MEDT8007

Simulering av ultralydsignal fra spredere i bevegelse

Hans Torp

Institutt for sirkulasjon og medisinsk bildediagnostikk

Hans TorpNTNU, Norway

Signal processing Signal processing for CW Dopplerfor CW Doppler

fofrequency

fo+fd0 frequencyfd0

Hans TorpNTNU, Norway

Matlab: cwdoppler.m

Blood velocity calculated from Blood velocity calculated from measured Doppler-shiftmeasured Doppler-shift

fd = 2 fo v cos() / c

v = c/2fo/cos() fd

fd : Dopplershiftfo : Transmitted frequencyv : blood velocity

: beam angle c : speed of sound (1540 m/s )Hans Torp

NTNU, Norway

Continous Wave DopplerContinous Wave Doppler

ø

Single transducerPW

Double transducerCW

ø

transmit

recieve

Velocity profile, v

Artery

Range cell

Observation region in overlap of beams

Signal from all scattererswithin the ultrasound beam

Pulsed Wave DopplerPulsed Wave Doppler

Signal from a limited sample volume

ø

Single transducerPW

Double transducerCW

ø

transmit

recieve

Velocity profile, v

Artery

Range cell

Observation region in overlap of beams

Hans TorpNTNU, Norway

Matlab: pwdoppler.m

a)

b)

10-08/12pt

Hans TorpNTNU, Norway

Signal from a large number of red blood cellsSignal from a large number of red blood cellsadd up to a Gaussian random processadd up to a Gaussian random process

G ()e

10-11/12pt

Hans TorpNTNU, Norway

Power spectrum of the Doppler signal Power spectrum of the Doppler signal represents the distribution of velocitiesrepresents the distribution of velocities

Definition of Complex Gaussian process Definition of Complex Gaussian process

0)()(:

average)(ensemblevalueexpectedmeans

)(*)(zmatrixCovariance

21 vector signal

)(

,

H

1 1H

nzkzthatNote

nzkzz

),...,z(N)),z(z(z

ep

nk

N

zzz

Stationary Complex Gaussian processStationary Complex Gaussian process

;)()(m

miemRGPower spectrum

Autocorrelation function ,......2,1,0)(*)()( mmnznzmR

mieGdmR )(2

1)(

Autocorrelation function = coeficients in Fourier series of G

Power spectrum estimatePower spectrum estimateStatistical propertiesStatistical properties

)()(2

1)(

2

GWdGN

m

miN emwW )()(

Expected value:

m

miNN emzmwZ )()()(

2)(

1)( NN Z

NG Power spectrum estimate:

Power spectrum estimatePower spectrum estimateStatistical propertiesStatistical properties

N

G

GWWdN

GG

N

NN

/1 when 0

0 when )(

)()()(2

1)(),(cov

2

2

*

Covariance:

m

miNN emzmwZ )()()(

2)(

1)( NN Z

NG Power spectrum estimate:

Computer simulation of Computer simulation of Complex Gaussian processComplex Gaussian process

1.Complex Gaussian white noise Zn(0),..,Zn(N-1)2. Shape with requested power spectrum: Z(k)=G(2k/N) Zn(k); k=0,..,N-13. Inverse FFT: z(n) = ifft(Z)

< |Z(w)|^2 >= G(w)|Zn(w)|^2 = G(w); for w= 2k/N

)()(2

1)(

2

GWdGZPower spectrum for z(n):(smoothed version of G(w) )

1

0

)(N

m

mieW Autocorrelation function: )()()( mRmTrimRz

• The power spectrum of the simulated signal is The power spectrum of the simulated signal is smoothed with a window given by the number of smoothed with a window given by the number of samples Nsamples N

• The autocorrelation function of the simulated signal The autocorrelation function of the simulated signal Rz(m)= 0 for m>|N| Rz(m)= 0 for m>|N|

Computer simulation of Computer simulation of Complex Gaussian processComplex Gaussian process

Matlab: CsignalDemo.m

Properties of power spectrum estimateProperties of power spectrum estimate

• Fractional variance = 1 independent of the window form and size

• GN(ω1) and GN(ω2) are uncorrelated when |ω1-ω2| > 1/N

• Increasing window length N gives better frequency resolution, but no decrease in variance

• Smooth window functions give lower side lobe level, but wider main lobe than the rectangular window

• Decrease in variance can be obtained by averaging spectral estimates from different data segments.

Doppler spektrumDoppler spektrum

frequ

ency

time

spec

tral

am

plit

ude

Hans TorpNTNU, Norway

Thermal noise

Signal from moving scattererSignal from moving scattererPulse no1 2 .. … N

2D Fouriertransform

Hans TorpNTNU, Norway

Doppler shift frequency [kHz]

Ult

raso

und

puls

e fr

eque

ncy

[MH

z]

Doppler shift frequency [kHz]

Pow

erSignal from one range

Fas

t ti

me

Slow time

Blood

Clutter

RF versus basebandRF versus baseband

Doppler shift frequency [kHz]

Blood signalClutter

Ultrasound frequency [MHz]

Doppler frequency [kHz]

Remove negative ultrasoundFrequencies by Hilbert transformor complex demodulation

• Skewed clutter filter (signal adaptive Skewed clutter filter (signal adaptive filter) can be implemented with 1D filter) can be implemented with 1D filteringfiltering

• Axial sampling frequency Axial sampling frequency reduced by a factor > 4reduced by a factor > 4

2D Spectrum Subclavian artery2D Spectrum Subclavian artery

Summary spectral DopplerSummary spectral Doppler

• Complex demodulation give direction information of blood Complex demodulation give direction information of blood flowflow

• Smooth window function removes sidelobes from clutter-Smooth window function removes sidelobes from clutter-signalsignal

• PW Doppler suffers from aliasing in many cardiac PW Doppler suffers from aliasing in many cardiac applicationsapplications

• SNR increases by the square of pulse length in PW Doppler SNR increases by the square of pulse length in PW Doppler