EE445S RealTime Digital Signal Processing Laboratory
Handout G: Raised Cosine Pulse
Section 7.5, pp. 431434,
Simon Haykin, Communication Systems, 4th ed.
We may overcome the practical difficulties encounted with the ideal Nyquist
channel by extending the bandwidth from the minimum value to
an adjustable value between and .
We now specify the frequency function to satisfy a condition
more elaborate than that for the ideal Nyquist channel; specifically,
we retain three terms of (7.53) and restrict the frequency band of
interest to , as shown by

(1) 
We may devise several bandlimited functions to satisfy
(1).
A particular form of that embodies many desirable features
is provided by a raised cosine spectrum.
This frequency characteristic consists of a flat portion and a rolloff
portion that has a sinusoidal form, as follows:

(2) 
The frequency parameter and bandwidth are related by

(3) 
The parameter is called the rolloff factor; it indicates the
excess bandwidth over the ideal solution, .
Specifically, the transmission bandwidth is defined by
.
The frequency response , normalized by multiplying it by ,
is shown plotted in Fig. 1 for three values
of , namely, 0, 0.5, and 1.
We see that for or 1, the function cuts off
gradually as compared with the ideal Nyquist channel (i.e., )
and is therefore easier to implement in practice.
Also the function exhibits odd symmetry with respect to the
Nyquist bandwidth , making it possible to satisfy the condition of
(1).
The time response is the inverse Fourier transform of the
function .
Hence, using the defined in (2),
we obtain the result (see Problem 7.9)

(4) 
which is shown plotted in Fig. 2 for
, 0.5, and 1.
The function consists of the product of two factors:
the factor
characterizing the ideal Nyquist channel
and a second factor that decreases as for large .
The first factor ensures zero crossings of at the desired
sampling instants of time with an integer (positive
and negative).
The second factor reduces the tails of the pulse considerably below
that obtained from the ideal Nyquist channel, so that the transmission
of binary waves using such pulses is relatively insensitive to sampling
time errors.
In fact, for , we have the most gradual rolloff in that
the amplitudes of the oscillatory tails of are smallest.
Thus, the amount of intersymbol interference resulting from timing
error decreases as the rolloff factor is increased from zero to unity.
Figure 1:
Frequency response for the raised cosine function.

The special case with (i.e., ) is known as the
fullcosine rolloff characteristic, for which the frequency
response of (2) simplifies to
Correspondingly, the time response simplifies to

(5) 
The time response exhibits two interesting properties:
 At
, we have ; that is,
the pulse width measured at half amplitude is exactly equal
to the bit duration .
 There are zero crossings at
, ,...
in addition to the usual zero crossings at the sampling times
These two properties are extremely useful in extracting a timing signal from
the received signal for the purpose of synchronization.
However, the price paid for this desirable property is the use of a
channel bandwidth double that required for the ideal Nyquist channel
corresponding to .
Figure 2:
Time response for the raised cosine function.

Placeholder  please ignore.
Brian L. Evans