Collaboration diagram for Tracking Loops:

Classes
struct	comp_tl_state_t
struct	simple_lf_state_t
struct	simple_tl_state_t
Functions
void	calc_loop_gains (float bw, float zeta, float k, float loop_freq, float pgain, float igain)
void	comp_tl_init (comp_tl_state_t *s, float loop_freq, float code_freq, float code_bw, float code_zeta, float code_k, float carr_freq, float carr_bw, float carr_zeta, float carr_k, float tau, float cpc, u32 sched)
void	comp_tl_update (comp_tl_state_t *s, correlation_t cs[3])
float	costas_discriminator (float I, float Q)
float	dll_discriminator (correlation_t cs[3])
void	simple_lf_init (simple_lf_state_t *s, float y0, float pgain, float igain)
float	simple_lf_update (simple_lf_state_t *s, float error)
void	simple_tl_init (simple_tl_state_t *s, float loop_freq, float code_freq, float code_bw, float code_zeta, float code_k, float carr_freq, float carr_bw, float carr_zeta, float carr_k)
void	simple_tl_update (simple_tl_state_t *s, correlation_t cs[3])

Detailed Description

Functions used by the tracking loops.

Function Documentation

void calc_loop_gains	(	float	bw,
		float	zeta,
		float	k,
		float	loop_freq,
		float *	pgain,
		float *	igain
	)

Calculate coefficients for a 2nd order digital PLL / DLL loop filter.

A second order digital PLL consists of a first-order filter and a Numerically Controlled Oscillator (NCO). A linearised model of a digital second order PLL is shown below:

Linearised digital second order PLL model.

Where $K_d$ is the discriminator gain and $F[z]$ and $N[z]$ are the loop filter and NCO transfer functions. The NCO is essentially an accumulator and hence has a transfer function:

$N[z] = \frac{K_0 z^{-1}}{1 - z^{-1}}$

The first-order loop filter is shown below:

Digital loop filter block diagram.

and has transfer function:

$F[z] = \frac{(k_p+k_i) - k_p z^{-1}}{1 - z^{-1}}$

It is useful to be able to calculate the loop filter coefficients, $k_p$ and $k_i$ by comparison to the parameters used in analog PLL design. By comparison between the digital and analog PLL transfer functions we can show the digital loop filter coefficients are related to the analog PLL natural frequency, $\omega_n$ and damping ratio, $\zeta$ by:

$k_p = \frac{1}{k} \frac{8 \zeta \omega_n T} {4 + 4 \zeta \omega_n T + \omega_n^2 T^2}$

$k_i = \frac{1}{k} \frac{4 \omega_n^2 T^2} {4 + 4 \zeta \omega_n T + \omega_n^2 T^2}$

Where $T$ is the loop update period and the overall loop gain, $k$ , is the product of the NCO and discriminator gains, $ k = K_0 K_d $ . The natural frequency is related to the loop noise bandwidth, $B_L$ by the following relationship:

$\omega_n = \frac{8 \zeta B_L}{4 \zeta^2 + 1}$

These coefficients are applicable to both the Carrier phase Costas loop and the Code phase DLL.

References:

Performance analysis of an all-digital BPSK direct-sequence spread-spectrum IF receiver architecture. B-Y. Chung, C. Chien, H. Samueli, and R. Jain. IEEE Journal on Selected Areas in Communications, 11:1096–1107, 1993.

Todo:: This math is all wrong, these slides show the analysis we want: http://www.compdsp.com/presentations/Jacobsen/abineau_dpll_analysis.pdf

Parameters:

bw	The loop noise bandwidth, .
zeta	The damping ratio, $\zeta$ .
k	The loop gain, .
loop_freq	The loop update frequency, .
pgain	Where to store the calculated proportional gain, .
igain	Where to store the calculated integral gain, .

Definition at line 100 of file track.c.

void comp_tl_init	(	comp_tl_state_t *	s,
		float	loop_freq,
		float	code_freq,
		float	code_bw,
		float	code_zeta,
		float	code_k,
		float	carr_freq,
		float	carr_bw,
		float	carr_zeta,
		float	carr_k,
		float	tau,
		float	cpc,
		u32	sched
	)

Initialise a code/carrier phase complimentary filter tracking loop.

For a full description of the loop filter parameters, see calc_loop_gains().

This filter implements gain scheduling. Before `sched` iterations of the loop filter the behaviour will be identical to the simple loop filter. After `sched` iterations, carrier phase information will be used in the code tracking loop.

Note, this filter requires that the code and carrier frequencies are expressed as a difference from the nominal frequncy (e.g. 1.023MHz nominal GPS C/A code phase rate, 4.092MHz IF for the carrier).

Parameters:

s	The tracking loop state struct to initialise.
code_freq	The initial code phase rate (i.e. frequency) difference from nominal.
code_bw	The code tracking loop noise bandwidth.
code_zeta	The code tracking loop damping ratio.
code_k	The code tracking loop gain.
carr_freq	The initial carrier frequency difference from nominal, i.e. Doppler shift.
carr_bw	The carrier tracking loop noise bandwidth.
carr_zeta	The carrier tracking loop damping ratio.
carr_k	The carrier tracking loop gain.
tau	The complimentary filter cross-over frequency.
cpc	The number of carrier cycles per complete code, or equivalently the ratio of the carrier frequency to the nominal code frequency.
sched	The gain scheduling count.

Definition at line 301 of file track.c.

void comp_tl_update	(	comp_tl_state_t *	s,
		correlation_t	cs[3]
	)

Update step for a code/carrier phase complimentary filter tracking loop.

The tracking loop output variables, i.e. code and carrier frequencies can be read out directly from the state struct.

Todo:: Write proper documentation with math and diagram.

Parameters:

s	The tracking loop state struct.
cs	An array [E, P, L] of correlation_t structs for the Early, Prompt and Late correlations.

Definition at line 337 of file track.c.

float costas_discriminator	(	float	I,
		float	Q
	)

Phase discriminator for a Costas loop.

Costas loop block diagram.

Implements the $\tan^{-1}$ Costas loop discriminator.

$\varepsilon_k = \tan^{-1} \left(\frac{I_k}{Q_k}\right)$

References:

Understanding GPS: Principles and Applications. Elliott D. Kaplan. Artech House, 1996.

Todo:: Fix potential divide by zero if Q is zero.

Parameters:

I	The prompt in-phase correlation, .
Q	The prompt quadrature correlation, .

Returns:: The discriminator value, $\varepsilon_k$ .

Definition at line 138 of file track.c.

float dll_discriminator ( correlation_t cs[3] )

Normalised non-coherent early-minus-late envelope discriminator.

Implements the normalised non-coherent early-minus-late envelope DLL discriminator.

$\varepsilon_k = \frac{1}{2} \frac{E - L}{E + L}$

where:

$E = \sqrt{I^2_E + Q^2_E}$

$L = \sqrt{I^2_L + Q^2_L}$

References:

Understanding GPS: Principles and Applications. Elliott D. Kaplan. Artech House, 1996.

Parameters:

cs	An array [E, P, L] of correlation_t structs for the Early, Prompt and Late correlations.

Returns:: The discriminator value, $\varepsilon_k$ .

Definition at line 169 of file track.c.

void simple_lf_init	(	simple_lf_state_t *	s,
		float	y0,
		float	pgain,
		float	igain
	)

Initialise a simple first-order loop filter. The gains can be calculated using calc_loop_gains().

Parameters:

s	The loop filter state struct to initialise.
y0	The initial value of the output variable, .
pgain	The proportional gain, .
igain	The integral gain, .

Definition at line 185 of file track.c.

float simple_lf_update	(	simple_lf_state_t *	s,
		float	error
	)

Update step for the simple first-order loop filter.

Implements the first-order loop filter as shown below:

Digital loop filter block diagram.

with transfer function:

$F[z] = \frac{(k_p+k_i) - k_p z^{-1}}{1 - z^{-1}}$

Parameters:

s	The loop filter state struct.
error	The error output from the discriminator, $\varepsilon_k$ .

Returns:: The updated output variable, .

Definition at line 210 of file track.c.

void simple_tl_init	(	simple_tl_state_t *	s,
		float	loop_freq,
		float	code_freq,
		float	code_bw,
		float	code_zeta,
		float	code_k,
		float	carr_freq,
		float	carr_bw,
		float	carr_zeta,
		float	carr_k
	)

Initialise a simple tracking loop.

For a full description of the loop filter parameters, see calc_loop_gains().

Parameters:

s	The tracking loop state struct to initialise.
code_freq	The initial code phase rate (i.e. frequency).
code_bw	The code tracking loop noise bandwidth.
code_zeta	The code tracking loop damping ratio.
code_k	The code tracking loop gain.
carr_freq	The initial carrier frequency.
carr_bw	The carrier tracking loop noise bandwidth.
carr_zeta	The carrier tracking loop damping ratio.
carr_k	The carrier tracking loop gain.

Definition at line 233 of file track.c.

void simple_tl_update	(	simple_tl_state_t *	s,
		correlation_t	cs[3]
	)

Update step for the simple tracking loop.

Implements a basic second-order tracking loop. The code tracking loop is a second-order DLL using dll_discriminator() as its discriminator function. The carrier phase tracking loop is a second-order Costas loop using costas_discriminator().

The tracking loop output variables, i.e. code and carrier frequencies can be read out directly from the state struct.

Parameters:

s	The tracking loop state struct.
cs	An array [E, P, L] of correlation_t structs for the Early, Prompt and Late correlations.

Definition at line 264 of file track.c.

Classes

Functions

Detailed Description

Function Documentation