MEDT8007 Simulering av ultralydsignal fra spredere i bevegelse Hans Torp Institutt for sirkulasjon og medisinsk bildediagnostikk Hans Torp NTNU, Norway

Signal processing for CW Doppler fo frequency fo+fd 0frequencyfd 0 Hans Torp NTNU, Norway Matlab: cwdoppler.m

Blood velocity calculated from measured Doppler-shift f d = 2 f o v cos(  ) / c v = c/ 2f o / cos (  ) f d fd : Dopplershift fo : Transmitted frequency v : blood velocity  : beam angle c : speed of sound (1540 m/s ) Hans Torp NTNU, Norway

Continous Wave Doppler Signal from all scatterers within the ultrasound beam Pulsed Wave Doppler Signal from a limited sample volume Hans Torp NTNU, Norway Matlab: pwdoppler.m

a) b) 10-08/12pt Hans Torp NTNU, Norway Signal from a large number of red blood cells add up to a Gaussian random process

 G (  ) e 10-11/12pt Hans Torp NTNU, Norway Power spectrum of the Doppler signal represents the distribution of velocities

Definition of Complex Gaussian process

Stationary Complex Gaussian process Power spectrum Autocorrelation function = coeficients in Fourier series of G

Power spectrum estimate Statistical properties Expected value: Power spectrum estimate:

Power spectrum estimate Statistical properties Covariance: Power spectrum estimate:

Computer simulation of Complex Gaussian process 1.Complex Gaussian white noise Zn(0),..,Zn(N-1) 2. Shape with requested power spectrum: Z(k)=  G(2  k/N) Zn(k); k=0,..,N-1 3. Inverse FFT: z(n) = ifft(Z) = G(w)|Zn(w)|^2 = G(w); for w= 2  k/N Power spectrum for z(n): (smoothed version of G(w) ) Autocorrelation function:

The power spectrum of the simulated signal is smoothed with a window given by the number of samples NThe power spectrum of the simulated signal is smoothed with a window given by the number of samples N The autocorrelation function of the simulated signal Rz(m)= 0 for m>|N|The autocorrelation function of the simulated signal Rz(m)= 0 for m>|N| Computer simulation of Complex Gaussian process Matlab: CsignalDemo.m

Properties of power spectrum estimate Fractional variance = 1 independent of the window form and size G N (ω 1 ) and G N (ω 2 ) are uncorrelated when |ω 1 -ω 2 | > 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 spektrum Hans Torp NTNU, Norway

Thermal noise Signal from moving scatterer Pulse no … N 2D Fourier transform Hans Torp NTNU, Norway Doppler shift frequency [kHz] Ultrasound pulse frequency [MHz] Doppler shift frequency [kHz] Power Signal from one range Fast time Slow time Blood Clutter

RF versus baseband Doppler shift frequency [kHz] Blood signal Clutter Ultrasound frequency [MHz] Doppler frequency [kHz] Remove negative ultrasound Frequencies by Hilbert transform or complex demodulation Skewed clutter filter (signal adaptive filter) can be implemented with 1D filtering Skewed clutter filter (signal adaptive filter) can be implemented with 1D filtering Axial sampling frequency reduced by a factor > 4 Axial sampling frequency reduced by a factor > 4

2D Spectrum Subclavian artery

Summary spectral Doppler Complex demodulation give direction information of blood flowComplex demodulation give direction information of blood flow Smooth window function removes sidelobes from clutter- signalSmooth window function removes sidelobes from clutter- signal PW Doppler suffers from aliasing in many cardiac applicationsPW Doppler suffers from aliasing in many cardiac applications SNR increases by the square of pulse length in PW DopplerSNR increases by the square of pulse length in PW Doppler