In this section, we first introduce the structure of MTNA, which can be used directly for VCAM method to obtain the SDCA with maximum continuous DOFs. Then, under the condition of maximizing the continuous DOFs, we give the positions of redundant sensors existed in MTNA. After eliminating the redundancy, NNAs are finally proposed. Compared with other sparse arrays, NNAs possess larger physical apertures and continuous DOFs.
3.1. Introduction of Modified Translational Nested Array (MTNA)
From Reference [
25], we know that the sensor positions of translational NA can be expressed as
where
and
represent subarray 1 and subarray 2 of translational NA, respectively.
and
respectively denote the sensor number of
and
, while
and
are the corresponding translation distance. Note that
denotes the positive integer set.
It is obvious that the prototype NA is a special case of translational NA with
. According to Theorem 1 in [
25], we know that
and
should satisfy the following relationships for making the SDCA of translational NA possess the maximum continuous DOFs.
R1: .
R2: .
R3: , .
R4: , .
For the above four cases,
and
denote the rounding to integer operations, where
and
. However, to make the SDCA keep the full continuous characteristic, only the values of
and
provided in R2 and R3 can be utilized. In addition, corollary 1 in [
25] indicates that translational NA structures in R2 and R3 are mirror symmetric about zero point. Therefore, we just need to consider R3 in this paper for the convenience of analysis. Then, Equation (14) can be denoted as:
where
. When the equivalent received array
is the union of
and
, it is clear from Definition 1 that SDCA can be expressed directly as:
where
. Obviously, SDCA expressed in Equation (16) is completely continuous. From Definition 2 we know that the number of continuous DOFs of SDCA is:
Although the translational NA denoted by (15) can generate a continuous SDCA, it cannot be used directly as the received array. The reason has been mentioned in
Section 2 that only the equivalent received array
has a direct connection with SDCA. Specifically, since the elements of
are obtained by performing the difference operation on those of
, it is obvious that at least one element of
should be selected as the subtrahend. Nevertheless, observing Equation (15), we can find that all of elements in
and
are always greater than or equal to one. Thus, when
is the union of
and
,
cannot have the same form as
regardless of what the index set
is selected as. In this way, the resulting SDCA can no longer possess the maximum continuous DOFs. To solve the above problem, we modify the translational NA in (15) as follows.
Definition 3. (
MTNA).
Let the physical sensor number of subarrays in translational NA are and , respectively, then the MTNA can be defined aswhere the elements of are sorted in ascending order. and are defined in Equation (15), and . It is obvious that MTNA is the union of
,
, and
. So, the total number of sensors in MTNA is
. In addition, based on the description in
Section 2, it is easy to define the index sets as
and
, where
contains
identical elements, i.e., 1. Combining (18) with its index sets, we have
, where the corresponding SDCA possesses the maximum continuous DOFs.
Next, we consider an example with to illustrate the above analysis. In this example, sensor positions of subarrays in translational NA can be expressed as and , respectively. From Definition 3, we can get . Then, letting and , we have and . It is obvious that the resulting SDCA is fully continuous, and the corresponding number of continuous DOFs is equal to 53.
3.2. Redundancy Analysis of MTNA
From the previous subsection, we know that SDCA of MTNA has the maximum continuous DOFs. However, to achieve this goal, the optimal selections of
and
need to be determined first when we know the total number of sensors
. Accordingly, we build the following optimization problem:
Since the specific value of is related with the parity of , we can obtain multiple different solutions of Equation (19) provided below.
S1: If , we have and . Then, .
S2: If , we have and , or and . Then, .
S3: If , we have and , or and . Then, .
S4: If , we have and . Then, .
Note that is a positive integer for the above solutions. Since both S2 and S3 can be divided into two different solutions, it is clear that there exist six different selections about and to maximize . Observing and in S1–S4 again, we find that they can also be divided into four different cases from the view of parity property. Accordingly, we derive the following property of MTNA involving redundant sensors.
Property 1. For MTNA, its redundant sensors can be analyzed under four different combinations ofand, i.e.,
C1: Ifandare even,or, andorin MTNA are redundant sensors. Then, the total number of redundant sensors is.
C2: Ifis even andis odd,or, andorin MTNA are redundant sensors. Then, the total number of redundant sensors is.
C3: Ifis odd andis even,or, andorin MTNA are redundant sensors. Then, the total number of redundant sensors is.
C4: Ifandare odd,or, andorin MTNA are redundant sensors. Then, the total number of redundant sensors is.
From Property 1, we can find that although there exist redundant sensors in C1–C4, the number of redundant sensors in C3 is the largest compared to the other three cases, which implies that we can remove more redundant sensors as long as
is odd and
is even. In order to visually illustrate this interesting phenomenon,
Figure 1 depicts two examples, where the total number of sensors is fixed to be 10. According to S3, we can confirm that there exist two different solutions, i.e.,
and
, or
and
. Then, based on Property 1, if
and
, the redundant sensors of MTNA as shown in
Figure 1a can be expressed as
or
, and
or
. It is clear that the total number of redundant sensors is 3. Conversely, if
and
, the redundant sensors of MTNA as shown in
Figure 1b can be denoted as
or
, and
or
, where the total number of redundant sensors is 4. Note that, the number of rest of sensors for MTNAs shown in
Figure 1a,b is 7 and 6, respectively. And, by constructing the time average vectors as described in
Section 2, the remaining sensors of MTNAs shown in
Figure 1a,b can generate a same SDCA with the maximum continuous DOFs. Hence, for a known total sensor number
, when
and
are respectively odd and even, sensors except for redundant ones in MTNA can generate the SDCA with the largest continuous DOFs.
3.3. The Proposed Novel Nested Arrays (NNAs)
As aforementioned, if is odd and is even, the number of removable sensors in MTNA becomes the largest. Observing C3 mentioned in Property 1, we know that both and contain two-part alternative redundant sensors. So, there exist four different combinations for the rest of sensors in and . Based on this, four kinds of novel nested arrays (NNAs) are defined below.
Definition 4. (NNAs). Given parameters and , where is odd and is even, then four kinds of NNAs are defined as follows:
(1): Ifandinare removed, then the first kind of NNA (i.e., NNA-I) can be expressed as:where,
.
(2): Ifandinare removed, then the second kind of NNA (i.e., NNA-II) can be expressed as:where,. (3): Ifandinare removed, then the third kind of NNA (i.e., NNA-III) can be expressed as:where,. (4): Ifandinare removed, then fourth kind of NNA (i.e., NNA-IV) can be expressed as:where,. It should be noted that, elements of the above NNAs are sorted in ascending order of their respective absolute values. Meanwhile, different index sets and corresponding to the above four kinds of NNAs are defined as follows.
Definition 5. (
Index Sets).
For NNA-I and NNA-II, the index sets are collectively defined as:whereFor NNA-III and NNA-IV, the index sets are collectively defined as:where Apparently, combining NNAs with their respective index sets, according to the construction principle in Definition 1, we can construct the specific time average vectors so as to obtain the equivalent received array
, and then the satisfying SDCA with maximum continuous DOFs can be obtained. Nevertheless, although NNAs and index sets are already given in Definitions 4 and 5, the relationship among
,
, and total number of sensors of NNAs is still indistinct. Hence, before using NNAs for DOA estimation, we need to address this problem first. Note that, it is apparent from Definition 4 that the sensor number of NNAs is
. From Equation (17), we know that the number of continuous DOFs of SDCA is
, where
and
. Hence, the optimization problem can be constructed as follows:
Combining C3 in Property 1 with Equation (26), it is easy to get the relationship among
,
,
, as well as
, which is as shown in
Table 1.
Then, according to
Table 1 and Definition 4, physical apertures of the proposed four kinds of NNAs can be summarized as follows.
Property 2. For NNA-I and NNA-II withsensors, their physical apertures are identical and can be expressed as: While for NNA-III and NNA-IV withsensors, their physical apertures are also identical and can be expressed as: It is obvious from Property 2 that NNA-III and NNA-IV possess larger physical aperture than NNA-I and NNA-II for the same sensor number, which means that the former can realize better DOA estimation performance than the latter.
Next, to illustrate the exploitation of the proposed NNAs and index sets more clearly, two examples of NNA-I and NNA-IV are provided as shown in
Figure 2. Let
be 10, then the optimal
and
are 9 and 8, respectively. According to Definition 4, the sensor position sets of NNA-I and NNA-IV can be given as
and
, respectively. It is evident that physical apertures of NNA-I and NNA-IV are respectively equal to 84 and 159. Shown in
Figure 2a is the physical structure of NNA-I, while
Figure 2b shows the physical structure of NNA-IV. Besides, from Definition 5, we can determine their respective index sets as
,
,
, and
. As shown in
Figure 2c, we then obtain the equivalent received arrays of NNA-I and NNA-IV, which are identical and can be expressed as
. Finally, according to Equation (12), we can easily obtain this fully continuous SDCA. It is obvious from
Figure 2d that the number of continuous DOFs of SDCA is equal to 337. However, in both of these examples, physical aperture of NNA-IV is larger than that of NNA-I. Thus, we can infer that NNA-IV has better DOA estimation performance than NNA-I.