Selected Topics in Applied Signal Processing - Digital Signal Processing: An Introduction with MATLAB and Applications - page 149

The sampling interval of the DCO at the kth sampling instant is given by:

T ð k Þ¼ T o y ð k 1 Þ ð 3 : 27 Þ

(see Figs. 3.12 and 3.14 ).
Note that the kth output sample of the DF, y(k), effectively determines the
sampling period, T(k ? 1). The kth sampling instant t(k) is given by the cumu-
lative sum of all sampling periods:

t ð k Þ¼ t ð 0 Þþ X

k

T ð i Þ:

0

For simplicity it is assumed that the initial time instant is zero, i.e., t(0) = 0.
Hence, the kth sampling instant is:

t ð k Þ¼ kT o X

k 1

y ð i Þ

ð 3 : 28 Þ

0

From Eqs. 3.26 and 3.28 , the input signal phase at the kth sampling instant can be
written as follows:

/ ð k Þ¼ h ð k Þ x o X

k 1

y ð i Þ: ð 3 : 29 Þ

0

Hence, at the (k ? 1)th sampling instant, the input signal phase is given by:

/ ð k þ 1 Þ¼ h ð k þ 1 Þ x o X

k

y ð i Þ: ð 3 : 30 Þ

0

The phase equations ( 3.29 ) and ( 3.30 ) will determine the system difference
equation for the SDPLL of any order.

3.4.2.3 The First-Order Noise-Free SDPLL

The first-order SDPLL is widely used, although it gives a non-zero steady-state
phase error. Its digital filter transfer function is H(z) = G 1 (constant), hence its
output is given by:

y ð k Þ¼ G 1 x ð k Þ¼ G 1 A sin ½ / ð k Þ ð 3 : 31 Þ

From Eqs. 3.29 and 3.30 the following phase difference equation is obtained:

/ ð k þ 1 Þ / ð k Þ¼ h ð k þ 1 Þ h ð k Þ x o y ð k Þ: ð 3 : 32 Þ