# Tracking

The Tracking block is continually receiving the data stream $x_\text{IN}$, but does nothing until it receives a “positive acquisition” message from the control plane, along with the coarse estimations $\hat{\tau}_{acq}$ and $\hat{f}_{d_{acq}}$. Then, its role is to refine such estimations and track their changes along the time.

The role of a Tracking block is to follow the evolution of the signal synchronization parameters: code phase $\tau$, Doppler shift $f_d$ and carrier phase $\phi$.

According to the Maximum Likelihood approach, obtaining the optimum estimators implies the maximization of the correlation of the incoming signal with its matched filter. This is usually achieved with closed-loop structures designed to minimize the difference between the code phase, carrier phase and frequency of the incoming signal with respect to a locally-generated replica.

In the case of code phase tracking, the cost function is driven to the maximum using feedback loops that employ the derivative $\frac{dR_{xd}(\tau)}{d\tau}$ zero-crossing as a timing error detector. This is the case of the Delay Lock Loop (DLL) architecture and its wide range of variants, where the receiver computes three samples of $R_{xd}$, usually referred to as Early $E=R_{xd}(\hat{\tau}-\epsilon)$, Prompt $P=R_{xd}(\hat{\tau})$ and Late $L=R_{xd}(\hat{\tau}-\epsilon)$, with $\epsilon$ ranging from $0.1T_c$ to $0.5T_c$, and then computes a timing error with some combination of those samples, known as discriminator functions. The result is low-pass filtered and reinjected back to the matched filter, as shown in the figure below:

Typical diagram of a tracking block. Colored boxes indicate functions implemented in the VOLK_GNSSSDR library.

GNSS-SDR’s Tracking implementations make heavy use of VOLK_GNSSSDR, an extension module of the original VOLK library which contains some functions that are specially useful in the context of a GNSS receiver (some examples in the figure above).

The VOLK_GNSSSDR library addresses Efficiency and Portability at the same time, by providing several implementations of the same functions in different SIMD technologies, benchmarking them and selecting the fastest in your machine at runtime.

## GPS L1 C/A signal tracking

### Implementation: GPS_L1_CA_DLL_PLL_Tracking

This implementation accepts the following parameters:

Global Parameter Description Required
GNSS-SDR.internal_fs_hz Input sample rate to the processing channels, in samples per second. Mandatory
Parameter Description Required
implementation GPS_L1_CA_DLL_PLL_Tracking Mandatory
item_type [gr_complex]: Set the sample data type expected at the block input. It defaults to gr_complex. Optional
if Intermediate frequency of the incoming signal, in Hz. It defaults to 0 (i.e., complex baseband signal). Optional
pll_bw_hz Bandwidth of the PLL low pass filter, in Hz. It defaults to 50 Hz. Optional
dll_bw_hz Bandwidth of the DLL low pass filter, in Hz. It defaults to 2 Hz. Optional
early_late_space_chips Spacing between Early and Prompt and between Prompt and Late correlators, normalized by the chip period $T_c$. It defaults to $0.5$. Optional
dump [true, false]: If set to true, it enables the Tracking internal binary data file logging. It defaults to false. Optional
dump_filename If dump is set to true, name of the file in which internal data will be stored. It defaults to ./track_ch Optional

Tracking implementation: GPS_L1_CA_DLL_PLL_Tracking.

Example:

;######### TRACKING GLOBAL CONFIG ############
Tracking_1C.implementation=GPS_L1_CA_DLL_PLL_Tracking
Tracking_1C.pll_bw_hz=30.0
Tracking_1C.dll_bw_hz=4.0
Tracking_1C.early_late_space_chips=0.5


### Implementation: GPS_L1_CA_DLL_PLL_C_Aid_Tracking

This implementation accepts the following parameters:

Global Parameter Description Required
GNSS-SDR.internal_fs_hz Input sample rate to the processing channels, in samples per second. Mandatory
Parameter Description Required
implementation GPS_L1_CA_DLL_PLL_C_Aid_Tracking Mandatory
item_type [gr_complex, cshort]. Set the sample data type expected at the block input. It defaults to gr_complex. Optional
if Intermediate frequency of the incoming signal, in Hz. It defaults to 0 (i.e., complex baseband signal). Optional
pll_bw_hz Bandwidth of the PLL low pass filter before bit synchronization, in Hz. It defaults to 50 Hz. Optional
dll_bw_hz Bandwidth of the DLL low pass filter before bit synchronization, in Hz. It defaults to 2 Hz. Optional
pll_bw_narrow_hz Bandwidth of the PLL low pass filter after bit synchronization, in Hz. It defaults to 20 Hz. Optional
dll_bw_narrow_hz Bandwidth of the DLL low pass filter after bit synchronization, in Hz. It defaults to 2 Hz. Optional
extend_correlation_ms Correlation length, in ms. It defaults to 1 ms. Optional
early_late_space_chips Spacing between Early and Prompt and between Prompt and Late correlators, normalized by the chip period $T_c$. It defaults to $0.5$. Optional
dump [true, false]: If set to true, it enables the Tracking internal binary data file logging. It defaults to false. Optional
dump_filename If dump is set to true, name of the file in which internal data will be stored. It defaults to ./track_ch Optional

Tracking implementation: GPS_L1_CA_DLL_PLL_C_Aid_Tracking.

Example:

;######### TRACKING GLOBAL CONFIG ############
Tracking_1C.implementation=GPS_L1_CA_DLL_PLL_C_Aid_Tracking
Tracking_1C.item_type=cshort
Tracking_1C.pll_bw_hz=40.0;
Tracking_1C.dll_bw_hz=4.0;


### Implementation: GPS_L1_CA_DLL_PLL_Tracking_GPU

GPU-accelerated computing consists in the use of a graphics processing unit (GPU) together with a CPU to accelerate the execution of a software application, by offloading computation-intensive portions of the application to the GPU, while the remainder of the code still runs on the CPU. The key idea is to utilize the computation power of both CPU cores and GPU execution units in tandem for better utilization of available computing power.

This implementation follows the CUDA programming model and targets NVIDIA’s GPU computing platform. Thus, you will need a CUDA-enabled GPU and the CUDA Toolkit installed. Moreover, it is only available if GNSS-SDR has been built from source and compiled with the flag ENABLE_CUDA set to ON:

\$ cmake -DENABLE_CUDA=ON ../ && make && sudo make install


This implementation accepts the following parameters:

Global Parameter Description Required
GNSS-SDR.internal_fs_hz Input sample rate to the processing channels, in samples per second. Mandatory
Parameter Description Required
implementation GPS_L1_CA_DLL_PLL_Tracking_GPU Mandatory
item_type [gr_complex]: Set the sample data type expected at the block input. It defaults to gr_complex. Optional
if Intermediate frequency of the incoming signal, in Hz. It defaults to 0 (i.e., complex baseband signal). Optional
pll_bw_hz Bandwidth of the PLL low pass filter, in Hz. It defaults to 50 Hz. Optional
dll_bw_hz Bandwidth of the DLL low pass filter, in Hz. It defaults to 2 Hz. Optional
early_late_space_chips Spacing between Early and Prompt and between Prompt and Late correlators, normalized by the chip period $T_c$. It defaults to $0.5$. Optional
dump [true, false]: If set to true, it enables the Tracking internal binary data file logging. It defaults to false. Optional
dump_filename If dump is set to true, name of the file in which internal data will be stored. It defaults to ./track_ch Optional

Tracking implementation: GPS_L1_CA_DLL_PLL_Tracking_GPU.

Example:

;######### TRACKING GLOBAL CONFIG ############
Tracking_1C.implementation=GPS_L1_CA_DLL_PLL_Tracking_GPU
Tracking_1C.pll_bw_hz=40.0;
Tracking_1C.dll_bw_hz=4.0;


## GPS L2CM signal tracking

### Implementation: GPS_L2_M_DLL_PLL_Tracking

This implementation accepts the following parameters:

Global Parameter Description Required
GNSS-SDR.internal_fs_hz Input sample rate to the processing channels, in samples per second. Mandatory
Parameter Description Required
implementation GPS_L2_M_DLL_PLL_Tracking Mandatory
item_type [gr_complex]: Set the sample data type expected at the block input. It defaults to gr_complex. Optional
if Intermediate frequency of the incoming signal, in Hz. It defaults to 0 (i.e., complex baseband signal). Optional
pll_bw_hz Bandwidth of the PLL low pass filter, in Hz. It defaults to 50 Hz. Optional
dll_bw_hz Bandwidth of the DLL low pass filter, in Hz. It defaults to 2 Hz. Optional
early_late_space_chips Spacing between Early and Prompt and between Prompt and Late correlators, normalized by the chip period $T_c$. It defaults to $0.5$. Optional
dump [true, false]: If set to true, it enables the Tracking internal binary data file logging. It defaults to false. Optional
dump_filename If dump is set to true, name of the file in which internal data will be stored. It defaults to ./track_ch Optional

Tracking implementation: GPS_L2_M_DLL_PLL_Tracking.

Example:

;######### TRACKING GLOBAL CONFIG ############
Tracking_2S.implementation=GPS_L2_M_DLL_PLL_Tracking
Tracking_2S.pll_bw_hz=40.0;
Tracking_2S.dll_bw_hz=4.0;
Tracking_2S.early_late_space_chips=0.4


## Galileo E1B signal tracking

### Implementation: Galileo_E1_DLL_PLL_VEML_Tracking

The Maximum likelihood (ML) estimates of $f_d$ and $\tau$ can be obtained by maximizing the function

where

$x_{\text{IN}}[n]$ is a complex vector containing I&Q samples of the received signal, $T_s$ is the sampling period, $\tau$ is the code phase of the received signal with respect to a local reference, $f_d$ is the Doppler shift, $N$ is the number of samples in a spreading code (4 ms for E1), and $d[n]$ is a locally generated reference. The user can also configure the shape of $d[n]$, allowing simplifications that reduce the computational load. For the E1B signal component, the reference signals available in our implementation are:

while for E1C, users can choose among:

Normalized $\left|R_{xd}\left(\check{f}_d=f_d, \tau \right) \right|^2$ for different sampling rates and local reference waveforms1.

In case of Galileo E1, the CBOC(6,1,$\frac{1}{11}$) modulation creates correlation ambiguities, as shown in the figure above. The possibility of tracking a local maximum instead of the global one can be avoided by using discriminators that consider two extra samples of the cost function, referred to as Very Early $\text{VE}=R_{xd}(\hat{\tau}-\epsilon^\prime)$ and Very Late $\text{VL}=R_{xd}(\hat{\tau}+\epsilon^\prime)$, with $\epsilon^\prime > \epsilon$.

In the case of carrier tracking loops, the pilot channel E1C can be used for the phase error estimation, since it does not contain data bit transitions, and, theoretically, coherent integration of several code periods can be used for as long as needed. As a consequence, a discriminator that is insensitive to phase jumps can be used. Using pure PLL tracking on the pilot channel as well as longer coherent integration improves the carrier tracking sensitivity, the minimum signal power at which the receiver can keep the tracking process in lock.

In addition to track the synchronization parameters, the Tracking block must also implement code and carrier lock detectors, providing indicators of the tracking performance.

The implementation of this block is described in Algorithm below. The computation of the complex values VE, E, P, L and VL in step $5$ was implemented using the VOLK_GNSSSDR library. The PLL discriminator implemented in step $6$ is the extended arctangent (four-quadrant) discriminator, and for the DLL we used the normalized Very Early Minus Late Power discriminator (step $10$ ). For code lock detection (step $13$ ), we used the Squared Signal-to-Noise Variance (SNV) estimator2. In the case of carrier lock detection (step $14$ ), we used the normalized estimate of the cosine of twice the carrier phase3. The values of the lock indicator range from $-1$, when the locally generated carrier is completely out of phase, to $1$, that indicates a perfect match. When either the code or the carrier detectors are below given thresholds during a consecutive number of code periods $\vartheta$, the Tracking block informs to control plane through the message queue.

• Require: Complex sample stream, $\mathbf{x}_{\text{IN}}$; estimations of code phase $\hat{\tau}_{acq}$ and Doppler shift $\hat{f}_{d_{acq}}$; buffer size for power estimation, $\mathcal{U}$; carrier lock detector threshold, $\mathcal{T}$; $CN0_{min}$; maximum value for the lock fail counter, $\vartheta$; correlators spacing $\epsilon$ and $\epsilon^\prime$; loop filters bandwidth $BW_{DLL}$ and $BW_{PLL}$; integration time $T_{int}$. Track signal’s synchronization parameters within a given lock margin. Inform about a loss of lock.
1. Initialization: Using $\hat{\tau}_{acq}$ and a sample counter $\mathcal{N}$, skip samples until $\mathbf{x}_{\text{IN}}$ is aligned with local PRN replica. Set $\upsilon=0$, $k=0$, $\hat{f}_{d_{0}}=\hat{f}_{d_{acq}}$, $\hat{\phi}_0=0$, $\psi_1=0$, $N_1=\text{round}(T_{int} f_{\text{IN}})$.

2. Increase the integration period counter: $k=k+1$.

3. Generate local code references: for $n=1...N_k$, $s[n]=d_{E1B/E1C_{p}}\left[\text{round}(\delta_{k} \cdot n + \psi_{k})\right]$, where $\delta_{k}= \frac{1}{T_{c,E1B} \cdot f_{\text{IN}} }\left( 1 + \frac{\hat{f}_{d_{k-1}}}{f^{\text{(Gal E1)}}_c} \right)$, and the Very Early, Early, Late, and Very Late versions with $\epsilon$ and $\epsilon^\prime$.
4. Generate local carrier: for $n=1...N_k$, $c[n]=e^{-j\left(2\pi \hat{f}_{d_{k-1}} \frac{n}{f_{\text{IN}}}+\text{mod}\left(\hat{\phi}_{k-1},2\pi \right) \right)}$.

5. Perform carrier wipe-off and compute the complex samples VE$_k$, E$_k$, P$_k$, L$_k$ and VL$_k$. Example: $\text{P}_k=\frac{1}{N_k} \sum_{n=0}^{N_k-1} x_{\text{IN}}[n] s[n] c[n]$.

6. Compute PLL discriminator: $\Delta \hat{\phi}_{k} = \mbox{atan2}\left( \frac{ \text{P}_{Q_{k}}}{\text{P}_{I_{k}}} \right)$

7. Filter $\Delta \hat{\phi}_{k}$ with a bandwidth $BW_{PLL}$: $h_{PLL}\left( \Delta \hat{\phi}_{k}\right)$.

8. Update carrier frequency estimation (in Hz): $\hat{f}_{d_{k}}=\hat{f}_{d_{acq}}+\frac{1}{ 2\pi T_{int} } h_{PLL}\left( \Delta \hat{\phi}_{k} \right)$.

9. Update carrier phase estimation (in rad): $\hat{\phi}_k=\hat{\phi}_{k-1}+ 2 \pi \hat{f}_{d_{k}} T_{int}+ h_{PLL}(\Delta \hat{\phi})$.

10. Compute DLL discriminator: $\Delta \hat{\tau}_{k}=\frac{\mathcal{E}_{k}-\mathcal{L}_{k}}{\mathcal{E}_{k}+\mathcal{L}_{k}}$, where: $\mathcal{E}_{k}=\sqrt{\text{VE}_{I_{k}}^2+\text{VE}_{Q_{k}}^2+E_{I_{k}}^2+E_{Q_{k}}^2}$, and $\mathcal{L}_{k}=\sqrt{\text{VL}_{I_{k}}^2+\text{VL}_{Q_{k}}^2+L_{I_{k}}^2+L_{Q_{k}}^2}$.

11. Filter $\Delta \hat{\tau}_{k}$ with a bandwidth $BW_{DLL}$: $h_{DLL}\left( \Delta \hat{\tau}_{k}\right)$.

12. Update code phase estimation (in samples): $N_{k+1}=\text{round}(S)$ and $\psi_{k+1}=S-N_{k+1}$, where $S = \frac{T_{int}f_{\text{IN} } }{\left( 1 + \frac{\hat{f}_{d_{k} } }{f^{\text{(Gal E1) } }_c} \right)} +\psi_{k} + h_{DLL}(\hat{\Delta \tau}_k)f_{\text{IN} }$.

13. Code lock indicator: $\hat{ \text{CN0} } = 10 \cdot \log_{10} ( \hat{\rho}) + 10 \cdot \log_{10}(\frac{ f_{ \text{IN} } }{2} )-10 \cdot \log_{10} (L_{ \text{PRN} })$, where: $\hat{\rho}=\frac{ \hat{P}_s }{ \hat{P}_n } = \frac{\hat{P}_s}{\hat{P}_{tot}-\hat{P}_s}$, $\hat{P}_s = \left(\frac{1}{\mathcal{U}}\sum^{\mathcal{U}-1}_{i=0}|\text{P}_{I_{k-i}} |\right)^2$, and $\hat{P}_{tot} = \frac{1}{\mathcal{U}}\sum^{\mathcal{U}-1}_{i=0}|\text{P}_{k-i}|^2$.

14. Phase lock indicator: $T_{carrier} = \frac{ \left( \sum^{\mathcal{U}-1}_{i=0} \text{P}_{ {I}_{k-i}}\right)^2 - \left( \sum^{\mathcal{U} -1}_{i=0} \text{P}_{Q_{k-i}}\right)^2}{\left(\sum^{\mathcal{U}-1}_{i=0} \text{P}_{ {I}_{k-i}}\right)^2 + \left( \sum^{\mathcal{U} -1}_{i=0} \text{P}_{Q_{k-i}}\right)^2}$.

15. if $% $ or $% $
• Increase lock fail counter $\upsilon \leftarrow \upsilon +1$.
16. else
• Decrease lock fail counter $\upsilon \leftarrow \max(\upsilon -1,0)$.
17. endif

18. if $\upsilon > \vartheta$
• Notify the loss of lock to the control plane through the message queue.
19. endif

20. Output: $\text{P}_k$, accumulated carrier phase error $\hat{\phi}_k$, code phase $\mathcal{N} \leftarrow \mathcal{N}+ N_k + \psi_k$, carrier-to-noise-density ratio $\hat{\text{CN0}}$.

This implementation accepts the following parameters:

Global Parameter Description Required
GNSS-SDR.internal_fs_hz Input sample rate to the processing channels, in samples per second. Mandatory
Parameter Description Required
implementation Galileo_E1_DLL_PLL_VEML_Tracking Mandatory
item_type [gr_complex]: Set the sample data type expected at the block input. It defaults to gr_complex. Optional
if Intermediate frequency of the incoming signal, in Hz. It defaults to 0 (i.e., complex baseband signal). Optional
pll_bw_hz Bandwidth of the PLL low pass filter, in Hz. It defaults to 50 Hz. Optional
dll_bw_hz Bandwidth of the DLL low pass filter, in Hz. It defaults to 2 Hz. Optional
early_late_space_chips Spacing between Early and Prompt and between Prompt and Late correlators, normalized by the chip period $T_c$. It defaults to $0.15$. Optional
very_early_late_space_chips Spacing between Very Early and Prompt and between Prompt and Very Late correlators, normalized by the chip period $T_c$ It defaults to $0.6$. Optional
dump [true, false]: If set to true, it enables the Tracking internal binary data file logging. It defaults to false. Optional
dump_filename If dump is set to true, name of the file in which internal data will be stored. It defaults to ./track_ch Optional

Tracking implementation: Galileo_E1_DLL_PLL_VEML_Tracking.

Example:

;######### TRACKING GLOBAL CONFIG ############
Tracking_1B.implementation=Galileo_E1_DLL_PLL_VEML_Tracking
Tracking_1B.pll_bw_hz=15.0;
Tracking_1B.dll_bw_hz=2.0;


## Galileo E5a signal tracking

### Implementation: Galileo_E5a_DLL_PLL_Tracking

The AltBOC modulation in the Galileo E5 band allows the approximation to two sub-bands, referred to as E5a and E5b, QPSK-modulated and centered at $f_{\text{Gal E5a}}=1176.450$ MHz and $f_{Gal E5b}=1207.140$ MHz, respectively.

The baseband signal at E5a can then be approximated by:

where the signal components are defined as:

where $T_{c,E5s}=1$ ms and $T_{c,E5p}=\frac{1}{10.23}$ $\mu$s.

This implementation accepts the following parameters:

Global Parameter Description Required
GNSS-SDR.internal_fs_hz Input sample rate to the processing channels, in samples per second. Mandatory
Parameter Description Required
implementation Galileo_E5a_DLL_PLL_Tracking Mandatory
item_type [gr_complex]: Set the sample data type expected at the block input. It defaults to gr_complex. Optional
if Intermediate frequency of the incoming signal, in Hz. It defaults to $0$ (i.e., complex baseband signal). Optional
pll_bw_init_hz Bandwidth of the PLL low pass filter before the secondary code lock, in Hz. It defaults to 20 Hz. Optional
dll_bw_init_hz Bandwidth of the DLL low pass filter before the secondary code lock, in Hz. It defaults to 20 Hz. Optional
dll_bw_hz Bandwidth of the DLL low pass filter after the secondary code lock, in Hz. It defaults to 5 Hz. Optional
pll_bw_hz Bandwidth of the PLL low pass filter after the secondary code lock, in Hz. It defaults to 2 Hz. Optional
ti_ms Correlation length after the secondary code lock, in ms. It defaults to 3 ms. Optional
early_late_space_chips Spacing between Early and Prompt and between Prompt and Late correlators, normalized by the chip period $T_c$. It defaults to $0.5$. Optional
dump [true, false]: If set to true, it enables the Tracking internal binary data file logging. It defaults to false. Optional
dump_filename If dump is set to true, name of the file in which internal data will be stored. It defaults to ./track_ch Optional

Tracking implementation: Galileo_E5a_DLL_PLL_Tracking.

Example:

;######### TRACKING GLOBAL CONFIG ############
Tracking_5X.implementation=Galileo_E5a_DLL_PLL_Tracking
Tracking_5X.item_type=gr_complex
Tracking_5X.pll_bw_hz_init=25.0
Tracking_5X.dll_bw_hz_init=4.0
Tracking_5X.ti_ms=1
Tracking_5X.pll_bw_hz=10.0
Tracking_5X.dll_bw_hz=4.0
Tracking_5X.early_late_space_chips=0.5


## References

1. C. Fernández-Prades, J. Arribas, L. Esteve-Elfau, D. Pubill, P. Closas, An Open Source Galileo E1 Software Receiver, in Proceedings of the 6th ESA Workshop on Satellite Navigation Technologies (NAVITEC 2012), 5-7 December 2012, ESTEC, Noordwijk (The Netherlands).

2. M. Petovello, E. Falletti, M. Pini, L. Lo Presti, Are Carrier-to-Noise algorithms equivalent in all situations?. Inside GNSS, Vol. 5, no. 1, pp. 20-27, Jan.-Feb. 2010.

3. A. J. Van Dierendonck, “GPS Receivers”, from “Global Positioning System: Theory and Applications”, Volume I, Edited by B. W. Parkinson, J. J. Spilker Jr.

Updated: