Signal Processing - September 2017 - 62
Channel Combining and/or
Integration G (.)
to-noise ratio (SNR) loss in Z 1di. To
I
-1
cope with the phase shift, the modi1 Nco . RP
Σ ()
fied general differential combination
Nco n = 0
I
(MGDC) technique [7] uses Z mgdc that
I
-1
N
y B [n]
1 co . RD
Σ ()
noncoherently accumulates the differNco n = 0
cos(2π [fI + λ∆f ]n)
ential coherently integrated detection
variables of all possible spans (from
Spreading
Carrier
Spreading
Code
Generator
Code
1 to N i -1 span). The MGDC shows a
κ
Z
y [n]
Generator
Generator
higher robustness against the phase
(Pilot)
(Data)
90°
shift than the general difference comQ
sin(2π [fI + λ∆f ]n)
-1
bination (GDC) technique [8], which
1 Nco . RD
Σ ()
Q
Nco n = 0
coherently accumulates the differential
y B [n]
Q
coherently integrated detection variNco-1
R
1
P
Σ (.)
ables of all possible spans. The MGDC
Nco n = 0
technique is useful for combining partial correlation outputs of GNSS signals
FIGURE 1. A diagram for the correlation, combining, and integration function to test a hypothesis at
that have long spreading codes, such as
[ l, m ]. The dotted lines are for the pilot channel, the solid lines are for the data channel, and the
GPS L2C, without degrading the persuperscripts I and Q represent the in-phase and quadrature-phase components, respectively.
formance due to the bit transitions [9].
The SNR improvement with Z mgdc is
higher than Z nc but much lower than
Table 2. Detection variables of various integration techniques.
Z co because of the noise amplification
Detection Variable for the l th Code Phase and m th Dopplercaused by the multiplication of noise
Integration Techniques
Frequency Hypothesis
2
N -1
components in the two multiplied corCoherent
Z co 6l, m@ = / R 6h, l, m@
relation outputs.
h=0
Note that the detection variables in
N -1
2
Noncoherent
Z nc 6l, m@ = / R 6h, l, m@
the aforementioned integration techh=0
niques are different and have different
2
N -1
1
Differentially coherent
Z di 6l, m@ = / R 6h, l, m@ # R * 6h - 1, l, m@
statistical distributions (in the presh=1
ence and absence of the signal), and,
N -1
N -1 N -1
m
Z mgdc 6l, m@ = / Z di 6l, m@ = / / R 6h, l, m@ # R * 6h - m, l, m@
as a result, each of these integration
m=1
m=1 h=m
MGDC
techniques has different probabilities
of detection ( Pd ), misdetection ( Pm ),
and false alarm ( P f ) in the individual
Tb and noncoherently combines N i consecutive coherent correhypothesis tests. Also, note that the detection variables in
lation outputs by taking the sum of squares of each coherent corTable 2 have magnitude-squared function but can also have
relation output to build a detection variable Z nc. As a result, Z nc
different nonlinear function [3].
is immune to both the bit transition problem and the phase shift
in the consecutive correlation outputs. However, the resulting
Search strategy for lower acquisition complexity
increase in the signal peak amplitude after the N i times noncoIn general, the code-phase search step size ∆ x is set to a half-chip
herent accumulation is much less than N i times, and the tech( Tc /2 ) for BPSK signals (1a) so that the total number of codenique suffers from the squaring loss [3] for weak GNSS signals.
phase hypotheses N Hc is two times the code length L c . On the
other hand, the total number of Doppler-frequency hypotheses
D
/∆ f h + 1, since the Doppler-frequency search is
Differential integration technique
N Hf is 2 ^ f max
Another possible scheme to cope with the bit transition problem
from the negative to positive maximum Doppler frequencies, i.e.,
D
D
- f max
, with a step size ∆ f , where ^^ $ hh is the ceiling
is to multiply the prompt (at time instant h ) coherent correlation
to + f max
D
= 5 kHz (or more)
∆
h
1
)
output with the conjugate of the previous one (at time instant
operator. Since f = 1/ ^2Tco h [3] and f max
and coherently accumulate the consecutive results to build a difare one of the common choices, the maximum number of hypothD
D
/∆ f h + 1h , 4L c ^2f max
Tcoh .
ferential coherently integrated detection variable Z 1di [6], where
esis to test is N H = 2L c # ^2 ^ f max
D
f
∆
=
2
/
^
3
T
h
the superscript 1 represents the one span. This technique exploits
Note that f
co can be used [2], max =10 kHz is posthe fact that the signal components in consecutive correlation
sible for dynamic receivers, and the choice of ∆ f 2 1/ ^2Tco h
outputs (e.g., R 6h 1, x, m@ and R 6h 2, x, m@) that are separated by
causes larger SNR loss in the correlation output but smaller N H
a unit correlation interval (i.e., h 2 - h 1 = 1 span) have the same
than the choice of ∆ f = 1/ ^2Tco h, for example, ∆ f = 2/ ^3Tco h
(or similar) phase, while the noise components are independent.
may cause 1 dB lower SNR than ∆ f = 1/ ^2Tco h . For an examHowever, there can be a monotonous phase shift between the
ple with a simple GPS L1 C/A signal acquisition function using
consecutive correlation outputs, which can result in a signala single correlator with Tco = 1 ms and N i = 1, each of the
i
i
i
i
62
i
i
IEEE SIGNAL PROCESSING MAGAZINE
|
September 2017
|
Table of Contents for the Digital Edition of Signal Processing - September 2017
Signal Processing - September 2017 - Cover1
Signal Processing - September 2017 - Cover2
Signal Processing - September 2017 - 1
Signal Processing - September 2017 - 2
Signal Processing - September 2017 - 3
Signal Processing - September 2017 - 4
Signal Processing - September 2017 - 5
Signal Processing - September 2017 - 6
Signal Processing - September 2017 - 7
Signal Processing - September 2017 - 8
Signal Processing - September 2017 - 9
Signal Processing - September 2017 - 10
Signal Processing - September 2017 - 11
Signal Processing - September 2017 - 12
Signal Processing - September 2017 - 13
Signal Processing - September 2017 - 14
Signal Processing - September 2017 - 15
Signal Processing - September 2017 - 16
Signal Processing - September 2017 - 17
Signal Processing - September 2017 - 18
Signal Processing - September 2017 - 19
Signal Processing - September 2017 - 20
Signal Processing - September 2017 - 21
Signal Processing - September 2017 - 22
Signal Processing - September 2017 - 23
Signal Processing - September 2017 - 24
Signal Processing - September 2017 - 25
Signal Processing - September 2017 - 26
Signal Processing - September 2017 - 27
Signal Processing - September 2017 - 28
Signal Processing - September 2017 - 29
Signal Processing - September 2017 - 30
Signal Processing - September 2017 - 31
Signal Processing - September 2017 - 32
Signal Processing - September 2017 - 33
Signal Processing - September 2017 - 34
Signal Processing - September 2017 - 35
Signal Processing - September 2017 - 36
Signal Processing - September 2017 - 37
Signal Processing - September 2017 - 38
Signal Processing - September 2017 - 39
Signal Processing - September 2017 - 40
Signal Processing - September 2017 - 41
Signal Processing - September 2017 - 42
Signal Processing - September 2017 - 43
Signal Processing - September 2017 - 44
Signal Processing - September 2017 - 45
Signal Processing - September 2017 - 46
Signal Processing - September 2017 - 47
Signal Processing - September 2017 - 48
Signal Processing - September 2017 - 49
Signal Processing - September 2017 - 50
Signal Processing - September 2017 - 51
Signal Processing - September 2017 - 52
Signal Processing - September 2017 - 53
Signal Processing - September 2017 - 54
Signal Processing - September 2017 - 55
Signal Processing - September 2017 - 56
Signal Processing - September 2017 - 57
Signal Processing - September 2017 - 58
Signal Processing - September 2017 - 59
Signal Processing - September 2017 - 60
Signal Processing - September 2017 - 61
Signal Processing - September 2017 - 62
Signal Processing - September 2017 - 63
Signal Processing - September 2017 - 64
Signal Processing - September 2017 - 65
Signal Processing - September 2017 - 66
Signal Processing - September 2017 - 67
Signal Processing - September 2017 - 68
Signal Processing - September 2017 - 69
Signal Processing - September 2017 - 70
Signal Processing - September 2017 - 71
Signal Processing - September 2017 - 72
Signal Processing - September 2017 - 73
Signal Processing - September 2017 - 74
Signal Processing - September 2017 - 75
Signal Processing - September 2017 - 76
Signal Processing - September 2017 - 77
Signal Processing - September 2017 - 78
Signal Processing - September 2017 - 79
Signal Processing - September 2017 - 80
Signal Processing - September 2017 - 81
Signal Processing - September 2017 - 82
Signal Processing - September 2017 - 83
Signal Processing - September 2017 - 84
Signal Processing - September 2017 - 85
Signal Processing - September 2017 - 86
Signal Processing - September 2017 - 87
Signal Processing - September 2017 - 88
Signal Processing - September 2017 - 89
Signal Processing - September 2017 - 90
Signal Processing - September 2017 - 91
Signal Processing - September 2017 - 92
Signal Processing - September 2017 - 93
Signal Processing - September 2017 - 94
Signal Processing - September 2017 - 95
Signal Processing - September 2017 - 96
Signal Processing - September 2017 - 97
Signal Processing - September 2017 - 98
Signal Processing - September 2017 - 99
Signal Processing - September 2017 - 100
Signal Processing - September 2017 - 101
Signal Processing - September 2017 - 102
Signal Processing - September 2017 - 103
Signal Processing - September 2017 - 104
Signal Processing - September 2017 - 105
Signal Processing - September 2017 - 106
Signal Processing - September 2017 - 107
Signal Processing - September 2017 - 108
Signal Processing - September 2017 - 109
Signal Processing - September 2017 - 110
Signal Processing - September 2017 - 111
Signal Processing - September 2017 - 112
Signal Processing - September 2017 - 113
Signal Processing - September 2017 - 114
Signal Processing - September 2017 - 115
Signal Processing - September 2017 - 116
Signal Processing - September 2017 - 117
Signal Processing - September 2017 - 118
Signal Processing - September 2017 - 119
Signal Processing - September 2017 - 120
Signal Processing - September 2017 - 121
Signal Processing - September 2017 - 122
Signal Processing - September 2017 - 123
Signal Processing - September 2017 - 124
Signal Processing - September 2017 - 125
Signal Processing - September 2017 - 126
Signal Processing - September 2017 - 127
Signal Processing - September 2017 - 128
Signal Processing - September 2017 - 129
Signal Processing - September 2017 - 130
Signal Processing - September 2017 - 131
Signal Processing - September 2017 - 132
Signal Processing - September 2017 - 133
Signal Processing - September 2017 - 134
Signal Processing - September 2017 - 135
Signal Processing - September 2017 - 136
Signal Processing - September 2017 - 137
Signal Processing - September 2017 - 138
Signal Processing - September 2017 - 139
Signal Processing - September 2017 - 140
Signal Processing - September 2017 - 141
Signal Processing - September 2017 - 142
Signal Processing - September 2017 - 143
Signal Processing - September 2017 - 144
Signal Processing - September 2017 - 145
Signal Processing - September 2017 - 146
Signal Processing - September 2017 - 147
Signal Processing - September 2017 - 148
Signal Processing - September 2017 - 149
Signal Processing - September 2017 - 150
Signal Processing - September 2017 - 151
Signal Processing - September 2017 - 152
Signal Processing - September 2017 - 153
Signal Processing - September 2017 - 154
Signal Processing - September 2017 - 155
Signal Processing - September 2017 - 156
Signal Processing - September 2017 - 157
Signal Processing - September 2017 - 158
Signal Processing - September 2017 - 159
Signal Processing - September 2017 - 160
Signal Processing - September 2017 - 161
Signal Processing - September 2017 - 162
Signal Processing - September 2017 - 163
Signal Processing - September 2017 - 164
Signal Processing - September 2017 - 165
Signal Processing - September 2017 - 166
Signal Processing - September 2017 - 167
Signal Processing - September 2017 - 168
Signal Processing - September 2017 - 169
Signal Processing - September 2017 - 170
Signal Processing - September 2017 - 171
Signal Processing - September 2017 - 172
Signal Processing - September 2017 - 173
Signal Processing - September 2017 - 174
Signal Processing - September 2017 - 175
Signal Processing - September 2017 - 176
Signal Processing - September 2017 - 177
Signal Processing - September 2017 - 178
Signal Processing - September 2017 - 179
Signal Processing - September 2017 - 180
Signal Processing - September 2017 - 181
Signal Processing - September 2017 - 182
Signal Processing - September 2017 - 183
Signal Processing - September 2017 - 184
Signal Processing - September 2017 - 185
Signal Processing - September 2017 - 186
Signal Processing - September 2017 - 187
Signal Processing - September 2017 - 188
Signal Processing - September 2017 - 189
Signal Processing - September 2017 - 190
Signal Processing - September 2017 - 191
Signal Processing - September 2017 - 192
Signal Processing - September 2017 - 193
Signal Processing - September 2017 - 194
Signal Processing - September 2017 - 195
Signal Processing - September 2017 - 196
Signal Processing - September 2017 - Cover3
Signal Processing - September 2017 - Cover4
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