B. P. Lathi, Zhi Ding - Modern Digital and Analog Communication Systems-Oxford University Press (2009)

7.2 Line Coding 333
and from Eq. (7.3b)
. 1
Ro = hm - " 2
L
a N➔oo N k
k
(7.5)
Observe that the summation is over N pulses. Hence, Ro is the time average of the square of
the pulse amplitudes a k . Using our time average notation, we can express Ro as
. 1 " 2 2
N➔oc N
Ro = hm - L
a k
= a k
k
(7.6)
We also know that R ;; (r) is an even function of r [see Eq. (3.83)). Hence, Eq. (7.3) can be
expressed as
Ro l r l
R ;; (r) =- ( 1 --)
ET b E
l r l < E (7.7)
This is a triangular pulse of height Ro/ET b and width 2E centered at r = 0 (Fig. 7.5d).
This is expected because as r increases beyond E, there is no overlap between the delayed
signal x(t - r) and .x(t); hence, R ;; (r) = 0, as seen from Fig. 7.5d. But as we increase r
further, we find that the kth pulse of .x(t - r) will start overlapping the (k + l)th pulse of .x(t)
as r approaches T b (Fig. 7 .5c ). Repeating the earlier argument, we see that R ;; ( r) will have
another triangular pulse of width 2E centered at r = T b and of height R1/ET b where
Observe that R1 is obtained by multiplying every pulse strength (a k ) by the strength of its
immediate neighbor (ak+1 ), adding all these products, and then dividing by the total number
of pulses. This is clearly the time average (mean) of the product a k a k+I and is, in our notation,
Asimilar thing happens around r = 2T b , 3T b , .... Hence, R;; (r) consists of a
sequence of triangular pulses of width 2E centered at r = 0, ±h, ±2T b , .... The height of
the pulses centered at ±nT b is R n / ET b , where
R n is essentially the discrete autocorrelation function of the line code symbols { a k }.