1 lecture 6: doppler techniques: physics, processing, interpretation

1

Lecture 6: Doppler Techniques:

Physics, processing, interpretation

2

Doppler US Techniques

• As an object emitting sound moves at a velocity v,

• the wavelength of the sound in the forward direction is compressed (λs) and

• the wavelength of the sound in the receding direction is elongated (λl).

• Since frequency (f) is inversely related to wavelength, the compression increases the perceived frequency and the elongation decreases the perceived frequency.

• c = sound speed.

3

Doppler US Techniques                                                        

•In Equations (1) and (2), f is the frequency of the sound emitted by the object and would be detected by the observer if the object were at rest. ±Δf represents a Doppler effect–induced frequency shift •The sign depends on the direction in which the object is traveling with respect to the observer. •These equations apply to the specific condition that the object is traveling either directly toward or directly away from the observer

4

Doppler US Techniques

f f fvf

cd t rt

2 cos

vf c

fd

t

2 cos

ft is transmitted frequencyfr is received frequencyv is the velocity of the target, θ is the angle between the ultrasound beam and the direction of the target's motion, and c is the velocity of sound in the medium

5

A general Doppler ultrasound signal measurement system

Transmission&

Reception ProcessingDisplay &ElectricalFurtherprocessing

in

out

acousticalenergy

electricalenergy

audio-visual displaystoreprint etc.

6

A simplified equivalent representation of an ultrasonic transducer

Electrical Mechanical

V FZe Zm

7

Block diagram of a non-directional continuous wave Doppler system

8

Block diagram of a non-directional pulsed wave Doppler system

9

Oscillator

7 1 4

856

0.1uF 0.1uF

0.1uF

xtal(4 MHz)

MC12061

3

2

sineout

-

+

Note: All unused pins should be connected to ground

totransmitter

tomixer

100pF

100pF

C

R

R

C

sin

cos

78L05+12.0 V

0.1uF0.1uF12

3

CfR

12

10

Transmitter

+12V 56k

9k1

680k

910R

2:1BC107

BFW11

0.1uF

100R

+12V

+12V

56k

9k1

910R

0.1uF

0.1uF

BC107

toprobe

fromosc.

+12V

11

Demodulator

1k

1k 1k

1k

3k 3k

1k3820

330

.1uF

.1uF

.1uF

.1uF

4.7nF

4.7nF 4.7nF

1uF

10k

8

10

1

4

2 3

6

12

14 5

100

MC1496

1k

1k 1k

1k

3k 3k

1k3820

330

.1uF

.1uF

.1uF

.1uF

4.7nF

4.7nF 4.7nF

1uF

10k

8

10

1

4

2 3

6

12

14 5

100

MC1496

1k

4.7nF

1uF

1k

4.7nF

1uF

+12 V

+12 V

DIFF.

OUTPUTsin

cosDIFF.OUTPUT

FROMPROBE

.1uF

.1uF

12

Two channel differential audio amplifier

10

11

1213 1

6

7

3

5

4

14

+12V

-12V

10k 1%

R

150k

5pF

C2 C1

30pF

50pFC3

0.1u0.1u

200pF

10k 10k 10k 1%

SSM-2015

R2 BIAS

RG

-IN

+IN

+RG

-RG

OUT

COMP2

COMP3

+V

-VBIAS

+IN

-IN

to BPF

R1

10

11

1213 1

6

7

3

5

4

14

+12V

-12V

10k 1%

R

150k

5pF

C2 C1

30pF

50pFC3

0.1u0.1u

200pF

10k 10k 10k 1%

SSM-2015

R2 BIAS

RG

-IN

+IN

+RG

-RG

OUT

COMP2

COMP3

+V

-VBIAS

+IN

-IN

to BPF

R1

9

9

13

Programmable bandpass filter and amplifier

14

the audio amplifier

10k

33k

10R

3

0.1uF

33k 0.1uF

0.1uF

LM386

+5 V

100uF

0.1uF

10uF

0.22uF 0.22uF

+

-2

18

65

4

7

8R

AUDIO

15

PC interface

16

17

PROCESSING OF DOPPLER ULTRASOUND SIGNALS

18

Processing of Doppler Ultrasound SignalsGated

transmiterMaster

osc.Band-pass

filterSample& hold

Demodulator

Receiveramplifier

RFfilter

Band-passfilter

Sample& hold

Demodulator

Logicunit

si

sq

V

Transducer

sincos

• Demodulation• Quadrature to directional signal conversion• Time-frequency/scale analysis• Data visualization• Detection and estimation• Derivation of diagnostic information

Further processing

19

Single side-band detection

USBF

LSBF

LPF

LPF

RF signal

y f

y r

cos t0

20

Heterodyne detection

LSBF

LPF

Master

RF signal

Transmitteroscillator

2-20 MHz

Heterodyneoscillator1-10 kHz

o h

o h

o h

o d h d

d hd h

:forward:reverse

21

Frequency translation and side-band filtering detection

RF signal

Trans.

Oscillator12-20 MHz

o

h

LPF1

o h

o d h d

h d

h d

Oscillator22-20 MHz

LPF5

HPF

LPF2

o

o h

Optional stage

yf

yr

LPF3

LPF4

22

direct sampling • Effect of the undersampling.

(a) before sampling; (b) after sampling

fs 2fs 3fs 4fs nfs

fs 2fs 3fs 4fs nfs

(a)

(b)

23

Quadrature phase detection

LPF

LPF

RF signal cos t0

sin t0

Quad.signaloscillator

y D

y Q

24

TOOLS FOR DIGITAL SIGNAL PROCESSING

25

Understanding the complex Fourier transform

• The Fourier transform pair is defined as

• In general the Fourier transform is a complex quantity:

• where R(f) is the real part of the FT, I(f) is the imaginary part, X(f) is the amplitude or Fourier spectrum of x(t) and is given by ,θ(f) is the phase angle of the Fourier transform and given by tan-1[I(f)/R(f)]

dfefXtxdtetxfX ftjftj 22 )()(,)()(

)()()()()( fjefXfjIfRfX

)()( 22 fIfR

26

• If x(t) is a complex time function, i.e. x(t)=xr(t)+jxi(t) where xr(t) and xi(t) are respectively the real part and imaginary part of the complex function x(t), then the Fourier integral becomes

)()(]cos)(sin)([

]sin)(cos)([)]()([)( 2

fjIfRdtttxttxj

dtttxttxdtetjxtxfX

ir

irftj

ir

)}()({}{cos 000 tF

)}()({}{sin 000 j

tF

27

Properties of the Fourier transform for complex time functions

Time domain (x(t)) Frequency domain (X(f))

Real Real part even, imaginary part odd

Imaginary Real part odd, imaginary part even

Real even, imaginary odd Real

Real odd, imaginary even Imaginary

Real and even Real and even

Real and odd Imaginary and odd

Imaginary and even Imaginary and even

Imaginary and odd Real and odd

Complex and even Complex and even

Complex and odd Complex and odd

28

Interpretation of the complex Fourier transform

• If an input of the complex Fourier transform is a complex quadrature time signal (specifically, a quadrature Doppler signal), it is possible to extract directional information by looking at its spectrum.

• Next, some results are obtained by calculating the complex Fourier transform for several combinations of the real and imaginary parts of the time signal (single frequency sine and cosine for simplicity).

• These results were confirmed by implementing simulations.

29

• Case (1).

• Case (2).

• Case (3).

• Case (4).

• Case (5).

• Case (6).

• Case (7).

• Case (8).

1

-1

real part of complex FFT

0 +Fs/2-Fs/2

1

-1

imaginary part of complex FFT(1)

1

-1

real part of complex FFT

1

-1

imaginary part of complex FFT(5)

1

-1

real part of complex FFT

1

-1

imaginary part of complex FFT(2)

1

-1

real part of complex FFT

1

-1

imaginary part of complex FFT(6)

1

-1

real part of complex FFT

1

-1

imaginary part of complex FFT(3)

1

-1

real part of complex FFT

1

-1

imaginary part of complex FFT(7)

1

-1

real part of complex FFT

1

-1

imaginary part of complex FFT(4)

1

-1

real part of complex FFT

1

-1

imaginary part of complex FFT(8)

0 +Fs/2-Fs/2

0 +Fs/2-Fs/2

0 +Fs/2-Fs/2

0 +Fs/2-Fs/2

0 +Fs/2-Fs/2

0 +Fs/2-Fs/2

0 +Fs/2-Fs/2 0 +Fs/2-Fs/2

0 +Fs/2-Fs/2

0 +Fs/2-Fs/2

0 +Fs/2-Fs/2

0 +Fs/2-Fs/2

0 +Fs/2-Fs/2

0 +Fs/2-Fs/2

0 +Fs/2-Fs/2

x t t x t t

X F t jF tr i( ) cos , ( ) sin ,

( ) {cos } {sin } ( )

0 0

0 0 02

x t t x t t

X F t jF tr i( ) cos , ( ) sin ,

( ) {cos } { sin } ( )

0 0

0 0 02

x t t x t t

X F t jF tr i( ) cos , ( ) sin ,

( ) { cos } {sin } ( )

0 0

0 0 02

x t t x t t

X F t jF tr i( ) cos , ( ) sin ,

( ) { cos } { sin } ( )

0 0

0 0 02

x t t x t t

X F t jF t jr i( ) sin , ( ) cos ,

( ) {sin } {cos } ( )

0 0

0 0 02

x t t x t t

X F t jF t jr i( ) sin , ( ) cos ,

( ) {sin } { cos } ( )

0 0

0 0 02

x t t x t t

X F t jF t jr i( ) sin , ( ) cos ,

( ) { sin } {cos } ( )

0 0

0 0 02

x t t x t t

X F t jF t jr i( ) sin , ( ) cos ,

( ) { sin } { cos } ( )

0 0

0 0 02

30

The discrete Fourier transform

• The discrete Fourier transform (DFT) is a special case of the continuous Fourier transform. To determine the Fourier transform of a continuous time function by means of digital analysis techniques, it is necessary to sample this time function. An infinite number of samples are not suitable for machine computation. It is necessary to truncate the sampled function so that a finite number of samples are considered

31

Discrete Fourier transform pair

otherwise

Nkenx

kX

N

n

knNj

,0

10,)(

)(

1

0

)/2(

otherwise

NnekXN

nx

N

n

knNj

,0

10,)(1

)(

1

0

)/2(

32

complex modulation

x t e X f fj tc

c( ) ( )

X(f)

-W Wf

0 0

X(f-f )

f -W f f +Wf

c

c c c

)}()({}{cos 000 tF

)}()({}{sin 000 j

tF

33

Hilbert transform

• The Hilbert transform (HT) is another widely used frequency domain transform.

• It shifts the phase of positive frequency components by -900 and negative frequency components by +900.

• The HT of a given function x(t) is defined by the convolution between this function and the impulse response of the HT (1/πt).

d

t

x

ttxtxH

)(11)()]([

34

Hilbert transform

• Specifically, if X(f) is the Fourier transform of x(t), its Hilbert transform is represented by XH(f), where

• A ± 900 phase shift is equivalent to multiplying by

ej900=±j, so the transfer function of the HT HH(f) can

be written as

)()sgn()()()]([)( fXfjfXfHfXHfX HH

0,

0,sgn)(

fj

fjfjfH H

35

impulse response of HT

0,)2/(sin20,0

)( 2

nn

nn

nhH

1

1/31/5

1 2 3 4 5 6 n

-1-2-3-4-5

-1

-1/3-1/5

-6

2

h (n)H

An ideal HT filter can be approximated using standard filter design techniques. If a FIR filter is to be used , only a finite number of samples of the impulse response suggested in the figure would be utilised.

36

• x(t)ejωct is not a real time function and cannot occur as a communication signal. However, signals of the form x(t)cos(ωt+θ) are common and the related modulation theorem can be given as

• So, multiplying a band limited signal by a sinusoidal signal translates its spectrum up and down in frequency by fc

x t te

X f fe

X f fc

j

c

j

c( ) cos( ) ( ) ( )

2 2

37

Digital filtering• Digital filtering is one of the most important DSP tools. • Its main objective is to eliminate or remove unwanted

signals and noise from the required signal. • Compared to analogue filters digital filters offer sharper

rolloffs, • require no calibration, and • have greater stability with time, temperature, and power

supply variations. • Adaptive filters can easily be created by simple software

modifications

h(k), k=0,1,...(impulse response)

x(n) y(n)(input) (output)

38

Digital Filters• Non-recursive (finite impulse response, FIR)

• Recursive (infinite impulse response, IIR).

• The input and the output signals of the filter are related by the convolution sum.

• Output of an FIR filter is a function of past and present values of the input,

• Output of an IIR filter is a function of past outputs as well as past and present values of the input

y n h k x n kk

N( ) ( ) ( )

0

1

y n h k x n k a x n k b y n kk

kk

N

kk

N( ) ( ) ( ) ( ) ( )

0 0 1

39

Basic IIR filter and FIR filter realisations

(a) (b)

Z-1

y(n)

x(n) a0

a1-b1

Z-1

-bN

Z-1

y(n)

x(n)Z

-1Z

-1

h(0) h(1) h(2) h(N-1)

Z-1

a2-b2

aN

x(n-1) x(n-2) x(n-N+1)

40

DSP for Quadrature to Directional Signal Conversion

• Time domain methods– Phasing filter technique (PFT) (time domain Hilbert

transform)– Weaver receiver technique

• Frequency domain methods – Frequency domain Hilbert transform – Complex FFT– Spectral translocation

• Scale domain methods (Complex wavelet)

• Complex neural network

41

GENERAL DEFINITION OF A QUADRATURE DOPPLER SIGNAL

• A general definition of a discrete quadrature Doppler signal equation can be given by

• D(n) and Q(n), each containing information concerning forward channel and reverse channel signals (sf(n) and sr(n) and their Hilbert transforms H[sf(n)] and H[sr(n)]), are real signals.

)()]([)(

)]([)()(

nsnsHnQ

nsHnsnD

rf

rf

42

Asymmetrical implementation of the PFT DSP Algorithm

D(n)

Q(n)

HILBERT

TRANSFORM

DELAY

FILTER

+

+

+

-

y (n)

y (n)

f

rH D n H s n H s n H s n s nf r f r[ ( )] [ ( ) [ ( )]] [ ( )] ( )

)()]([)()]([)]([)()(

)()]([)()]([)]([)()(

nsnsHnsnsHnDHnQny

nsnsHnsnsHnDHnQny

rfrfr

rfrff

)(2)(

)]([2)(

nsny

nsHny

rr

ff

)()]([)(

)]([)()(

nsnsHnQ

nsHnsnD

rf

rf

43

Symmetrical implementation of the PFT

)()]([)(

)]([)()(

nsnsHnQ

nsHnsnD

rf

rf

HILBERTTRANS.

DELAYFILTER

HILBERTTRANS.

DELAYFILTER

DSP Algorithm

D(n)

Q(n)

y (n)

y (n)

f

r

H Q n H H s n s n s n H s nf r f r[ ( )] [ [ ( )] ( )] ( ) [ ( )] H D n H s n H s n H s n s nf r f r[ ( )] [ ( ) [ ( )]] [ ( )] ( )

)]([2)]([)()(

)]([2)]([)()(

nsHnQHnDny

nsHnDHnQny

rr

ff

)(2)]([)()(

)(2)]([)()(

nsnDHnQny

nsnQHnDny

rr

ff

44

• An alternative algorithm is to implement the HT using phase splitting networks

• A phase splitter is an all-pass filter which produces a quadrature signal pair from a single input

• The main advantage of this algorithm over the single filter HT is that the two filters have almost identical pass-band ripple characteristics

Q(n)

yf(n)

Phase splitter

yr(n)

DSP Algorithm

D(n)

Phase splitter

+450

+450

-45 0

-45 0

(900)

(900)

45

Weaver Receiver Technique (WRT)

• For a theoretical description of the system consider the quadrature Doppler signal defined by

which is band limited to fs/4, and a pair of quadrature pilot frequency signals given by

where ωc/2π=fs/4. • The LPF is assumed to be an ideal LPF

having a cut-off frequency of fs/4.

)()]([)(

)]([)()(

nsnsHnQ

nsHnsnD

rf

rf

p n n p n nd c q c( ) sin , ( ) cos

46

Asymmetrical implementation of the WRT

+

+

+

-

D(n)

Q(n)

DSP Algorithm

yf(n)

yr(n)

LPF

LPF

LPF

LPF

fp=fc=fs/4

X1

Y1 Y2

X2 X3

Y3

pd

(n)

pq(n)

p n n p n nd c q c( ) sin , ( ) cos

)()]([)(

)]([)()(

nsnsHnQ

nsHnsnD

rf

rf

47

X Y D n p n Q n p nd q2 2, ( ). ( ) ( ). ( )

{ ( ).sin [ ( )].sin } { [ ( )].cos ( ) cos }s n n H s n n H s n n s n nf c r c f c r c

F s n S and F s n Sf f r r{ ( )} ( ) { ( )} ( )

0),(

0),()]([)]}([{

f

fff jS

jSSHnsHF

0),(

0),()]([)]}([{

r

rrr jS

jSSHnsHF

0),()(

0),()(

ff

ff

SS

SS

0),()(

0),()(

rr

rr

SS

SS

)()()(

)()()(

rrr

fff

SSS

SSS

)()()]([

)()()]([

rrr

fff

jSjSSH

jSjSSH

48

F X jS jS S Sf c f c r c r c{ } { ( ) ( )} { ( ) ( )}2 H S Sf c r c[ ( )] ( )

F Y jS jS S Sf c f c r c r c{ } { ( ) ( )} { ( ) ( )}2 H S Sf c r c[ ( )] ( )

F X H S jS jSf c f c f c{ } [ ( )] ( ) ( )2

F Y S S Sr c r c r c{ } ( ) ( ) ( )2

F X S S S Sf f f c f c{ } ( ) ( ) ( ) ( )31

2

1

2

1

22

1

22

1

2

1

22S Sf f c( ) ( )

F Y S S S Sr r r c r c{ } ( ) ( ) ( ) ( )31

2

1

2

1

22

1

22

1

2

1

22S Sr r c( ) ( )

).(2

1)}({

),(2

1)}({

rr

ff

SnyF

SnyF

).(2

1)}(

2

1{)(

),(2

1)}(

2

1{)(

1

1

nsSFny

nsSFny

rrr

fff

49

1

2

3

4

5

6

7

8

9

10

0 +f-f

Lowpass filter cut-off frequency=f , f =f /4

Stop-band region Pass-band region

D

Q

X1

Y1

X2

Y2

yf

yr

pd

pq

ff +fr-fr +fc +2fc-fc-2fc

c c s

sin cos cos sin f r f rt t j t j t

50

Symmetrical implementation

LPF

LPF

LPF

LPF

D(n)

Q(n)

DSP Algorithm

pd(n)p

q(n)

yf(n)

yr(n)

A1

B1

A2

B2

51

Implementation of the WRT algorithm using low-pass/high-pass filter pair

D(n)

Q(n)

DSP Algorithm

yf(n)

yr(n)

LPF

HPF

LPF

LPF

fp=fc=fs/4

X1

X2

X3

pd(n)

pq(n)

52

FREQUENCY DOMAIN PROCESSING

• These algorithms are almost entirely implemented in the frequency domain (after fast Fourier transform),

• They are based on the complex FFT process. • The common steps for the all these implementations

are the complex FFT, the inverse FFT and overlapping techniques to avoid Gibbs phenomena

• Three types of frequency domain algorithm will be described:

Hilbert transform method, Complex FFT method, and Spectral translocation method.

53

frequency domain Hilbert transform algorithm

D(n)

Q(n)

CFFT IFFT

S R

S I

H[S]R

I

yf(n)

yr(n)

DSP Algorithm

Q'(n)

D'(n)

Frequency domain complex HT algorithm

-jS, w>0

+jS, w<0 H[S]

54

Complex FFT Method (CFFT)

• The complex FFT has been used to separate the directional signal information from quadrature signals so that the spectra of the directional signals can be estimated and displayed as sonograms.

• It can be shown that the phase information of the directional signals is well preserved and can be used to recover these signals.

55

D(n)

Q(n)

CFFT

IFFTS R

S I

y (n)

y (n)

DSP Algorithm

IFFT

S (w)

S (w)

S fR

S fI

S rR

S rI

f

r

f

r

56

s n D n jQ n s n H s n j H s n s n

s n jH s n j s n jH s n

f r f r

f f r r

( ) ( ) ( ) { ( ) [ ( )]} { [ ( )] ( )}

{ ( ) [ ( )]} { ( ) [ ( )]}

0)},()({)}()({

0)},()({)}()({)}({[

rrff

rrff

SSjSS

SSjSSnsF

0),(2

0),(2)()}({

r

f

Sj

SSnsF

0,0

0),()(

S

S

0),(

0,0)(

S

S

0)},({

0)},({)}({

S

SS f

0)},({

0)},({)}({

S

SS f

0)},({

0)},({)}({

S

SSr

0)},({

0)},({)}({

S

SSr

57

Spectral Translocation Method (STM)

XR-

= S I+

XR+

= -S I-

XI-

= -S R+

XI+

= S R-

FFT

IFFT

D(n)

Q(n)

yf(n)

yr(n)

S R

S I

X R

X I

Y R

Y I

DSP Agorithm

58

S S j S S S j S Sf r( ) ( ) ( ) { ( ) ( )} { ( ) ( )} 2 2

S S S S ( ) ( ), ( ) ( )

X S S j S S

S S j S S

( ) { ( ) ( )} { ( ) ( )}

{ ( ) ( )} { ( ) ( )}

2 2 2 2

2 2

2 2

S j S S j S

S S j S S

S j S

f r f r

f f r r

f r

( ) ( ) ( ) ( )

{ ( ) ( )} { ( ) ( )}

( ) ( )

Y S X

S X j S X

( ) ( ) ( )

{ { ( )} { ( )}} { { ( )} { ( )}}

y n s n j s nf r( ) ( ) ( ) 2 2

0),(2

0),(2)()}({

r

f

Sj

SSnsF