Question

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Drill Problem 6.2 Show that for positive frequencies, the area under the normalized raised-cosine curve of P /VE/2B.) versus f/Bo is equal to unity for all values of the roll- off factor in the range 0 Sas 1. A similar statement holds for negative frequencies.

Answer 1

The pulse spectrum P(f) is, VE 2B, VE 1+ cos fi <\5|<2B, - fi P(f)= 2(B, - fi) 4B 2B, - f <|f| 0, 1, P(f) JE fi s\$|<2B, - fi 1+cos | 2(B,- fi)] 2B, - f <|f| 2B, 0, Here, a =1- B. For a =0, %3! fi 0=1- B. f=1 B. fi = B, Now,

0<|f\SB, [1, P(f) VE 2B, B, <\f< B, 1+cos !! 2(B, - fi) B, <\1 0, 0. 0<|1\< B, B, <f| P(f) _ [1. VE 2B, 0, 1, P(f) VE 0< <1 0, Bo 2B, The plot of the normalized raised-cosine curve for a = 0 is, %3D P(f) VE 2B, 1 -1 B. VI

Hence, the normalized raised-cosine curve reduces to idealized Nyquist channel. For positive frequencies, the area under the curve is 1. A =1x1 =1 For non-zero values of a in the range 0<a<1, the normalized raised-cosine curve is as shown below. P(f) VE 2B. a = 0.5 0.5 -2 -1 1 B.

The maximum value of the normalized raised-cosine curve P(f) VE = 0.5 is 1 and it exhibits odd symmetry about the axis 2B, as shown below. P(f) VE 2B a = 0 a = 0.5 a = 1 P(f) = 0.5 VE 2B, 1 Bo

Hence, the area under the normalized raised-cosine curve is equal to unity for positive frequencies. Therefore, the area under the normalized raised-cosine curve P(f) is equal to unity for all values of the Bo of versus VE 2B, roll- off factor in the range 0sasl. A similar statement holds for negative frequencies because the normalized raised-cosine curve is symmetric about y- axis. So, the curve covers same area on positive and negative frequencies.