Sparse Sensing with Semi-Coprime Arrays

1 Sparse Sensing with Semi-Coprime Arrays Kaushallya Adhikari Abstract —A semi-coprime array (SCA) interlea ves two under - sampled uniform linear arrays (ULAs) and a Q element standard ULA. The undersampling factors of the first two arrays are QM and QN respecti vely where M and N are coprime. The resulting non-uniform linear array is highly sparse. T aking the minimum of the absolute values of the con ventional beampatterns of the three arrays results in a beampatter n free of grating lobes. The SCA offers more savings in the number of sensors than other popular sparse arrays like coprime arrays, nested arrays, and minimum redundant arrays. Also, the SCA exhibits better side lobe patterns than other sparse arrays. An example of direction of arriv al estimation with the SCA illustrates SCA ’s pr omising potential in reducing number of sensors, decr easing system cost and complexity in various signal sensing and processing applications. Index T erms —Semi-coprime arrays, spare arrays, super - resolution, coprime arrays, nested arrays, minimum redundant arrays. I . I N T RO D U C T I O N M ANY linear sparse array designs achiev e the resolu- tion of a fully populated uniform linear array (ULA), hereafter called full ULA [1]–[14]. Some of these designs, [1]–[5], use con ventional beamforming (CBF) to process the receiv ed signal, while some other designs, [6]–[14], split the sparse array into two subarrays and multiply the subarrays’ individual CBF outputs. The sparse array designs that existed prior to 2010 suffered from a common limitation — the lack of concrete design criteria or the analytical expressions for the sensor locations. The nested arrays [11], [12] and coprime sensor arrays (CSAs) [13], [14] overcome the limitation by providing analytical expressions for the sensor locations. The generalized coprime array configuration treats the coprime and nested arrays as its special cases [15]. A nested array interlea ves a short full ULA with an un- dersampled ULA having the same undersampling factor as the number of sensors in the short full ULA. A CSA inter- leav es two undersampled subarrays where the undersampling factors are coprime integers. In addition to having con venient analytical expressions for the sensor locations, the NSA and CSA also ha ve a clear mechanism to disambiguate aliasing that occurs due to undersampling. Moreover , the CSAs can also match the peak side lobe height of a full ULA of equal resolution [16] thereby dominating the field of sparse arrays in recent years. The reduction of number of sensors in these sparse arrays translates to lower system cost, and decreased system complexity . This paper introduces a nov el sparse array called semi- coprime array (SCA). The SCA has the potential to offer more reduction in the number of sensors than even the CSA and This material is based upon research supported by the Louisiana T ech Univ ersity . The author is with the Louisiana T ech Uni versity , Ruston, LA, 71270 USA. e-mail: adhikari@latech.edu. the NSA, while still offering the crucial advantages of the CSA and the NSA which are the analytical expressions for the sensor locations, equal resolution as the full ULA with equiv alent aperture, and prudent cancellation of the grating lobes resulting from undersampling. The SCA can achiev e the peak side lobe height of a full ULA with much less extension than the CSA and the NSA. Since the SCA is sparser than the CSA and the NSA, the SCA suffers less from mutual coupling effect between adjacent sensors than the CSA and the NSA. Section II describes the SCA in detail. Section III compares the SCA with other popular sparse arrays (CSA, NSA, and minimum redundant arrays). Section IV demonstrates ho w SCA ’ s inherent super-resolution characteristic can be used in direction of arriv al estimation and compares the SCA with other sparse arrays. Con ventions: Bold-faced letters represent vectors; x ∗ de- notes complex conjugate of x ; x H denotes Hermitian of x ; GC D ( a, b ) denotes the greatest common divisor (GCD) of the integers a and b ; min ( a, b, c ) denotes the minimum of a , b , and c . I I . S E M I - C O P R I M E A R R A Y S This section is going to describe the novel array design and its associated processor in detail. A. Semi-Coprime Arrays Structur e A semi-coprime array (SCA) is a sparse array that inter - leav es three ULAs, hereafter called Subarray 1, Subarray 2, and Subarray 3. Each SCA has underlying coprime integers M , and N . Subarray 1 has P M sensors (Symbol 4 ) and QN λ 2 intersensor spacing, and Subarray 2 has P N sensors (Symbol 5 ) and QM λ 2 intersensor spacing, where P and Q are integers greater than 1 , and λ is the wavelength of the signal to be sampled. The Subarray 3 has λ 2 intersensor spacing and the number of sensors (Symbol C ) is equal to the GCD of the undersampling factors in the Subarray 1 and Subarray 2, i.e., Q = GC D ( QM , QN ) . Figure 1 depicts the formation of an SCA for M = 3 , N = 4 , P = 2 , and Q = 2 . The three subarrays always share the first sensor . The Subarray 1 and the Subarray 2 always share P sensors. Hence, the total number of sensors in an SCA is P M + P N + Q − 1 − P but it can achiev e the resolution of a full ULA with P QM N sensors. For example, if P = 2 , Q = 6 , M = 4 , and N = 5 , the SCA has only 21 sensors and it achieves the resolution of a full ULA with 240 sensors. B. Min Pr ocessing The proposed processor for the SCA is a min processor as depicted in Figure 2. The vectors x 1 , x 2 , and x 3 represent the 2 Subarray 2 (2N sensors) 2M λ/2 Subarray 3 (2 Sensors) λ/2 SCA (2M+2N−1 sensors) Subarray 1 (2M sensors) 2N λ/2 (d) (a) (c) (b) Fig. 1. (a). Subarray 1 with 2 M sensors and undersampling factor 2 N (b). Subarray 2 with 2 N sensors and undersampling factor 2 M (c). Subarray 3 with 2 sensors and undersampling factor 1 (d). The SCA resulting from interleaving Subarray 1, Subarray 2, and Subarray 3 signal receiv ed by the three subarrays. The SCA processor con ventionally beamforms each of the three subarrays’ re- ceiv ed signal by using the weight v ectors w 1 , w 2 , and w 3 . Assuming uniform weighting, the vectors w 1 , w 2 , and w 3 are P M , P N , and Q element vectors and their i th elements for direction cosine u = cos( θ ) are 1 P M exp( j π u ( i − 1) QN ) , 1 P N exp( j π u ( i − 1) QM ) , and 1 Q exp( j π u ( i − 1)) respec- tiv ely . The CBF outputs for the three subarrays are y 1 = w H 1 x 1 , y 2 = w H 2 x 2 , and y 3 = w H 3 x 3 . The final SCA output, y , is the minimum of the absolute values of the three CBF outputs, i.e., y = min ( | y 1 | , | y 2 | , | y 3 | ) . Although both Subarray 1 and Subarray 2 are undersam- pled, the SCA output disambiguates aliasing by appropriately combining the Subarray 1, Subarray 2 and Subarray 3 outputs. The Subarray 1 beampattern has the undersampling factor QN , and as a result, it has QN major lobes at integer multiples of 2 / ( QN ) . Assuming the array is steered to broadside, i.e. u = cos  π 2  = 0 , the major lobe at u = 0 is the main lobe and the other QN − 1 major lobes are grating lobes resulting from undersampling. The Subarray 2 beampattern has the undersampling f actor QM , and consequently , it has QM major lobes at inte ger multiples of 2 / ( QM ) . The major lobe at u = 0 is the main lobe and the other QM − 1 major lobes are grating lobes due to undersampling. The Subarray 3 beampattern has one major lobe at u = 0 which is the main lobe and it has nulls at integer multiples of 2 /Q . Since the undersampling factors of Subarray 1 and Subarray 2 have GC D ( QM , QN ) = Q, only Q major lobes of Subarray 1 overlap with Q major lobes of Subarray 2. One of the overlapping major lobes from each of the Subarray 1 and Subarray 2 is the main lobe at u = 0 , and the other Q − 1 overlapping major lobes from each of the Subarray 1 and Subarray 2 cause aliasing. Howe ver , the Subarray 3 has nulls exactly at the locations where the other two subarrays have overlapping grating lobes. As a result, taking the minimum of the three beampatterns generates a beampattern with no grating lobes at all. Figure 3 elucidates the beampattern formation mechanism for the SCA sho wn in Figure 1, assuming the array is steered to the direction u = 0 . The Subarray 1 beampattern (blue dashed in Figure 3) has the undersampling factor 2 N = 8 , and as a result, it has 8 major lobes. The major lobe locations are inte ger multiples of 1 / N , i.e., u = − 0 . 75 , u = − 0 . 5 , u = − 0 . 25 , u = 0 , u = 0 . 25 , u = 0 . 5 , u = 0 . 75 , and u = ± 1 . The major lobe at u = 0 is the main lobe and the other 7 major lobes are grating lobes resulting from undersampling. The Subarray 2 beampattern (red dashed-dot in Figure 3) has the undersampling factor 2 M = 6 , and consequently , it has 6 major lobes. The major lobe locations are integer multiples of 1 / M , i.e., u = − 0 . 67 , u = − 0 . 33 , u = 0 , u = 0 . 33 , u = 0 . 67 , and u = ± 1 . The major lobe at u = 0 is the main lobe and the other 5 major lobes are grating lobes due to undersampling. The Subarray 3 beampattern (purple dot in Figure 3) has one major lobe at u = 0 which is the main lobe. Since the undersampling factors of Subarray 1 and Subarray 2 hav e GC D ( QM , QN ) = 2 , two major lobes of Subarray 1 o verlap with two major lobes of Subarray 2. One of the ov erlapping major lobes from each of the Subarray 1 and Subarray 2 is the main lobe at u = 0 , and the other o verlapping major lobe from each of the Subarray 1 and Subarray 2 is the grating lobe at u = ± 1 . Howe ver , the Subarray 3 has nulls exactly at u = ± 1 . Therefore, taking the minimum of the three beampatterns shown in Figure 3 results in a beampattern (black solid in Figure 3) that is dev oid of grating lobes. Figure 4 compares an SCA beampattern with a standard ULA that offers the same resolution. The standard ULA (green dashed in Figure 4) has 48 sensors while the SCA (black solid 3 x 2 : input x 2 H : array weights w 2 : input x 1 x w H + Noise 1 1 + H Noise w 2 x 2 + H Noise w 2 y 3 y : array weights H : array weights 2 : output y 3 y : SCA output : input x 3 w 3 y | | | | | | y 1 y 2 y 1 y 2 : output y 1 1 w H : output y 2 min( , , ) Fig. 2. The SCA min processor conventionally beamforms the Subarray 1 (blue), Subarray 2 (red) and Subarray 3 (purple) data and finds the minimum of the absolute values of the CBF outputs to produce the final output. in Figure 4) has only 13 sensors. The SCA and the standard ULA have equal main lobe widths and hence equal resolution, and almost equal peak side lobe (PSL) heights. Using only about 27% of the sensors in the standard ULA, the SCA is able to match both resolution and PSL height of the standard ULA, offering a substantial saving in the number of sensors. I I I . C O M PAR I S O N W I T H O T H E R S PA R S E A R R A Y S As noted in the introduction, many authors hav e proposed various sparse array schemes ov er the course of the last fe w decades. Howe ver , the introduction of the NSA and CSA in 2010 and 2011 has sparked rene wed interest in sparse arrays. The CSA and NSA have drawn great attention from researchers since they possess concrete expressions for sensor locations and their exists a clear mechanism to remov e pru- dently any grating lobes that arise due to undersampling. An- other category of arrays, minimum redundant arrays (MRAs), has also been a relev ant topic of research and reference in the field of sparse arrays for se veral decades. This section summarizes three different versions of the CSA (basic CSA, extended CSA, and min-processing CSA), the NSA, and the MRA and compare them with the novel array , SCA. 1) Basic Coprime Sensor Array: A basic CSA is a sparse array that interleaves two ULAs, hereafter called Subarray 1, and Subarray 2. Each CSA has underlying coprime integers M , and N . The Subarray 1 has M sensors (Symbol 4 in −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −30 −25 −20 −15 −10 −5 0 u=cos( θ ) Beampattern, dB Subarray 1 Subarray 2 Subarray 3 SCA Fig. 3. Formation of the SCA beampattern. T aking the minimum of the absolute values of the Subarray 1 (blue dashed-dot), Subarray 2 (red dashed- dot) and Subarray 3 (purple dashed-dot) beampatterns results in a beampattern (black solid) free of grating lobes. −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −30 −25 −20 −15 −10 −5 0 u=cos( θ ) Beampattern, dB SCA Full ULA Fig. 4. Comparison of the SCA (black solid), and ULA (green dashed) beampatterns. The two beampatterns have equal main lobe with and PSL height. Figure 5) and N λ 2 intersensor spacing, and the Subarray 2 has N sensors (Symbol 5 in Figure 5) and M λ 2 intersensor spacing. The total number of sensors in a basic CSA is M + N − 1 but it can achiev e the resolution of a full ULA with M N sensors [13], [14]. Figure 5 depicts the formation of a basic CSA for M = 4 , and N = 5 . The basic CSA has only 8 sensors and it achiev es the resolution of a full ULA with 20 sensors. For a CSA with a gi ven aperture, the coprime pair M and N = M + 1 minimizes the total number of sensors 4 needed to span that aperture [16], [17]. This paper assumes that N = M + 1 . λ/2 N M λ/2 CSA (M+N−1 sensors) (a) Subarray 1 (M sensors) Subarray 2 (N sensors) (b) (c) Fig. 5. (a) Subarray 1 (blue) of a CSA with coprime pair (4 , 5) (2) Subarray 2 (red) of a CSA with coprime pair (4 , 5) (c) The resulting CSA formed by interleaving Subarray 1 and Subarray 2. A CSA processor eliminates the grating lobes by taking the product of the CBF beampatterns of the two subarrays. The Subarray 1 beampattern has the undersampling factor N , and as a result, it has N major lobes at integer multiples of 2 / N . The major lobe at u = 0 is the main lobe and the other N − 1 major lobes are grating lobes due to undersampling. The Subarray 2 beampattern has the undersampling factor M , and as a result, it has M major lobes at integer multiples of 2 / M . The major lobe at u = 0 is the main lobe and the other M − 1 major lobes are grating lobes due to undersampling. Since M and N are coprime, all the grating lobe locations are unique while the main lobes of the Subarray 1 and Subarray 2 are exactly at the same location. T aking the product of the two beampatterns removes the grating lobes. Figure 6 illustrates the formation of a CSA beampattern for the arrays shown in Figure 5. The CSA has total 8 sensors and the ULA has 20 sensors. Ho wever , they hav e the equal main lobe width and therefore, have equal resolution. Comparing the CSA and the ULA beampatterns also shows that the CSA peak side lobe (PSL) height is much higher than the ULA PSL height. The ULA PSL height is − 13 dB while the CSA PSL height is about − 4 dB in Figure 6. 2) Extended Coprime Sensor Array: As noted in Section III-1, the PSL height of a basic CSA is too high for it to be considered useful in major applications. Keeping the intersensor spacings of the subarrays fixed and adding more sensors to both subarrays causes the CSA PSL to decrease and match the ULA PSL height [13], [14]. The number of additional sensors required to be added to the subarrays to match the PSL height of a standard ULA depends on the coprime factors and the weighting functions used. The deriv ation of the number of additional sensors for v arious standard uniform and non-uniform weighting functions exists in [17]. The coprime sensor array where the subarrays hav e been extended to match the PSL height of the full ULA with the equiv alent aperture is called the extended coprime sensor array (ECSA). The numbers of sensors in the Subarray 1 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −30 −25 −20 −15 −10 −5 0 u=cos( θ ) Beampattern, dB Subarray 1 Subarray 2 CSA Full ULA Fig. 6. Subarrays, CSA and full ULA beampatterns. and Subarray 2 are M e = d cN e − 1 and N e = d cN e . The constant c is called the extension factor which is the number of repetitions of the basic CSA in an ECSA. F or uniform shading, the total number of sensors in an ECSA is L = 13 M + 6 and the ECSA has the resolution of a full ULA with M e N sensors (See Appendix A). 3) Min-processing Coprime Sensor Array: T aking the mini- mum of the two CBF beampatterns in a basic CSA or an ECSA also removes the grating lobes since the grating lobes are at different locations [18]–[20]. A CSA where each subarray has two periods of the basic subarrays and the associated processor is the min-processor explained in [18]–[20] is, subsequently , called min-processing comprime sensor array (MCSA). An MCSA has 4 M sensors and it achiev es the resolution of a full ULA with 2 M N sensors (See Appendix B). 4) Nested Sensor Array: A nested sensor array (NSA) is a sparse array that interleaves two ULAs, hereafter called Subarray 1, and Subarray 2. The Subarray 1 has M sensors (Symbol 4 in Figure 7) and λ 2 intersensor spacing, and Subarray 2 has N sensors (Symbol 5 in Figure 7) and M λ 2 intersensor spacing. The total number of sensors in an NSA is M + N − 1 b ut it can achiev e the resolution of a full ULA with M N sensors. Figure 7 depicts the formation of an NSA for M = 5 , and N = 3 . The NSA has only 7 sensors and it achiev es the resolution of a full ULA with 15 sensors. An NSA processor eliminates the grating lobes by taking the product of the CBF beampatterns of the two subarrays. The Subarray 1 beampattern has only one major lobe at u = 0 and it is the main lobe. The Subarray 1 has nulls at integer multiples of 2 / M . The Subarray 2 beampattern has the undersampling factor M , and as a result, it has M major lobes at integer multiples of 2 / M . The major lobe at u = 0 is the main lobe and the other M − 1 major lobes are grating lobes due to undersampling. Since the grating lobes of the Subarray 2 are exactly at the nulls of the Subarray 1, taking the product 5 λ/2 Subarray 1 (M sensors) M λ/2 Subarray 2 (N sensors) NSA (M+N−1 sensors) (a) (b) (c) Fig. 7. (a) Subarray 1 (blue) of an NSA with M = 5 and N = 3 (2) Subarray 2 (red) of an NSA with M = 5 and N = 3 (c) The resulting NSA. of the two beampatterns removes the grating lobes. Figure 8 illustrates the formation of an NSA beampattern for the arrays shown in Figure 7. The NSA has total 7 sensors and the ULA has 15 sensors. Ho wever , they hav e the equal main lobe width and therefore, ha ve equal resolution. Comparing the NSA and the ULA beampatterns also shows that the NSA PSL height is much higher than the ULA PSL height. Extending the NSA subarrays like in CSA does not decrease the NSA PSL height to the level of the full ULA. −1 −0.5 0 0.5 1 −30 −25 −20 −15 −10 −5 0 u=cos( θ ) Beampattern, dB Subarray 1 Subarray 2 NSA Full ULA Fig. 8. Subarrays, nested and full ULA beampatterns 5) Minimum Redundant Array: Minimum redundant arrays is a class of sparse arrays designed to have the least redun- dancy in the coarray [21, page 179]. For a gi ven number of sensors, an MRA attains the largest possible hole-free corrary while at the same time, minimizes the number of sensor pairs leading to the same spatial correlation lag. MRAs are the sparsest arrays among arrays that achie ve hole-free coarrays. [21, page 182] lists the sensor locations for up to 17 sensors. These sensor locations were obtained through exhausti ve search routines and the analytical expressions for the sensor locations do not exist. T ABLE I Comparison of the numbers of sensors relative to the full ULA with the equal resolution Array Number of sensors Matches ULA ’s PSL height? ECSA 2 N  13 M + 6 13 M + 11  Y es Basic CSA 2 N No MCSA 2 N Y es NSA 2 N No SCA 2 N  P M + 0 . 5 Q − 0 . 5 P QM  Y es 6) Comparison of SCA with CSA, NSA, and MRA: A sparse array achieves the resolution of a full ULA using fewer sensors. In addition to matching the resolution, some sparse arrays can match the PSL height of a full ULA. For a giv en resolution, the ratio of the number of sensors in the sparse array to the number of sensors in the full ULA provides a measure of the savings in sensors. The lower ratios indicate more savings in sensors. T able I lists the ratios of the total numbers of sensors of the sparse arrays to the total number of sensors of the full ULA with equal resolution (See Appendix C). The achiev able ratio of the numbers of sensors is equal to 2 N for the basic CSA, MCSA, and NSA. F or the ECSA, the achiev able ratio is 2 N  13 M + 6 13 M + 11  which is less than 2 N for any M . For the SCA, the achiev able ratio is 2 N  P M + 0 . 5 Q − 0 . 5 P QM  which is also less than 2 N for any M . T o realize the super-resolution feature embedded in the SCA, consider fixing the number of sensors to L = 32 . An ECSA with M = 2 has 32 sensors and it can match both the resolution and the PSL height of a 57 sensor full ULA using only 32 sensors. An MCSA with M = 8 has 32 sensors and it can match both the resolution and the PSL height of a 144 sensor full ULA using only 32 sensors. A basic CSA with M = 16 has 32 sensors and it can match the resolution of a 272 sensor full ULA, but its PSL height is about − 5 . 5 dB which is much higher than the full ULA ( − 13 dB). An NSA with M = 16 and N = 17 also has 32 sensors and it can match the resolution of a 272 sensor full ULA, but its PSL height is much higher than the full ULA. On the other hand, an SCA with P = 4 , Q = 9 , M = 3 and N = 4 has 32 sensors and it can match both the resolution and the PSL of a 432 sensor full ULA using only 32 sensors. Hence, for a fixed number of sensors, the SCA achie ves higher resolution than the other sparse arrays and at the same time, matches the PSL height of the full ULA. Consider a minimum redundant array with number of sen- 6 sors L = 17 . This MRA can achiev e the resolution of a 102 sensor full array . An SCA with P = 3 , Q = 6 , M = 2 , and N = 3 has 17 sensors and it can achieve the resolution of a 102 sensor full ULA. Hence, the SCA of fers ev en more sparsity than the MRA. Moreover , the SCA exhibits better side lobe patterns than the MRA. The beampatterns of the SCA and MRA with 17 sensors are shown in Figure 9. The PSL height of the SCA is − 13 db whereas the PSL height of the MRA is − 6 . 1 dB as evident in Figure 9. −1 −0.5 0 0.5 1 −30 −25 −20 −15 −10 −5 0 u Beampattern, dB MRA SCA Fig. 9. Comparison of the MRA (red) and SCA (black) beampatterns. The two beampatterns exhibit equal main lobe width, howe ver MRA ’ s PSL height is much higher . I V . E S T I M A T I O N W I T H I N C R E A S E D D E G R E E S O F F R E E D O M −1 −0.5 0 0.5 1 −8 −7 −6 −5 −4 −3 −2 −1 0 u=cos( θ ) Output, dB Number of Sensors = 32 and Number of Sources = 54 Full ULA SCA Source Locations Fig. 10. DoA estimation comparison between an SCA (black solid) with parameters M = 3 , N = 4 , P = 5 , Q = 3 and a ULA (green dashed) with the equal resolution. Number of Sensors = 32 , Number of Sources = 54 , Number of Snapshots = 100 S N R = 0 dB This section demonstrates SCA ’ s ability to detect remark- ably more sources than the number of sensors, and compares SCA ’ s direction-of-arriv al (DoA) estimation with the other spare arrays briefed in Section III. In all examples, the total −1 −0.5 0 0.5 1 −10 −8 −6 −4 −2 0 u=cos( θ ) Output, dB Number of Sensors = 32 and Number of Sources = 54 SCA ECSA Basic CSA Source Locations Fig. 11. DoA estimation comparison between an SCA (black solid) with parameters M = 3 , N = 4 , P = 5 , Q = 3 and an ECSA (brown solid) with parameters M e = 19 , N e = 20 , M = 2 , N = 3 , c = 6 . 5 and a basic CSA (purple solid) with parameters M = 16 , N = 17 . Number of Sensors = 32 , Number of Sources = 54 , Number of Snapshots = 100 S N R = 0 dB −1 −0.5 0 0.5 1 −10 −8 −6 −4 −2 0 u=cos( θ ) Output, dB Number of Sensors = 32 and Number of Sources = 54 SCA NSA Source Locations Fig. 12. DoA estimation comparison between an SCA with parameters M = 3 , N = 4 , P = 5 , Q = 3 and an NSA (cyan solid) with parameters M = 10 , N = 23 . Number of Sensors = 32 , Number of Sources = 54 , Number of Snapshots = 100 S N R = 0 dB number of sensors L is 32 , the number of sources is 54 , the number of snapshots is 100 , and S N R is 0 dB. Figure 10 compares DoA estimation by an SCA (black solid line) with parameters M = 3 , N = 4 , P = 5 , Q = 3 with a full ULA (green dashed line). The 54 source locations are marked by red circles in the figure. The SCA is able to detect all 54 sources correctly . The full ULA cannot detect any source clearly in this example. Similarly , Figure 11 compares DoA estimation by an SCA with an ECSA ( M e = 19 , N e = 20 , M = 2 , N = 3 , c = 6 . 5 ), and a basic CSA ( M = 16 , N = 17 ). The SCA is able to identify all 54 sources, whereas the CSAs fail to do so. Finally , Figure 12 compares DoA estimation by an SCA with an NSA with M = 10 and N = 23 . The SCA detects 54 sources using only 32 sensors, whereas the NSA does not. Hence, the degrees of freedom offered relative to the numbers of sensors is much higher in 7 the SCA than in the other sparse arrays. V . C O N C L U S I O N This paper introduced a no vel sparse array called semi- coprime array , formed by interleaving three subarrays — Subarray 1, Subarray 2, and Subarray 3. Subarray 1 and Subarray 2 have undersampling factors QN and QM , where M and N are coprime integers and Q is the number of sensors in Subarray 3. Min-processing the CBF beampatterns of the three subarrays remov es the grating lobes from Subarray 1 and Subarray 2 that are at different locations, while the grating lobes that are at the same locations are suppressed by the nulls of Subarray 3. The resulting non-uniform sparse linear array offers more savings in the number of sensors than other sparse counterparts like coprime arrays, nested arrays, and minimum redundant arrays. Moreover , the SCA matches the PSL height of a full ULA more easily . The super -resolution feature offered by the SCA can be exploited in applications that in volv e sensing and processing signals, for example signal estimation and detection. The examples presented in Section IV shows that the SCA has the potential to of fer much higher degrees of freedom relativ e to its number of sensors than possible with other existing sparse arrays, and this could lead to significant reduction in system cost and complexity . A P P E N D I X A T OTA L N U M B E R O F S E N S O R S A N D R E S O L U T I O N O F A U N I F O R M L Y S H A D E D E C S A For an ECSA, the numbers of sensors in the Subarray 1 and Subarray 2 are M e = d cN e − 1 and N e = d cN e , where c is the smallest positiv e number that guarantees ECSA ’ s PSL is less than or equal to − 13 dB. Since the subarrays share d c e sensors, the total number of sensors in the array is L = d 2 cN − 1 − c e = d 13 N − 1 − 6 . 5 e since c = 6 . 5 for uniform shading [17] = d 13 N − 7 . 5 e = d 13 M + 13 − 7 . 5 e since N = M + 1 . Hence, the total number of sensors is L = 13 M + 6 . Since the Subarray 1 has M e sensors and the intersensor spacing N λ 2 , the main lobe width of its CBF beampattern is 4 M e N . Since the Subarray 2 has M e sensors and the intersen- sor spacing M λ 2 , the main lobe width of its CBF beampattern is 4 N e M . The product of the two CBF beampatterns has the main lobe width M LW = min  4 M e N , 4 N e M  = min  4 ( cN − 1)( M + 1) , 4 cN M  = min  4 cN M + cN − ( M + 1) , 4 cN M  = min  4 cN M + c ( M + 1) − ( M + 1) , 4 cN M  = 4 cN M + c ( M + 1) − ( M + 1) since c > 1 . Hence, the main lobe width of the ECSA beampattern is equal to the main lobe width of the Subarray 1, 4 M e N . A P P E N D I X B T OTA L N U M B E R O F S E N S O R S A N D R E S O L U T I O N O F A N M C S A For an MCSA, when the numbers of sensors in the Subarray 1 and Subarray 2 are M e = 2 M and N e = 2 N , the PSL height is close to − 13 dB. The total number of sensors in an MCSA with N = M + 1 , M e = 2 M and N e = 2 N is L = 2 M + 2 N − 2 = 4 M . Each subarray has the main lobe width of 4 2 M N . Hence, the MCSA has the resolution equal to a full ULA with 2 M N sensors. A P P E N D I X C R A T I O S O F T H E N U M B E R S O F S E N S O R S I N T H E S PA R S E A R R A Y S T O T H E F U L L U L A For an ECSA, the total number of sensors is 13 M + 6 and it has the resolution of a full ULA with M e N sensors. Therefore, the ratio of the numbers of sensors in the ECSA to the full ULA is R = 13 M + 6 M e N = 2 N 13 M + 6 2 M e = 2 N 13 M + 6 (13 N − 2) since M e = d 6 . 5 N e − 1 = 2 N 13 M + 6 (13 M + 11) since N = M + 1 For a basic CSA, the total number of sensors is M + N − 1 = 2 M and it has the resolution of a full ULA with M N sensors. Therefore, the ratio of the numbers of sensors in the basic CSA to the full ULA is R = 2 M M N = 2 N . For an MCSA, the total number of sensors is 2 M + 2 N − 2 = 4 M and it has the resolution of a full ULA with 2 M N sensors. Therefore, the ratio of the numbers of sensors in the MCSA to the full ULA is R = 4 M 2 M N = 2 N . For an NSA, the total number of sensors is M + N − 1 and it has the resolution of a full ULA with M N sensors. Therefore, the ratio of the numbers of the sensors in the NSA to the full ULA is R = M + N − 1 M N . 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