# Angle-of-Arrival Estimation Using Difference Beams in Localized Hybrid Arrays
## 3 Proposed AOA Estimation Approach
### Estimation of \(u\)
Performing \(K/2\)-point IDFT of the even and odd samples of \(s_{m}(t)e^{-jm\widehat{Nu}}\), respectively, and then calculating the cross-correlation between adjacent IDFToutputs, denoted by \(d^{v}_{k}(t)\) and \(d^{v}_{k}(t)\), \(k=0,...,K/2-2\), we have \(\hat{u}\)\(=\arg\Bigl{\{}d^{e}_{k}(t)e^{j\frac{2\pi t}{L}}+d^{o}_{k}(t)e^{j\left(\frac{2 \pi t}{L}+\frac{2\pi}{K}\right)}\Bigr{\}}\). The estimation of \(u\) is summarized in Algorithm 2, where \(\tilde{\mathbf{S}}_{q}(t)_{(k_{1}:k_{2})}\) denotes a vector consisting of the \(k_{1}\)th to \(k_{2}\)th elements of \(\tilde{\mathbf{S}}_{q}(t)\) and \((\cdot)^{T}\) stands for the transpose of \((\cdot)\). Note that in Step 12, the samples from the \((M-\lfloor M/K\rfloor K)\)th to the \((K-1)\)th are concatenated after the samples from the \((\lfloor M/K\rfloor K)\)th to the \((M-1)\)th to constructively estimate \(u\), exploiting the periodicity of the phase shifts designed in (4). As the proposed approach is based on cross-correlation and IDFT computation, its computational complexity is similar to that in [5] given by \(\mathcal{O}(N(3+\log_{2}M))\), which is much lower than the subspace-based methods, e.g., MUSIC or ESPRIT in [10; 11] given by \(\mathcal{O}(M^{3}N^{3})\). ```
1:\(\widehat{N}\hat{u}\), \(s_{m}(t)\), \(m=0:M-1\), \(t=0:T-1\);
2:for\(t=0:T-1\)do
3:\(\hat{s}_{m}(t)\gets s_{m}(t)e^{-jm\widehat{N}\hat{u}}\), \(m=0:M-1\);//Calibrate \(s_{m}(t)\)
4:for\(q=0:\lfloor M/K\rfloor-1\)do//\(\lfloor M/K\rfloor\) non-overlapping groups
5:if\(Q\) is even then
6:\(\tilde{\mathbf{s}}_{q}(t)\leftarrow\)\(\Bigl{\{}\tilde{s}_{qK}(t),\tilde{s}_{qK+1}(t),...,\tilde{s}_{(q+1)K-1}(t)\Bigr{\}}\), \(\tilde{\mathbf{S}}_{q}(t)\leftarrow\)IDFT\(\bigl{\{}\tilde{\mathbf{s}}_{q}(t)\bigr{\}}\);//\(K\)-point IDFT
7:else
8:\(\tilde{\mathbf{s}}^{e}_{q}(t)\leftarrow\)the even samples of \(\tilde{\mathbf{s}}_{q}(t)\), \(\tilde{\mathbf{S}}^{e}_{q}(t)\leftarrow\)IDFT\(\bigl{\{}\tilde{\mathbf{s}}^{e}_{q}(t)\bigr{\}}\);//\(K\)/2-point IDFT
9:endif
10:endfor
11:if\(Q\) is even then
12:\(\tilde{\mathbf{s}}_{\lfloor M/K\rfloor}(t)\leftarrow\)\(\Bigl{\{}\tilde{s}_{\lfloor M/K\rfloor K}(t),...,\tilde{s}_{M-1}(t), \tilde{s}_{M-\lfloor M/K\rfloor K}(t),...,\tilde{s}_{K-1}(t)\Bigr{\}}\);
13:\(\tilde{\mathbf{S}}_{\lfloor M/K\rfloor}(t)\leftarrow\)IDFT\(\bigl{\{}\tilde{\mathbf{s}}_{\lfloor M/K\rfloor}(t)\Bigr{\}}\);
14:else
15:\(\tilde{\mathbf{s}}^{e}_{\lfloor M/K\rfloor}(t)\leftarrow\)the even samples of \(\tilde{\mathbf{s}}_{\lfloor M/K\rfloor}(t)\), \(\tilde{\mathbf{S}}^{e}_{\lfloor M/K\rfloor}(t)\leftarrow\)IDFT\(\bigl{\{}\tilde{\mathbf{s}}^{e}_{\lfloor M/K\rfloor}(t)\Bigr{\}}\);
16:\(\tilde{\mathbf{s}}^{e}_{\lfloor M/K\rfloor}(t)\leftarrow\)the odd samples of \(\tilde{\mathbf{s}}_{\lfloor M/K\rfloor}(t)\), \(\tilde{\mathbf{S}}^{e}_{\lfloor M/K\rfloor}(t)\leftarrow\)IDFT\(\bigl{\{}\tilde{\mathbf{s}}^{e}_{\lfloor M/K\rfloor}(t)\Bigr{\}}\);
17:endif
18:endfor
19:if\(Q\) is even then
20:\(\hat{u}\leftarrow\arg\biggl{\{}\sum\limits_{t=0}^{T-1}e^{j\frac{2\pi t}{L}} \sum\limits_{q=0}^{\lfloor M/K\rfloor}\tilde{\mathbf{S}}^{*}_{q}(t)_{(1:K-1)} \tilde{\mathbf{S}}^{T}_{q}(t)_{(2:K)}\biggr{\}}\);//Coherent combination
21:else
22:endif
```  
**Algorithm 2** Estimation of \(u\)