MEDT 8007 Simulering av ultralydsignal fra spredere i

MEDT 8007 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 0 Hans Torp NTNU, Norway fo+fd frequency 0 fd frequency Matlab: cwdoppler. m

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

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

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

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

Definition of Complex Gaussian process

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

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

Power spectrum estimate Statistical properties Power spectrum estimate: Covariance:

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) < |Z(w)|^2 >= G(w)|Zn(w)|^2 = G(w); for w= 2 k/N Power spectrum for z(n): (smoothed version of G(w) ) Autocorrelation function:

Computer simulation of Complex Gaussian process • The 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| Matlab: Csignal. Demo. m

Properties 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 spektrum Hans Torp NTNU, Norway

N Slow time Fast time Pulse no 1 2. . … 2 D Fourier transform Ultrasound pulse frequency [MHz] Signal from moving scatterer Clutter Blood Thermal noise Power Doppler shift frequency [k. Hz] Signal from one range Hans Torp NTNU, Norway Doppler shift frequency [k. Hz]

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

2 D Spectrum Subclavian artery

Summary spectral Doppler • Complex demodulation give direction information of blood flow • Smooth window function removes sidelobes from cluttersignal • PW Doppler suffers from aliasing in many cardiac applications • SNR increases by the square of pulse length in PW Doppler