Signal Processing - September 2017 - 55

The L2C signal, received from satellite pseudorandom noise
(PRN) k, applied to the first processing stage, is modeled as:

rate of 1/Tc sps (Tc being smaller than CM period). The averaged
I and Q samples are expressed as

2Pk 6d k ^t - x k h cm k ^t - x k + hT h + cl k ^t - x k h@
(1)
# cos 62r ^ fIF + fD, k h t + { k@ + n ^ t h,

I s = 8 " r ^t + t 0h cos 62r f IF ^t + t 0h + { f@ $ cm i ^t - x ih, dt (2)

r^ t h =

where t = pTs for discrete domain representation with sampling
interval Ts = 1/ fs, fs being the full sampling rate and p is an
integer sequence " 1, 2, 3, f , . (For sake of simplicity, we adopt
continuous-domain representation and specify the sampling rate
for discrete equivalent where necessary.) Pk is the received signal power, d k ^ t h ! " + 1 - 1 , denotes the L2C CNAV navigation data symbols, cm k ^ t h and cl k ^ t h represent the CM and
CL code components (including the pulse shape necessary for
time multiplexing), respectively, T denotes the CM code period
(20  ms), and 0 # h # 74 is the index of this CM code epoch
within the CL period. The IF carrier frequency is designated as
fIF while fD, k signifies the carrier Doppler, { k represents the
carrier phase and n (t) is the thermal noise in the receiver, modeled as additive white Gaussian noise with two-sided power spectral density of N 0 / 2. The signal is mixed with two orthogonal IF
carriers to produce the in-phase ^ I h and quadrature ^Qh signal
components, each to be correlated with the local replica CM code
cm i (t) at the traditional processing rates ($2.046 Msps). While
CM code, being 75 times shorter than CL code, is the obvious
choice for local replica code in L2C acquisition, the time multiplexed structure of L2C allows for multiple designs of local CM
code replica, each offering a different relative advantage [18]. For
local replica code, the return-to-zero (RZ) CM design is considered for performance evaluation as it offers the highest detection
performance. For each delay of local replica code, after the code
correlation, I and Q signal components are down-sampled by
accumulating the signal samples over each Tc milliseconds containing N c samples, resulting in averaged signals sampled at the

≥ 2.046 Msps

Tc

0
Tc

Q s = 8 " r ^t + t 0 h sin 62r f IF ^t + t 0 h + { f@ $ cm i ^t - x i h, dt , (3)
0

where t 0 represents the time-instant for start of an averaging epoch, x i is the local code delay, and { f represents the
phase of the local IF carrier. The processing for the Doppler
search is then carried out with the averaged signals at the rate
of 1/Tc sps. For the correct local code delay, the averaged I
and Q signals preserve the carrier Doppler frequency ^ fD, k h
and phase ^{ D, k h information. To extract this information,
the signals are multiplied with a local carrier for a number of
expected Doppler frequencies, and the product samples are
coherently accumulated over each CM period T (i.e., N product samples) to produce the final I and Q outputs, given by:
I = Pk / 2 6R ^∆xh@ cos ^2r∆{h

sin ^r∆fD T h sin ^rfD, k Tch
$
r∆fD T
rfD, k Tc
(4)

+ nI
Q = Pk /2 6R ^∆xh@ sin ^2r∆{h
+ nQ

sin ^r∆fD T h sin ^rfD, k Tch
$
,
r∆fD T
rfD, k Tc
(5)

where R ^∆xh represents the correlation between the re ceived and the local replica code sequences as a function of
relative code delay ∆x = x k - x i. Similarly, ∆{ = { k - { i
and ∆fD = fD, k - fD, i denote the relative carrier phase offset and relative Doppler offset, respectively, between the local
and received signals. Each pair of the final I and Q samples

1/Tc sps
NC

I

N

∑

∑
(.)2

Sampled
IF Signal

IF
Carrier
90°

Carrier
Doppler
Generator

CM Code
Generator

M

∑

Ds

<ν → H0
>ν → H1

(.)2
NC

N

∑

∑

Q

FIGURE 3. The system architecture for the low-rate Doppler search. IF: intermediate frequency.
IEEE SIGNAL PROCESSING MAGAZINE

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September 2017

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55
Table of Contents for the Digital Edition of Signal Processing - September 2017
