Digital communications
Pulse shaping
fi
lters
Up to now, we have emphasized the fact that digital constellations have a
fi
nite
number of allowed states, but we have not discussed how the signal transitions
from one allowed state to the next. The simplest method would be to jump as
quickly as possible from state to state. Although simple, this method turns out to
be undesirable in most cases, because it creates spurious energy at large offsets
from carrier. This is important because the RF spectrum is a limited resource
that has to be shared by many people cooperatively at the same time. Lots of
people are trying to transmit data simultaneously. Without cooperation, all of
these transmissions would interfere with one another and nobody would be able
to communicate.
One of the most common means of sharing the RF spectrum is with frequency
division multiple access (FDMA). In this scheme the RF spectrum is divided into
many small frequency bands. Each user is assigned one band and may transmit
at will as long as his transmission is con
fi
ned to his assigned band. If this basic
rule is obeyed, everyone can communicate simultaneously without interference.
Unfortunately, transmissions that jump from symbol to symbol as quickly as
possible invariably violate this rule. Thus, almost all communication protocols
stipulate pulse shaping
fi
lters to overcome this problem.
Pulse shaping
fi
lters limit the bandwidth of a digital transmission by converting
the sharp transitions into gradual transitions with much lower bandwidth. They
are essentially low pass
fi
lters, which
fi
lter out all the high frequency components
of the sharp transitions.
Three different pulse shaping
fi
lters are commonly used in digital communications:
the raised cosine
fi
lter, the root-raised cosine
fi
lter, and the Gaussian
fi
lter. Each
addresses the problem of ISI differently.
Raised cosine.
The
fi
rst strategy for dealing with ISI is to remove it with a
cleverly designed
fi
lter that has zero intersymbol interference. The raised cosine
fi
lter meets this criterion. It is de
fi
ned by the following frequency response:
where f is the frequency, T is the symbol period and
α
is a dimensionless
parameter controlling the excess bandwidth of the
fi
lter. When
α
= 0, the
fi
lter
approximates a brick wall. When
α
= 1.0 the
fi
lter has 100 % excess bandwidth
over the brick wall
fi
lter, i.e. it is twice as wide.
The impulse response of the raised cosine
fi
lter is given by
TSG4100A Series RF Signal Generators User Manual
41
