In pulsed-wave ultrasound Doppler velocimetry (PW–UDV), short bursts are emitted periodically with a pulse repetition frequency fPR . The emission times ts=nb/fPR span the so-called slow-time axis ts, with nb=0…NEPP being the bursts number. The emitted bursts usually consist of Nperiods periods of a sinusoidal wave, with the frequency f0. As the bursts travel through the fluid, scattering particles reflect a fraction of the signal back to the ultrasound transceiver. The received echo signal z(tf,ts) is acquired, starting from the emission time along the fast-time axis tf. Figure  depicts an example of the echo signal for a single moving scattering particle.

The movement of a scattering particle leads to a phase shift of the echo signal between multiple burst emissions . The mean phase shift per time unit, expressed as mean frequency fd, is related to the velocity v for a given speed of sound c by the following: 1v=-12fdftxc, with ftx denoting the mean frequency of the received signal burst and c≫v. The mean phase shift per time unit fd can be interpreted as a Doppler frequency shift fd ; hence the name Doppler velocimetry.

The time since the burst emission tf corresponds to the distance d between the scattering particle and the transducer according to the following equation: 2d=12tfc. This allows a spatially resolved flow measurement along the axis of the transducer, given that the scattering particles follow the motion of the fluid with negligible slip. The axial resolution can be estimated with the following : 3Δd=12Nperiodscf0. The lateral resolution Δx is given by the width of the ultrasound beam, which is a result of the transducer geometry, the frequency f0, and the speed of sound c in the fluid. The temporal resolution Δt of the velocity measurement is determined through the following: 4Δt=NEPPfPR.

The ultrasound array Doppler velocimeter (UADV) is a modular research platform developed at the Laboratory of Measurement and Sensor System Technique (MST) for flow imaging in opaque liquids with PW–UDV. It is flexible and especially well suited for instrumenting a wide range of experiments in the field of MHD. The hardware of the UADV consists of individually configurable modules driving 25 ultrasound transducers each. It can be scaled to support up to 200 transducers in various configurations; for instance, in four linear arrays which can be individually parameterized regarding ultrasound frequency, pulse shape and length, and pulse-repetition frequency .

A module of the UADV consists of an arbitrary function generator and a power amplifier for generating parameterizable burst signals which are routed through a programmable switching matrix and a transmit/receive switch to the transducers. The received echo signals are amplified with a parametric gain and routed to the digitization unit. A single microcontroller-driven control unit provides the overall synchronization and the communication with the host PC. Using a combined spatial- and time-division multiplexing scheme, an ultrasound transducer array can scan a measurement plane at higher rates than a strict sequential scan. The UADV supports four independent digitization channels per module. The detailed description of the measurement system is given in .

The signal processing for PW–UDV can generally be classified into wideband and narrowband techniques; a comprehensive comparison is given by . While parts of the signal processing in the radio frequency (RF) band can be realized in analog circuitry , fully digital implementations have found widespread use in the last decades because of the availability of fast digitizers and the increased flexibility and robustness of such approaches. To simultaneously handle a high number (e.g., 32) of channels through a fully digital signal-processing chain, very large data bandwidths have to be processed. This can be achieved by utilizing the parallel-processing capability of a field-programmable gate array (FPGA). Especially narrowband algorithms are very suitable for FPGA-based implementations, due to their low computational complexity . Therefore, this paper focuses on investigating the most common narrowband method, the velocity estimator by  and the extensions proposed by .

A typical narrowband signal-processing chain is shown in Fig. . In this fully digital realization, the slow time ts is sampled for each burst nb at ts=nb1fPR and the fast time tf is sampled with a frequency fs as follows: 5zraw′(k,nb)=z(tf=k/fs,ts=nb/fpr),k=0,1,…K,nb=0,1,…NEPP. The signals are then bandpass filtered to reduce noise contributions outside of the bandwidth of the transmitted ultrasound signal. A quadrature demodulation is performed, consisting of a Hilbert transform and a subsequent down sampling. Static echoes are removed through a clutter reduction filter (CRF) and the velocities are estimated by an autocorrelation.

In order to meet the assumptions of the narrowband signal processing and to reduce the influence of noise, a bandpass filtering is performed as follows: 6z′(k,nb)=∑n=0Nperiodsci⋅zraw′(k-n,nb), with the filter coefficients ci. In order to maximize the SNR for signals with additive white Gaussian noise, a matched filter is used  as follows: 7ci=stx(Ntx-i),i∈[1,Ntx], with the transmitted signal stx with Ntx samples.

The result of the quadrature demodulation is a complex signal h′unfilt(k,nb) in the baseband, which can be sampled at a lower rate (reduction by a factor of nsub) than the raw signal, as follows: 8h′unfilt(k/nsub,nb)=z′(k,nb)+j⋅z′^(k,nb), with Nb=0,1,…Nepp,k=0,1,…K and the Hilbert transform signal z′^(k,nb) (with 90∘ phase shift with respect to z′).

A common problem of ultrasound Doppler flow measurements is distinguishing between static echoes originating from the walls (the so-called clutter) and echoes originating from scattering particles. Multiple reflections from the transmitted burst inside the wall superimpose the signal from scatter particles in the vicinity of the wall. For this problem, a multitude of signal-processing methods were proposed, most of them based on digital filters (finite impulse response (FIR) or infinite impulse response (IIR) filters) with various initialization techniques . With these methods, the clutter is distinguished from the particle echoes by a velocity close to zero, respectively, by a Doppler frequency shift close to zero. Because filtering will influence the spectrum of the signal, a bias may be introduced to the subsequent velocity estimation, depending on the frequency cutoff. For typical MHD experimental setups, the wall can be assumed to be completely stationary (in contrast to, e.g., medical applications where clutter is often constituted by slowly moving tissue; cf. ); therefore, a steep cutoff at a frequency of zero is desirable. The simplest and computationally most efficient approach is to filter the constant component of the demodulated IQ signal by subtracting its mean value, which is the equivalent of applying a very narrow-band, high-pass filter  as follows:

9h′(k/nsub,nb)=h′unfilt(k/nsub,nb)-1NEPP∑nb′=0NEPP-1h′unfilt(k/nsub,nb′). As the filter is noncausal, all NEPP samples have to be acquired before the result can be computed.

A widely used approach for velocity estimation is the autocorrelation method proposed by , which operates solely in the domain of IQ-demodulated echo signals and therefore can be implemented very efficiently . It uses the properties of the signals' discrete autocorrelation function as follows: 10R′(Δk,Δnb)=∑m=0K/nsub-Δk-1∑n=0NEPP-Δnb-1h′(m,n)⋅h′*(m+Δk,n+Δnb), where its values at a lag of 1 relate to the center of mass of the signal's power density spectrum through the Wiener–Khinchin theorem. As shown by  and , the mean Doppler shift fd can be approximated through evaluating the autocorrelation function at a lag of Δnb=1 slow-time samples as follows: 11fd≈1/TPRarg(R(Δk=0,Δnb=1)). This autocorrelation computation can be expressed solely by repeatedly multiplying accumulate operations and therefore can be implemented very efficiently. Kasai's method approximates the center frequency fd of the received signal with the frequency of the emitted signal as follows: 12frx≈f0. Being based on a phase estimation, the Kasai algorithm is inherently limited in the maximum measurable velocity. Given the 2π-phase ambiguity in Eq. (), the measurable velocity range resulting from Eq. () is as follows: 13v∈[±vmax];vmax=cfPR4f0.

An extension of Kasai's autocorrelation method is proposed by Loupas et al. to improve its performance in the following two regards :

The assumption of an unchanged center frequency of an ultrasound burst throughout emission, propagation inside the fluid and reception is discarded. This allows one to account for the effect of frequency-dependent attenuation, which is present in most relevant fluids. By explicitly estimating the center frequency of the received signal, a systematic velocity error stemming from the relationship in Eq. () v∝1/f‾tx is avoided.

Both aspects are addressed by increasing the dimensionality of Kasai's autocorrelation; instead of just correlating along the slow-time axis, a 2D autocorrelation along the slow- and fast-time axis is performed. An autocorrelation with a lag of one fast-time sample yields the estimate of the center frequency as follows: 14frx≈fsarg∑mR(Δk=1,Δnb=0)≈12πfsnsub2π⌊1/2+nsubf0/fs⌋+argR′(1,0)15frx∈(fsnsub⌊1/2+nsubf0/fs⌋±12fsnsub].

Furthermore, the estimation of the frequencies ftx and fd can be performed using M samples per gate, as follows:

16fd≈12πfPRarg⁡R′(0,1).

The extension of the Kasai autocorrelation algorithm potentially improves the estimation performance while still preserving a low computational complexity.

In order to characterize the performance of a signal-processing algorithm, it is not only helpful to have relative data compared to other algorithms but also to relate it to a fundamental limit of attainable precision. This absolute limit of uncertainty can be provided by means of the estimation theory using the Cramér–Rao bound CRB;. Given a suitable signal model, the CRB represents the lowest possible variance for estimating a parameter from the signal with an unbiased estimator. In the following, a simple signal model for a discrete time idealized ultrasound echo is described and a derivation of the CRB for velocity estimation is given.

A simple approximation of the ultrasound echo signal realizations x(k,nb,θ,σn2) consists of a sinusoidal signal s(k,nb,θ) superimposed with additive white Gaussian noise n(k,nb,σn2) sparsely and periodically sampled in the fast- (k) and slow-time (nb) axis as follows:

21x(k,nb,θ,σn2)=s(k,nb,θ)+n(σn2,k,nb),22withs(k,nb,θ)=Acos⁡(2π(f0+fd)kfs+nbfPR+φ0), and A being the amplitude of the scattering particles' echo, φ0 a constant phase, and n(k,nb,σn2) Gaussian white noise, with a variance σn2 and zero mean.

The unknown quantities are as follows: 23θ=Afdφ0.

The CRB provides the lower boundary for the variance of an estimator θ^i according to the inequality, as follows: 24var(θ^i)≥CRB(θ^i)=I-1(θ)ii, with I(θ) being the Fisher information matrix, as follows: 25I(θ)ij=-Eδ2ln⁡(p(x,θ))δθiδθj. provided a formula for the case when the probability density function p(x,θ) of the signal model x, Eq. (), is a Gaussian joint probability function as follows: 26I(θ)ij=1σn2∑k∑nbδs(k,nb,θ)δθiδs(k,nb,θ)δθj.

The differentiation of s(k,nb,θ) with respect to the unknown quantities is performed analytically, while the matrix inversion was performed numerically using MATLAB (The MathWorks, Inc., Natick, Massachusetts, USA). The resulting CRB for the velocity uncertainty as a function of the signal-to-noise ratio (SNR) is given in Fig. c–d. It has a slope of -20 dB/decade, which is consistent with the CRB of other Doppler-based signal-processing problems .

For an experimental characterization of the measurement performance of the UADV system, a test rig based on the linear translation of a single scattering object is used (Fig. ). It consists of a linear stage (41.121.102E; OWIS GmbH, Staufen, Germany) that is mounted over a glass tank with the dimensions of 212×81×135mm3. It moves a scattering object (glass fiber with a spherical tip, and diameter of 0.6mm, mounted in a hollow needle) with a constant velocity through water (ϑ=20∘C; c=1480ms-1). The ultrasound sensor array is mounted on the front wall of the tank and therefore insonates through an 8mm glass wall and a water-based ultrasound couplant.

In order to trace back the measurement results of the UADV to the definitions of the respective units in the SI system, a simultaneous measurement of the relative position and velocity was done with a vibrometer (OFV-503; Polytech, Waldbronn, Germany; displacement decoder DD-900 and velocity decoder VD-09). A retroreflective tape (3M Scotchlite) was attached to the shaft of the scattering object's mount. For a velocity set point of 10mms-1, a standard deviation of the velocity σv,ref,rel=0.178% was determined for the linear stage–vibrometer combination (for the same averaging time as the UADV system).

A total of 130 measurement cycles were conducted, consisting of a constant translation away from the front wall of the tank with a velocity set-point vref=10mms-1 and the respective backward motion. Of the continuously obtained UADV measurements, only those that originate from two defined positions near to and far from the wall during the movement away from the ultrasound transducer (Richter Sensor and Transducer Technology, Germany) are selected in the postprocessing. The clutter-to-signal ratio (CSR) is CSR1=-7.3dB and CSR2=-19.0dB, respectively. To ensure a common time base for vibrometer and UADV measurements, the trigger signal of the UADV is acquired simultaneously with the velocity and position signals. To test the performance under different SNR conditions, white Gaussian noise was added to the raw digitized signals to achieve SNR=-6,-3,…,12dB. Four algorithmic variants were compared as follows:

(DEF) – the 1D Kasai velocity estimator without clutter filtering, as described in Sect.