The aliasing is inherent to the fact that the ultrasonic Doppler velocimetry uses a pulsed emission.
This implies that the maximum measurable Doppler frequency is the half of the pulsed repetition frequency
(Nyquist limit). Nevertheless, UDOP offers two ways to overcome that limitation.
An often used method is based on the assumption that the velocity can't change more than a define amount
between two adjacent gates. This method implies that:
We developed a new method to overcome the above limitations. This new method uses two different pulsed repetition
frequency and a special algorithm. This method allows to measure Doppler frequencies many times higher than the Nyquist
limit and does not require any a priori knowledge. The basic idea of this method is to take into account the phase difference
resulting from a change in the PRF between two successive emissions.
The figure below shows an example of how the new developed method corrects, here up to 3 times, the aliasing contained
in the aliased red curve.
The interfaces reflect and modify the acoustic field. The intensity of the acoustic field received in a point inside the
liquid depends on the material, the shape and the number of these interfaces. This intensity is most of the time very difficult
to evaluate. This lack of knowledge does not allow a precise determination of the size of the measuring volume.
These interfaces may generate, in certain situations, artifacts and induce modifications in the velocity profiles as presented
in the figures below.
The ultrasonic beam BC reflected by the far interface of the figure above transforms this interface in a transmitter. The same
particles contained in the liquid will backscatter a second time energy in the direction to the transducer. The depth
associated to the path ABC is located outside the flowing liquid. Imaginary velocity components are added to the real velocity
profile. The measurement of velocities near the far interface is affected by this phenomena. The size of the ultrasonic beam
determines mainly the level of this artifact.
The figure above displays another situation. The reflected ultrasonic waves inside a wall enlarge the ultrasonic beam inside the liquid and modify its shape. These reflections disturb the determination of the size and the shape of the measuring volume. The thickness, the acoustical impedance and the attenuation coefficient of the interface determine the level of this phenomena. This situation appears when measuring for instance through steel walls.
The interfaces often give strong reflections. Despite of the many reflections which are necessary
to reach the transducer, the energy reflected by these interfaces is often
stronger than the energy coming from the particles flowing with the liquid.
Most of the algorithms used to compute the Doppler frequency shift do not
allowed stationary components. The elimination of these stationary components
by high-pass filtering implies an increase in the dynamic range of the analyzed
echoes and a reduction in the sensitivity in the measurement of low velocities.
When some interfaces are in movement the correct estimation of all the velocity
field is very difficult. The echoes generated by such interfaces may affect
the velocity profile in many places due to the combination of many reflections.
The Doppler frequency shift induced by these movable interfaces can not be
removed if their values have the same values as the flowing particles.
When an ultrasonic beam encounters a wall, a part of its energy is reflected and an other part is refracted. The intensity of both beams, the reflected and the refracted beams, can be computed when the two following parameters are known:
The simple equations below assume a plane infinite wall or interface. This is of course never the case. Nevertheless, these three equations could really help because:
where:
Zi is the acoustic impedance of the medium i, which is equal to the product of the sound speed by the density
of the medium i. R is the ratio of the reflected intensity to the incident intensity and D is the ratio of
the refracted intensity to the incident intensity.
From these equations, it is possible to compute the value of the angle for which all the ultrasonic waves
will be reflected. By setting a value of 90 degrees to the refracted angle, we can compute the value of the
total reflection angle. For instance:
We can also see the very strong influence of the wall material when the angle of incidence is different from the perpendicular. The two examples below show this influence very well.
In pulsed ultrasound Doppler velocimetry the sampling volume contains not a unique particle but a lot
of small particles having most of the time different shapes, different sizes and different acoustic impedances.
As all of these particles contribute to generate a unique value of the amplitude of the echo for each emission,
the evolution of the echo amplitude will fluctuate. The reason comes from the fact that some particles enter in
the sampling volume and others leave it. This fluctuation in amplitude will remain after the demodulation process
and will be present on the demodulated Doppler signal, often called I and Q.
All the Doppler frequency shifts induced by the movement of the particles are combined together. If all the
particles do not have the same velocity (in amplitude and in direction) the demodulated signals will contain many
Doppler frequencies. The resulting demodulated signals I and Q contain therefore the combination of these two phenomena.
In order to have an idea of the aspect of a real demodulated Doppler signal corresponding to one gate, the figure below
illustrates its evolution.
The best way to analyze the frequency content of the demodulated echo signal issued from one gate is to compute its
power spectrum. The power spectrum gives information on the distribution of the measured Doppler frequencies and their
relative influences on the computed mean Doppler frequency, which is computed when the velocimeter displays the velocity profile.
The DOP can compute the power spectrum of a data series by means of an FFT algorithm. The data series is formed of
samples taken on the demodulated I and Q signals for a selected depth which corresponds to one gate. Before the
computation of the power spectrum, the data series is filtered by a high pass filter which removes all the stationary
components contained in the demodulated signals. In order to make the reading of the frequency scale more easy, the
frequency scale is converted in velocity by using the standard Doppler formula. The ordinate gives the relative amplitude,
in a logarithmic scale, of the power spectrum.
When once looks at the velocity profile, all the gates give a single value of velocity. These values are the result of a computation of the mean Doppler frequency, which is the real mean value (non biased) of the power spectrum. This means that no information is given about the distribution of the Doppler energy. The same value of velocity can result from many different frequency distributions. The display of the complete power spectrum is a good method to increase the knowledge on the measured velocity values.
The following examples will display different situations where the power spectrum could increase the knowledge in the measured velocity values.
The first example (see figure above), shows the computed power spectrum from a gate placed in the middle of a tube where a liquid is flowing. As shown in the power spectrum the sampling volume does not contain a single Doppler frequency. The width of the peak is related to the number of Doppler frequencies present in the sampling volume.
In the second example (see figure above), the sampling volume has been moved closer to the wall of the tube. The width of the Doppler peak is now much larger which means that much more different velocities are present in the sampling volume. The power spectrum reveals also a small influence of the movements of the walls. As the power spectrum is computed from the high-passed filtered data values the amplitude of the power spectrum at the origin is always zero.
The third example (see figure above), considers a rotating cylinder filled of liquid. The ultrasonic beam crosses the
cylinder perpendicularly to the axis of the cylinder. The sampling volume is placed inside the cylinder in a position near
the wall.
The power spectrum reveals a very strong influence of the movements of the walls of the cylinder despite the sampling volume
does not touch a wall. This situation may appear if some of the multiple ultrasonic reflections may coincide with the
sampling time of the echo. In such a case the velocity value is corrupted by the Doppler effect induced by the movement
of the walls. The display of the power spectrum indicates clearly that the mean Doppler frequency computed is in fact the
result of the mean value of two different velocity components, one coming from the movements of the walls and the other
coming from the movements of the particles contained in the liquid.
The amplitudes of the echoes reflected by the particles within the flowing fluid are somewhat random in nature, corresponding to the random distribution of the particles in the fluid medium. Thus, the Doppler signals may be treated as random processes, and characterized by different moments. In order to be able to determine the probability of occurrence of this process, one must have access to a great number of actual occurrences of the process. In practice, it is difficult to obtain measurements of the exact same process under the exact same conditions at several different times. Therefore, a temporal average is preferable to an ensemble average. The temporal average and the ensemble average will not be the same unless the process is stationary and the analysis time is very long (tending to infinity). Considering the Doppler process as stationary, the average frequency may be expressed as the normalized first moment, or:
where S(f) is the spectral density or probability density of the Doppler signal.
The Doppler frequency calculation algorithm is based on the fact that the inverse Fourier transform of the probability density
of a stationary process is equal to the auto-correlation function. The mean Doppler frequency may be expressed in terms of the
time derivatives of the auto-correlation function at the origin:
Transient flows are characterized by variations in velocities versus time. The non-stationary propriety of transient flows implies that a bias will be introduced in the time derivatives of the auto-correlation function. The computation of the auto-correlation could only be realized during a finite time interval and only samples of the analytical signal are available (space in time by PRF).
In order to reduce the bias induced by the non-stationary propriety of transient flows, the number of emissions used to compute
the mean Doppler frequency must be reduced. Unfortunately reducing the number of emissions will decrease the quality of
the estimation of the auto-correlation function. A comprise must be therefore realized between the number of emissions used and
the quality of the estimated mean Doppler frequency.
A low number of emissions pro profile increases the variance in the measured mean Doppler frequency. This variance can be
reduced by using a moving average filter.
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