Transcript Document

Lecture 6:
Doppler Techniques:
Physics, processing, interpretation
1
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.
2
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
3
Doppler US Techniques
fd  f t  f r 
2vf t cos 
c
fd c
v
2 ft cos 
ft is transmitted frequency
fr is received frequency
v 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
4
A general Doppler ultrasound signal
measurement system
acoustical
energy
electrical
energy
audio-visual display
store
print etc.
out
in
Transmission
&
Reception
Electrical
Processing
Display &
Further
processing
5
A simplified equivalent representation of
an ultrasonic transducer
Electrical
V
Ze
Mechanical
Zm
F
6
Block diagram of a non-directional
continuous wave Doppler system
7
Block diagram of a non-directional pulsed
wave Doppler system
8
Oscillator
2
0.1uF
0.1uF
0.1uF
78L05
1
3
+12.0 V
0.1uF
R
7
1
0.1uF
4
sin
-
3
C
sine
out
MC12061
+
C
6
5
1
2 fR
to
mixer
2
to
transmitter
100pF
R
100pF
8
cos
C
xtal(4 MHz)
Note: All unused pins should be connected to ground
9
Transmitter
+12V
+12V
56k
910R
BC107
0.1uF
9k1
BFW11
from
osc.
100R
0.1uF
+12V
2:1
to
probe
0.1uF
680k
9k1
910R
BC107
56k
+12V
10
Demodulator
1k
820
+12 V
1k3
.1uF
100
.1uF
3k
2
8
10
sin
4.7nF
MC1496
1
.1uF
4
1k
.1uF
1k
FROM
PROBE
1k
6
1uF
330
.1uF
1k
3k
3
4.7nF
12
14
5
10k
1uF
4.7nF
4.7nF
+12 V
1k3
820
1k
DIFF.
OUTPUT
.1uF
100
.1uF
2
8
.1uF
cos
1
.1uF
1k
4.7nF
MC1496
4
1k
.1uF
12
14
3k
1k
6
330
10
3k
3
5
10k
1uF
4.7nF DIFF.
OUTPUT
1k
1uF
4.7nF
4.7nF
11
Two channel differential audio amplifier
10k 1%
R1
C2
13
+IN
12
11
200pF
RG
-IN
R2
10k
30pF
5pF
1
+IN
+RG
10
OUT
-R G
-IN
BIAS
14
10k 1%
50pF
C3
6
COMP2
7
COMP3
SSM-2015
9
10k
C1
to BPF
3
5
+V
4
-V
+12V
-12V
0.1u
RBIAS
0.1u
150k
150k
10k
10k
R2
10k 1%
9
-IN
RG
10
0.1u
RBIAS
-V
-R G
0.1u
11
12
+RG
13
COMP2
1
5pF
6
+12V
5
OUT
COMP3
+IN
-12V
4
+V
SSM-2015
200pF
+IN
-IN
14
BIAS
7
3
to BPF
C3
50pF
30pF
C2
C1
R1
10k 1%
12
Programmable bandpass filter and amplifier
13
the audio amplifier
+5 V
10uF
AUDIO
33k
33k
0.1uF
3
10k
0.22uF
2
1
8
+
LM386
-
0.1uF
6
7
0.22uF
4
100uF
5
10R
0.1uF
0.1uF
8R
14
PC interface
15
16
PROCESSING OF DOPPLER
ULTRASOUND SIGNALS
17
Processing of Doppler Ultrasound Signals
Gated
transmiter
sin
Master
cos
osc.
Demodulator
Sample
& hold
Band-pass
filter
si
Demodulator
Sample
& hold
Band-pass
filter
sq
Further
processing
Logic
unit
Receiver
amplifier
RF
filter
Transducer
V
•
•
•
•
•
•
Demodulation
Quadrature to directional signal conversion
Time-frequency/scale analysis
Data visualization
Detection and estimation
Derivation of diagnostic information
18
Single side-band detection
yf
USBF
LPF
cos 0t
RF signal
yr
LSBF
LPF
19
Heterodyne detection
Transmitter
h
o
Master
oscillator
2-20 MHz
Heterodyne
oscillator
1-10 kHz
o  h
LSBF
o  h
RF signal
o  d
d  h :forward
d  h :reverse
h  d
LPF
20
Frequency translation and side-band filtering detection
Optional stage
o Oscillator1 o
2-20 MHz
Trans.
LPF5
h
Oscillator2 o  h
2-20 MHz
HPF
h  d
o  h
o  d
RF signal
LPF1
LPF3
yf
h  d
LPF2
h  d
yr
LPF4
21
direct sampling
• Effect of the undersampling.
(a) before sampling; (b) after sampling
(a)
fs
2fs
3fs
4fs
nfs
fs
2fs
3fs
4fs
nfs
(b)
22
Quadrature phase detection
RF signal
Quad.
signal
oscillator
LPF
yD
LPF
yQ
cos 0t
sin 0t
23
TOOLS FOR DIGITAL
SIGNAL PROCESSING
24
Understanding the complex Fourier transform
• The Fourier transform pair is defined as

X ( f )   x(t )e

 j 2ft

dt, x(t )   X ( f )e

j 2ft
df
• In general the Fourier transform is a complex quantity:
X ( f )  R( f )  jI ( f )  X ( f ) e
j ( f )
• 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 R ( f )  I ( f ) ,θ(f) is the phase angle of the Fourier
transform and given by tan-1[I(f)/R(f)]
2
2
25
• 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

X ( f )   [ xr (t )  jxi (t )]e
 j 2ft


dt   [ xr (t ) cost  xi (t ) sin t ]dt


 j  [ xr (t ) sin t  xi (t ) cost ]dt  R( f )  jI ( f )

F{cos 0t}   { (   0 )   (   0 )}

F{sin  0t}  { (   0 )   (   0 )}
j
26
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
27
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.
28
real part of complex FFT
1
real part of complex FFT
1
•
Case (1).
xr ( t )  cos  0 t , xi ( t )  sin 0 t ,
-1
-Fs/2
(1)
X (  )  F {cos  0 t }  jF {sin 0 t }  2(    0 )
1
•
Case (2).
xr ( t )  cos  0 t , xi ( t )   sin 0 t ,
-1
-Fs/2
X (  )  F {cos  0 t }  jF {  sin 0 t }  2(    0 )
•
Case (3).
1
xr ( t )   cos  0 t , xi ( t )  sin 0 t ,
X (  )  F {  cos  0 t }  jF {sin 0 t }  2(    0 ) -1
(2) -Fs/2
•
Case (4).
1
xr ( t )   cos  0 t , xi ( t )   sin 0 t ,
X (  )  F {  cos  0 t }  jF {  sin 0 t }  2(    0 )
-1
-Fs/2
•
Case (5).
xr ( t )  sin 0 t , xi ( t )  cos  0 t ,
1
X (  )  F {sin 0 t }  jF {cos  0 t }  j 2(    0 )
•
Case (6).
-1
-Fs/2
xr ( t )  sin 0 t , xi ( t )   cos  0 t ,
(3)
1
X (  )  F {sin 0 t }  jF {  cos  0 t }   j 2(    0 )
•
Case (7).
-1
xr ( t )   sin 0 t , xi ( t )  cos  0 t ,
-Fs/2
X (  )  F {  sin 0 t }  jF {cos  0 t }  j 2(    0 )
1
•
Case (8).
xr ( t )   sin 0 t ,
xi ( t )   cos  0 t ,
X (  )  F {  sin 0 t }  jF {  cos  0 t }   j 2(    0 )
-1
0
imaginary part of complex FFT
+Fs/2
-1
0
0
imaginary part of complex FFT
+Fs/2
1
-Fs/2
+Fs/2
0
+Fs/2
real part of complex FFT
real part of complex FFT
1
-1
0
imaginary part of complex FFT
+Fs/2
-Fs/2
(6)
0
+Fs/2
imaginary part of complex FFT
1
-1
0
-Fs/2
+Fs/2
0
+Fs/2
real part of complex FFT
real part of complex FFT
1
-1
0
imaginary part of complex FFT
+Fs/2
-Fs/2
(7)
0
imaginary part of complex FFT
+Fs/2
1
-1
0
-Fs/2
+Fs/2
0
+Fs/2
real part of complex FFT
real part of complex FFT
1
-1
-1
-Fs/2
(4)
-Fs/2
(5)
0
imaginary part of complex FFT
+Fs/2
-Fs/2
(8)
1
1
0
+Fs/2
imaginary part of complex FFT
-1
-1
-Fs/2
0
+Fs/2
-Fs/2
0
+Fs/2
29
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
30
Discrete Fourier transform pair
 N 1
 j ( 2 / N ) kn
x
(
n
)
e
,0  k  N 1

 n 0
X (k )  

0, otherwise


1

N
x ( n)  



N 1
j ( 2 / N ) kn
X
(
k
)
e
,0  n  N 1

n 0
0, otherwise
31
complex modulation
x ( t )e
j c t
 X ( f  fc )
X(f)
-W 0
W
X(f-fc )
f
0
fc -W
fc
fc +W
f
F{cos 0t}   { (   0 )   (   0 )}

F{sin  0t}  { (   0 )   (   0 )}
j
32
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).
1 1  x( )
H [ x(t )]  x(t )   
d
t   t  
33
Hilbert transform
• Specifically, if X(f) is the Fourier transform of x(t), its
Hilbert transform is represented by XH(f), where
X H ( f )  H [ X ( f )]  H H ( f ) X ( f )  ( j sgn f ) X ( f )
• A ± 0900 phase shift is equivalent to multiplying by
ej90 =±j, so the transfer function of the HT HH(f) can
be written as
 j , f  0
H H ( f )   j sgn f  
 j , f  0
34
impulse response of HT
0, n  0


hH (n)   2 sin 2 (n / 2)
,n  0

n
 h (n)
H
2
1
1/3
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.
1/5
-6
-5
-4
-3 -2
-1
1
2
3
4
5
6
n
-1/5
-1/3
-1
35
• 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
e j
e  j
x ( t ) cos(  c t   ) 
X ( f  fc ) 
X ( f  fc )
2
2
• So, multiplying a band limited signal by a
sinusoidal signal translates its spectrum up
and down in frequency by fc
36
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
x(n)
(input)
h(k), k=0,1,...
(impulse response)
y(n)
(output)
37
Digital Filters
• Non-recursive (finite impulse response, FIR)
y ( n) 
N 1
 h( k ) x ( n  k )
k 0
• Recursive (infinite impulse response, IIR).
y ( n) 

N
N
k 0
k 0
k 1
 h( k ) x ( n  k )   ak x ( n  k )   bk y ( n  k )
• 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
38
Basic IIR filter and FIR filter realisations
x(n)
a0


y(n)
x(n)
-1
Z
x(n-1)
-1
Z
x(n-2)
-1
Z
x(n-N+1)
-1
Z

h(0)
a1
-b1
h(N-1)

-1
a2
-b2
h(2)

Z

h(1)

y(n)
-1
Z
-bN
aN
(a)
(b)
39
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
40
GENERAL DEFINITION OF A QUADRATURE DOPPLER SIGNAL
• A general definition of a discrete quadrature
Doppler signal equation can be given by
 D(n)  s f (n)  H [ s r (n)]

Q(n)  H [ s f (n)]  s r (n)
• 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.
41
Asymmetrical implementation of the PFT
DSP Algorithm
D(n)
HILBERT
TRANSFORM
+
 D(n)  s f (n)  H [ s r (n)]

Q(n)  H [ s f (n)]  s r (n)
yf(n)
+
-
H[ D( n)]  H[ s f ( n)  H[ sr ( n)]]  H[ s f ( n)]  sr ( n)
Q(n)
DELAY
FILTER
+
yr(n)

 y f (n)  Q(n)  H [ D(n)]  H [ s f (n)]  sr (n)  H [ s f (n)]  sr (n)


 y r (n)  Q(n)  H [ D(n)]  H [ s f (n)]  sr (n)  H [ s f (n)]  sr (n)
 y f (n)  2 H [ s f (n)]

 y r ( n)  2 s r ( n)
42
Symmetrical implementation of the PFT
DSP Algorithm
D(n)
 D(n)  s f (n)  H [ s r (n)]

Q(n)  H [ s f (n)]  s r (n)
HILBERT
TRANS.
yf(n)
DELAY
FILTER
DELAY
FILTER
Q(n)
HILBERT
TRANS.
yr(n)
H[ D( n)]  H[ s f ( n)  H[ sr ( n)]]  H[ s f ( n)]  sr ( n)
H[Q( n)]  H[ H[ s f ( n)]  sr ( n)]   s f ( n)  H[ sr ( n)]
 y f (n)  Q(n)  H [ D(n)]  2 H [ s f (n)]

 y r (n)  D(n)  H [Q(n)]  2 H [ s r (n)]
 y f (n)  D(n)  H [Q(n)]  2s f (n)

 y r (n)  Q(n)  H [ D(n)]  2sr (n)
43
•
•
•
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
DSP Algorithm
+450
D(n)
Phase splitter
(90 0 )
y (n)
f
-45 0
-45 0
Q(n)
Phase splitter
(90 0 )
+450
y (n)
r
44
Weaver Receiver Technique (WRT)
• For a theoretical description of the system
consider the quadrature Doppler signal
defined by
 D(n)  s f (n)  H [ s r (n)]

Q(n)  H [ s f (n)]  s r (n)
which is band limited to fs/4, and a pair of
quadrature pilot frequency signals given by
pd ( n)  sin c n, pq ( n)  cos c n
where ωc/2π=fs/4.
• The LPF is assumed to be an ideal LPF
having a cut-off frequency of fs/4.
45
Asymmetrical implementation of the WRT
DSP Algorithm
p (n)
d
D(n)
X1
X2
+
 D(n)  s f (n)  H [ s r (n)]

Q(n)  H [ s f (n)]  s r (n)
y (n)
X3
LPF
LPF
f
+
+
Q(n)
Y1
-
Y3
Y2
LPF
y (n)
LPF
r
p (n)
q
fp=fc=fs/4
pd ( n)  sin c n, pq ( n)  cos c n
46
X 2, Y 2  D( n). pd ( n)  Q( n). pq ( n)
 { s f ( n).sin c n  H[ sr ( n)].sin c n}  { H[ s f ( n)].cos c n  sr ( n) cos c n}
F{ s f ( n)}  S f (  ) and F{ sr ( n)}  Sr (  )


S f ( )  S f ( ),0    
 

S f ( )  S f ( ),      0
 jS f ( ), 0    
F{H [ s f (n)]}  H [ S f ( )]  
 jS f ( ),      0
 jS r ( ), 0    
F{H [ s r (n)]}  H [ S r ( )]  
 jS r ( ),      0


 S r ( )  S r ( ),0    
 

 S r ( )  S r ( ),     0
S f ( )  S f ( )  S f ( )

 S r ( )  S r ( )  S r ( )



H [ S f ( )]   jS f ( )  jS f ( )




H [S r ( )]   jS r ( )  jS r ( )
47
F { X 2}  {  jS f (    c )  jS f (    c )}  { Sr (    c )  Sr (    c )}
 H [ S f (  c   )]  Sr (  c   )
F {Y 2}  { jS f (    c )  jS f (    c )}  {  Sr (    c )  Sr (    c )}
  H[ S f (  c   )]  Sr (  c   )
F { X 2}  H [ S f (  c   )]   jS f (    c )  jS f (    c )
F{Y 2}   Sr (  c   )   Sr (    c )  Sr (    c )
F { X 3} 
1 
1
1
1
S f (  )  S f (  )  S f (   2 c )  S f (   2 c )
2
2
2
2
1
1
1
1
F {Y 3}   Sr (  )  Sr (  )  Sr (   2 c )  Sr (   2 c )
2
2
2
2
1

F
{
y
(
n
)}

S f ( ),
f

2

 F { y ( n)}   1 S ( ).
r
r

2

1
1
S f (  )  S f ( 2 c   )
2
2
1
1
  Sr (  )  Sr ( 2 c   )
2
2
1

1 1
y
(
n
)

F
{
S
(

)}

s f (n),
f
 f
2
2

 y (n)  F 1{ 1 S ( )}   1 s (n).
r
r
 r
2
2
48
1
D
2
Q
3
pd
4
pq
5
X1
6
Y1
7
X2
8
Y2
9
yf
10
yr
sin  f t  cos  r t  j cos  f t  j sin  r t
-2fc
-fc
-fr -ff 0 +ff +fr
Lowpass filter cut-off frequency=f , fc =fc /4s
Stop-band region
+fc
+2fc
Pass-band region
49
Symmetrical implementation
DSP Algorithm
A1
y (n)
A2
LPF
f
LPF
D(n)
p(n)
q
p (n)
d
Q(n)
B1
y (n)
B2
LPF
r
LPF
50
Implementation of the WRT algorithm using low-pass/high-pass filter pair
DSP Algorithm
p (n)
d
D(n)
y (n)
X2
LPF
LPF
f
X1
Q(n)
HPF
X3
y (n)
LPF
r
p (n)
q
fp=fc=fs/4
51
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.
52
frequency domain Hilbert transform algorithm
DSP Algorithm
Frequency domain complex HT algorithm
SR
Q(n)
H[S] R
D'(n)
y f(n)
Q'(n)
y r(n)
-jS, w>0
CFFT
D(n)
SI
+jS, w<0
IFFT
H[S] I
53
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.
54
DSP Algorithm
S fR
D(n)
SR
Sf (w)
S fI
IFFT
y f (n)
CFFT
Q(n)
S rR
SI
Sr (w)
S rI
IFFT
y r (n)
55
s ( n)  D( n)  jQ ( n)  { s f ( n)  H [ sr ( n)]}  j{ H [ s f ( n)]  sr ( n)}
 { s f ( n)  jH [ s f ( n)]}  j{ sr ( n)  jH [ sr ( n)]}

{S f ( )  S f ( )}  j{S r ( )  S r ( )},0    
F{[ s(n)}  

{S f ( )  S f ( )}  j{S r ( )  S r ( )},      0
S ( ), 0    
S ( )  
0,     0



{S ( )}, 0    
{S f ( )}  


{S ( )},      0


{S ( )}, 0    
{S r ( )}  


{S ( )},      0
2S f ( ), 0    
F{s(n)}  S ( )  
 j 2S r ( ),     0
0, 0    
S  ( )  
S ( ),     0


{S ( )}, 0    
{S f ( )}  


 {S ( )},      0


{S ( )},0    
{S r ( )}  


 {S ( )},      0
56
Spectral Translocation Method (STM)
DSP Agorithm
SR
Q(n)
FFT
D(n)
SI
y f(n)
YR
IFFT
YI
X =
RX =
R+
X =
IX =
I+
y r(n)
S I+ X
R
-S I-S R+ X I
S R-
57
2S f ( ), 0    
F{s(n)}  S ( )  
 j 2S r ( ),     0
S (  )  2 S f (  )  j 2 Sr (  )  { S  (  )  S  (  )}  j { S  (  )  S  (  )}
S  (  )  S  (  ), S  (  )  S  (  )
X (  )  { S  (   )  S  (   )}  j {  S  (   )  S  (   )}
 {  S  (  )  S  (  )}  j { S  (  )  S  (  )}
Y ( )  S ( )  X ( )
 { { S (  )}  { X (  )}}  j{ { S (  )}  { X (  )}}
 2 S f (  )  j 2 Sr (  )  2 S f (  )  j 2 Sr (  )
 2{ S f (  )  S f (  )}  j 2{ Sr (  )  Sr (  )}
 2 S f (  )  j 2 Sr (  )
y ( n)  2s f ( n)  j 2sr ( n)
58