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Given samples of the baseband signal exiting the RF front‐end defined in Eq. (38.5), the cellular CDMA receiver first wipes off the residual carrier phase and match‐filters the resulting signal. The output of the matched filter can be expressed as

(38.6)

where is the beat carrier phase estimate, and h is a pulse shaping filter, which is a discrete‐time version of the one used to shape the spectrum of the transmitted signal, with a finite‐impulse response (FIR) given in Table 38.3. The samples m′ of the FIR in Table 38.3 are spaced by .

Table 38.3 FIR of the pulse shaping filter used in cdma2000 [50]

Source: 3GPP2, “Physical layer standard for cdma2000 spread spectrum systems (C.S0002‐E),” 3rd Generation Partnership Project 2 (3GPP2), TS C.S0002‐E, June 2011. Reproduced with permission of 3CPP2.

m′	h[m′]	m′	h[m′]	m′	h[m′]
0, 47	−0.02528832	8, 39	0.03707116	16, 31	−0.01283966
1, 46	−0.03416793	9, 38	−0.02199807	17, 30	−0.14347703
2, 45	−0.03575232	10, 37	−0.06071628	18, 29	−0.21182909
3, 44	−0.01673370	11, 36	−0.05117866	19, 28	−0.14051313
4, 43	0.02160251	12, 35	0.00787453	20, 27	0.09460192
5, 42	0.06493849	13, 34	0.08436873	21, 26	0.44138714
6, 41	0.09100214	14, 33	0.12686931	22, 25	0.78587564
7, 40	0.08189497	15, 32	0.09452834	23, 24	1.0

Next, x[m] is correlated with a local replica of the spreading PN sequence. In a digital receiver, the correlation operation is expressed as

(38.7)

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.