Physics - Pitch-Catch

This exercise is just a small variation on the previous one, but serves us to introduce the first deliverable of the course. Don't blink

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by Alfonso Rodriguez-Molares (alfonso.r.molares@ntnu.no) 21.07.2016

Contents

Two transducers

Based on the model we introduced in the previous exercise now we would like to examine the impulse response of a system with two transducers in a pitch-catch configuration, this is, one acting only as transmitter and the other acting only as receiver. Let us denote $\mathbf{t}$ the position where the transmitting element is and $\mathbf{r}$ the position where the receiving element is. Let us assume there is a point reflector at point $\mathbf{p}$ with a reflection coefficient $\Gamma$.

% Easy peasy.
%
% $h(t,\mathbf{s},\mathbf{r}) = \Gamma \frac{\delta(t-(r_T+r_R)/c_0)}{4\pi r_T r_R} \, D(\theta_T,\phi_T) D(\theta_R,\phi_R)$
%
% where
%
% $r_T=||\mathbf{t}$ - $\mathbf{p}||$,
% $r_R=||\mathbf{p}$ - $\mathbf{r}||$,
% $\theta_T=\arctan\left(\frac{P(1)-T(1)}{P(3)-T(3)}\right)$
% $\phi_T=  \arctan\left(\frac{P(2)-T(2)}{P(3)-T(3)}\right)$
% $\theta_R=\arctan\left(\frac{P(1)-R(1)}{P(3)-R(3)}\right)$
% $\phi_R=  \arctan\left(\frac{P(2)-R(2)}{P(3)-R(3)}\right)$

Using a Gaussian-modulated RF signal

$v(t) = \cos \left(2\pi f t \right) e^{-2.77 \left(1.1364 \, t \, \Delta f\right)^2}$.

where $f$= 4 MHz, $\Delta f=0.6 f$, and $F_s$=80 MHz. Consider two identical transducers $a$=1 mm wide, $b$=5 mm tall. One transducer is at the origin of coordinates with its face pointing downwards (towards positive z). The other is placed at (10,0,0) mm also pointed downwards. A point scatterer exists at $\mathbf{r}$=(0,0,20) with reflection coefficient $\Gamma$=1.

f=4e6;                              % transducer frequency [MHz]
Deltaf=0.6*f;                       % pulse bandwidth [MHz]
c0=1540;                            % speed of sound [m/s]
lambda=c0/f;                        % wavelength [m]
k=2*pi*f/c0;                        % wavenumber [rad/m]
Fs=80e6;                            % sampling frequency [Hz]
t=0:(1/Fs):2*40e-3/c0;              % time vector [s]
depth=t*c0/2;                       % depth vector [m]
a=1e-3;                             % element width [m]
b=5e-3;                             % element height [m]
Gamma=1;                            % reflaction coefficient [unitless]

% some geometrical variables
T=[0 0 0];              % transmitted position [m]
P=[0 0 20e-3];          % point position [m]
R=[5e-3 0 0];          % receiver position [m]

theta_T=atan2(P(1)-T(1),P(3)-T(3));
phi_T=atan2(P(2)-T(2),P(3)-T(3));
distance_T=norm(T-P,2);
directivity_T = sinc(k*a/2.*sin(theta_T).*cos(phi_T)/pi).*sinc(k*b/2.*sin(theta_T).*sin(phi_T)/pi);

theta_R=atan2(P(1)-R(1),P(3)-R(3));
phi_R=atan2(P(2)-R(2),P(3)-R(3));
distance_R=norm(R-P,2);
directivity_R = sinc(k*a/2.*sin(theta_R).*cos(phi_R)/pi).*sinc(k*b/2.*sin(theta_R).*sin(phi_R)/pi);

% the 2-way Fresnel model for rectangular elements
s_r=directivity_T.*directivity_R.*Gamma/(4*pi*distance_T*distance_R)*cos(2*pi*f*(t-(distance_T+distance_R)/c0)).*exp(-2.77*(1.1364*(t-(distance_T+distance_R)/c0)*Deltaf).^2);

figure3 = figure('Name','background','Color',[1 1 1]);
plot(depth*1e3,s_r,'b','linewidth',1); grid on; axis tight; box('on'); hold on;
xlabel('z [mm]');
ylabel('signal [unitless]')
title({'Received signal'});
set(figure3,'InvertHardcopy','off');
xlim((distance_T+distance_R)/2*[0.9 1.1]*1e3);

Now let us assume that the receiving transducer is tilted -12 degrees in the azimuth direction.

theta_R=atan2(P(1)-R(1),P(3)-R(3))-(-12*pi/180);
phi_R=atan2(P(2)-R(2),P(3)-R(3));
distance_R=norm(R-P,2);
directivity_R = sinc(k*a/2.*sin(theta_R).*cos(phi_R)/pi).*sinc(k*b/2.*sin(theta_R).*sin(phi_R)/pi);

% the 2-way Fresnel model for rectangular elements
s_r_tilted=directivity_T.*directivity_R.*Gamma/(4*pi*distance_T*distance_R)*cos(2*pi*f*(t-(distance_T+distance_R)/c0)).*exp(-2.77*(1.1364*(t-(distance_T+distance_R)/c0)*Deltaf).^2);

figure3 = figure('Name','background','Color',[1 1 1]);

plot(depth*1e3,s_r,'b','linewidth',1);
grid on; axis tight; hold on; box('on');
plot(depth*1e3,s_r_tilted,'r--','linewidth',1);
xlabel('z [mm]');
ylabel('signal [unitless]')
title({'Received signal'});
set(figure3,'InvertHardcopy','off');
xlim((distance_T+distance_R)/2*[0.9 1.1]*1e3);
legend('Normal','-12^\circ tilted');


Published with MATLAB® R2015a