping

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. This pulse shaping will make the spectrum smooth, leading to a time limited signal again. Usually the transmitted symbols are represented as a time sequence of dirac delta pulses multiplied with the symbol. This is the formal transition from the digital to the analog domain. At this point, the bandwidth of the signal is unlimited. This theoretical signal is then filtered with the pulse shaping filter, producing the transmitted signal. If the pulse shaping filter is a rectangular in the time domain (like this is usually done when drawing it), this would lead to an unlimited spectrum.

Not every filter can be used as a pulse shaping filter. The filter itself must not introduce intersymbol interference — it needs to satisfy certain criteria. The Nyquist ISI criterion is a commonly used criterion for evaluation, because it relates the frequency spectrum of the transmitter signal to intersymbol interference.

Examples of pulse shaping filters that are commonly found in communication systems are:

Sender side pulse shaping is often combined with a receiver side matched filter to achieve optimum tolerance for noise in the system. In this case the pulse shaping is equally distributed between the sender and receiver filters. The filters' amplitude responses are thus pointwise square roots of the system filters.

Other approaches that eliminate complex pulse shaping filters have been invented. In OFDM, the carriers are modulated so slowly that each carrier is virtually unaffected by the bandwidth limitation of the channel.

It is also called as Boxcar filter as its frequency domain equivalent is a rectangular shape. Theoretically the best pulse shaping filter would be the sinc filter, but it cannot be implemented precisely. It is a non-causal filter with relatively slowly decaying tails. 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. Raised-cosine filters are practical to implement and they are in wide use. They have a configurable excess bandwidth, so communication systems can choose a trade off between a simpler filter and spectral efficiency.