N. Roosnek / Roosnek Research & Development
To estimate the sound and flow velocities of fluid in pipes the transit times of ultrasonic pulses travelling upstream and downstream are used. These transit times are classically measured by using zero crossings in the received signals.
The estimation errors in the flow are directly
related to the signal to noise ratio S/N , as
(1)
wherebyis the
resonance frequency of the transducer. A much greater problem
is the false alarm and detection problem with zero crossings.
A parameter for this problem is the quantity
(2)
whereby is the factor of merit withandthe mean frequency and bandwidth.
Fig. 1 shows the alarm and detection probabilities as functions
of this parameter. The noise is assumed to be Gaussian and the
signal a band filtered Dirac function.
It is clear that the zero crossing technique
does not work with low S/N ‘s. By using instead of zero crossings a number of data of the phase signals, the latter obtained from the real signals, in a nonlinear leastsquare method, the accuracy and robustness of estimation of the flow are increased an order, see Roosnek [1], [2]. 

Fig. 1. The false alarm and detection characteristics for filtered Dirac pulses as function of Q and S/N for the detection of zero crossing; see text. 
We can write the phase function for the
nth transducer for example as
(3)
whereby t,, c andare
respectively the time relative to the arrival time of the pulse,
the resonance frequency of the nth transducer, the sound velocity
and the vector describing the flow pattern.
The time of arrival is defined by letting
the phase function to be
(4)
leading to the removal of the 2ambiguity.
We can rewrite the phase function as
(5)
making it suitable for digital signal processing.
The phase signal, constructed from the real and imaginary signal,
the latter via Hilbert or Fourier transforms, is written as
(6)
Linearization leads to
(7)
Eq. (7) is used to approximate the measured phase functions and
for calculating the’s. These steps
are used to minimize the following cost function
(8)
withthe weight of the kth phase point
of the signal received by the nth transducer.
By calculating in subsequent steps the’s for, c andfor
the mth measurement cycle, the estimated parameters , c
and are
obtained for that cycle.
This method is known as the nonlinear least square method (LSQ).
Fig. 2 shows the results of the simulation of a twochannel flow meter for a pipe diameter of 16 cm, a mean signal frequency of 100 kHz with a crude model in the estimation and air as the gas, see Roosnek [1].  
Fig. 2. Flow error as function of S/N and rapid flow fluctuations for nonlinear least square estimation. 
With modelling of the flow, determination of the S/N from the measured signals and the errors of the previous parameters and applying optimal or Kalman estimation the resulting algorithm is very robust, accurate and without any out of lock condition for the S/N range 12 to 12 dB. Fig. 3 shows the results of the simulation. 

Fig. 3. Flow error as function of S/N and rapid fluctuations for Kalman estimation. 
These figures show clearly that estimation
and robustness can be increased more then an order by using many
more points of the measured signal in a proper way, using a model
for the flow and applying optimal estimation.
These optimal estimation techniques are also used in different
fields, for instance in aircraft noise tracking and monitoring
and offshore pipeline detection and tracking.
Literature:
[1] Roosnek, N., “Novel digital signal processing techniques for ultrasonic gas flow measurements,” Flow. Meas. Instrumen. July 2000, 8898
[2] Roosnek, N., Corrigendum to ''Novel digital signal processing techniques for ultrasonic gas flow measurements'' Flow. Meas. Instrumen. June 2001, 231