Numerical Simulation

A numerical simulation is run in order to compare the performance attained by using different discriminators. The input signal (data and pilot channel) is generated with a constant frequency offset (Δ𝐸𝐸) of 20 Hz, and an acceleration offset of 1 Hz/s2. The former is an arbitrary frequency offset to allow the FLL to gain lock, and the latter is to test the ability of the FLL to track a signal with maximum frequency rate as faced by a static GPS receiver (Tsui 2000). Four different FLL implementations are allowed to track the input signal simultaneously. The implementations differ by the choice of discriminator:

i. data channel only with 𝐷𝐷_{𝑑𝑑𝐸𝐸}, ii. pilot channel only with 𝐷𝐷_{𝑐𝑐𝑃𝑃𝑐𝑐𝐿𝐿𝐿𝐿},

iii. standard discriminator combination with 𝐷𝐷_{𝑑𝑑𝐸𝐸} and 𝐷𝐷_{𝑐𝑐𝑃𝑃𝑐𝑐𝐿𝐿𝐿𝐿}, and

iv. Costas discriminator combination with 𝐷𝐷_{𝑑𝑑𝐸𝐸} on both data and pilot channels.

4.4.1.2 Test Methodology

The noise bandwidth of all the implementations is fixed to 4 Hz for fair comparisons and the loop filters designed in the analog domain are implemented using bilinear transformations. The predetection interval is set to 5 ms. Since this is a numerical simulation, instantaneous values of original input frequency and acceleration are known. These are compared with the frequency estimates used by the local oscillator in order to

measure the average frequency error of each predetection interval. A total of 3000 such frequency error values, measured in steady state, are then used to compute the frequency jitter due to thermal noise (𝜎𝜎_{𝑃𝑃ℎ𝑅𝑅𝑃𝑃𝑚𝑚𝑃𝑃𝑙𝑙}) and the steady-state error (Δ𝐸𝐸_{𝐿𝐿𝐿𝐿}). As defined by Kaplan (2006), frequency lock is declared if the frequency jitter (𝜎𝜎_{𝑛𝑛𝑅𝑅𝑃𝑃}) satisfies the condition

3σnet = 3σFLL + Δfss ≤_4T1

coh (Hz) (4.23)

where the threshold of 1

4𝑇𝑇𝑐𝑐𝑐𝑐 ℎ corresponds to the pull-in range of the cross product

discriminator with decision feedback (𝐷𝐷_{𝑑𝑑𝐸𝐸}). Although the pull-in range of the cross product discriminator (𝐷𝐷_{𝑐𝑐𝑃𝑃𝑐𝑐𝐿𝐿𝐿𝐿}) is twice as that of 𝐷𝐷_{𝑑𝑑𝐸𝐸}, this threshold is chosen based on the minimum pull-in range in the combination. For statistical reliability of the results, 𝜎𝜎𝑃𝑃ℎ𝑅𝑅𝑃𝑃𝑚𝑚𝑃𝑃𝑙𝑙 and Δ𝐸𝐸𝐿𝐿𝐿𝐿 are averaged across 100 such independent runs of the FLL for each

considered C/N0. For each run, the FLL transient is allowed to settle, and the errors are

measured at steady state. Only those points that have a probability of frequency lock of at least 75% for a given C/N0 are used in the analysis. The frequency jitter is measured for a

C/N0 in the 15 to 40 dB-Hz range.

4.4.1.3 Analysis of Results

Figure 4-7 shows the simulation results. For C/N0 greater than 33 dB-Hz, the

performance of 𝐷𝐷_{𝑐𝑐𝑃𝑃𝑐𝑐𝐿𝐿𝐿𝐿} on the pilot channel and 𝐷𝐷_{𝑑𝑑𝐸𝐸} on the data channel are the same. Data bit predictions are more reliable under this situation; hence, there is no significant difference in performance. Under these C/N0 conditions, the standard and Costas

For C/N0 lower than 33 dB-Hz, increased noise is introduced by the data bit

decision process (Natali 1986). This is observed on the curve corresponding to the FLL on the data channel (𝐷𝐷_{𝑑𝑑𝐸𝐸}). The Costas discriminator combination, which suffers from the same drawback, continues to provide nearly a 3 dB noise variance reduction with respect to 𝐷𝐷_{𝑑𝑑𝐸𝐸}. The standard discriminator combination shows a slightly degraded performance in comparison with the pilot channel only for low C/N0. The reason for this degradation

in performance is explained by analyzing the noise performance of 𝐷𝐷_{𝑑𝑑𝐸𝐸} used in the combination.

The 𝐷𝐷_{𝑑𝑑𝐸𝐸} relies on a differential data bit decision (𝐿𝐿𝑙𝑙𝑛𝑛(ℜ[. ])) to account for the effect of the data bit presence. The noise �𝜖𝜖(𝑃𝑃)� introduced by the data bit decision part of the discriminator can be given by a model approximated to the first order as (Natali 1986):

Figure 4-7: Comparison of frequency jitter across different implementations to track data and pilot channels

ϵ(t) = E[ϵ(t)] + ϵ1(t) = (1 − 2Pe) + ϵ1(t) (4.24)

Pe=1_{2 exp}�−_𝑁𝑁𝐶𝐶

0𝑇𝑇𝑐𝑐𝑐𝑐ℎ� (4.25)

where 𝑃𝑃_𝑅𝑅 is the probability of data bit error (P_e), 𝐸𝐸[𝜖𝜖(𝑃𝑃)] = (1 − 2𝑃𝑃_𝑅𝑅) accounts for the change in discriminator gain due to data bit errors, and 𝜖𝜖₁(𝑃𝑃) accounts for the noise introduced by faulty decisions. The autocorrelation function �𝑅𝑅_𝜖𝜖1(𝜏𝜏)� of 𝜖𝜖₁(𝑃𝑃) is given by (Natali 1986): 𝑅𝑅𝜖𝜖1(𝜏𝜏) = ⎩ ⎪ ⎨ ⎪ ⎧2𝑃𝑃𝑅𝑅�[2(1 − 𝑃𝑃𝑅𝑅)] − �_𝑇𝑇𝜏𝜏 𝑐𝑐𝑐𝑐ℎ� � 0 < 𝜏𝜏 < 𝑇𝑇𝑐𝑐𝑐𝑐ℎ 2𝑃𝑃𝑅𝑅(1 − 2𝑃𝑃𝑅𝑅) �2 − �_𝑇𝑇𝜏𝜏 𝑐𝑐𝑐𝑐ℎ�� 𝑇𝑇𝑐𝑐𝑐𝑐ℎ < 𝜏𝜏 < 2𝑇𝑇𝑐𝑐𝑐𝑐ℎ 0 𝑐𝑐𝑃𝑃ℎ𝑅𝑅𝑃𝑃𝑃𝑃𝑅𝑅𝐿𝐿𝑅𝑅 . (4.26)

Although the difference in noise variance at the discriminator output of 𝐷𝐷_{𝑑𝑑𝐸𝐸} and 𝐷𝐷_{𝑐𝑐𝑃𝑃𝑐𝑐𝐿𝐿𝐿𝐿} in a standard combination is accounted for by the weighting approach, the change in gain across 𝐷𝐷_{𝑑𝑑𝐸𝐸} and 𝐷𝐷_{𝑐𝑐𝑃𝑃𝑐𝑐𝐿𝐿𝐿𝐿} at lower C/N0, as given by 𝐸𝐸[𝜖𝜖(𝑃𝑃)] in Eq.(4.24), is unaccounted

for in the combination. Hence, a difference in performance is observed.

In terms of the minimum C/N0 required for tracking, as observed from Figure

4-7, the pilot channel has a frequency tracking threshold that is 6 dB lower than the data channel. The standard discriminator combination performs quite close (1 dB difference in minimum required C/N0) to the pilot-channel-only frequency tracking, whereas the

Costas discriminator combination performs close to the data-channel-only frequency tracking.

4.4.2 Validation of Results with Live Signals