2013-03-06
항법 및 측위 수신기 신호처리의 기초
2013.3.5
한국항공대학교 항공전자및정보통신공학부
이형근
2013-03-06
Basics
Sine/Cosine 관련 법칙
cos sinje j
sin( ) sin( )cos( ) cos( )sin( )
cos( ) cos( )cos( ) sin( )sin( )
A B A B A B
A B A B A B
1 1
sin , cos2 2
j j j je e e ej
Sine/Cosine 법칙
Euler 법칙
복소수와 삼각함수 사이의 관계
Time Domain Signal vs Phasor
Assumption : Only known, fixed, and single frequency component exists
Acos(2 ) sin(2 )jAe A ft jA ft
cos(2 )A A ft
Auto/Cross Correlation
( ) ( ) *( )xR t x x t d
( ) ( ) *( )yxR t y x t d
Auto-correlation on infinite-horizon (aperiodic)
Cross-correlation on infinite-horizon (aperiodic)
/ 2
/ 2
1( ) lim ( ) *( )
TT
xTT
R t x x t dT
/ 2
1( ) lim ( ) *( )
TT
yxTT
R t y x t dT
Auto-correlation on finite-horizon (periodic)
Cross-correlation on finite-horizon (periodic)
Sine/Cosine Formulae for Correlations
/ 2
/ 2
/ 2 / 22
/ 2 / 2
1sin(2 )sin(2 )
1 1 1 cos(4 ) 1 sin (2 )
2 2
T
T
T T
T T
mft mft dT
mftmft d d
T T
/ 2
/ 2
/ 2 / 22
/ 2 / 2
1cos(2 )cos(2 )
1 1 1 cos(4 ) 1 cos (2 )
2 2
T
T
T T
T T
mft mft dT
mftmft d d
T T
/ 2
/ 2
/ 2
/ 2
1sin(2 )cos(2 )
1 sin(4 ) 0
2
T
T
T
T
mft mft dT
mftd
T
1T
mf
/ 2
/ 2
1sin(2 )sin(2 ) 0
T
Tmft nft d
T
/ 2
/ 2
1sin(2 )cos(2 ) 0
T
Tmft nft d
T
/ 2
/ 2
1cos(2 )cos(2 ) 0
T
Tmft nft d
T
1, least common multiple of and T p m n
pf
Fourier Series with Real Coefficients
0
1 1
1 1
( ) cos 2 sin 2 2
n n
n n
ax t a nf t b nf t
1
0
1
0
2( )cos 2 0,1,2,...
2( )sin 2 0,1,2,...
T
n
T
n
a x t nf t dt nT
b x t nf t dt nT
Fourier Series with Complex Coefficients
0
1
1
2 12
cos 2 1, 2,3,...2
where: and tan
n n
n
nn n n n
n
Cx(t) C nf t n
bC a ba
Fourier Transform
2( ) ( ) ( ) j ftX f F x t x t e dt
1 2( ) ( ) ( )
1 ( )
2
j ft
j t
x t F X f X f e df
X j e d
2 f
Laplace Transform
( ) ( ) ( ) stX s L x t x t e dt
1( ) ( ) ( ) stx t L X s X s e ds
- Laplace transform is applicable to any signal - Fourier transform is applicable only to periodic signal
Parseval’s Theorem
2 2| ( ) | | ( ) |x t dt X f df
Energy/Power in Time Domain
/ 2 2
/ 2
1lim ( )
T
TTP x t dt
T
(Average) Power In Time Domain
Energy In Time Domain
2( )E x t dt
Energy/Power Spectral Density in Frequency Domain
( ) [ ( ) *( ) ] [ ( )]x xESD f F x x t d F R t
Energy Spectral Density
Power Spectral Density
/ 2
/ 2
1( ) [lim ( ) *( ) ] [ ( )]
TT
x xTT
PSD f F x x t d F R tT
Time Domain Response by Convolution Integral
( ) ( ) ( ) ( ) ( ) ( ) ( )y t h t x t h t x d h x t d
Linear Time Invariant System A
( )t ( )h t
Linear Time Invariant System A
( )x t ( )?y t
Impulse input Impulse response
Arbitrary input output
Frequency Domain Response by Transfer Function
( ) ( ) ( )Y f H f X f
Linear Time Invariant System A
1 [ ( )]F t ( ) ( )H f F h t
Linear Time Invariant System A
( ) ( )X f F x t ( ) ( ) ?Y f F y t
Impulse input Transfer Function
Arbitrary input output
Input/Output Cross-Correlation and Transfer Function
/ 2
/ 2
/ 2
/ 2
/ 2
/ 2
1( ) lim ( ) *( )
1 lim ( ) *( ) *( )
1 *( ) lim ( ) *( )
*( ) ( ) *( ) [ ( )]
TT
xyTT
T
TT
T
TT
T T
x x
R t x y t dT
x x t h d dT
h x x t d dT
h R t d h R t d
*( ) ( )T
xh t R t
( ) *( ) ( )T T
xy xR t h t R t ( ) *( ) ( )T T
xy xR t h t R t
In summary,
Output Auto-Correlation and Transfer Function
/ 2
/ 2
/ 2
/ 2
/ 2
/ 2
1( ) lim ( ) *( )
1 lim *( ) ( ) ( )
1 ( ) lim ( ) *( )
( ) ( ) ( ) [ ( )]
TT
yTT
T
TT
T
TT
T T
xy xy
R t y y t dT
y t x h d dT
h x y t d dT
h R t d h R t d
( ) ( ) ( ) *( ) ( )T T
xy xh t R t h t h t R t
( ) ( ) *( ) ( )T T
y xR t h t h t R t
In summary,
Input/Output Power Spectral Density and Transfer Function
2
( ) ( ) ( ) *( ) ( )
( ) *( ) ( )
( ) *( ) ( )
( ) ( )
T T
y y x
T
x
x
x
PSD f F R t F h t h t R t
F h t F h t F R t
H f H f PSD f
H f PSD f
2( ) ( ) ( )y xPSD f H f PSD f
In summary,
Frequency Response : Bode Plot of Transfer Function
2 : angular frequencyf
Magnitude Plot
Phase Plot
Frequency Response Examples
low-pass filter high-pass filter
band-reject filter band-pass filter
Modulation (Transmitter)
( ) cos(2 )c cX t A f t
''( ) cos 2 : sum2
cos 2 : difference2
c m
c m
ABY t f f t
ABf f t
Modulating Signal
(ex: Music)
Carrier Signal (ex: LF band)
( ) cos(2 )m mX t B f t
Band-Pass Filter
'( ) cos 22
c m
ABY t f f t
( ) cos 2 c mY t K f f t
modulator
( )c mf f
Demodulation (Receiver)
''( ) cos 2 ( ) : sum 2
cos 2 ( ) : difference2
osc amp
c m c
osc amp
c m c
K KS t f f f t
K Kf f f t
'( ) cos 2amp c mY t K f f t Local Oscillator
( ) cos 2osc osc cX t K f t
( ) cos 2 c mY t K f f t
demodulator
Band-Pass Filter ( )mf
' ) cos 22
osc amp
m
K KS t f t
( ) cos 2 mS t M f t
2013-03-06
GNSS Receiver
25
Receiver Functions
□ Receives RF signal
□ RF (Radio freq.) / IF (Intermediate freq.) down conversion
□ Signal Acquisition
– 2 dimensional search in time and freq. domains
□ Signal Tracking
– DLL for code tracking
– FLL or PLL for carrier tracking
□ Bit & Frame Synchronization
□ Generation of PRs, CRs, and DPPLs
□ Computation of position, velocity, and time
26
Example : GPS Receiver
27
More Detailed Receiver Architecture
28
Basic Crystal Oscillator
Tuning
Voltage
Crystal
resonator
Amplifier
Output
Frequency
29
SIGNAL CONDITIONING
Signal processing portions of a receiver
30
□ Super-Heterodyne
– A "heterodyne" refers to a beat or "difference" frequency produced when two
or more external radio frequency carrier waves are fed to a detector. The term
was coined by Canadian Engineer Reginald Fessenden in early 1900s.
– Later, when vacuum triodes became available, the same result could be
achieved more conveniently by incorporating a "local oscillator" in the receiver.
– In December, 1919, Major E. H. Armstrong gave publicity to an indirect method
of obtaining short-wave amplification, called the super-heterodyne.
31
Frequency down conversion (heterodyning)
( cos )( cos ) cos( ) cos( )2
where
cos 2 ( ) ( ) cos 2 ( ) : received signal
cos 2 cos 2 ( ) : receiver-internally-generated signal
: elapesed time for the propagation from
L D
L IF IF
ABA B
A Cx t D t f f t
B f f t
transmitter to receiver
: carrier frequency
: doppler frequency
: intermediate frequency ( )
, : phase offsets
L
D
IF IF L
IF
f
f
f f f
32
1 to 4bit ADC
Baseband
LNA AGC BPF
SiGe/Bipolar CMOS
Frequecy Synthesizer
BPF
wLO1 wLO2
IF Freq.: 1.023 MHz
RF+
RF-
Block Diagram of RF/IF Down Conversion
33
□ Mixing – Combination of muliplying and filtering – Generates a signal modulated by the intermediate freqeuncy
cos( ) ( ) ( ) cos 2 (2 )2
(removed by a low-pass filter)
cos( ) ( ) ( ) cos 2 ( )2
(down-coverted signa
L IF D IF
IF D IF
ABCx t D t f f f t
ABCx t D t f f t
l)
34
Multiple mixers and intermediate frequency stages are often employed
35
□ Sampling
– Many receivers use a simple AD converter and these have four levels of
quantization
– Such receivers do not suffer much SNR loss relative to a receiver without
quantization but they require automatic gain control (AGC)
– The AGC ensures that the signal amplitude is spread among the quatization
boundaries of the A/D
– If the sample rate is high enough, then the samples can be used to reconstruct
the original signal
36
Signal
1-bit quantized signal
* Misconception: One-bit sampling does not provide information on
signal amplitude.
–A/D Conversion: One-bit sampling
37
Signal with noise 1-bit quantized signal [average of many samples]
* In Reality: With help of noise, ensemble average of one-bit samples gives
signal amplitude.
38
Qaudrature Signals □ Inphase and Quadrature Processing and Doppler Removal
– With respect to the input signal
– The receiver internally generates two reference signals
– After low-pass filtering, the outputs from the inphase and quadrature channels are (carrier wipeoff)
( ) ( )cos 2 ( )IF DCD t x t f f t
2 cos 2 ( ) : inphase reference
2 sin 2 ( ) : quadrature reference
IF D
IF D
f f t
f f t
( ) ( )cos(2 ) : inphase
( ) ( )sin(2 ) : quadrature
where
D
D
D D D
CD t x t f t
CD t x t f t
f f f
39
Numerically Controlled Oscillator
Integrate & Dump
Integrate & Dump
)cos( tIF
)cos()( ttD IF
)sin( tIF
Quadrature-phase arm
In-phase arm
Incoming Signal Loop Filter Discriminator
o90
40
Locked State Unlocked State
Correlator Outputs at (Un)Locked States
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