Request for clarification in the paper: stability analysis of GPS carrier tracking loops by phase margin approach

Appendix

Test of NCO and predetection filter transfer functions in z-domain

This test applies a simple arbitrary input sequence into the NCO and predetection filter transfer functions using the directly derived equations of the hardware in the z-domain, then applies the same input sequence into the G(z) implementation, and compares both outputs to verify they give the same result.

The arbitrary input sequence for $\omega_{i}$ is shown in the second column in the Table below. It is assumed that the starting phase and frequency both are zero for all past times.

Hardware equations of NCO and predetection filter data flow

Iteration number “I”	Assumed NCO input at t i $(\omega_{i} )$	NCO output at t i , (θ i ) using Eq. 3	Predetection filter output at t i , $(\emptyset {\text{out}}_{i} )$ using Eq. 6
0	0	0	0
1	1	0	0
2	1	T	0.5T
3	0	2T	1.5T
4	0	2T	2T
5	1	2T	2T
6	2	3T	2.5T
7	3	5T	4T
8	2	8T	6.5T
9	1	10T	9T
10	0	11T	10.5T

Now, do the same for G(z). An implementation of G(z) is as follows:

Equations for this data flow are

$$ x_{i} = x_{i - 1} + T\omega_{i - 1} $$

(16)

and

$$ \emptyset {\text{out}}_{i} = \frac{{x_{i} + x_{i - 1} }}{2} $$

(17)

Domain data flow of NCO and Predetection filter

Iteration number “I”	Assumed NCO input at t i $(\omega_{i} )$	NCO output at t i , $(x_{i} )$ using Eq. 16	Predetection filter output at t i , $(\emptyset {\text{out}}_{i} )$ using Eq. 17
0	0	0	0
1	1	0	0
2	1	T	0.5T
3	0	2T	1.5T
4	0	2T	2T
5	1	2T	2T
6	2	3T	2.5T
7	3	5T	4T
8	2	8T	6.5T
9	1	10T	9T
10	0	11T	10.5T

Note that these tables have identical entries throughout. Thus, in the z-domain implementation above, $x_{i}$ is identical to the NCO output, $\theta_{i}$. Likewise the predetection filter output, $\emptyset {\text{out}}_{i}$ is identical in both tables. Also note the equivalence between Eqs. 16 and 3, as well as the equivalence between Eqs. 17 and 6.

This also means that one could have broken down G(z) as follows:

$$ {\text{NCO}}\left( z \right) = \frac{{Tz^{ - 1} }}{{1 - z^{ - 1} }} $$

(18)

and

$$ {\text{Pre}}\;{\text{Detect}}\left( z \right) = \frac{{1 + z^{ - 1} }}{2} $$

(19)

giving

$$ G\left( z \right) = {\text{NCO}}\left( z \right){\text{Pre}}\;{\text{Detect}}(z) = \frac{T}{2}\frac{{(1 + z^{ - 1} )z^{ - 1} }}{{1 - z^{ - 1} }} $$

(20)