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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