#### 3.1. GIBF Principle

In order to eliminate unnecessary phase components and reduce sidelobes, this paper proposes an improved IBF method, denoted as GIBF. Because sidelobes come from the term

${R}_{cross1}$, we improve the IBF from this one. In order to make the signal component only contain the phases

${\phi}_{1}$ and

${\phi}_{2}$ corresponding to the two targets azimuth, we let

i = 0. Thus, the cross term

${R}_{cross1}$ becomes:

It was found from the phase of Equation (15) that only phase ${\phi}_{1}$ remains at this time. In the same way, only phase ${\phi}_{2}$ is left in ${A}_{1}{A}_{2}{e}^{j({w}_{2}{t}_{2}-{w}_{1}{t}_{1}+i{\phi}_{2}-(i+n){\phi}_{1})}$. At this time, only $R(i,i+n)$ of i = 0 in $C(n)$ is retained. Further, since –N < i + n < N, the number of terms of $R(i,i+n)$ corresponding to each i value is 2N – 1 − i. Therefore, when i = 0, the obtained array aperture extensions are the most. Equivalent to the number of array element number changed from N to 2N − 1, so the azimuth resolution is also improved. We record the improved $C(n)$ is $G(n)$.

After that, the DOA estimation can be given by:

Also, we use the assumption in Equation (10), and two situations are discussed here: (1) signals of two targets which are coherent; and (2) signals of two targets which are incoherent. In the first situation, if we define

${w}_{1}{t}_{1}-{w}_{2}{t}_{2}={\phi}_{\mathrm{s}}$,

${\phi}_{\mathrm{s}}$ should be a constant. Then, Formulas (16) and (17) can be written as:

For simplification, consider

${A}_{1}{}^{2}+{A}_{1}{A}_{2}{e}^{j({w}_{2}{t}_{2}-{w}_{1}{t}_{1})}$ as one item with the amplitude of

${A}_{a}$ and the phase of

${\phi}_{\mathrm{a}}$,

${A}_{1}{A}_{2}{e}^{j({w}_{1}{t}_{1}-{w}_{2}{t}_{2})}+{A}_{2}{}^{2}$ as the item with the amplitude of

${A}_{b}$ and the phase of

${\phi}_{\mathrm{b}}$,

${A}_{1}{A}_{2}{e}^{j({w}_{1}{t}_{1}-{w}_{2}{t}_{2})}+{A}_{2}{}^{2}$ as the item with the amplitude of

${A}_{c}$ and the phase of

${\phi}_{\mathrm{c}}$ and

${A}_{1}{A}_{2}{e}^{j({w}_{2}{t}_{2}-{w}_{1}{t}_{1})}+{A}_{2}{}^{2}$ as the item with the amplitude of

${A}_{d}$ and the phase of

${\phi}_{\mathrm{d}}$. Formulas (19) and (20) can be rewritten as:

As can be seen from the Formulas (19) and (20), the output signals of the reconstructed N-1-element array when $n>0$ and the reconstructed N-element array when $n\le 0$ are both corresponding to the target azimuth. When the two sets of array data are put together for CBF processing, the components of each target signal in the two set of arrays cannot be superimposed on the same phase because ${\phi}_{\mathrm{a}}\ne {\phi}_{\mathrm{c}}$, ${\phi}_{\mathrm{b}}\ne {\phi}_{\mathrm{d}}$. Therefore, the method does not extend the number of elements from N to 2N − 1 but is equivalent to superimposing the beamforming outputs of two N − 1 elements arrays. The beam pattern may be smoother, and the main lobe will be narrower. Because the effective array length was not improved, the azimuth resolution was not improved.

When dealing with two incoherent sources, the cross term in Formulas (16) and (17) is expected to be zero after accumulation of time.

where,

T is the signal processing time. Therefore,

$G(n)$ can be simplified as:

Formula (24) tells us that the signals of all the virtual array elements can be in-phase superimposed. The number of effective array elements is expanded from N to 2N − 1 so that the azimuth resolution is improved.

#### 3.2. Extension of GIBF Method

The GIBF is equivalent to the expansion of the array elements under the condition of the incoherent signal sources. The reconstructed phase information of the signal is consistent with the phase information of the actual signal. Therefore, after the generation of the new array output by GIBF, a high-resolution algorithm, such as MVDR, can still be used for subsequent processing. We can even apply the GIBF algorithm to each output signal obtained by GIBF. The following is a theoretical analysis of the second order GIBF: Assuming

${\phi}_{a}=n{\phi}_{1}$,

${\phi}_{b}=n{\phi}_{2}$, Formula (24) can be rewritten as:

Regarding

$G(n)$ as the new array element output signal, the corresponding

$R(0,{n}_{2})$ and

$R(-{n}_{2},0)$ can be obtained by using the method above, where

${n}_{2}=-2N+1,-2N+2,\cdots 2N-1$. As shown in the derivation above, the output

${G}_{G}$ of the array after two GIBF processing is:

In this case ${\phi}_{a}$ and ${\phi}_{b}$ are basically independent of time. Unlike incoherent signal processing, time integration cannot remove the phase difference introduced by ${A}_{1}{A}_{2}{e}^{j({\phi}_{b}-{\phi}_{a})}$. Therefore, high-order GIBF can increases the array gain and narrow the main lobe width while it cannot improve the azimuth resolution again.