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ieee transactions on ultrasonics, ferroelectrics, and frequency control, vol. 52, no. 1, january 2005 37

Coherent-Array Imaging Using PhasedSubarrays. Part I: Basic Principles

Jeremy A. Johnson, Student Member, IEEE, Mustafa Karaman, Member, IEEE,and Butrus T. Khuri-Yakub, Fellow, IEEE

Abstract—The front-end hardware complexity of a co-herent array imaging system scales with the number of ac-tive array elements that are simultaneously used for trans-mission or reception of signals. Different imaging methodsuse different numbers of active channels and data collectionstrategies. Conventional full phased array (FPA) imagingproduces the best image quality using all elements for bothtransmission and reception, and it has high front-end hard-ware complexity. In contrast, classical synthetic aperture(CSA) imaging only transmits on and receives from a sin-gle element at a time, minimizing the hardware complex-ity but achieving poor image quality. We propose a newcoherent array imaging method—phased subarray (PSA)imaging—that performs partial transmit and receive beam-forming using a subset of adjacent elements at each firingstep. This method reduces the number of active channels tothe number of subarray elements; these channels are multi-plexed across the full array and a reduced number of beamsare acquired from each subarray. The low-resolution subar-ray images are laterally upsampled, interpolated, weighted,and coherently summed to form the final high-resolutionPSA image. The PSA imaging reduces the complexity ofthe front-end hardware while achieving image quality ap-proaching that of FPA imaging.

I. Introduction

Array-based imaging systems have been used for var-ious applications, including radar, sonar, and medi-

cal ultrasound [1], [2]. In general, a collection of transmit-ters emits electromagnetic or mechanical waves into themedium to be imaged. The waves scattered or reflectedby different parts of the medium then are received by aset of sensors. The received signals are processed to forman image with intensities that represent some physical pa-rameter of the medium. Although applicable to a rangeof array-based imaging system, the work presented in thispaper will be presented from the perspective of pulse-echoultrasound imaging using a phased array.

A typical phased-array ultrasound imaging system con-sists of a one-dimensional (1-D) linear array of transduc-ers that are capable of both transmitting and receivingacoustic energy [3]. A short modulated pulse is transmit-ted from each of the elements. The timing of the trans-

Manuscript received July 6, 2004; accepted August 11, 2004.J. A. Johnson and B. T. Khuri-Yakub are with Stanford Uni-

versity, Image Guidance Laboratory, Stanford, CA (e-mail: [email protected]).

J. A. Johnson is also with Stanford University, Image GuidanceLaboratory, Stanford, CA.

M. Karaman is with Isik University, Department of Electrical En-gineering, Istanbul, Turkey.

mission from each element can be adjusted to steer thetransmitted waveform to a desired direction. Additionally,the timing can be set to cause the transmitted pulses toconverge, forming a beam that is narrowest at the desiredfocal depth. When the wavefront crosses boundaries be-tween materials with different acoustic impedances, a por-tion of the pulse is reflected or scattered back toward thetransducer array. These reflected signals are received bythe elements of the array. In the same way that the trans-mitted pulses can be timed to achieve beam steering andfocusing, the received signals can be delayed such that areflection from a single point along the beam will add con-structively. The process of steering and focusing the arraysignals is called beamforming [4]. A simplified diagram il-lustrating the concept is shown in Fig. 1.

Current ultrasound systems use a highly parallel hard-ware architecture for signal conditioning between thetransducer array and the beamformer. For every trans-ducer element that is active for a given firing event, anindependent front-end electronic processing channel per-forms transmit/receiveswitching, amplification, filtering,time-gain compensation, and digital-to-analog conversion.A high-level block diagram of a phased array imaging sys-tem is shown in Fig. 2, and Fig. 3 illustrates the front-end hardware components. The cost and complexity of thefront-end hardware scales with the number of active chan-nels [5]–[7]. The beamforming architecture determines thenumber of active channels for a particular system design.The image quality produced by an ultrasound system gen-erally improves with the number of active channels, mean-ing that better images correspond to the systems withlarger arrays and thus of greater cost and size.

One of the most space- and power-consuming parts ofa typical ultrasound imaging system is the front-end hard-ware [5], [7]. This is especially true since the advent of dig-ital beamforming, which has greatly reduced the back-endhardware requirements [8], [9]. Early beamforming hard-ware was primarily analog. The increase in performanceand decrease in size of digital and mixed-signal electronicseventually enabled the implementation of digital beam-formers. Unfortunately, the analog nature of the front-endhardware has not experienced an equal reduction in costand size. High-end commercial ultrasound machines stillhouse the analog and mixed-signal, front-end electronicswithin the base unit, requiring costly and bulky probe ca-bles that contain dedicated coaxial transmission lines foreach transducer element. Although acceptable for today’s1-D arrays containing on the order of 100 elements, al-

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38 ieee transactions on ultrasonics, ferroelectrics, and frequency control, vol. 52, no. 1, january 2005

Fig. 1. Delay-and-sum beamforming in a phased array ultrasound imaging system. The set of beamforming delays defines the steering angleand focal distance of the transmit and receive beams.

Fig. 2. Block diagram of a phased array ultrasound imaging system.

ternate configurations will be needed for tomorrow’s 2-Darrays consisting of thousands of elements.

Classical full phased array (FPA) imaging sweeps out asector image by electronically steering beams across imagespace [10]–[14]. All N transducer elements are active dur-ing both transmit and receive formation of each beam,requiring N dedicated front-end processing channels asshown in Fig. 4(a). In addition to high front-end hardwarecomplexity, the large number of received signals requiredto form each beam causes a significant increase in beam-former complexity. One benefit of using all N elements isthat high signal-to-noise ratio (SNR) is achieved.

A variety of beamforming methods have been devel-oped, each requiring a different number of active chan-nels [15], [16]. Thomenius [17] has provided an overviewand history of beamformers. Classical synthetic aperture(CSA) techniques using a single channel for transmit andreceive minimize the hardware complexity [18]. Syntheticaperture techniques were first developed for radar [19] andlater adapted to ultrasound imaging [20]. Synthetic aper-ture was first used with linear arrays with reconstruction inthe spatial domain [21], but it since has been modified foruse with circular arrays [22], [23], and frequency-domainreconstruction methods also have been developed [24]. Forthe standard linear array method, a single processing chan-nel is time multiplexed across all N transducer elements[Fig. 4(b)]. Because only a single element is used for bothtransmit and receive, the transmitted power and receivesensitivity are minimal and lead to a low SNR.

Array imaging techniques have continued to strike com-promises between CSA and FPA, aiming to improve theSNR of CSA methods and reduce the number of chan-

Fig. 3. Front-end hardware for a phased array imaging system. T/R:transmit/receive switch; TX: transmit driver; LNA: low-noise ampli-fier; TGC: time-gain compensation; LPF: low-pass anti-alias filter;ADC: analog-to-digital converter.

nels required for FPA imaging. Karaman et al. [25] havesuggested transmitting from multiple elements to emulatea more powerful transmit element for synthetic apertureimaging, and later described how to correct for motion andphase aberration [26]. Lockwood et al. [27] used a similarmethod by transmitting from five virtual elements and us-ing the full aperture in receive in order to achieve higherframe rates needed for 3-D imaging with a 1-D transducerarray. Frazier and O’Brien [28] proposed a synthetic aper-ture method for increasing the lateral resolution by consid-ering virtual elements located at the focal point. Nikolovand Jensen [29] combined the previous two techniques toachieve increased frame rate and improved resolution inelevation for 3-D imaging with a mechanically scanned 1-D transducer array. Nikolov et al. [30] introduced recur-sive ultrasound imaging as a method for increasing framerate. Synthesizing an effective aperture for 3-D imagingusing 2-D transducer arrays using only the outermost el-ements was described in [31]. The use of coded excitationis a current area of study that has benefits of improvedframe rates, increased SNR, and improved depth pene-tration [32]–[35]. Multielement synthetic aperture using afrequency-modulated signal also has been proposed as amethod for increasing the SNR [36]. Early proposals forreducing the number of active channels in phased-arrayimaging systems did so by transmitting on a single centralportion of the array and receiving on a number of overlap-ping [37] or adjacent [38] subarrays. Later developments

johnson et al.: proposed phased subarray imaging method 39

Fig. 4. Front-end hardware configuration for (a) phased array,(b) synthetic aperture, and (c) proposed phased subarray imagingmethods.

improved the frame rate of subarray imaging by acquiringa subset of the beam lines and interpolating the others[39]–[41]. These methods are limited in use to narrowbandimaging systems and did not achieve FPA-equivalent im-age quality due to a smaller effective aperture.

This paper presents a new method that combines theprinciples of phased-array and synthetic-aperture imagingmethods to reduce the system cost and size by decreas-ing the number of active channels while maintaining high-image quality. The method presented is most useful forcost- or size-constrained real-time acoustic imaging sys-tems. The phased-subarray technique described here ex-tends the capabilities of the earlier subarray imaging ap-proach [39]–[41] by contributing in two ways. First, thetransmit subarray is not fixed in the center of the arraybut is placed at a number of positions across the entire

array. This allows the realization of a significantly widereffective aperture compared to the earlier methods using afixed-transmit subarray, resulting in improved lateral res-olution. Second, a new reconstruction method is presentedthat uses a set of subarray-dependent, 2-D filters for wide-band imaging. This extends the capabilities of previousreconstruction methods that used 1-D lateral filters suit-able for narrowband imaging.

This paper presents the basic principles and theory ofphased subarray (PSA) imaging. The companion paperpresents the performance of the PSA method on simu-lated and experimental data [42]. The paper is organizedas follows. In Section II basic terminology is described thatwill provide the framework for the presentation of the PSAimaging technique. Section III describes the data acquisi-tion and image reconstruction methods for PSA imagingin beamspace. Sections IV and V present the same processdescribed in the frequency domain along with reconstruc-tion filtering. Section VI includes a generalization of thePSA method. The theoretical performance of PSA imagingis provided in Section VII and compared to FPA imaging.

II. Basic Definitions

A. Spatial and Spatial Frequency Responses

The response of a phased-array imaging system at thefar-field or focal depth can be approximated using linearsystems theory. The performance of the system then ischaracterized by its two-way point spread function (PSF)in the spatial domain, or equivalently, its transfer functionin the spatial frequency domain [43]–[45]. There is a one-to-one relationship between these two functions given bythe 2-D discrete Fourier transform (DFT):

u[m,n] = �2D{U [p, q]}, (1)

where U [p, q] is the PSF, and p and q are the axial and lat-eral indices, respectively; u[m,n] is the transfer function,and m and n are the axial and lateral spatial frequency in-dices, respectively; and �2D{·} is the 2-D DFT operator.The 1-D lateral spatial frequency response at a given axialspatial frequency is referred to as the coarray or the effec-tive aperture, and can be computed as the convolution ofthe transmit and receive aperture functions [46], [47]:

u[n] = aT [n] ⊗ aR[n]. (2)

The axial PSF of a phased-array imaging system isdetermined by the transmit pulse, s(t), centered at timet = 0:

U [p] = s

((p − P

2

)∆r

), (3)

where P is the number of axial samples and ∆r is the axialdistance between beam samples. The 2-D spatial frequency


response can be expressed as the product of the lateral andaxial spatial frequency responses:

u[m,n] = u[m] · u[n], (4)

where u[m] is the 1-D axial spatial frequency responsegiven by the 1-D DFT of the axial PSF. Note that thetemporal frequency response of the excitation pulse deter-mines the axial spatial frequency response of the system,and are related by a scaling factor.

B. Comatrix Representation

A useful technique for visualizing the relationship be-tween the aperture function and the coarray is by meansof the comatrix. Comatrix representation was introducedin [25] and formalized as the transmit-receive apodizationmatrix in [48]. Each transmit/receive element pair con-tributes to a specific bin of the coarray representing a spe-cific lateral spatial frequency. Each entry in the comatrixrepresents a specific pair of single transmit element andreceive element. For example, the topmost comatrix entryrepresents transmitting on array element 16 and receiv-ing from array element 1. The transmit and receive arraysshown above each comatrix in Fig. 5 are the same phys-ical array, and are used to index into the comatrix withthe given element numbers. The numbers in the comatrixindicate the corresponding coarray bin. All entries in a co-matrix column contribute to the same coarray bin. For theFPA example in Fig. 5(a), all TX/RX pairs are acquiredat once as all array elements are used during each firingevent. This results in a comatrix with all elements equalto unity, and a coarray equal to a triangle function.

The comatrix and coarray are also useful for evaluatingmethods in which individual TX/RX pairs are obtainedthrough separate firings. In contrast to FPA imaging, CSAimaging transmits and receives to/from a single element ata time. As shown in Fig. 5(b), each of these firing eventscontributes to a single comatrix and coarray entry. Thecorresponding comatrix structure is in the form of theidentity matrix, resulting in a coarray equal to the shah[49] or comb function. The resulting coarray is zero valuedat every other bin, which results in grating lobes.

In the previous subaperture imaging work done byKaraman and O’Donnell [40], a single transmit subaper-ture located at the center of the array fires multiple times,and a receive subaperture is scanned across the full array.The comatrix and coarray representation of this modal-ity is shown in Fig. 5(c). The width of the coarray is lessthan that of both FPA and CSA imaging, and thus thelateral resolution is poorer. The PSA imaging method ad-vances this approach by translating both the transmit andreceive subapertures to obtain an FPA-equivalent effectiveaperture.

III. Phased Subarray Image Formation

A 1-D, N -element array is subdivided into K overlap-ping subarrays, each consisting of M elements. The sub-

array pitch is J elements, thus the amount of subarrayoverlap is M–J . A general condition that is satisfied for avalid subarray configuration is:

J(K − 1) = N − M. (5)

The first M -element subarray can be used as a referencesubarray and expressed by its aperture function:

a0[n] =

{1, n ∈ [0,M − 1]0, otherwise

. (6)

The aperture functions of all K subarrays then can beexpressed in terms of this reference subarray:

ak[n] = a0[n − kJ ], k ∈ [0,K − 1], (7)

where k is the subarray index.For the sake of simplicity, the examples in this paper

and results in the companion paper [42] assume that thesubarray pitch is equal to half the width of a subarray(J = M/2). In this case, the full array response—a trianglefunction—is equal to a linear combination of the subarrayresponses, each a narrower triangle function with its basealigned with the peak of the adjacent triangle function.As a result, the interpolation filters only need to suppressaliases and do not need to reshape the amplitude responseof each subarray function. The interpolation filters there-fore simplify to bandpass filters. When this condition isnot met, the filters also must modify the amplitude re-sponse in order to reconstruct an FPA-equivalent image.Methods for designing these filters are beyond the scopeof this paper and are presented in [50].

As shown earlier in Fig. 4(c), the number of front-endtransmit and receive processing channels is equal to thenumber of elements in each subarray. The hardware maybe designed to allow the active transmit subarray to dif-fer from the active receive subarray—as is the case inFig. 4(c)—or may be designed such that the same subarrayis used for transmit and receive. Both cases are discussed,but the discussion primarily focuses on the latter case dueto simplicity of presentation.

Fig. 6 illustrates all acquisition and processing steps in-volved in PSA image formation as compared to FPA. Theobject of PSA imaging is to generate an FPA-equivalentset of QFPA beams by transmitting and receiving on onlyM elements at a time. This full set of QFPA beams formsa high-resolution image. Although each of the K subarrayscould acquire a high-resolution image, the frame rate is in-creased by acquiring only a subset of the QFPA beams overthe same sector angle. Each of the K subarrays acquiresa total of QPSA subarray beams, forming a low-resolutionimage.

With the exception of limiting the transmit and receiveelements to those of the transmit and receive subarrays,the beamforming used to generate each subarray image isidentical to that used for FPA imaging. The beamformerdelays are calculated in the same manner as for FPA imag-ing; for example, the delays shown in Fig. 1 required to


Fig. 5. Comatrices and coarrays for different imaging methods. Shaded comatrix elements correspond to TX/RX element or subarray pairsthat contribute to the image. The diagonal sum of the comatrix produces the coarray that represents the lateral spatial frequency responseof the imaging system. All figures correspond to a 16-element array with half-wavelength element pitch. (a) Phased array imaging uses allelements in TX and RX. (b) Synthetic aperture imaging transmits and receives on the same single element, resulting in an undersampledcoarray. (c) Synthetic receive aperture imaging uses a single transmit subarray and several receive subarrays. (d–f) Three representativephased subarray methods. (d) Seven overlapping subarrays are used for both transmit and receive (configuration used by examples in text).(e) Four adjacent subarrays acquire some redundant spatial frequency information. (f) Four adjacent subarrays use all TX/RX subarraycombinations.


Fig. 6. Image formation and reconstruction in the spatial domain. Right: FPA imaging directly acquires QFPA beams using the full array.Left: K phased subarrays sequentially acquire a subset (QPSA) of the beams, forming K subarray images. These images are laterallyupsampled and interpolated to reconstruct the unacquired beams, then weighted and summed to form and FPA-equivalent image.

steer and focus the beam to the desire point would be thesame for FPA and for all of the subarray images. The sub-array beam origins are located at the center of the fullarray, not at the center of each subarray, as illustrated inFig. 7.

The next step is to form high-resolution subarray im-ages from each of the low-resolution subarray images. Eachof the subarray images is first upsampled laterally by a fac-tor of L by inserting L–1 zero-valued beams between theacquired beams. An interpolation filter is applied to thisset of upsampled beams, reconstructing the empty beamsand leaving the originals intact. Two methods for designof the reconstruction filter are presented in the next twosections. The resulting K high-resolution subarray images

are then coherently weighted and summed to form the finalPSA image. At this point, the final PSA image is equiva-lent to the FPA image.

Note that the equivalence of PSA imaging to FPA imag-ing holds true in the far-field or at the focal distance asit depends on the Fourier transform relationship betweenthe aperture function and the beam pattern. The extent towhich the equivalence holds away from the focal distanceis shown experimentally in the companion paper [42].

The beam profiles for each subarray differ from eachother and from the FPA beam profile; however, the equiv-alent beam profile representing the PSA system at thefocal distance are equivalent to that of FPA imaging asillustrated in Fig. 7. Although not obvious in the spatial


Fig. 7. Approximate beam profiles for PSA and FPA imaging when acquiring the center beam. (a)–(c) The beam profiles for a single subarrayare wider at the focal point than the FPA beam profile. (d) The combined response of PSA is equivalent to a beam profile equal to that ofFPA imaging.

domain, this observation is clearer when considering thatthe coarrays of PSA systems are equivalent to those ofFPA systems [compare Fig. 5(a) to Figs. 5(d)–(f)].

IV. Narrowband PSA Processing

Due to the minimal support in the temporal frequencydomain of a narrowband system, its array response can beapproximated by the 1-D cross section through the coar-ray at the center operating frequency. We refer to the lat-eral spatial frequency response of a single subarray as acosubarray. In order to understand the function of the re-construction filter, we outline the PSA procedure by illus-trating the resulting cosubarrays at each step in Fig. 8.The drawings represent a system with a 16-element array,7 subarrays, each with 4 elements, and an upsampling rateof 4 (N = 16, K = 7, M = 4, L = 4). The array in thesedrawings is sampled with an element pitch of d = λ/2 toavoid grating lobes, and QPSA = 2M sin(Θ/2) beams areacquired to avoid aliasing in the lateral spatial frequencydomain.

The subarray aperture functions are shown in Fig. 8(a).The corresponding cosubarrays—equal to the convolutionof the transmit and receive aperture functions—are trian-gle functions of width 2M—as shown in Fig. 8(b). Thesecosubarrays correspond to acquiring all QFPA beams witheach subarray. Because the subarray pitch, J , is equal tohalf the subarray size, M/2, then these triangle-functioncosubarrays overlap one another by half the width of thetriangle, and an FPA-equivalent subarray can be recon-structed as a linear combination of the cosubarrays.

The cosubarrays corresponding to the acquisition ofonly QPSA beams by each subarray are shown in Fig. 8(c).Due to the band-limited nature of the ideal cosubarrays,no information is lost when the beam sampling rate is suf-ficiently large (equivalently the subsampling ratio, L, suf-ficiently small):

QPSA > 2M sin(

Θ2

). (8)

The upsampling and filtering steps performed in the spa-tial domain correspond to reconstruction of the ideal co-

subarrays. The resulting cosubarrays after upsampling areshown in Fig. 8(d). At this point, the cosubarrays con-tain the lateral spatial frequency components of the origi-nal cosubarrays in addition to periodic replicates of thesecomponents shifted by multiples of 2N/L. The purpose ofthe interpolation filter is to suppress the aliases while pre-serving the original cosubarray response. The lateral spa-tial frequency response of the ideal interpolation filters isshown in black behind the original cosubarray componentsin Fig. 8(d). The spatial frequency domain representationof the ideal filter for each subarray can be written as:

Hk[n] =

⎧⎨⎩1,

(k − 1)M2

+ 1 ≤ n ≤ (k + 1)M2

0 otherwise.

(9)

Because this is a bandpass filter, hk[q] is sinc-like andthus the energy decreases further from the filter center.The size of the filter is reduced by truncation. The resul-tant rippling in the spatial frequency response is minimizedby applying a window function.

The reconstructed cosubarrays are found by applyingthe filter transfer function to the upsampled cosubarrays,and are shown in Fig. 8(e). These upsampled cosubarraysare equivalent to the high beam rate cosubarrays shownin Fig. 8(b). The final PSA coarray [Fig. 8(h)] is formedby linearly combining the upsampled coarrays [Fig. 8(e)]using a triangular weighting function:

wk =K + 1

2−

∣∣∣∣k − K + 12

∣∣∣∣ . (10)

The final PSA coarray is equivalent to the FPA coarray[Fig. 8(g)]. The closeness of this approximation is depen-dent on the performance of the reconstruction filter.

V. Wideband PSA Processing

The PSA imaging for a wideband system differs fromnarrowband imaging in that the 2-D spatial frequency re-sponse must be taken into account. The 2-D spatial fre-quency response is the sum of the coarrays over all tem-


Fig. 8. PSA image formation and reconstruction in the lateral spatial freqency (coarray) domain. (a) Subarray aperture functions in arrayspace. (b) Ideal cosubarrays if full set (QFPA) of beams were acquired. (c) Cosubarrays when a subset (QPSA) of the beams is acquired byeach subarray. (d) Upsampling leads to periodic replication of the cosubarrays in (c). Reconstruction filters are designed as bandpass filtersto suppress the unwanted aliases. (e) Cosubarrays of the reconstructed subarray system should approximate ideal cosubarrays shown in (b).By properly weighting and summing these responses, the final coarray (e) is equivalent to the FPA coarray (g) corresponding to phasedarray imaging with the full aperture (f).

poral frequencies contained in the transmit pulse. The lat-eral spatial frequency width of the response varies linearlywith the temporal frequency of operation; the axial spatialfrequency extent of the 2-D spatial frequency response isdetermined by the minimum and maximum temporal fre-quencies present, fmin and fmax. The nonzero portion ofthe 2-D spatial frequency response for an N -element arraytherefore is limited to a trapezoidal region.

For example, consider the wideband cosubarrays illus-trated in Fig. 9. The 2-D spatial frequency response of thecenter M -element subarray acquiring QFPA beams (highbeam rate) is shown in Fig. 9(a). A horizontal cross sec-tion through any part of the nonzero spatial frequencyresponse yields a triangular function whose peak locationis indicated in Fig. 9(a) by the dashed line. For off-centersubarrays, the spatial frequency response is sheared pro-portional to the distance of the subarray from the centerof the full array. Fig. 9(b) illustrates the spatial frequencyresponse of the end subarray.

The 2-D spatial frequency responses corresponding toeach stage of the PSA imaging system are outlined inFigs. 9(c)–(h). Because the beams are treated as analyticsignals, the negative temporal frequency response is zero;therefore, the illustrations only show the response for pos-itive temporal frequencies. The illustrations are similar tothose given in Fig. 8, except that only the responses for thecenter and end subarrays are shown. With the exceptionof using a 2-D rather than a 1-D filter kernel, the pro-cessing steps are identical. The drawings are based on thesame system as in Figs. 7 and 8 (N = 16, K = 7, M = 4,L = 4). The center frequency, f0, is identical to that used

in the narrowband example. However, the temporal fre-quencies now range from fmin to fmax. This increase intemporal frequency causes a corresponding increase in thelateral spatial frequency bandwidth. Assuming that the el-ement pitch is d = λmin

2 , the number of beams that mustbe acquired to avoid aliasing of the 2-D spatial frequencyresponse is:

QPSA > 2M sin(

Θ2

). (11)

The drawings in Fig. 9 correspond to a system that isslightly oversampled.

The spatial frequency responses of the 1-D filters used inthe previous section correspond to lateral frequency bandsthat are constant over all temporal frequencies. Althoughthe 1-D filter is suitable for reconstructing the center sub-array image, a 2-D filter kernel is needed for all othersubarrays because the lateral spatial frequency passbandvaries with temporal frequency. The ideal passbands forthe center and end subarrays are indicated as dark grayregions behind the original subarray spatial frequency re-sponses in Figs. 9(e) and (f). After the appropriate 2-Dfilters are applied to each subarray image, the result ap-proximates the high beam rate spatial frequency response[Figs. 9(a) and (b)]. The reconstructed subarray imagesare then coherently weighted and summed, using the sameweights as given in (10). The nonzero portions of the sub-array spatial frequency responses are all shown overlappingone another in Fig. 9(g). A lateral cross section throughthese subarray spatial frequency responses would revealthat adjacent cosubarrays overlap one another by half their


Fig. 9. PSA image formation and restoration in the 2-D spatial frequency domain for wideband system consideration. Coarrays shown inFig. 8 correspond to horizontal cross-sections through these figures at the center frequency, f0. Shaded areas correspond to regions of nonzeroresponse. (a) Response of center subarray when full set (QFPA) beams are acquired. (b) Subarray shift along the array causes a shearedresponse relative to (a). (c) Critical beamsampling (QPSA beams) leads to a laterally compressed frequency domain. (d) The wrappingof the response in (b) is evident due to the subsampling of beamspace. (e), (f) Upsampling in beamspace leads to periodic replication ofthe response in (c), (d), and requires reconstruction filtering to suppress the aliases. (g) The overlapping individual subarray responses areweighted and summed to form (h) the overall response of the system, equivalent to FPA imaging.


width independent of the temporal frequency. Fig. 9(h)shows the 2-D spatial frequency response of the overallPSA system, equivalent to that of FPA imaging.

The 2-D interpolation filter is applied in both the lat-eral and axial dimensions by 2-D convolution. The sub-array filters illustrated are best described in the spatialfrequency domain as shown in (12) (see next page). Thisexpression describes the filters shown in Figs. 9(e) and (f).The filter for the center subarray is independent of tem-poral frequency and, thus, is a sinc function in the spatialdomain. The spatial-frequency-domain representations ofthe other subarray filters are sheared versions of the cen-ter array filter. Just as for the 1-D case, the spatial filtermust be truncated and windowed. In this case, the filter istruncated to a finite 2-D kernel over the axial and lateraldimensions. A window function is applied to the ideal filterto obtain the final filter.

VI. Alternate Acquisition Methods

The PSA acquisition method presented earlier assumedthat the same subarray is used in transmit and receive.Although this is the simplest method to describe and il-lustrate, the reconstruction filters presented are equallyuseful when reconstructing subarray images acquired byother pairs of transmit and receive subarrays. The sim-ilarities between these different methods of acquisitionare best visualized using the comatrix representation de-scribed in Section II-B. Figs. 5(d)–(f) illustrate the co-matrices and resultant coarrays for three different PSAacquisition methods. The comatrix for the example usedthroughout this paper (N = 16, K = 7, M = 4, L = 4) isshown in Fig. 5(d). Each of the small, gray diamonds cor-responds to a firing event from one of the seven subarrays.The dashed line indicates the overall coarray without anyweighting; applying the weights shown in Fig. 5(d) restoresan FPA-equivalent response.

The same overall coarray response can be acquired withfewer subarrays, although with additional subarray imageacquisitions; Fig. 5(e) illustrates a configuration that usesfour subarrays (K = 4) to acquire 10 subarray images.As an example, the cosubarray formed by transmittingand receiving from the second subarray in PSA-1 is iden-tical to that formed by PSA-2 by transmitting on the firstsubarray and receiving on the second. In order to avoiddistortion due to phase asymmetry, the opposite subar-ray pair also must be used—that is, transmitting on thesecond and receiving on the first. Because some of the co-subarrays have been acquired twice using this method, theweights for achieving an FPA-equivalent response are dif-ferent. A third PSA example with only four subarrays usesall TX/RX subarray pairs and is shown in Fig. 5(f). Noweighting is necessary for this method. The subsamplingand reconstruction are not illustrated using the comatrixrepresentation. However, the same principles apply regard-less of the subarray pair used for acquisition. The subsam-pling rate, the beam sampling requirement, and the recon-

struction filters—both 1-D and 2-D—remain the same forthese alternate acquisition methods. The only part of re-construction that differs are the weights used prior to sum-mation. The primary differences between these alternatemethods are due to the difference in the total number offirings. The two alternate PSA schemes (PSA-2 and PSA-3) require an increased number of firings. The frame rateis inversely proportional to the number of firings. Whilethe frame rate decreases, the SNR increases.

VII. Results

The reduction of front-end hardware complexity can berepresented by the ratio of the number of front-end hard-ware channels required for PSA versus FPA imaging:

Complexity of PSAComplexity of FPA

=M

N. (13)

Other theoretical performance metrics of FPA, CSA, andPSA imaging are summarized in Table I and describedhere in detail.

The number of firings per frame required for FPA andPSA imaging is equal to the total number of beams formed:

BFPA = 2N sin(

Θ2

), and (14)

BPSA = 2KM sin(

Θ2

). (15)

CSA imaging does not form beams directly; rather, theimage is formed from pulse-echo scans acquired from indi-vidual array elements. Therefore, the number of firings issimply equal to the number of elements:

BCSA = N. (16)

The ratio of the number of firings required for the twophased array methods is:

BPSA

BFPA=

KM

N=

(2N

M− 1

)M

N=

2N − M

N,

(17)

where the substitution for K is based on (5) with J =M/2. Note that this ratio is bounded by:

BPSA

BFPA< 2, (18)

meaning that the number of firings required for any PSAconfiguration will never require more than twice the num-ber of firings required for FPA imaging.

The frame rate of an ultrasound imaging system is de-termined by the number of scans required per frame (B),the velocity of sound in the medium (c), and the desiredimaging depth (R):

F ≤ c

2RB. (19)


Hk[m,n] =

⎧⎨⎩1,

∣∣∣∣(

K + 12

− k

)(2P

m − 1)

−(

K + 12

)(1

2Nn − 1

)∣∣∣∣ < 1

0 otherwise. (12)

TABLE I

FPA CSA PSAExact O(·) Example1 Exact O(·) Example Exact O(·) Example

Number of front-endN N 128 1 1 1 M M 32

hardware channels

Number of firings2N sin(Θ

2 ) N 181 N N 128 2KM sin(

Θ2

)KM 317

per frame (B)

Frame Rate (F ) c

4NR sin(Θ2 )

1N

39 c2RN

1N

58 c

4KMR sin(Θ2 )

1KM

23

Normalized SNR (dB) 20 log10(N√

N) — 63.2 20 log10(√

N) — 21.0 20 login(M√

KM) — 53.6

1The numerical results are given for the following example setup: number of array elements (N), 128; number of subarray elements (M),32; number of subarrays (K), 7; scan angle (Θ), 90◦; velocity of sound (c), 1430 m/s; imaging depth (R), 10 cm; and half-wavelengtharray spacing (d = λmin/2).

This is the basis for the results presented in Table I.The SNR of a single point reconstructed from multiple

pulse-echo scans generally is accepted to be:

SNR = 20 log10

(NT

√NR

)+ SNR0, (20)

where NT is the number of active transmit elements foreach scan, NR is the number of receive scans used to con-struct each image pixel, and SNR0 is the SNR of the pulse-echo signal generated by transmitting and receiving on asingle array element. The basis for the equation is thatthe total signal power is proportional to the number ofelements firing simultaneously (NT ), and uncorrelated ad-ditive noise on receive, reducing the noise by the root ofthe number of independent measurements (

√NR). For a

given firing event in PSA imaging, there are M transmitelements and M receive elements, resulting in an SNR gainof 20 log10(M

√M) for each subarray image. An additional

gain is realized by combining the K subarray images. If allK subarray images are weighted uniformly, as they are inPSA-3 [Fig. 5(f)], then the SNR becomes:

SNRPSA = 20 log10(M√

KM) + SNR0. (21)

However, in cases in which the subarray images are nonuni-formly weighted when combined to form the final image,the SNR becomes:

SNRPSA = 20 log10

(M

√M

Σwk√Σw2

k

)+ SNR0.

(22)

When uniform weights are used, (22) simplifies to (21).One of the primary advantages of PSA imaging over

FPA imaging is the reduced cost and size of the front-end hardware. For the example illustrated in Table I, the

FPA front-end requires 128 buffers, time-gain compensa-tion circuits, low-pass filters, and analog-to-digital convert-ers (ADCs). In contrast, the PSA front-end adds M multi-plexers (one K : 1 multiplexer for each subarray channel)and reduces the number of buffers, time-gain compensation(TGC) circuits, low-pass filters, and ADCs to 32. In ad-dition to the cost savings this represents, the reduction insize increases the feasibility of incorporating the ADCs intothe transducer head or integrating them with the beam-former on a single application specific integrated circuit(ASIC).

Compared to the previous work by Karaman andO’Donnell [40] the method described here has improvedlateral spatial resolution and more closely approximatesphased array imaging using the full array. This can be ob-served by comparing the coarray of the previous methodshown in Fig. 5(c) to any of the PSA method in Figs. 5(d)–(f). By translating both the transmit and receive subaper-tures across the full array as in PSA, the FPA responsecan be achieved.

There are multiple factors that influence the motion ar-tifacts present in a PSA image. First, the order in whichthe subarray beams are acquired will affect motion arti-facts. As illustrated in Fig. 6, all subarrays acquire thefirst beam in the subarray image before acquiring the sec-ond, and so on. Although it also is possible to acquireall beams from each subarray consecutively, the order il-lustrated minimizes the time between acquisitions of thesame beam by different subarrays, thus reducing motionartifacts. Second, the total acquisition time for acquiringthe beams used to form a single PSA image is longer thanthat required by FPA imaging. The size of this artifactdepends on the total number of beams acquired to forma single image, and, therefore, will be less than twice thesize of the FPA artifact (18). Third, application of the in-


terpolation filter will influence motion artifacts. Movingobjects detected by any of the acquired beams used in theinterpolation will effect the reconstructed beam. The im-pact on the reconstructed beam will depend on both thenumber of beams used in the interpolation—determinedby the interpolation filter length—and the filter weightof each beam. In general, beams further from the recon-structed beam will have less impact due to the sinc-likenature of the interpolation filter.

VIII. Discussion and Conclusions

The proposed PSA imaging method provides a balancebetween conventional FPA imaging—having a high framerate and good SNR but high hardware complexity—andclassical synthetic aperture imaging—having a high framerate and low hardware complexity, but with poor SNR.The PSA imaging significantly reduces the front-end hard-ware complexity compared to FPA imaging at the expenseof a slight decrease in frame rate and SNR, in which theSNR of the proposed method is significantly higher thanCSA imaging.

The aperture functions for both the subarrays and thefull array were assumed to be uniform with no apodization.In addition, the subarray spacing was assumed to be halfthe subarray size (J = M/2). These two requirements al-low the full FPA coarray to be a linear combination of thePSA cosubarrays. Using arbitrary apodization or differentsubarray spacing requires a more generalized restorationfilter. Methods for calculating such generalized subarrayfilters are presented in [50].

One of the most exciting potential applications of thesynthetic aperture and subarray imaging methods is theirapplication to 3-D imaging using 2-D transducer arrays[51]–[53]. By extending the theory presented in this pa-per to include a 2-D aperture function to form beams in3-D image space, the channel count can be significantlyreduced to address the latter limitation.

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Jeremy A. Johnson (S’92) received his B.S.in electrical engineering and a minor in math-ematics with honors from Walla Walla Col-lege, College Place, Washington, in 1997. Hereceived his M.S. and Ph.D. in 1999 and 2003,respectively, in electrical engineering fromStanford University, Stanford, California. Hecompleted the Biodesign Innovation Fellow-ship at Stanford University in 2004. Jeremy isa Senior R&D Engineer at Medtronic Vascu-lar, Santa Rosa, CA.

He has several summers of industry experi-ence. He worked as a software engineer at Interactive Northwest, Inc.,Tualatin, OR, during the summer of 1995; worked as an ASIC designengineer at Intel, Inc., Hillsboro, OR, during the summers of 1996and 1997; performed research in color science at Sony Research Lab-oratories, San Jose, CA, during the summer of 1998; performed re-search in computer vision at Hughes Research Laboratories, Malibu,CA, during the summer of 1999; and developed an endoscopic cali-bration routine for image-enhanced endoscopy at Cbyon, Inc., PaloAlto, CA, during the summer of 2000. His research interests includemedical imaging, computer-aided diagnosis, and surgical navigationsystems.

Mustafa Karaman (S’88–S’89–M’89–M’93–M’97) was born in Balıkesir, Turkey, in 1964.He received the B.Sc. degree from the MiddleEast Technical University, Ankara, Turkey,and the M.Sc. and Ph.D. degrees from BilkentUniversity, Ankara, Turkey, in 1986, 1988, and1992, respectively, all in electrical and elec-tronics engineering.

From 1993 to 1994, he was a post-doctoralfellow in the Biomedical Ultrasonics Labora-tory in the Bioengineering Department, Uni-versity of Michigan, Ann Arbor. From 1995

to 1996, he was on the faculty with the Electrical and ElectronicsEngineering Department of Kırıkkale University, Turkey, first as As-sistant Professor and later as Associate Professor. In 1996, he joinedBaskent University, Ankara, Turkey, as the Chairman of Electricaland Electronics Engineering and Acting Chairman of the ComputerEngineering Department and served in founding these departments.He was a visiting scholar in the Biomedical Ultrasonics Laboratoryat the University of Michigan, Ann Arbor, and in the E. L. GinztonLaboratory at Stanford University, Stanford, California, in the sum-mer terms of 1996–1997 and 1999, respectively.

Between 2000–2002, he was with the E. L. Ginzton Laboratoryat Stanford University, Stanford, California, as a visiting faculty inelectrical engineering. In 2002, he joined Isık University, Istanbul,Turkey, where he is currently working as a faculty member in Elec-tronics Engineering.

In 1996, he was awarded H. Tugac Foundation Research Award ofTurkish Scientific and Technical Research Council for his contribu-tions to ultrasonic imaging. His research interests include signal andimage processing, ultrasonic imaging and integrated circuit design.Dr. Karaman is a member of the IEEE.

Butrus T. Khuri-Yakub (S’70–S’73–M’76–SM’87–F’95) was born in Beirut, Lebanon.He received the B.S. degree in 1970 fromthe American University of Beirut, Beirut,Lebanon, the M.S. degree in 1972 from Dart-mouth College, Hanover, NH, and the Ph.D.degree in 1975 from Stanford University, Stan-ford, CA, all in elecrical engineering.

He joined the research staff at the E. L.Ginzton Laboratory of Stanford University in1976 as a research associate. He was promotedto a senior research associate in 1978 and to a

professor of electrical engineering (research) in 1982. He has served onmany university committees in the School of Engineering and the De-partment of Electrical Engineering at Stanford University. Presently,he is the Deputy Director of the E. L. Ginzton Laboratory.


Dr. Khuri-Yakub has been teaching at both the graduate and un-dergraduate levels for over 15 years. His current research interestsinclude in situ acoustic sensors (temperature, .1m thickness, resistcure, etc.) for monitoring and control of integrated circuits manu-facturing processes, micromaching silicon to make acoustic materialsand devices such as airborne and water immersion ultrasonic trans-ducers and arrays, and fluid ejectors, and in the field of ultrasonicnondestructive evaluation and acoustic imaging and microscopy.

Dr. Khuri-Yakub is a fellow of the IEEE, a senior member of theAcoustic Society of America, and a member of Tau Beta Pi. He isassociate editor of Research in Nondestructive Evaluation, a Journalof the American Society for Nondestructive Testing. He has authoredover 400 publications and has been principal inventor or coinventorof 61 issued patents. He received the Stanford University School ofEngineering Distinguished Advisor Award, June 1987, and the Medalof the City of Bordeaux for contributions to NDE, 1983.

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