where Z_k is the k‐th subaccumulation, N_s is the number of samples per subaccumulation, and is the code start time estimate over the k‐th subaccumulation. The code phase can be assumed to be approximately constant over a short subaccumulation interval T_sub = N_sT_s; hence, . It is worth mentioning that theoretically, T_sub can be made arbitrarily large since no data is transmitted on the pilot channel. Practically, T_sub is mainly limited by the stability of the BTS and receiver oscillators. In the following, T_sub is set to one PN code period. The carrier phase estimate is modeled as , where is the apparent Doppler frequency estimate over the i‐th subaccumulation, and θ₀ is the initial beat carrier phase of the received signal. As in a GPS receiver, the value of θ₀ is set to zero in the acquisition stage and is subsequently updated in the tracking stage. The apparent Doppler frequency is assumed to be constant over a short T_sub. Substituting for r[m] and x[m], defined in Eqs. (38.5)–(38.6), into Eq. (38.7), it can be shown that

(38.8)

      where R_c is the autocorrelation function of the PN sequences c′_I, and is the code phase error, is the carrier phase error, and with and being independent and identically distributed Gaussian random sequences with zero mean and variance .

The expression for Z_k in Eq. (38.8) assumes that the locally generated c_I and c_Q have the same code phase. To ensure this, both sequences must begin with the first binary “1” that occurs after 15 consecutive zeros; otherwise, ∣Z_k∣ will be halved. Figure 38.9 shows |Z_k|² for unsynchronized and synchronized c_I and c_Q code phases (i.e. shifted by 34 chips). The correlation peak of the synchronized codes is four times the peak of the unsynchronized case.

The carrier wipe‐off and correlation stages are illustrated in Figure 38.10.

      38.5.2.2 Acquisition

The objective of this stage is to determine which BTSs are in the receiver’s proximity and to obtain a coarse estimate of their corresponding code start times and Doppler frequencies. For a particular PN offset, a search over the code start time and Doppler frequency is performed to detect the presence of a signal. To determine the range of Doppler frequencies to search over, one must consider the relative motion between the receiver and the BTS and the stability of the receiver’s oscillator. For instance, a Doppler shift of 122 Hz will be observed for a cellular CDMA carrier frequency of 882.75 MHz at a mobile receiver with a receiver‐to‐BTS line‐of‐sight velocity of 150 km/h. Therefore, to account for this Doppler (at a carrier frequency of 882.75 MHz) as well as oscillator‐induced Doppler, the Doppler frequency search window is chosen to lie between −500 and 500 Hz. The frequency spacing Δf_D must be a fraction of 1/T_sub, which implies that Δf_D ≪ 37.5 Hz, if T_sub is assumed to be one PN code period (e.g. a Δf_D between 8 and 12 Hz can be chosen). The code start time search window is naturally chosen to be one PN code interval with a delay spacing of one sample.

Figure 38.9 |Z_k|² for (a) unsynchronized and (b) synchronized c_I and c_Q codes (Khalife et al. [18]).

      Source: Reproduced with permission of IEEE.

Figure 38.10 Carrier wipe‐off and correlator. Thick lines indicate a complex‐valued variable (Khalife et al. [18]).

      Source: Reproduced with permission of IEEE.

Similar to GPS signal acquisition, the search could be implemented either serially or in parallel, which in turn could be performed over the code phase or the Doppler frequency. The receiver presented here performs a parallel code phase search by exploiting the optimized efficiency of the fast Fourier transform (FFT) [53]. If a signal is present, a plot of |Z_k|² will show a high peak at the corresponding code start time and Doppler frequency estimates. A hypothesis test could be performed to decide whether the peak corresponds to a desired signal or noise. Since there is only one PN sequence, the search needs to be performed once. Then, the resulting surface is subdivided in the time axis into intervals of 64 chips, each division corresponding to a particular PN offset. The PN sequences for the pilot, sync, and paging channels could be generated off‐line and stored in a binary file to speed up the processing. Figure 38.11 depicts the acquisition stage of a cellular CDMA signal with a software‐defined receiver (SDR) developed in LabVIEW, showing |Z_k|² along with , the PN offset, and the carrier‐to‐noise ratio C/N₀ for a particular BTS [18].

      38.5.2.3 Tracking

After obtaining an initial coarse estimate of the code start time and Doppler frequency Скачать книгу