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{{Article Infobox2
|Category=Receivers
|Category=Receivers
|Title={{PAGENAME}}
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|Authors=GMV
|Level=Advanced
|Level=Advanced
|YearOfPublication=2011
|YearOfPublication=2011
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Frequency Lock Loops are part of the receiver's signal tracking loops, and aim at tracking the frequency of the incoming GNSS signal. The FLL provides frequency corrections in a continuous loop, generating a frequency error signal.


==Principle ==
'''Frequency Lock Loops (FLL)''' track the Doppler frequency of the incoming signal (with respect to Intermediate Frequency), without necessarily correcting for phase. For that purpose, the FLL uses integrations, filters and Numerical Control Oscillators (NCO) – described [[Tracking Loops|here]] – as any other typical loop. In this case, the specificity relies on the discriminator used to assess the current frequency error estimated at the receiver.


Frequency Lock Loops are part of the Tracking Loops and aim at tracking the frequency of the incoming GNSS signal, providing corrections in a continuous loop.
The principle behind most common FLL implementation is to retrieve frequency information, by using the derivative of the phase between two consecutive sets of data (samples). As such, the common FLL is a differential-phase tracking loop. The FLL is expected to work well if the coherent integration time <math>T\,</math> is small, in order to avoid aliasing:




==Principle ==
<math>\frac{-1}{2T}< \hat {f}< \frac{1}{2T}\,</math>


Frequency Lock Loops (FLL) track the Doppler frequency of the incoming signal (with respect to Intermediate Frequency), without necessarily correcting for phase.  
Other tracking strategies exist which operate on a different principles, for example, they may directly observe the frequency information via a DFT, or by observing the relative power in correlator values which are computed at various frequency offsets. <ref name="Juang">J. C. Juang and Y. H. Chen, “Phase/frequency tracking in a GNSS software receiver,” Selected Topics in Signal Processing, IEEE Journal of, vol. 3, no. 4, pp. 651 –660, August 2009.</ref>
For that purpose, the PLL uses integrations, filters and Numerical Control Oscillators (NCO) – described [[Tracking Loops|here]] – as any other loop.  
The specificity relies on the discriminator used to assess the current phase error estimated at the receiver.


The principle behind the FLL is to retrieve frequency information by using the derivative of the phase between two consecutive samples.
==Discriminators ==


The FLL is expected to work well if the coherent integration time, T, is small, in order to avoid aliasing:  
First, let us define ''cross product'' and ''dot product'', in terms of In-Phase ''I'' and Quadrature ''Q'' components, as:


<math>\frac{-1}{2T}< \hat {f}< \frac{1}{2T}</math>


<math>cross=I_{P1} \times Q_{P2} - I_{P2} \times Q_{P1}\,</math>


==Discriminators ==
<math>dot=I_{P1} \times I_{P2} + Q_{P1} \times Q_{P2}\,</math>


Defining cross and dot product as:


<math>cross=I_{P1} \times Q_{P2} - I_{P2} \times Q_{P1}</math>
where the indices 1 and 2 refer to samples taken at consecutive time instants, <math>t_1\,</math> and <math>t_2\,</math>.


<math>dot=I_{P1} \times I_{P2} + Q_{P1} \times Q_{P2}</math>
This way, the most commonly used FLL discriminators are (ordered from lowest to highest computational burden)<ref name="kaplan">Kaplan E.D., Hegarty C.J., "Understanding GPS: Principles and Applications", second edition.</ref>:


where the indexes 1 and 2 refer to samples taken at consecutive time instants: <math>t_1</math> and <math>t_2</math>.


*<math>\frac{cross}{t_2-t_1}\,</math>


Then the most commonly used FLL discriminators are (ordered by computational burden) <ref name="kaplan">Kaplan E.D., Hegarty C.J.,"Understanding GPS: Principles and Applications", second edition</ref>:
*<math>\frac{cross \times sign(dot)}{t_2-t_1}\,</math>


*<math>\frac{cross}{t_2-t_1}</math>
*<math>\frac{atan2 (cross, dot)}{t_2-t_1}\,</math>
*<math>\frac{cross \times sign(dot)}{t_2-t_1}</math>
*<math>\frac{ATAN2 (cross, dot)}{t_2-t_1}</math>


The ATAN2 discriminator is optimal although it has the greater computational requirements. Finally, the FLL is insensitive to data transitions as long as the consecutive measurements (correlator outputs) refer to the same data bit.


The <math>atan2\,</math> discriminator is optimal, although it presents the greatest computational requirements. Note that the FLL is insensitive to data transitions, as long as the consecutive measurements (correlator outputs) refer to the same data bit.


==Performance ==
==Performance ==
The main sources of errors in the FLL are frequency jitter and dynamic stress error<ref name="kaplan"></ref>, which are mostly caused by thermal noise. The FLL thermal noise jitter can be written (in m/s) as:


The main sources of errors in the FLL are frequency jitter and dynamic stress error <ref name="kaplan"></ref>, which are greatly produced by thermal noise.
The FLL thermal noise jitter can be written (in m/s) as:


<math>\sigma_{th}=\frac{\lambda}{2 \pi T} \sqrt{\frac{4FB_n}{C/N_0}(1+\frac{1}{TC/N_0})}</math>
<math>\sigma_{th}=\frac{\lambda}{2 \pi T} \sqrt{\frac{4FB_n}{C/N_0}(1+\frac{1}{TC/N_0})}\,</math>


Where
*<math>\lambda</math> is wavelength of the carrier signal [m]
*<math>B_n</math> is the loop bandwidth [Hz]
*<math>C/N_0</math> is the carrier to noise ratio [dB-Hz]
*T is the integration time [s]
*F=1 at high carrier to noise ration and 2 otherwise


where:


The performance of the FLL depends mainly on the loop bandwidth and the integration times used to accumulate the correlator outputs.  
*<math>\lambda\,</math> is the wavelength of the carrier signal [m].
The impact of these parameters is shown in Figure 1, and they are visible mainly for low carrier to noise ratio.
*<math>B_n\,</math> is the loop bandwidth [Hz].
*<math>C/N_0\,</math> is the carrier-to-noise ratio [dB-Hz].
*<math>T\,</math> is the integration time [s].
*<math>F\,</math> = 1, at high carrier to noise ration; <math>F\,</math> = 2, otherwise.


The performance of the FLL depends mainly on the loop bandwidth and the integration times used to accumulate the correlator outputs. The impact of these parameters is shown in Figure 1, and they are visible mainly for low carrier to noise ratio.


[[File:fll_performance.png|center|thumb|600px|'''''Figure 1:''''' Impact of loop bandwidth (left) and integration time (right) on the FLL thermal noise jitter performance.]]
[[File:fll_performance.png|center|thumb|600px|'''''Figure 1:''''' Impact of loop bandwidth (left) and integration time (right) on the FLL thermal noise jitter performance.]]


 
These results also illustrate the fact that longer integration times (and lower bandwidths) lead to noise reduction, hence higher performances, as discussed [[Tracking Loops|here]]. In addition, lower loop noise bandwidths allow coping with higher dynamic environments.
These results also illustrate the fact that longer integration times (and lower bandwidths) lead to noise reduction and hence higher performances, as discussed [[Tracking Loops|previously]].  
In addition, lower loop noise bandwidths allow coping with higher dynamic environments.
 


==Related articles==
==Related articles==
Line 78: Line 72:
*[[Phase Lock Loop (PLL)]]
*[[Phase Lock Loop (PLL)]]
*[[Delay Lock Loop (DLL)]]
*[[Delay Lock Loop (DLL)]]


==References==
==References==

Latest revision as of 16:36, 18 September 2014


ReceiversReceivers
Title Frequency Lock Loop (FLL)
Edited by GMV
Level Advanced
Year of Publication 2011
Logo GMV.png

Frequency Lock Loops are part of the receiver's signal tracking loops, and aim at tracking the frequency of the incoming GNSS signal. The FLL provides frequency corrections in a continuous loop, generating a frequency error signal.

Principle

Frequency Lock Loops (FLL) track the Doppler frequency of the incoming signal (with respect to Intermediate Frequency), without necessarily correcting for phase. For that purpose, the FLL uses integrations, filters and Numerical Control Oscillators (NCO) – described here – as any other typical loop. In this case, the specificity relies on the discriminator used to assess the current frequency error estimated at the receiver.

The principle behind most common FLL implementation is to retrieve frequency information, by using the derivative of the phase between two consecutive sets of data (samples). As such, the common FLL is a differential-phase tracking loop. The FLL is expected to work well if the coherent integration time [math]\displaystyle{ T\, }[/math] is small, in order to avoid aliasing:


[math]\displaystyle{ \frac{-1}{2T}\lt \hat {f}\lt \frac{1}{2T}\, }[/math]

Other tracking strategies exist which operate on a different principles, for example, they may directly observe the frequency information via a DFT, or by observing the relative power in correlator values which are computed at various frequency offsets. [1]

Discriminators

First, let us define cross product and dot product, in terms of In-Phase I and Quadrature Q components, as:


[math]\displaystyle{ cross=I_{P1} \times Q_{P2} - I_{P2} \times Q_{P1}\, }[/math]

[math]\displaystyle{ dot=I_{P1} \times I_{P2} + Q_{P1} \times Q_{P2}\, }[/math]


where the indices 1 and 2 refer to samples taken at consecutive time instants, [math]\displaystyle{ t_1\, }[/math] and [math]\displaystyle{ t_2\, }[/math].

This way, the most commonly used FLL discriminators are (ordered from lowest to highest computational burden)[2]:


  • [math]\displaystyle{ \frac{cross}{t_2-t_1}\, }[/math]
  • [math]\displaystyle{ \frac{cross \times sign(dot)}{t_2-t_1}\, }[/math]
  • [math]\displaystyle{ \frac{atan2 (cross, dot)}{t_2-t_1}\, }[/math]


The [math]\displaystyle{ atan2\, }[/math] discriminator is optimal, although it presents the greatest computational requirements. Note that the FLL is insensitive to data transitions, as long as the consecutive measurements (correlator outputs) refer to the same data bit.

Performance

The main sources of errors in the FLL are frequency jitter and dynamic stress error[2], which are mostly caused by thermal noise. The FLL thermal noise jitter can be written (in m/s) as:


[math]\displaystyle{ \sigma_{th}=\frac{\lambda}{2 \pi T} \sqrt{\frac{4FB_n}{C/N_0}(1+\frac{1}{TC/N_0})}\, }[/math]


where:

  • [math]\displaystyle{ \lambda\, }[/math] is the wavelength of the carrier signal [m].
  • [math]\displaystyle{ B_n\, }[/math] is the loop bandwidth [Hz].
  • [math]\displaystyle{ C/N_0\, }[/math] is the carrier-to-noise ratio [dB-Hz].
  • [math]\displaystyle{ T\, }[/math] is the integration time [s].
  • [math]\displaystyle{ F\, }[/math] = 1, at high carrier to noise ration; [math]\displaystyle{ F\, }[/math] = 2, otherwise.


The performance of the FLL depends mainly on the loop bandwidth and the integration times used to accumulate the correlator outputs. The impact of these parameters is shown in Figure 1, and they are visible mainly for low carrier to noise ratio.

Figure 1: Impact of loop bandwidth (left) and integration time (right) on the FLL thermal noise jitter performance.

These results also illustrate the fact that longer integration times (and lower bandwidths) lead to noise reduction, hence higher performances, as discussed here. In addition, lower loop noise bandwidths allow coping with higher dynamic environments.

Related articles

References

  1. ^ J. C. Juang and Y. H. Chen, “Phase/frequency tracking in a GNSS software receiver,” Selected Topics in Signal Processing, IEEE Journal of, vol. 3, no. 4, pp. 651 –660, August 2009.
  2. ^ a b Kaplan E.D., Hegarty C.J., "Understanding GPS: Principles and Applications", second edition.
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