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