In order to measure the computation complexity of above methods, some assumptions are made. The length of FFT must be of radix 2. In the following discussions, the number of operations required to an FFT of length N is equal to (N log2N )/2 multiplications and N log2N additions. All methods are compared with each other under the same frequency resolution and the equal range of Doppler frequency. The frequency resolution is ∆f = fr/(Rw − 1), where fr is the range of the Doppler frequency and Rw is the number of frequency bins. Define Rd be the number of code delays to be searched, N is the length of input signal in FFT methods, and Ni is the length of input signal in CTA method. Let Nc = Ni + M − 1. Assume that one multiplication and one addition is regarded as one operation.
In S-P method, the whole process of acquisition needs 2N Rd + RdRw+ Rd multiplications, N Rd additions, and Rd FFT operations.
The computation complexity of S-P method is Csp = Rd[3N + Rw+ 1 + 3(N log2N )/2]
In P-S method, it needs 3RwN multiplications and 2Rw FFT operations.
The computation complexity of P-S method is Cps = 3Rw(N + 3N log2N )
In CTA method using FFT, it needs 3RdNi + Rd + RwRd multiplications, RdNi additions, and Rd CTA operations; one CTA operation needs Ni+ Nc+ Rw multiplications, and 2 FFT operations.
Thus, its computation complexity of CTA method is Cc = Rd(5Ni+ 1 + Nc+ Rw + 3Nclog2Nc)
In two-stage method with R0w = (Rw− 1)/2 + 1 frequency bins, its computation complexity is
Ct = 3R0w(N + 3N log2N ) + Ni+ Nc+ Rw+ 3Nclog2Nc)
The computation complexity of all discussed approaches are shown in Table 3.3. Here, we
CHAPTER 3. C/A CODE ACQUISITION
Table 3.1: Computation complexity of all
Common parameters The sampling frequency fs 2.1518 MHz The resolution of code delay Ts = 1.023T2.1518p The range of Doppler frequency fr 20 kHz The frequency resolution ∆f 131.33 Hz
Table 3.2: Common parameters
give a GPS application as a demonstrative example using the parameters shown in Table 3.3.
In S-P method, the length of input signal, N, must be chosen 16384, because N = fs/∆f . The frequency bins is Rw = dfr/∆f e = 153 , where d·e is the ceiling function. The number of code delays, Rd, is chosen as 2152 for less computation complexity. Thus, the computation complexity of S-P method is
Csp = 2152 × (3 × 16384 + 153 + 1 + 3 × 16384 × 14/2)
= 846, 532, 240
In P-S method, the length of input signal, N, is chosen as 4096 to be of radix 2 and over one period of C/A code. The number of frequency bins is Rw=153. The computation complexity of P-S method is
Cps = 3 × 153 × (4096 + 3 × 4096 × 12)
= 69, 562, 368
In CTA method , the length of FFT in CTA, Nc, must be of radix 2 and over one period of C/A code. Thus it can be chosen as 4096. Ni = Nc− Rw+ 1 = 4096 − 153 + 1 = 3944. Rd
CHAPTER 3. C/A CODE ACQUISITION
can be chosen as 2152. The computation complexity of CTA method is Cc = 2152 × [5 × 3944 + 1 + 4096 + 153 + 3 × 4096 × 12]
= 368, 908, 752
In two-stage method, at the first stage N can be 4096 and R0w =77. At the second stage, Ni
is the same, 3944, so Nc is 4096. The computation complexity of method two-stage is
Ct = 3 × 77 × (4096 + 3 × 4096 × 12) + 3944 + 4096 + 153 + 3 × 4096 × 12
= 35, 164, 161 (3.8)
From the above example, one can find the order of computation complexity is method S-P
> CTA > P-S > two-stage. As the frequency resolution increases, FFT methods will have more computation complexity than our methods. In comparison with S-P and CTA method, CTA does not only have the less computation complexity, but also has a shorter integration time than S-P method. In this example, the integration time of CTA is 3944Ts, and that of S-P is 16384Ts. It shows that the S-P method needs more time to collect the adequate data to make the frequency resolution high. However, in this way the SNR will be increased.
Under lower SNR environments, the S-P method works more sensitively than CTA method to detect the wanted signal at the cost of a long collected data time. In comparison with P-S and two-stage method, two-stage method can have the less computation complexity.
Chapter 4
SIMULATIONS AND DISCUSSIONS
In this chapter, we show our simulations with the received signal, and 3-dimensional fig-ures and 2-dimensional figfig-ures of uncertainty region. We use the same parameters in section 3.3. According to the characteristics of the real received signal, we use Matlab to simulate it. In the simulated received signal, the baseband frequency is set at fb = 17.248 kHz, the Doppler frequency is fd = 3200 Hz, fc= fb+ fd = 20.448 kHz, and the code phase is set as 1000Ts. As expected, the received signal looks like noise in Fig. 4.1.
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 0
Fig. 4.1: Received signal in baseband from the satellite 12 with C/N0 = 40 dB-Hz
Three Figures 4.2, 4.3, and 4.4 are to show the simulation results of calculating the crosscor-relation of each cell to find the position of the maximum value (conventional method) with the 131.33 Hz frequency resolution and N = 2152. From Fig. 4.2, one may find a maximum peak at (102,1000). (102-1)×131.33 + 7.248 kHz (the initial frequency) ≈ 20.51 kHz. The
CHAPTER 4. SIMULATIONS AND DISCUSSIONS
Y direction and X direction are shown in Fig. 4.3, and 4.4.
Fig. 4.2: Using the conventional method from the satellite 12 with C/N0 = 40 dB-Hz : 3-dimensional view
Fig. 4.3: Using the conventional method from the satellite 12 with C/N0 = 40 dB-Hz : lateral view
CHAPTER 4. SIMULATIONS AND DISCUSSIONS
Fig. 4.4: Using the conventional method from the satellite 12 with C/N0 = 40 dB-Hz : lateral view 2
Three Figures 4.5, 4.6, and 4.7 are the simulation results of using FFT to find the Doppler frequency (S-P Method) with the 131.33 Hz frequency resolution, N = 16384, and Rd = 2152. According to these figures, the code delay can be found at 1000 in Fig. 4.7, and the frequency bin of the largest magnitude is located at 157 in Fig. 4.6. (157-1)×131.33 Hz
≈ 20.487 kHz. It is very close to fc.
Fig. 4.5: Using FFT to estimate the Doppler frequency from the satellite 12 with C/N0 = 40 dB-Hz : 3-dimensional view
CHAPTER 4. SIMULATIONS AND DISCUSSIONS
Fig. 4.6: Using FFT to estimate the Doppler frequency from the satellite 12 with C/N0 = 40 dB-Hz : lateral view
Fig. 4.7: Using FFT to estimate the Doppler frequency from the satellite 12 with C/N0 = 40 dB-Hz : lateral view 2
Three Figures 4.8, 4.9, and 4.10 are the simulation results of using FFT to estimate the code delay (P-S Method) with the 131.33 Hz frequency resolution, N = 4096, Rw = 153. In Fig. 4.10, it is obvious to find the code delay located at 1000. In Fig. 4.9, the frequency bin of the largest magnitude is at 101. (102-1)× 131.33 Hz + 7.248 kHz ≈ 20.512 kHz. The signal is also detected. One can find they are the same to Figures 4.2, 4.3, and 4.4. It is
CHAPTER 4. SIMULATIONS AND DISCUSSIONS
similar to what we have discussed in section 3.1.2.
Fig. 4.8: Using FFT to estimate the code delay from the satellite 12 with C/N0 = 40 dB-Hz : 3-dimensional view
Fig. 4.9: Using FFT to estimate the code delay from the satellite 12 with C/N0 = 40 dB-Hz : lateral view
CHAPTER 4. SIMULATIONS AND DISCUSSIONS
Fig. 4.10: Using FFT to estimate the code delay from the satellite 12 with C/N0 = 40 dB-Hz : lateral view 2
Three Figures 4.11, 4.12, and 4.13 are the simulation results of using CTA to estimate the Doppler frequency (CTA Method) with the 131.33 Hz frequency resolution, Rw = 153, Nc = 4096, and Rd = 2152. One can find the frequency bin of the largest magnitude are at 102. (102 − 1)×131.33 Hz + 7.248 kHz≈ 20.512 kHz. The code delay of the largest magnitude is at 1000. Therefore, we can claim the signal is detected.
Fig. 4.11: Using CTA to estimate the Doppler frequency from the satellite 12 with C/N0 = 40 dB-Hz : 3-dimensional view
CHAPTER 4. SIMULATIONS AND DISCUSSIONS
Fig. 4.12: Using CTA to estimate the Doppler frequency from the satellite 12 with C/N0 = 40 dB-Hz : lateral view
Fig. 4.13: Using CTA to estimate the Doppler frequency from the satellite 12 with C/N0 = 40 dB-Hz : lateral view 2
Four Figures 4.14, 4.15, 4.16, and 4.17 are using two-stage search method with the 131.33 Hz frequency resolution, Rw = 153. At first stage, N=4096, and R0w=77. At second stage, Nc=4096 and Rd = 2152. The first three Figures 4.14, 4.15, and 4.16 are the simulation results of first stage with coarse 77 frequency bins. In Fig. 4.16, the code delay is found at 1000. In the final Fig. 4.17, one can find there is a maximum peak at 102. (102-1)×131.33
CHAPTER 4. SIMULATIONS AND DISCUSSIONS
Hz + 7.248 kHz ≈ 20.512 kHz. Hence, two-stage successfully detects the signal.
Fig. 4.14: Using two-stage search from the satellite 12 with 77 coarse frequency bins C/N0 = 40 dB-Hz : 3-dimensional view
Fig. 4.15: Using Two-Stage search from the satellite 12 with 77 coarse frequency bins C/N0 = 40 dB-Hz : lateral view
CHAPTER 4. SIMULATIONS AND DISCUSSIONS
Fig. 4.16: Using Two-Stage search from the satellite 12 with 77 coarse frequency bins C/N0 = 40 dB-Hz : lateral view 2
0 20 40 60 80 100 120 140 160
0 500 1000 1500 2000 2500 3000 3500
Doppler frequency bins
Magnitude
Fig. 4.17: Using CTA to estimate the Doppler frequency with the correct code delay from the satellite 12 C/N0 = 40 dB-Hz
Chapter 5
CONCLUSIONS
In this thesis, the CTA is employed to develop an effective frequency-acquisition algo-rithm. Then, the two-stage method is presented such that the complexity for acquisition realization can be reduced. Computer simulations demonstrate further that for GPS applica-tions the proposed methods perform satisfactorily with lower computation complexity under a wider range of the Doppler frequency. The proposed approach, naturally, is not limited to GPS applications, it can be applied to the acquisition applications of any communications via the direct sequence spread spectrum.
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