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The RC filter frequency response is given as
(2.2)
which can also be written as
(2.3)
where   the roll-off factor, BW  the Nyquist bandwidth, and fS  symbol rate. The roll-off factor
 is also used to represent the excess bandwidth with respect to the theoretical minimum BW. For
example,   0.5 can be interpreted to mean that the filter bandwidth is 50% larger than the Nyquist
bandwidth BW; in other words, the filter bandwidth  Nyquist BW(1  ).
The RC frequency responses shown in Fig. 2.4 correspond to roll-off factors of   0 (ideal brick
wall),   0.1,   0.5, and   1.0. As mentioned, the roll-off factor not only defines the response
in the transition region, but also defines the BW expansion beyond what Nyquist defined. For example,
with   0.5, the occupied BW is 1.5 times the Nyquist bandwidth.
H(w)  f 1 , 0  w 
p
Ts
(1  a)
cos 2a Ts
4a cw 
p(1  a)
Ts db , p
Ts
(1  a)  w 
p
Ts
(1  a)
0 , w 
p
Ts
(1  a)
H( f )  e 1 , 0  f  BW  fS
1
2 e1  sin c p
2aa f
BW
 1bd f , BW  fS  f  BW  fS
0 , f  BW  fS
MODULATION THEORY 53
Raised Cosine (RC) Frequency Response, H( f )
0
1
0 0.1
0.1
0.2
0.2
0.3
0.3
0.4
0.4
0.5
0.5
0.6
0.6
0.7
0.7
0.


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