Discrete-time representations - Signal Processing and Optimization for Transceiver Systems - page 124

P ( j ω)

T

2

T cos θ

ω

π /T

0

(1 − β )π /T (1 + β )π /T

Figure 4.12 . The raised-cosine function P ( jω ) .

The function P ( jω ) can be mathematically expressed as follows:

⎧
⎨

T for | ω |≤ (1 − β ) π/T

T
4 β

β ) π
T

T cos 2 P ( jω )= | ω |− (1 − for (1 − β ) π/T < | ω | < (1 + β ) π/T

⎩

0 for | ω |≥ (1 + β ) π/T.

(4 . 31)

The parameter β is chosen to be in the range 0 1. Figure 4.13 shows the

function P ( jω ) for the two extreme cases β = 0 and β =1.

≤ β ≤

1. For β =0 , the function reduces to an ideal lowpass filter with cutoff
frequency π/T.

2. For β =1 , the function has twice as much bandwidth, and is given by
P ( jω )= cos 2 ( ωT/ 4)

for | ω |≤ 2 π/T

(4 . 32)

0 otherwise.

3. For 0 <β< 1 , the function has intermediate bandwidth as seen from Fig.
4.12.

The name raised cosine comes from the fact that cos 2 θ can be written as
cos 2 θ = 1+cos2 θ
2

(4 . 33)

A plot of this would be the cosine function raised vertically by one, and then
divided by two.

4.4.1 Nyquist property of the raised cosine

The function P ( jω ) has the additional property that

∞

P ( j ( ω + kω s )) = T, for all ω, (4 . 34)

k = −∞