Adaptation of Range-Doppler Algorithm for Efficient Beamforming of Monostatic and Multistatic Ultrasound Signals

Abstract

Algorithmic changes that increase beamforming speed have become increasingly relevant to processing synthetic aperture (SA) ultrasound data. In particular, beamforming SA data in a spatio-temporal frequency domain using F-k (Stolt) migration has been shown to reduce beamforming time by up to two orders of magnitude compared to conventional delay-and-sum (DAS) beamforming, and it has been used in applications where large amounts of raw data make real-time frame rates difficult to attain, such as multistatic SA imaging and plane-wave Doppler imaging with large ensemble lengths. However, beamforming signals in a spatio-temporal Fourier space can require loading large blocks of data at once, making it memory intensive and less suited for parallel (i.e. multi-threaded) processing. As an alternative, we propose beamforming in a range-Doppler (RD) frequency domain using the range-Doppler algorithm (RDA) that has originally been developed for SA radar imaging. Through simulation and phantom experiments we show that RDA achieves similar lateral resolution and contrast compared to DAS and F-k migration. At the same time, higher axial sidelobes in RDA images can be reduced via (temporal) frequency binning. Like F-k migration, RDA significantly reduces the overall number of computations relative to DAS, and it achieves ten times lower processing time on a single CPU. Because RDA utilizes only a spatial Fourier Transform, it requires two times less memory than F-k migration to process the simulated multistatic data, and can be executed on as many as a thousand parallel threads (compared to eight parallel threads for F-k migration), making it more suitable for implementation on modern graphics processing units (GPUs). While RDA is not as parallelizable as DAS, it is expected to hold a significant speed advantage on devices with moderate parallel processing capabilities (up to several thousand cores), such as point-of-care and low-cost ultrasound devices.

I. Introduction

Synthetic aperture (SA) acquisition of ultrasound signals [1, 2] has been used to cover a wide field-of-view (FOV) and achieve uniform resolution of ultrasound images, as well as to improve image resolution at large depths [3]. SA reconstruction typically involves coherent summation of data from different transmit and/or receive locations after the data capture, so the effective aperture can be ”synthesized” and dynamically focused across the FOV. With the development of ultrasound systems that utilize fully programmable software-based beamformers over the past decade, it became easier to create SA images online and in real-time. However, traditional SA imaging relies on time-domain beamforming approaches such as delay-and-sum (DAS), which can be prohibitively expensive because it computes each pixel in the image independently, and can compromise system’s real-time imaging capabilities. This is especially true in SA applications that require storing and processing of large amounts of raw channel data, such as multistatic SA acquisitions [1], imaging with large swept synthetic apertures [3] and matrix arrays [4], and plane-wave Doppler imaging of microvasculature using large ensembles [5, 6].

Beamforming SA data in the frequency domain can offer a significant increase in computational speed compared to DAS-based SA reconstruction, as it enables focusing of the recorded signals at many pixels simultaneously. Frequency-domain beamforming methods have been used extensively in geophysics for rapid formation of images of Earth’s surface and subsurface [7], and in underwater acoustics [8, 9], and can be classified into two groups [10].

The first group of beamformers operate in a spatio-temporal frequency domain (such as F-k (Stolt) migration, phase migration, and chirp-scaling algorithm), and apply a Fourier transform to channel data in both aperture and time dimensions. These methods are well suited for broadband signals typically encountered in medical ultrasound, and as such have been adapted to beamform various kinds of ultrasound SA acquisitions, including monostatic acquisitions [11], multistatic acquisitions [12], plane-wave acquisitions [13–16], and ultrasound acquisitions with virtual sources [17, 18]. In this work, the terms monostatic and multistatic are used to describe two common SA acquisition geometries. For monostatic acquisitions, the transmitter and the receiver are at the same location, while for multistatic acquisitions, different combinations of transmit and receive elements are used to collect the data (Figure 1).

Fig. 1:

Synthetic aperture (SA) acquisition geometries used in this work. Monostatic acquisition (a) implies that the transmitter and receiver are at the same location. Monostatic data can be collected using an array of elements or by physically sliding a single element transducer. Multistatic acquisition (b) means that transmitting and receiving elements are in different locations. Multistatic acquisition can use all combinations of transmit and receive elements in the array to collect a complete SA dataset.

In particular, F-k migration has been shown to reduce the beamforming time of multistatic ultrasound signals by up to two orders of magnitude compared to multistatic delay-and-sum (DAS) beamforming [12]. However, processing signals in a multi-dimensional Fourier space can require loading large blocks of data at once, which can significantly increase the memory requirements and can make parallel (i.e. multi-threaded) implementation difficult. Taking a Fourier Transform in the time dimension can be especially memory consuming due to the large number of time samples in acquired data (on the order of thousands), and significant zero-padding (on the order of data size) that is required to avoid interpolation errors in the frequency domain and to achieve high reconstruction accuracy. As a result, spatio-temporal frequency-domain beamformers can be difficult to implement on ultrasound devices that have limited memory and/or processing power, such as point-of-care and handheld ultrasound devices.

The second group of frequency-domain beamforming methods assumes monochromatic signals and utilizes a Fourier Transform in the aperture dimension only, which allows signal processing in the time and space dimensions independently and with less memory constraints. The range-Doppler algorithm (RDA), originally developed for synthetic aperture radar (SAR) [7, 10] and that we propose to modify here is an example of such a method. RDA has been already applied to monostatic ultrasound signals from a sliding piston transducer [19], and to monostatic data from an ultrasound array of elements [20]. In [20], frequency binning was utilized to adapt RDA to the broadband nature of ultrasound signals, while keeping its memory requirements low and its beamforming speed higher than that of DAS. There is a potential for RDA in real-time imaging applications and to create high resolution ultrasound images on handheld devices.

Here, we adapt RDA to multistatic ultrasound signals, so they can be beamformed in a rapid and memory-efficient way. In Section II, we present the theory and physics behind beamforming in the range-Doppler domain and derive the expressions for (asymptotic) computational cost of RDA for both monostatic and multistatic cases. In Section III, we describe the simulations and phantom experiments carried out to validate the monostatic and multistatic versions of RDA against the DAS beamformer and F-k migration. In Section IV, we compare the resulting B-mode images, as well as reconstruction speeds and memory usage of the three beamforming approaches. We also estimate potential improvements in RDA speed that could be achieved using multiple processing cores. The discussion of the results and concluding remarks are offered in Sections V and VI, respectively. Implementations in MATLAB of monostatic and multistatic RDA are provided under https://gitlab.com/mj66/bentobox.

II. Range Doppler Algorithm (RDA)

A. RDA for Monostatic Ultrasound Signals

The RDA has been originally developed to beamform monostatic baseband signals [7]. The starting model for echoes originating from a point scatterer located at azimuth origin and depth R₀ is given by the following equation:

$$s(t,x)=A_0{w}_r\left(t-\frac{2R(x)}{c}\right)\exp \left[-j4\pi f_0\frac{R(x)}{c}\right]$$ (1)

where

$$R(x)=\sqrt{R_0^2+x^2},$$ (2)

and x is the transmit/receive aperture location, t is the receive-echo time, w_r is the signal envelope in range/time dimension, f₀ is the (center) transmit frequency, and c is the wavespeed in the medium. The signal in (1) is monochromatic, and is expressed in the so-called time-space domain. With the conventional DAS beamformer, focusing is performed entirely in this domain, on a pixel-by-pixel bases. Within RDA, aperture data is first converted to a range-Doppler (RD) domain by taking a Fourier Transform (FT) in the aperture dimension. In the RD domain, wavefronts from targets at the same depth overlap, and can therefore be migrated together, improving beamforming efficiency. The migrated data is then converted to the image space by taking an Inverse Fourier Transform in the spatial-frequency (also known as Doppler) dimension.

The key steps of RDA adapted to monostatic ultrasound data are outlined in Figure 2 and are explained in greater detail below. Because ultrasound scanners typically do not transmit chirps (but rather a Gaussian-weighted sinusoid), the range compression and the secondary compression steps are omitted from the original version of the method in [7].

Fig. 2:

Key processing steps of the Range Doppler Algorithm applied to monostatic ultrasound signals. Data space at each step is labeled below the operation.

1. Azimuth FFT. Starting with (1), the point-target signal in the RD domain can be written as

   $$\begin{gathered} S_0\left(t,k_x\right)=A_0{w}_r\left(t-t\left(k_x\right)\right)\times \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\times \exp \left[-j\pi \frac{4f_0{R}_0}{c}\sqrt{1-{\left(\frac{k_{x}c}{2f_0}\right)}^2}\right] \end{gathered}$$ (3)

   where

   $$t\left(k_x\right)=\frac{2}{c}\frac{R_0}{\sqrt{1-{\left(\frac{k_{x}c}{2f_0}\right)}^2}}.$$ (4)

   and k_x is the azimuth (so-called Doppler) frequency. In both time-space and RD domains, the travel-times from a point target have the shape of a hyperbola (equations (2) and (4)). The expressions in (3) and (4) are derived using the principle of stationary phase (POSP) [7, 21], and this derivation is presented in Appendix A.

2. Range Cell Migration Correction (RCMC). The range cell migration correction is performed in the RD domain, and it is used to migrate (i.e. align) the range envelopes w_r in equation (3). In the RD domain, targets located at the same depth have overlapping wavefronts whose envelopes can be migrated together, providing a significant speedup compared to beamforming in time-space (such as with DAS), where the wavefront from each target has to be migrated separately. This point is illustrated in Figure 3, which shows the wavefronts from three simulated point targets at the same depth, in time-space and RD domains. Migration (alignment) is typically implemented as a 1-D interpolation of signals along time/range axis, where the interpolation times are computed using expression (4).

3. Matched filter along Doppler (k_x) dimension. A matched filter removes the azimuth phase of received signals after RCMC, as illustrated in Figure 4. The filter is usually designed by taking a complex conjugate of the phaseterm in equation (3).¹ An example azimuth spectrum of a simulated point target and the spectrum model based on equation (3) are shown in Figure 5.

4. IFFT in Doppler (k_x) dimension. This last step of the RDA converts the migrated and filtered data into the image domain, as shown in the bottom image of Figure 4.

Fig. 3:

Wavefront migration in range-Doppler domain. Wavefront magnitude from three simulated point targets at the same depth is shown in the time-space domain (top row), in the range-Doppler domain before wavefront migration (middle row), and in the range-Doppler domain after wavefront migration (bottom row). In both domains, the wavefronts have the shape of a hyperbola. In the RD domain, the wavefronts from three point targets overlap and create an interference pattern. Since these wavefronts can be migrated together, RDA reduces computation time compared to DAS, which requires a separate wavefront migration at each pixel location.

Fig. 4:

Matched filtering and azimuth IFFT steps of RDA. Simulated signal from a point target after wavefront migration, but before matched filtering (top row), after matched filtering (middle row), and after IFFT in image space (bottom row). The matched filter removes phase oscillation in the k_x dimension.

Fig. 5:

Azimuth spectrum of a simulated point target (i.e. cross-section of the top image in Figure 4), and the model of the spectrum based on equation (3) that is used to create a matched filter. Phase of the spectrum looks nearly quadratic, indicating that the spectrum can be also modeled as a chirp.

An example implementation of monostatic RDA in MATLAB is included in Appendix C.

B. RDA for Multistatic Ultrasound Signals

For a multistatic acquisition, the pulse-echo travel time can be separated into transmit and receive components. The starting signal model then becomes:

$$\begin{gathered} s\left(t,x_{tx},x_{rx}\right)=A_0{w}_r\left(t-\frac{R_{tx}\left(x_{tx}\right)}{c}-\frac{R_{rx}\left(x_{rx}\right)}{c}\right)\times \\ \hspace{0.33em}\hspace{0.33em}\times \exp \left[-j2\pi f_0\frac{R_{tx}\left(x_{tx}\right)}{c}\right]\exp \left[-j2\pi f_0\frac{R_{rx}\left(x_{rx}\right)}{c}\right] \end{gathered}$$ (5)

where

$$R_{tx}\left(x_{tx}\right)=\sqrt{R_0^2+x_{tx}^2},\hspace{1em}{R}_{rx}\left(x_{rx}\right)=\sqrt{R_0^2+x_{rx}^2},$$ (6)

and x_tx and x_rx are transmit and receive element positions, respectively. For the multistatic version of RDA, having transmit and receive elements at different locations introduces an additional spatial/Doppler frequency dimension in the RD domain.

The key steps of multistatic RDA are similar to the monostatic version of the method, but extended along the additional Doppler dimension, as detailed below:

1. 2-D FFT in transmit and receive aperture dimensions. The multistatic signal in RD domain can be written as

   $$\begin{gathered} S_0\left(t,k_{tx},k_{rx}\right)=A_0{w}_r\left(t-\frac{R\left(k_{tx}\right)}{c}-\frac{R\left(k_{rx}\right)}{c}\right)\times \\ \hspace{0.33em}\times \exp \left[-j\pi \frac{2f_0{R}_0}{c}\sqrt{1-{\left(\frac{k_{tx}c}{f_0}\right)}^2}\right] \\ \hspace{0.33em}\times \exp \left[-j\pi \frac{2f_0{R}_0}{c}\sqrt{1-{\left(\frac{k_{rx}c}{f_0}\right)}^2}\right] \end{gathered}$$ (7)

   where

   $$\begin{gathered} R_{tx}\left(k_{tx}\right)=\frac{R_0}{\sqrt{1-{\left(\frac{k_{tx}c}{f_0}\right)}^2}}, \\ {R}_{rx}\left(k_{rx}\right)=\frac{R_0}{\sqrt{1-{\left(\frac{k_{rx}c}{f_0}\right)}^2}}. \end{gathered}$$ (8)

   and k_tx and k_rx are the transmit and receive aperture spatial frequencies. The travel-times from a point target have the shape of a hyperbola in both the time-space and RD domains (equations (6) and (8), respectively). Similar to equations (3) and (4) for the monostatic case, the expressions in (7) and (8) are also derived using the POSP described in Appendix A.

2. Range Cell Migration Correction (RCMC). The range cell migration correction migrates the range envelopes w_r across the transmit and receive spatial frequencies. In the RD domain, scatterers that are located at the same depth will have overlapping wavefronts, and can be migrated together.

3. 2-D Matched filter in k_tx and k_rx frequency dimensions. A matched filter removes the azimuth phase of migrated signals.

4. 2-D IFFT in k_tx and k_rx dimensions. An Inverse Fourier Transform converts the migrated and filtered 3-D data from the RD domain to a 3-D image space (r, x_tx, x_rx). To reduce the number of dimensions of beamformed data and obtain a standard (r, x) image, a diagonal plane is selected for the transmit and receive element axes.²

C. Computational Complexity

The computational complexity of an algorithm can be described using the ”big O” (denoted $𝓞$) notation [22]. This analysis counts the number of operations as a function of input and output array sizes. To find the asymptotic performance, only the dominant terms are kept and operations that are performed a fixed number of times are not counted towards the overall computation cost.

1. Monostatic case: For monostatic acquisitions, let us consider an input array of size T -by-N, where N is the number of transducer elements and T is the number of time samples recorded on each element. The beamformer output is a P-by-M B-mode image, where P and M are the numbers of pixels in the axial and lateral dimensions, respectively. A DAS beamformer performs NPM interpolations, so its computational complexity is ${𝓞}_{\text{DAS}}\left(NPM\right)$. The computational complexity of monostatic RDA is given by:

   $$\begin{gathered} 𝓞(\underset{\text{1-D FFT}}{\underbracket{TN\hspace{0.17em}{\log }_2\hspace{0.17em}N}}+\underset{\text{RCMC}}{\underbracket{MP}}+\underset{\text{Matched}\hspace{0.17em}\text{filter}}{\underbracket{MP}}+\underset{\text{1-D IFFT}}{\underbracket{PM{\log }_{2}M)}}~ \\ {𝓞}_{\text{RDA}}\left(2TN{\log }_{2}N\right), \end{gathered}$$ (9)

   where under-brackets are used to denote the term for each step in RDA (Figure 2). For large channel datasets and large B-mode images the FFT and IFFT terms dominate the second-order terms of the RCMC and the matched filter. In the final approximation of equation (9), we also assume that the number of time samples is similar to the number of depth pixels, and that the number of transducer elements is similar to the number of lateral pixels. By making the same assumptions for DAS we get ${𝓞}_{\text{DAS}}\left(T{N}^2\right)$, meaning that RDA reduces computational complexity by a factor of N/(2 log₂N).
   The computational complexity of RDA is similar to the computational complexity of spatio-temporal frequency-domain beamformers, such as F-k migration. For the monostatic case, the computational complexity of F-k migration is given by

   $$\begin{gathered} {𝓞}_{F-k}\left(TN{\log }_{2}TN+MP+PM{\log }_{2}PM\right)\sim \\ {𝓞}_{F-k}\left(2TN{\log }_{2}TN\right), \end{gathered}$$ (10)

   where the same assumptions about the data and image sizes are considered as in equation (9).
2. Multistatic case: In a multistatic acquisition, data is captured for each transmit-receive element pair and the input array size is TNN. The multistatic DAS beamformer performs PMN² interpolations, so its computational complexity is given by ${𝓞}_{\text{DAS}}\left(PM{N}^2\right)\sim {𝓞}_{\text{DAS}}\left(T{N}^3\right)$. The computational complexity of the multistatic RDA is given by:

   $$\begin{gathered} 𝓞(\underset{\text{2-D}\hspace{0.17em}\text{FFT}}{\underbracket{T{N}^2{\log }_2{N}^2}}+\underset{\text{RCMC}}{\underbracket{P{M}^2}}+\underset{\text{Matched}\hspace{0.17em}\text{filter}}{\underbracket{P{M}^2}}+\underset{\text{2-D}\hspace{0.17em}\text{IFFT}}{\underbracket{P{M}^2{\log }_2{M}^2}})~ \\ {𝓞}_{\text{RDA}}\left(2T{N}^2{\log }_2{N}^2\right). \end{gathered}$$ (11)

   Multistatic RDA reduces computational complexity relative to multistatic DAS by a factor of $N/(2{\log }_2{N}^2)$.

   The computational complexity of multistatic F-k migration is derived in [12] and is shown here for convenience:

   $$\begin{gathered} {𝓞}_{F-k}\left(T{N}^2{\log }_{2}T{N}^2+M^{2}P+PM{\log }_{2}PM\right)\sim \\ {𝓞}_{F-k}\left(T{N}^2{\log }_{2}T{N}^2\right). \end{gathered}$$ (12)

   Computational complexities of RDA, DAS, and F-k migration are plotted as functions of N in Figure 6. For both monostatic and multistatic cases, RDA and F-k migration show reduced computational complexity relative the DAS. Vertical black lines are used to denote computational complexity for the array size used in simulation and phantom experiments (128 elements). In this case, RDA is expected to reduce the number of operations (relative to DAS) nine times for the monostatic acquisitions, and five times for the multistatic acquisitions.

Fig. 6:

Computational complexity of RDA, DAS, and F-k beamformers expressed in ”big O” notation for monostatic acquisition (a), and multistatic acquisition (b). Vertical black lines indicate computational complexity for the array size used in simulation and phantom experiments (128 elements). The RDA plots are created using complete expressions in (9) and (11). RDA reduction in computational complexity (relative to DAS) can be approximated by a factor of N/(2 log₂ N) for monostatic data, and by a factor of N/(2 log₂ N²) for multistatic data. RDA shows similar computational complexity to F-k migration.

III. Methods

A. FIELD II Simulations of Synthetic Aperture Data

Multistatic synthetic aperture (SA) data from a 128-element linear array was simulated using FIELD II software [23]. The simulated acquisition was on a point target phantom designed to measure the resolution of the RDA beamformer as a function of depth. The transducer properties used in simulation are listed in Table I. B-mode images were created from the complete multistatic dataset and from the monostatic subset of data using the versions of RDA outlined in Sections II-B and II-A, respectively. Sub-band frequency binning was used in both versions of RDA to account for the wideband nature of ultrasound signals, similar to [20]. In particular, 1) the signals were divided into three frequency bins, 2) RDA was applied to data from each bin separately, and 3) the resulting radio-frequency (RF) images were summed coherently. For comparison, B-mode images were also reconstructed using monostatic and multistatic DAS, and using the multistatic version of F-k migration as proposed in [12].

TABLE I:

Simulated transducer properties.

Array Width	19.9 mm
Number of Elements	128
Element Pitch	0.15 mm (λ/2)
Center Frequency	5 MHz
Bandwidth	60 %

B. Phantom Acquisitions

Multistatic SA data was acquired off an L12–3v linear array that was attached to a Verasonics Vantage-256 scanner (Verasonics, Inc., Redmond, WA). The array has 128 elements with an element pitch of 0.2 mm. A 6 MHz center frequency was used. A CIRS calibrated phantom (CIRS Model 040GSE, Norfolk, VA) was scanned in the regions containing closely spaced point targets and anechoic lesions in the field of diffuse scatterers. B-mode images were beamformed from the multistatic and monostatic data using DAS, F-k migration, and RDA in a similar manner as in Section III-A.

C. Beamformer Performance Metrics

To evaluate beamformers’ resolution and contrast, lateral profiles were plotted from the simulated point target images, allowing for a direct assessment of the mainlobe width and the sidelobe levels at each point target depth. Beamforming times were also measured on the simulated data, with all the beamforming methods implemented using a custom MATLAB code (MATLAB, Natick, MA) on a single CPU. Peak memory requirements were estimated for the multistatic beamformers as the size of the largest data variable stored during runtime. For DAS, the largest data variable was a 3-D array containing multistatic channel data in the space-time domain, for RDA, it was a 3-D array with multistatic channel data in the range-Doppler domain, and for F-k migration, it was a 3-D array with aperture data in the spatio-temporal frequency domain.

To assess memory management within each beamformer and a potential speedup that could be achieved with multiple processing cores two additional metrics were computed from the simulated images: 1) minimum per-thread memory (computed as the memory required to reconstruct the smallest section of the image), and 2) the maximum number of potential threads that could be run in parallel for image reconstruction. To compute these metrics we used the fact that the simulated images had 1000-by-256 pixels, and that an algorithm would be executed on each thread independently from start to finish. Under such conditions, the size of the smallest image section that could be reconstructed on a single thread would depend on the nature of the beamforming method (pixel-based v.s. blockbased), and was a single pixel for DAS, a row of pixels for RDA, and a 128-by-256 block of pixels for F-k migration. For F-k migration, a minimum of 128 depth samples was assumed to ensure the accuracy of FFT and IFFT along that dimension.

Beamformer performance was also evaluated on anechoic targets in the tissue-mimicking phantom. To that end, the generalized contrast-to-noise ratio (gCNR) [24] was computed as follows:

$$\text{gCNR}=1-\text{OVL},$$ (13)

$$\text{OVL}={\int }_{-\infty }^{\infty }\underset{x}{\min }\left\{p_s(x)p_b(x)\right\}dx.$$ (14)

In (14), OVL is the area of overlap between the probability density functions (pdf’s) for the signal and background, p_s(x) and p_b(x), respectively. The pdf s p_s(x) and p_b(x) were approximated by the histograms over the speckle and background regions, respectively. The pixel values were grouped in 100 bins distributed on a logarithmic scale from −50 to 0 dB.

IV. Results

A. Simulated Point Targets

B-mode images of simulated point targets are shown in Figure 7. The images are created using monostatic and multistatic versions of DAS and RDA and the multistatic version of F-k migration. The images are displayed over 50 dB of dynamic range. Lateral profiles (i.e. beamplots) are also shown at each point target depth. For both multistatic and monostatic datasets, images created with different beamforming methods show similar point target width and lateral sidelobe levels. In the axial dimension, RDA causes higher sidelobes compared to DAS and F-k migration. Overall, the images created from multistatic data show lower lateral sidelobes compared to the images beamformed from monostatic data.

Fig. 7:

B-mode images of simulated point targets reconstructed from the monostatic (indicated as mono) and multistatic (indicated as multi) SA data using DAS, RDA, and F-k migration methods (a). The scale bar in the leftmost B-mode image is 5 mm. All images are log compressed and displayed over 50 dB of dynamic range. The corresponding beamplots at each point-target depth (b - e). The point target images and plots created from the multistatic data show reduced sidelobe levels compared to their monostatic counterparts. For both multistatic and monostatic acquisitions, the targets created with different beamforming methods have similar lateral profiles. The RDA images show higher axial sidelobes compared to the DAS and F-k images.

Running times and memory required to beamform images in Figure 7 are reported in Tables II and III, respectively. As implemented on a single CPU, RDA with three frequency bins reduces beamforming time approximately by a factor of nine compared to DAS, for both monostatic and multistatic data. In addition, multistatic RDA using a single temporal frequency bin achieves a similar computing time as multistatic F-k migration. Regarding memory consumption, RDA requires about four times more memory to process multistatic data than DAS, and two times less memory than F-k migration. Because the RDA beamformer produces a row of pixels together, it makes it possible to beamform the images in Figure 7 by running as many as 1000 threads in parallel, enabling a potential 1000-fold reduction in beamforming time. This is substantially more than F-k migration, which allows up to eight parallel threads, but also significantly less than DAS, which allows as many as 256, 000 threads, each beamforming a single pixel of the final image.

TABLE II:

Runtimes for RDA (using 1 and 3 freq. bins), F-k migration, and DAS (for 256 A-lines) measured on simulated data in Figure 7.

Beamforming runtimes (s)	RDA		F-k	DAS
	1 freq. bin	3 freq. bins
Monostatic	0.17	0.52	/	4.79
Multistatic	8.03	21.86	9.42	194.65

TABLE III:

Memory requirements for RDA, F-k, and DAS beamformers. Number estimates provided for multistatic images in Figure 7.

Memory estimates	DAS	RDA	F-k
Largest variable stored (MB)	120	482	1074
Minimum memory per thread a (MB)	0.67 b	4.40 c	134.30 d
Maximum no. of threads	256,000	1000	8

^aEstimated as the smallest portion of aperture data needed to reconstruct a part of a B-mode image.

^bMemory needed to reconstruct a single pixel.

^cMemory needed to reconstruct a row of pixels.

^dMemory needed to reconstruct a 128-by-256-pixel section of the image.

In Figure 8, we explore the effects of frequency binning on RDA image quality and show multistatic RDA images created with one, three, five, and ten frequency bins. All images are log compressed and displayed using a dynamic range of 50 dB. As the number of frequency bins increases, axial sidelobes are reduced, but the beamforming times also increase, as RDA is applied to signals from each frequency bin separately. As a compromise between axial sidelobe levels and beamforming time, three frequency bins are used to reconstruct RDA images in the rest of the paper.

Fig. 8:

RDA images created from simulated multistatic data using different numbers of frequency bins to account for the wideband nature of ultrasound signals. Specifically, RDA is applied to data at each frequency bin and beamformed images are coherently summed together. As the number of frequency bins increases, axial sidelobes are reduced. Going from left to right, the maximum axial sidelobe levels in RDA images are −7, −16, −23, and −28 dB. To reach a compromise between axial sidelobe levels and beamforming time, three frequency bins are used to reconstruct RDA images in this work. All images are log compressed and displayed using a dynamic range of 50 dB.

B. Phantom Acquisitions

To illustrate the frequency-binning process used in RDA, Figure 9 shows spectra of the individual (pre-beamformed) channel signals captured on the CIRS phantom, including a full-band spectrum of the original RF data, and bandpass-filtered spectra from each of the frequency bins used in the proposed implementation of RDA.

Fig. 9:

Frequency binning of ultrasound signals from a point-target phantom. The temporal spectra are shown for the original RF signal (i.e. the full-band spectrum), and for the signals band-pass-filtered over three frequency sub-bands. Applying RDA over each sub-band separately and coherently summing the results alleviates the violation of the monochromatic assumption.

B-mode images from two phantom acquisitions are shown in Figures 10 and 11. The images show point targets (Figure 10) and anechoic lesions (Figure 11) in the field of diffuse scatterers, and are beamformed from monostatic and multistatic SA data using DAS, RDA, and F-k migration in a similar manner as in Figure 7. All images are displayed over 50 dB of dynamic range, and the scale bar in the leftmost image in both figures is 5 mm. For both acquisitions, different beamforming methods create a similar speckle pattern with a similar speckle size. In Figure 10, the width of point targets is similar between the images, except for the point target located in the lower part of the field-of-view (FOV), which looks narrower in the multistatic images than in the monostatic images. In addition, the point targets in the multistatic RDA image have higher axial sidelobes than the point targets in the multistatic DAS and F-k images.

Fig. 10:

Point-target phantom images created with monostatic (abbreviated as mono) and multistatic (abbreviated as multi) versions of DAS and RDA and with multistatic F-k migration. The scale bar in the leftmost image is 5 mm. All images are log compressed and displayed over 50 dB of dynamic range. Point targets clustered in the upper part of FOV look similar between the images, except in the multistatic RDA image where they show higher axial sidelobes. The point target located in the lower part of FOV appears wider in the monostatic images than in the multistatic images.

Fig. 11:

B-mode images of the lesion phantom created from the monostatic (abbreviated as mono) and multistatic (abbreviated as multi) SA data. The images are reconstructed using DAS, RDA, and F-k migration in a similar manner as in Figures 7 and 10. The scale bar in the the leftmost B-mode is 5 mm, and all images are log compressed and displayed using a dynamic range of 50 dB. Two lesions appear in all images, but are more visible in the multistatic images than in their monostatic counterparts. Generalized contrast-to-noise ratio (gCNR) of the large lesion in the lower part of FOV is measured to be (going from left to right) 0.36, 0.43, 0.68, 0.67, and 0.65 indicating that RDA achieves a similar contrast compared to DAS and F-k migration. The regions inside and outside the lesion used to compute gCNR are denoted with white and red dashed lines, respectively.

The B-mode images in Figure 11 show a small anechoic lesion in the upper part of the FOV, and a large lesion in the lower part of the FOV. The images created from multistatic data show reduced clutter inside the lesions compared to the images from monostatic data; this is especially true for the smaller lesion, which is hardly visible in the monostatic images, but appears clear in the multistatic images for all beamforming methods. The generalized contrast-to-noise ratio (gCNR) values are reported for the larger lesion (in the caption of Figure 11), and they indicate that RDA achieves a comparable contrast to DAS and F-k migration.

V. Discussion

A. Processing Signals in range-Doppler Domain

The RDA signal model was designed to account for echoes coming from both near-field and far-field regions. The starting model in equations (1) and (2) describes the wavefront from a point target as a hyperbola in the space-time domain. The phase in (1) can be approximated by a quadratic polynomial (following the Taylor series expansion), which causes the frequency to change linearly as a function of space resulting in a chirp waveform. A similar model for near-field ultrasound signals in the time-space domain has been presented (with and without a parabolic approximation) in [25, 26], for the purpose of decomposing the signal and removing the contributions from off-axis scatterers to reduce clutter noise (a method called ADMIRE). The azimuth spectrum of a point target in equation (3) also has a hyperbolic phase, and can be approximated by a chirp (Figure 5). For plane waves that originate in the farfield, the quadratic phase term is zero and the signal oscillates at a constant frequency across the aperture (i.e. the signal is a sine wave).

Analyzing wavefronts in the RD domain helps establish the required sampling (i.e. Nyquist) rate across the aperture. In particular, because near-field targets have a chirp-like signature across the aperture, the required azimuth sampling frequency increases with aperture size. The largest acceptable pitch to avoid aliasing is given by $d_{Nyquist}=\frac{\lambda R_0}{2L}$, where λ is the signal wavelength at center frequency, R₀ is a target depth, and L is the aperture size. In other words, larger apertures require a smaller pitch in order to avoid beamforming errors in the near-field. A detailed derivation of the azimuth Nyquist frequency for near-field echoes is provided in Appendix B. To avoid aliasing for far-field echoes (i.e. plane-waves), which have a sinusoidal signature across the aperture the element pitch needs to be at least d_Nyquist = λ/2, regardless of the aperture size. This value is commonly taken into account when designing clinical ultrasound transducers.

B. RDA Image Quality

RDA forms images that are similar to B-mode images created with DAS and F-k migration. The three beamforming methods have approximately the same lateral resolution (as indicated by the beamplots in Figure 7), and achieve comparable contrast of anechoic/hypoechoic targets (as observed in the lesion-phantom images in Figure 11). The three methods also create similar speckle patterns (Figures 10 and 11).

The most visible artifact in RDA images is axial ringing around point targets. Because RDA assumes that the received signals are narrowband (monochromatic), the RDA point spread function (PSF) shows increased axial sidelobes compared to the PSF’s of DAS and F-k migration, which are developed for wideband signals. In particular, the narrowband signal is assumed in two steps of RDA: 1) during RCMC, which is guided by equations (4) and (8) for monostatic and multistatic cases, respectively, and 2) during matched filtering/azimuth compression, where the filter is a complex conjugate of the phase terms in equations (3) and (7). In the expressions for both steps, the square-root terms assume a single center frequency f₀. While the narrowband signal model closely matches the ultrasound signal in the azimuth dimension (Figure 5), it also limits the temporal frequency band, which effectively causes the data to oscillate in the time/depth dimension.

The axial ringing is successfully mitigated by frequency binning, as shown in Figure 8. It is worth noting that as the number of frequency bins increases and the bins get narrower, the narrowband signal model more closely resembles the signal within each bin, and the RDA PSF approaches the PSF’s of wideband-frequency-domain beamformers, such as F-k migration.³

C. RDA Speed on Single and Multiple Cores

When beamforming is performed using a single CPU core (so that computations are carried out serially), RDA is able to achieve a beamforming speed significantly higher than that of DAS and similar to that of F-k migration. Specifically, RDA with three frequency bins reduces beamforming time for the simulated images in Figure 7 by a factor of nine relative to DAS, for both monostatic and multistatic cases (Table II). An increase in beamforming speed by RDA and F-k migration is feasible because they both reduce the overall number of operations relative to DAS. The expressions in section II-C demonstrate that RDA reduces the overall number of operations relative to DAS approximately by factors of N/(2 log₂ N) and N/(2 log₂ N²) for monostatic and multistatic cases, respectively, where N is the number of transducer elements (Figure 6).⁴

Time required to reconstruct a full B-mode image can be further reduced by executing RDA on multiple processing units (or threads) in parallel. Estimates in Table III indicate that RDA is highly parallelizable, allowing for the use of as many as one thousand independent threads to beamform the multistatic B-mode image in Figure 7, which could result in a thousand-fold improvement in beamforming speed.

RDA is highly parallelizable due to it focusing the data in range-Doppler domain, which allows it to have a reasonably small output per thread and many threads running in parallel. Specifically, executing a beamforming algorithm on multiple threads independently, from start to finish means that each thread produces a portion of the final image, and the increase in beamforming speed is directly proportional to the number of threads. In such a case, the speedup is limited by the minimum beamformer output size, with smaller outputs allowing for more threads and greater speedup. Because RDA focuses the data in the range-Doppler domain, and thus utilizes only an azimuth Fourier Transform, its smallest output size is a single row of pixels, meaning in theory, each row of pixels can be beamformed on a separate thread. In this regard, RDA stands between DAS– which creates each pixel independently–and F-k migration–which focuses the data in a spatio-temporal frequency domain and therefore always produces a 2-D block of pixels. As a result, RDA requires less memory per thread and allows for a larger number of threads than F-k migration (Table III), making it more suitable for implementation on modern graphics processing units (GPUs) that contain thousands of processing cores and a limited amount of fast-access memory (i.e. local memory and cache). In the presence of such hardware acceleration, RDA has a potential to outperform F-k migration in terms of speed as the two methods involve a similar number of operations.

D. Adaptive Imaging with RDA

Beamforming in the range-Doppler domain is readily amenable to the presence of acoustic noise and/or signals from inhomogeneous media. For example, the adaptive filters designed for far-field signals (such as Minimum Variance Distortionless Response (MVDR) beamformer [27, 28]) can be added to the RDA processing chain before the azimuth IFFT step (Figure 2) to suppress off-axis scattering and clutter. Such a degree of compatibility between RDA and classic adaptive filters is due to the fact that the first three steps of RDA (namely azimuth FFT, RCMC, and azimuth compression) can be viewed together as a transformation of echoes from the near-field to the far-field.⁵ In addition, because only a spatial Fourier Transform is used to convert aperture data to the range-Doppler domain, the models in equations (3) and (7) can be easily adapted to account for sound-speed inhomogeneities in layered media. These adaptive techniques are currently being investigated and will be included in future adaptations of RDA.

VI. Conclusion

We proposed efficient beamforming of ultrasound signals in the range-Doppler frequency domain using the range-Doppler algorithm (RDA). We presented the models of ultrasound signals in RD domain for monostatic and multistatic SA acquisitions. We demonstrated the RDA on ultrasound signals from simulation and phantom experiments and compared RDA performance to that of F-k migration (which beamforms data in a spatio-temporal frequency domain) and to a conventional DAS beamformer. We showed that RDA achieves comparable image quality to DAS and F-k migration in terms of lateral resolution and lesion contrast, while higher axial sidelobes in RDA images (due to the narrowband signal assumption) can be alleviated using temporal frequency binning. Similar to F-k migration, RDA significantly reduces the overall number of computations relative to DAS beamforming, and achieves similar beamforming time as F-k migration on a single CPU core. At the same time, RDA requires less memory than F-k migration, and is more parallelizable because it utilizes only a spatial Fourier transform, meaning it can be implemented on a GPU device to further reduce beamforming time. Because RDA is not as parallelizable as DAS, it holds a speed advantage in low-power and low-compute-needs beamforming applications, where modest to mid-tier parallel hardware capabilities are available (i.e. up to several thousands of processing cores). In particular, RDA has the potential to allow for real-time and high-resolution imaging on cheaper and portable ultrasound devices. In the context of point-of-care ultrasound, we are also planning to explore the opportunities to adapt RDA to other applications that would benefit from efficient beamforming of large data, such as imaging with matrix arrays and plane-wave power Doppler ultrasound.

APPENDIX A. Principle of Stationary Phase (POSP)

To derive the signal model in (3) we must evaluate the Fourier Transform of the signal in the time-space domain, namely:

$$\begin{gathered} S_0\left(t,k_x\right)={\int }_{-\infty }^{\infty }{A}_0{w}_r\left(t-\frac{2R(x)}{c}\right)\times \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\times \exp \left[-j4\pi f_0\frac{R(x)}{c}\right]\exp \left(j2\pi x{k}_x\right)dx. \end{gathered}$$ (15)

Because it is difficult to evaluate (15) exactly, we approximate the Fourier integral using the principle of stationary phase (POSP) [7, 21]. According to POSP, when the integrand is an oscillating function, the value of the integral can be approximated by the value of the function at the point where the phase does not change (in other words, the phase is stationary and its derivative is zero). The POSP assumes that away from the stationary point, the function oscillates rapidly and the regions of positive and negative phases cancel each other. The limitations of such approximation are discussed in greater detail in Appendix of [21].

The phase of the Fourier integral in (15) is given by

$$\text{Θ}=-j4\pi f_0\frac{1}{c}\sqrt{R_0^2+x^2}+j2\pi x{k}_x.$$ (16)

The stationary point can be found by setting the derivative of phase to zero, and solving for x:

$$\begin{gathered} \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\frac{\text{dΘ}}{\text{d}x}=0, \\ \frac{2f_0}{c}\frac{x}{\sqrt{R_0^2+1}}-k_x=0, \\ \frac{2f_0}{c}\frac{1}{\sqrt{R_0^2/x^2+1}}=k_x, \\ {x}_0=\frac{R_0}{\sqrt{{\left(\frac{2f_0}{k_{x}c}\right)}^2-1}}. \end{gathered}$$ (17)

The Fourier integral can then be approximated by the value of the integrand at x₀

$${\int }_{-\infty }^{\infty }{e}^{\text{Θ}(x)}dx=e^{{\text{Θ}}_{x=x_0}\left(k_x\right)},$$

where

$${\text{Θ}}_{x=x_0}\left(k_x\right)=-j4\pi \frac{R_0}{\lambda }\sqrt{1-{\left(\frac{k_{x}c}{2f_0}\right)}^2}.$$ (18)

The expression in (17) can also be used in combination with (2) to derive the expression for echo travel-time as a function of spatial frequency k_x

$$t\left(k_x\right)=\frac{2}{c}\frac{R_0}{\sqrt{1-{\left(\frac{k_{x}c}{2f_0}\right)}^2}}.$$

Finally, using the expressions for echo travel time and phase, the Fourier integral in (15) can be evaluated as

$$S_0\left(t,k_x\right)=A_0{w}_r\left(t-t\left(k_x\right)\right)\exp \left[-j\pi \frac{4f_0{R}_0}{c}\sqrt{1-{\left(\frac{k_{x}c}{2f_0}\right)}^2}\right].$$

The expressions above for echo travel time and the signal model in range-Doppler domain are originally presented in Section II-A, equations (4) and (3), but are rewritten here for convenience.

APPENDIX B. Chirp Approximation and Sampling Requirements in Azimuth

To simplify the signal model and to derive an intuitive expression for the required azimuth sampling frequency, we can apply a Taylor series expansion (TSE) to the echo travel-distance in (20) ⁶:

$$R(x)=\sqrt{R_0^2+x^2}$$ (19)

$$=R_0+\frac{x^2}{2R_0}+O\left(x^3\right)$$ (20)

In (20), O(x³) denotes the higher order terms that can be ignored. The phase of the signal in the time-space domain in equation (16) can then be approximated as

$$\text{Θ}=-j4\pi f_0\frac{1}{c}\left(R_0+\frac{x^2}{2R_0}\right)+j2\pi x{k}_x.$$ (21)

Applying the POSP to (21) we can obtain a simplified expression for the azimuth frequency k_x. Specifically,

$$\begin{gathered} \hspace{1em}\hspace{1em}\hspace{1em}\frac{\text{dΘ}}{\text{d}x}=0, \\ \frac{2f_0}{R_{0}c}{x}_0-k_x=0, \\ \frac{2f_0}{R_{0}c}{x}_0=k_x, \end{gathered}$$

which can be written as

$$K_a\hspace{0.17em}{x}_0=k_x,\hspace{1em}{K}_a=\frac{2f_0}{c{R}_0}.$$ (22)

In (22), the constant K_a is the rate of change of frequency, or chirp rate. The Nyquist frequency can then be derived as

$$k_{Nyquist}=2\cdot k_{x,max}$$ (23)

$$=\frac{2}{\frac{\lambda }{2}{R}_0}\cdot \frac{L}{2}$$ (24)

$$=\frac{L}{\frac{\lambda }{2}{R}_0},$$ (25)

where L is the aperture size and R₀ is imaging depth. The largest pitch that would prevent aliasing is $d_{Nyquist}=\lambda R_0/(2L)$.

Following the TSE and POSP, the phase of the chirp-approximated spectrum is given by

$${\text{Θ}}_{x=x_0}\left(k_x\right)=-j4\pi \frac{f_0{R}_0}{c}+j\pi \frac{k_x^2}{K_a},$$ (26)

and the echo travel-time as a function of k becomes

$$t\left(k_x\right)=\frac{2R_0}{c}\left(1+\frac{1}{2}{\left(\frac{k_{x}c}{2f_0}\right)}^2\right).$$ (27)

Finally, the simplified signal model in the range-Doppler domain is a chirp waveform given by

$$S_0\left(t,k_x\right)=A_0{w}_r\left(t-t\left(k_x\right)\right)\exp \left[-j\pi \frac{4f_0{R}_0}{c}\right]\exp \left[j\pi \frac{k_x^2}{K_a}\right].$$ (28)

APPENDIX C. MATLAB Implementation of Monostatic RDA

MATLAB implementations of both monostatic and multistatic RDA beamformers are available under https://gitlab.com/mj66/bentobox. The repository includes several common frequency-domain beamformers, and examples that illustrate their use.

function [foc_data, t, x] = rda_mono(t, signal, ...
    fs, f0, La, dx, varargin)
 
%
% Beamforms monostatic data using RDA
% INPUTS:
% t - T x 1 vector containing sample times for ...
    the signal matrix
% signal - T x N matrix containing input RF data ...
    to be interpolated
% fs - temporal sampling frequency (scalar)
% f0 - center transmit frequency (scalar)
% La - size of the monostatic (moving) aperture [m]
% dx - distance between the two subsequent ...
    aperture centers (i.e. motion increment) [m]
%
% c - speed of sounds [m/s]; default 1540 m/s
%
% OUTPUT:
% foc_data - T x N vector with interpolated (RF) ...
    data points
% t - T x 1 time vector for output samples
% x - N x 1 lateral pixel positions in the ...
    reconstructed image
%
% written by Marko Jakovljevic
% last updated on 07/22/2022
 
% check for inputs  
switch nargin
    case 6
        c = 1540;
    case 7
        c = varargin{1};
    otherwise
        error(‘Improper argument list’);
end
 
[nT, nX] = size(signal);
if nT‰length(t)
    error(‘Time vector must match the first ...
        dimension of the input RF data.’)
end
lambda = c/f0;
 
%%%%%%%%%%%%%%%%%%%%%
% I-Q DEMODULATION %%
%%%%%%%%%%%%%%%%%%%%%
f_demod = f0;
[i_data, q_data] = iq_demod(signal, f_demod, fs);
baseband_data = i_data + 1i*q_data;
 
%%%%%%%%%%%%%%%%%%%%%%%%%
% RANGE DOPPLER DOMAIN %%
%%%%%%%%%%%%%%%%%%%%%%%%%
pad_factor = 2;
pad_length = round(nX*pad_factor/2);
if mod(pad_length,2) ≠ 0
    pad_length = pad_length +1;
end 
 
% ensure even number of samples 
if mod(2*pad_length+nX,2) ≠ 0
    baseband_data = [zeros(nT,pad_length+1), ...
        baseband_data, zeros(nT,pad_length)];
else
    baseband_data = [zeros(nT,pad_length), ...
        baseband_data, zeros(nT,pad_length)];
end
 
nX_FFT = size(baseband_data,2);
 
% range-Doppler domain
rd_data = ...
    fftshift(fft(fftshift(baseband_data,2),[],2),2);
 
real_rd = real(rd_data);
imag_rd = imag(rd_data);
 
%%%%%%%%%
% RCMC %%
%%%%%%%%%
range_axes = c*t/2; % range axis
k_x = [-nX_FFT/2:nX_FFT/2–1] * 1/dx/(nX_FFT);
 
% remove evanescent parts
isevanescent = (1 - (k_x*lambda/2).ˆ2 < 0);
isnotev = find(isevanescent);
 
real_rd(:,isevanescent) = 0;
imag_rd(:,isevanescent) = 0;
 
% migration factor
D = sqrt(1 - (lambda * k_x(isnotev)/2).ˆ2);
r_ad = bsxfun(@rdivide,range_axes’,D);
 
foc_rd_real = zeros(length(range_axes),nX_FFT);
foc_rd_imag = zeros(size(foc_rd_real));
 
for ii=1:length(isnotev)
 
    kx_id = isnotev(ii);
    foc_rd_real(:,kx_id) = ...
        interp1(range_axes,real_rd(:,kx_id),r_ad ...
        (:,ii),’pchip’,0);
    foc_rd_imag(:,kx_id) = ...
        interp1(range_axes,imag_rd(:,kx_id),r_ad ...
        (:,ii),’pchip’,0);
end
rd_migrated = foc_rd_real + 1i*foc_rd_imag;
 
%%%%%%%%%%%%%%%%%%%
% MATCHED FILTER %%
%%%%%%%%%%%%%%%%%%%
D_tmp = zeros(1,nX_FFT);
D_tmp(isnotev) = D;
 
azimuth_phi = -(4/lambda) * ...
    bsxfun(@times,range_axes’,D_tmp);
azimuth_response = exp(1i*pi*azimuth_phi);
 
% window the frequency response
azimuth_response = ...
    bsxfun(@times,azimuth_response,sinc(k_x*La));
% azimuth_response(:,abs(k_x) > 1/La/2) = 0;
 
matched_filter = conj(azimuth_response);
rd_filtered = rd_migrated .* matched_filter;
 
%%%%%%%%%
% IFFT %%
%%%%%%%%%
foc_data = ...
    fftshift(ifft(ifftshift(rd_filtered,2),[],2),2);
 
% change back coordinates 
if mod((pad_length*2+nX),2) ≠ 0
    foc_data = ...
        foc_data(:,pad_length+2:end-pad_length);
else
    foc_data = ...
        foc_data(:,pad_length+1:end-pad_length);
end
 
foc_data = real(foc_data); % return the real ...
    part of the beamformed data
 
x = (0:1:size(foc_data,2)-1)*dx;
x = x - mean(x);
 
return 
 
 
function [i_data, q_data] = iq_demod(rf_data, ...
    f_demod, f_s)
%
% Demodulates RF signals and returns I and Q ...
    components
%
% INUPTS:
% - rf_data: input rf signal T x Y (time x chan_no)
% - f_demod: demodulation frequency (Hz)
% - f_s: sampling frequency (Hz)
%
% last updated on 12/04/2014
%
% by Marko Jakovljevic
 
time = [0:1:size(rf_data,1)-1]*1/f_s;
 
i_data_term = 2*rf_data .* ...
    repmat(cos(2*pi*f_demod*time’), ...
[1,size(rf_data,2)]);
q_data_term = 2*rf_data .* ...
    repmat(-sin(2*pi*f_demod*time’), ...
[1,size(rf_data,2)]);
 
% IIR filter design
n_taps = 10;
f_cutoff = f_demod / (f_s/2); % cuto-off at RF freq
[b,a] = butter(n_taps,f_cutoff);
 
i_data = filtfilt(b,a,i_data_term);
q_data = filtfilt(b,a,q_data_term);
 
return