When compared with EDE, TDAEDE achieves improvements in three aspects. First, it reduces computation efforts of the estimation to around 1/3 of that of EDE. Second, it improves the accuracy of I/Q imbalance estimation within the whole SNR range of interest. Third, it significantly improves the accuracy of DCO estimation with lowtomedium SNR.
4.1 Coarse estimation and compensation of DCO by time domain average
The first improvement of TDAEDE comes from compensation of DCO before estimation of CFO and I/Q imbalance. Inspired by [11], we first make a coarse estimation for DCO by time domain average as
\begin{array}{ll}\hfill \widehat{d}\left(m\right)& =\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\frac{1}{N}\sum _{n=0}^{N1}r\left(n,\phantom{\rule{2.77695pt}{0ex}}m\right)\phantom{\rule{2em}{0ex}}\\ =\frac{1}{N}{1}_{N}^{T}{\mathbf{r}}_{N}\left(m\right),\phantom{\rule{2em}{0ex}}\end{array}
(6)
Then we cancel it from the received signal according to
{\mathbf{y}}_{N}\left(m\right)\triangleq {\mathbf{r}}_{N}\left(m\right){1}_{N}\widehat{d}\left(m\right).
(7)
By substituting (6) into (7), we have
\begin{array}{ll}\hfill {\mathbf{y}}_{N}\left(m\right)& ={\mathbf{r}}_{N}\left(m\right)\frac{1}{N}{1}_{N}{1}_{N}^{T}{\mathbf{r}}_{N}\left(m\right)\phantom{\rule{2em}{0ex}}\\ ={\mathbf{E}}_{N}{\mathbf{r}}_{N}\left(m\right),\phantom{\rule{2em}{0ex}}\end{array}
(8)
where {\mathbf{E}}_{N}\triangleq \phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}{\mathbf{I}}_{N}\frac{1}{N}{1}_{N}{1}_{N}^{T} with I_{
N
} being the N×N identity matrix. Due to the fact that E_{
N
}l_{
N
} = 0, substituting (2) into (8) yields
{\mathbf{y}}_{N}\left(m\right)={\mathbf{E}}_{N}{\mathbf{P}}_{N}\left(\epsilon \right){\mathbf{U}}_{N}\mathbf{x}\left(m\right)\alpha +{\mathbf{E}}_{N}{\mathbf{P}}_{N}\left(\epsilon \right){U}_{N}^{*}{\mathbf{x}}^{*}\left(m\right){\beta}^{*}+{\mathbf{E}}_{N}{\mathbf{w}}_{N}\left(m\right).
(9)
4.2 Estimation of CFO by eigendecomposition
We can find in (9) that not only DCO is completely removed but also a key property of (2) that the desired signal component with CFO ε, i.e., the first term on the righthand side (RHS), and its mirror component with CFO ε introduced by I/Q imbalance, i.e., the second term on the RHS, are conjugates to each other up to a scaling factor is remained. As analyzed in [13], either of the two signal components can be removed by linear combination
{\mathbf{b}}_{N}\left(m\right)\triangleq {\mathbf{Y}}_{2}\left(m\right){\mathbf{g}}_{2},
(10)
where
{\mathbf{Y}}_{2}\left(m\right)\triangleq \left[{\mathbf{y}}_{N}\left(m\right),\phantom{\rule{2.77695pt}{0ex}}{\mathbf{y}}_{N}^{*}\left(m\right)\right].
(11)
The optimal weighting vector denoted by g_{2} ≜ [g(0), g(1)] ^{T} should satisfy either
g\left(0\right){\beta}^{*}+\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}g\left(1\right){\alpha}^{*}=0,
(12)
to mitigate the component with CFO ε, or
g\left(0\right)\alpha +g\left(1\right)\beta =0,
(13)
to suppress the component with CFO ε. In absence of I/Q imbalance, we can directly apply NBE to the DCO compensated signal y_{
N
} (m). The cost function should be constructed as [11]
\begin{array}{l}\hfill {f}_{NBE}\left(v\right)=\sum _{m}{\u2225{\stackrel{\u0303}{\mathbf{V}}}_{N}^{H}\left(v\right){\mathbf{y}}_{N}\left(m\right)\u2225}^{2}\\ \hfill =\sum {\u2225\underset{N}{\overset{H}{\stackrel{\u0303}{\mathbf{V}}}}\left(v\right)\phantom{\rule{0.3em}{0ex}}{\mathbf{E}}_{N}{\mathbf{r}}_{N}\left(m\right)\u2225}^{2},\end{array}
(14)
where the columns of {\stackrel{\u0303}{\mathbf{V}}}_{N}\left(v\right) are eigenvectors corresponding to the zero eigenvalues of {\mathbf{Q}}_{N}\left(v\right)\triangleq {\mathbf{E}}_{N}{\mathbf{P}}_{N}\left(v\right){\mathbf{U}}_{N}{\mathbf{U}}_{N}^{H}{\mathbf{P}}_{N}\left(v\right){\mathbf{E}}_{N}. As E_{
N
} is of rank N  1, Q_{
N
} (ν) has at least one zero eigenvalue. In order to involve the compensation of I/Q imbalance, a new cost function is constructed for TDAEDE as
{{f}_{\mathsf{\text{TDA}}}}_{\mathsf{\text{EDE}}}\left(v,{\mathbf{a}}_{2}\right)=\sum _{m}{\u2225{\stackrel{\u0303}{\mathbf{V}}}_{N}^{H}\left(v\right){\mathbf{Y}}_{2}\left(m\right){\mathbf{a}}_{2}\u2225}^{2},
(15)
where a_{2} ≜ [a(0), a(1)] ^{T} . By substituting (8) into (11), we have
{\mathbf{Y}}_{2}\left(m\right)={\mathbf{E}}_{N}{\mathbf{r}}_{2}\left(m\right),
(16)
where {\mathbf{R}}_{2}\left(m\right)\triangleq \left[{\mathbf{r}}_{N}\left(m\right),\phantom{\rule{2.77695pt}{0ex}}{\mathbf{r}}_{N}^{*}\left(m\right)\right]. Consequently, (15) can be rewritten as
{{f}_{\mathsf{\text{TDA}}}}_{EDE}\left(v,{\mathbf{a}}_{2}\right)=\sum _{m}{\u2225{\stackrel{\u0303}{\mathbf{V}}}_{N}^{H}\left(v\right){\mathbf{E}}_{N}{\mathbf{r}}_{2}\left(m\right){\mathbf{a}}_{2}\u2225}^{2}.
(17)
Based on the constructed cost function, estimation of CFO and optimal weighting factor can be achieved by the following optimization approach as
\begin{array}{ll}\hfill \left(\widehat{\epsilon},{\mathbf{\u011d}}_{2}\right)& =\underset{\left(\nu ,{\mathbf{a}}_{2}\right)}{\text{arg}\text{min}}{{f}_{\mathsf{\text{TDA}}}}_{EDE}\left(\nu ,{\mathbf{a}}_{2}\right)\phantom{\rule{2em}{0ex}}\\ =\underset{\left(\nu ,{\mathbf{a}}_{2}\right)}{\text{arg}\text{min}}\sum _{m}{\u2225{\stackrel{\u0303}{\mathbf{V}}}_{N}^{H}\left(\nu \right){\mathbf{E}}_{N}{\mathbf{R}}_{2}\left(m\right){\mathbf{a}}_{2}\u2225}^{2}\phantom{\rule{2em}{0ex}}\\ =\underset{\left(\nu ,{\mathbf{a}}_{2}\right)}{\text{arg}\text{min}}\left\{{\mathbf{a}}_{2}^{H}{\mathbf{\Omega}}_{2}\left(\nu \right){\mathbf{a}}_{2}\right\},\phantom{\rule{2em}{0ex}}\end{array}
(18)
where {\mathbf{\Omega}}_{2}\left(\nu \right)\triangleq {\sum}_{m}{\mathbf{r}}_{2}^{H}\left(m\right){\mathbf{E}}_{N}{\stackrel{\u0303}{\mathbf{V}}}_{N}\left(v\right){\stackrel{\u0303}{\mathbf{V}}}_{N}^{H}\left(v\right){\mathbf{E}}_{N}{\mathbf{r}}_{2}\left(m\right). To avoid homogenous solution {\mathbf{\u011d}}_{\mathsf{\text{2}}}={0}_{2}, we impose a constraint ∥a_{2}∥^{2} = 1 to (17). The wellknown solution to this constrained minimization is [25]
{\widehat{\epsilon}}_{a,TDAEDE}=\underset{v}{\text{arg}\text{min}}{\lambda}_{min}\left\{{\mathbf{\Omega}}_{2}\left(v\right)\right\},
(19)
and {\widehat{\mathbf{g}}}_{2} equals to the eigenvector corresponding to the smallest eigenvalue of {\mathbf{\Omega}}_{2}\left({\widehat{\epsilon}}_{a,\mathsf{\text{TDAEDE}}}\right). In case that {\mathbf{\Omega}}_{2}\left({\widehat{\epsilon}}_{a,\mathsf{\text{TDAEDE}}}\right). has multiple minimum eigenvalues, {\widehat{\mathbf{g}}}_{2} can be randomly selected from the eigenvectors corresponding to the minimum eigenvalue. Actually, since {\mathbf{\Omega}}_{2}\left({\widehat{\epsilon}}_{a,\mathsf{\text{TDAEDE}}}\right). constitutes random data, it is of little possibility to have multiple minimum eigenvalues. Consequently, {\widehat{\mathbf{g}}}_{2} obtained under such case has little influence of average performance.
4.3 Sign ambiguity of CFO estimation and its solution
It should be noted that the CFO estimate {\widehat{\epsilon}}_{a,\mathsf{\text{TDA{EDE}}} obtained by (19) will involve sign ambiguity, i.e., {\widehat{\epsilon}}_{a,\mathsf{\text{TDAEDE}}} may approaches either ε or ε. As mentioned in Section 3, similar problem was also encountered by EDE [13]. From physical point of view, the ambiguity comes from the fact that the original received signal (2) contains component with CFO ε as well as component with CFO ε. From mathematical point of view, it can be deduced that {\widehat{\mathbf{V}}}_{N}^{H}\left(\epsilon \right){\mathbf{E}}_{N}{\mathbf{P}}_{N}\left(\epsilon \right){\mathbf{U}}_{N}={\widehat{\mathbf{V}}}_{N}^{H}\left(\epsilon \right){\mathbf{E}}_{N}{\mathbf{P}}_{N}\left(\epsilon \right){\mathbf{U}}_{N}^{*}={0}_{N}. Consequently, both Ω_{2}(ε) and Ω_{2}(ε) are rank deficient and therefore have eigenvalues of zero in absence of noise. To distinguish from sign ambiguity in CFO estimation, we take a similar approach as EDE with however a different metric {T}_{\mathsf{\text{CFO}},\mathsf{\text{TDAEDE}}}\triangleq {\sum}_{m}\left\{{\u2225{\stackrel{\u0303}{\mathbf{V}}}_{N}^{H}\left({\widehat{\epsilon}}_{a,\mathsf{\text{TDA}}\mathsf{\text{EDE}}}\right){\mathbf{y}}_{N}\left(m\right)\u2225}^{2}{\u2225{\stackrel{\u0303}{\mathbf{V}}}_{N}^{H}\left({\widehat{\epsilon}}_{a,\mathsf{\text{TDA}}\mathsf{\text{EDE}}}\right){\mathbf{y}}_{N}\left(m\right)\u2225}^{2}\right\}. Substituting (9) into T_{CFO,TDAEDE} yields that
\begin{array}{ll}\hfill {T}_{\mathsf{\text{CFO}},\mathsf{\text{TDAEDE}}}& ={\sum}_{m}\left\{\u2225{\stackrel{\u0303}{\mathbf{V}}}_{N}^{H}\left({\widehat{\epsilon}}_{a,\mathsf{\text{TDA}}\mathsf{\text{EDE}}}\right){\mathbf{E}}_{N}{\mathbf{P}}_{N}\left(\epsilon \right){U}_{N}\mathbf{x}\left(m\right)\alpha \right.\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{1em}{0ex}}+\phantom{\rule{0.3em}{0ex}}{\stackrel{\u0303}{\mathbf{V}}}_{N}^{H}\left({\widehat{\epsilon}}_{a,\mathsf{\text{TDA}}\mathsf{\text{EDE}}}\right){\mathbf{E}}_{N}{\mathbf{P}}_{N}\left(\epsilon \right){\mathbf{U}}_{N}^{*}{\mathbf{x}}^{*}\left(m\right){\beta}^{*}\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{1em}{0ex}}{\left(\right)close="\parallel ">+\phantom{\rule{0.3em}{0ex}}{\stackrel{\u0303}{\mathbf{V}}}_{N}^{H}\left({\widehat{\epsilon}}_{a,\mathsf{\text{TDA}}\mathsf{\text{EDE}}}\right){\mathbf{E}}_{N}{\mathbf{w}}_{N}\left(m\right)}^{}2& \phantom{\rule{2em}{0ex}}\end{array}\n \n \n \n \n \n \n \n \n \n \n V\n \n \u0303\n \n \n \n N\n \n \n H\n \n \n \n \n \n \n \n \n \n \epsilon \n \n ^\n \n \n \n a\n ,\n \n TDA\n \n \n \n EDE\n \n \n \n \n \n \n \n E\n \n \n N\n \n \n \n \n P\n \n \n N\n \n \n \n (\n \n \epsilon \n \n )\n \n \n \n U\n \n \n N\n \n \n x\n \n (\n \n m\n \n )\n \n \alpha \n \n \n \n \n \n \n \n \n +\n \n \n \n \n \n V\n \n \u0303\n \n \n \n N\n \n \n H\n \n \n \n \n \n \n \n \n \n \epsilon \n \n ^\n \n \n \n a\n ,\n \n TDA\n \n \n \n EDE\n \n \n \n \n \n \n \n E\n \n \n N\n \n \n \n \n P\n \n \n N\n \n \n \n (\n \n \n \epsilon \n \n )\n \n \n \n U\n \n \n N\n \n \n *\n \n \n \n \n x\n \n \n *\n \n \n \n (\n \n m\n \n )\n \n \n \n \beta \n \n \n *\n \n \n \n \n \n \n \n \n close="}">\n \n \n \n close="\u2225">\n \n +\n \n \n \n \n V\n \n \u0303\n \n \n \n N\n \n \n H\n \n \n \n \n \n \n \n \n \n \epsilon \n \n ^\n \n \n \n a\n ,\n \n TDA\n \n \n \n EDE\n \n \n \n \n \n \n \n E\n \n \n N\n \n \n \n \n w\n \n \n N\n \n \n \n (\n \n m\n \n )\n \n \n \n \n \n 2\n \n \n \n \n .\n \n
(20)
Recall that {\stackrel{\u0303}{\mathbf{V}}}_{N}\left(v\right) consists of eigenvectors corresponding to the zero eigenvalues of Q_{
N
} (ν), we have {\stackrel{\u0303}{\mathbf{V}}}_{N}^{H}\left(v\right){\mathbf{E}}_{N}{\mathbf{P}}_{N}\left(v\right){\mathbf{U}}_{N}={0}_{N}. In absence of noise, (20) can be simplified to
{T}_{\mathsf{\text{CFO}},\mathsf{\text{TDA}}\mathsf{\text{EDE}}}\left({\widehat{\epsilon}}_{a,\mathsf{\text{TDA}}\mathsf{\text{EDE}}}=\epsilon \right)=p\beta {}^{2}q\alpha {}^{2},
(21)
or
{T}_{\mathsf{\text{CFO}},\mathsf{\text{TDA}}\mathsf{\text{EDE}}}\left({\widehat{\epsilon}}_{a,\mathsf{\text{TDA}}\mathsf{\text{EDE}}}=\epsilon \right)=q\alpha {}^{2}p\beta {}^{2},
(22)
where p\triangleq {\sum}_{m}{\u2225{\stackrel{\u0303}{\mathbf{V}}}_{N}^{H}\left(\epsilon \right){\mathbf{E}}_{N}{\mathbf{P}}_{N}\left(\epsilon \right){\mathbf{U}}_{N}^{*}{\mathbf{x}}^{*}\left(m\right)\u2225}^{2}={\sum}_{m}{\u2225{\stackrel{\u0303}{\mathbf{V}}}_{N}^{T}\left(\epsilon \right){\mathbf{E}}_{N}{\mathbf{P}}_{N}\left(\epsilon \right){\mathbf{U}}_{N}\mathbf{x}\left(m\right)\u2225}^{2} and q\triangleq {\sum}_{m}{\u2225{\stackrel{\u0303}{\mathbf{V}}}_{N}^{H}\left(\epsilon \right){\mathbf{E}}_{N}{\mathbf{P}}_{N}\left(\epsilon \right){\mathbf{U}}_{N}\mathbf{x}\left(m\right)\u2225}^{2}. For most practical cases, p and q are comparable. If the real subcarriers are located symmetrically about dc, or mathematically if any column of U_{
N
} is also a column of {\mathbf{U}}_{N}^{*}, we have {\mathbf{\Omega}}_{2}^{*}\left(\epsilon \right)={\mathbf{E}}_{N}{\mathbf{P}}_{N}\left(\epsilon \right){\mathbf{U}}_{N}^{*}{\mathbf{U}}_{N}^{T}{\mathbf{P}}_{N}\left(\epsilon \right){\mathbf{E}}_{N}={\mathbf{E}}_{N}{\mathbf{P}}_{N}\left(\epsilon \right){\mathbf{U}}_{N}{\mathbf{U}}_{N}^{H}{\mathbf{P}}_{N}\left(\epsilon \right){\mathbf{E}}_{N}={\mathbf{\Omega}}_{2}\left(\epsilon \right) Recall that {\stackrel{\u0303}{\mathbf{V}}}_{N}^{T}\left(\epsilon \right) and {\stackrel{\u0303}{\mathbf{V}}}_{N}^{H}\left(\epsilon \right) consist of the eigenvectors of {\mathbf{\Omega}}_{2}^{*}\left(\epsilon \right) and Ω_{2}(ε), respectively, we have {\stackrel{\u0303}{\mathbf{V}}}_{N}^{*}\left(\epsilon \right){\stackrel{\u0303}{\mathbf{V}}}_{N}^{T}\left(\epsilon \right)={\stackrel{\u0303}{\mathbf{V}}}_{N}\left(\epsilon \right){\stackrel{\u0303}{\mathbf{V}}}_{N}^{H}\left(\epsilon \right) which leads to p = q. And because α^{2}≫β^{2}[14–16] in practice, the sign ambiguity can be distinguished by
{T}_{CFO,TDAEDE\phantom{\rule{0.3em}{0ex}}}\phantom{\rule{0.3em}{0ex}}\begin{array}{c}\hfill {\widehat{\epsilon}}_{a,TDAEDE}=\epsilon \hfill \\ \hfill \phantom{\rule{1em}{0ex}}\lessgtr \hfill \\ \hfill {\widehat{\epsilon}}_{a,TDAEDE}=\epsilon \hfill \end{array}0.
(23)
The final CFO estimate is obtained as
{\widehat{\epsilon}}_{\mathsf{\text{TDA}}\mathsf{\text{EDE}}}=sign\left({T}_{\mathsf{\text{CFO}},\mathsf{\text{TDA}}\mathsf{\text{EDE}}}\right){\widehat{\epsilon}}_{a,\mathsf{\text{TDA}}\mathsf{\text{EDE}}}.
(24)
4.4 Estimation and validation of I/Q imbalance
Integrated with the CFO estimation, the eigenvector corresponding to the smallest eigenvalue of {\mathbf{\Omega}}_{2}\left({\widehat{\epsilon}}_{a,\mathsf{\text{TDAEDE}}}\right). is taken as {\widehat{\mathbf{g}}}_{2}, which is the estimate of the optimal weighting vector for the linear combination. Consequently, I/Q imbalance can be derived either from (12) as
{\widehat{\gamma}}_{\mathsf{\text{TDA}}\mathsf{\text{EDE}}}=\frac{{\u011d}^{*}\left(1\right)}{{\u011d}^{*}\left(0\right)},
(25)
for the cases of T_{CFO,TDAEDE}< 0, or otherwise from (13) as
{\widehat{\gamma}}_{\mathsf{\text{TDA}}\mathsf{\text{EDE}}}=\frac{\u011d\left(0\right)}{\u011d\left(1\right)}.
(26)
The second and third improvements of TDAEDE are achieved mainly by validation of estimated I/Q imbalance according to its practical range. Recall the fact that α^{2}≫β^{2}[14–16] in practice, which equivalently gives the range that γ^{2} ≪ 1. A too large {\widehat{\gamma}}_{TDAEDE}usually indicates failure of I/Q imbalance estimation. In TDAEDE, we validate the estimated I/Q imbalance according to a controllable threshold T_{IQI,TDAEDE} ∈ (0, 1). If the estimated result exceeds this threshold, i.e., {\widehat{\gamma}}_{TDAEDE}>{T}_{\mathsf{\text{IQI,TDAEDE}}}, it will be reset to{\widehat{\gamma}}_{TDAEDE}=0, which means we would prefer to ignore I/Q imbalance for such cases rather than compensate it based on the unreasonable estimation results.
4.5 Compensation of CFO and I/Q imbalance
Given the estimates of CFO and I/Q imbalance, their impacts can be compensated in the original received signal by
{\mathbf{z}}_{N}\left(m\right)\triangleq {\mathbf{r}}_{N}\left(m\right){\mathbf{r}}_{N}^{*}\left(m\right){\widehat{\gamma}}_{\mathsf{\text{TDA}}\mathsf{\text{EDE}}}^{*}.
(27)
Suppose that {\widehat{\gamma}}_{\mathsf{\text{TDA}}\mathsf{\text{EDE}}}=\gamma and then substitute (2) into (27), we have
{\mathbf{z}}_{N}\left(m\right)={\mathbf{P}}_{N}\left(\epsilon \right){\mathbf{U}}_{N}{x}_{K}\left(m\right)\left(\alpha \beta {\widehat{\gamma}}_{\mathsf{\text{TDA}}\mathsf{\text{EDE}}}^{*}\right)+{\mathbf{1}}_{N}c+{\stackrel{\u0303}{\mathbf{w}}}_{N}\left(m\right),
(28)
where
c\triangleq d{d}^{*}{\widehat{\gamma}}_{\mathsf{\text{TDA}}EDE}^{*},
(29)
and {\stackrel{\u0303}{w}}_{N}\left(m\right)\triangleq {w}_{N}\left(m\right){w}_{N}^{*}\left(m\right){\widehat{\gamma}}_{\mathsf{\text{TDA}}\mathsf{\text{EDE}}}^{*}. It can be observed from the RHS of (28) that only the desired signal component with CFO ε is retained.
4.6 Fine estimation of DCO
Suppose that {\widehat{\epsilon}}_{TDAEDE}=\epsilon, it can be derived from (28) that
\begin{array}{c}{\mathbf{V}}_{N}^{H}{\mathbf{P}}_{N}\left({\widehat{\epsilon}}_{\mathsf{\text{TDA}}\mathsf{\text{EDE}}}\right){z}_{N}\left(m\right)={\mathbf{V}}_{N}^{H}{\mathbf{P}}_{N}\left({\widehat{\epsilon}}_{\mathsf{\text{TDA}}\mathsf{\text{EDE}}}\right){\mathbf{1}}_{N}c\\ +{\mathbf{V}}_{N}^{H}{\mathbf{P}}_{N}\left({\widehat{\epsilon}}_{\mathsf{\text{TDA}}\mathsf{\text{EDE}}}\right){\stackrel{\u0303}{\mathbf{w}}}_{N}\left(m\right).\end{array}
(30)
Through least square (LS) approach, estimate of c can be obtained by
\begin{array}{ll}\hfill \u0109& ={\left[{\mathbf{V}}_{N}^{H}{\mathbf{P}}_{N}\left({\widehat{\epsilon}}_{\mathsf{\text{TDA}}\mathsf{\text{EDE}}}\right){\mathbf{1}}_{N}\right]}^{+}{\mathbf{V}}_{N}^{H}{\mathbf{P}}_{N}\left({\widehat{\epsilon}}_{\mathsf{\text{TDA}}\mathsf{\text{EDE}}}\right){\mathbf{z}}_{N}\left(m\right)\phantom{\rule{2em}{0ex}}\\ =\frac{{\mathbf{1}}_{N}^{T}{\mathbf{G}}_{N}\left({\widehat{\epsilon}}_{\mathsf{\text{TDA}}\mathsf{\text{EDE}}}\right){\mathbf{z}}_{N}\left(m\right)}{{\mathbf{1}}_{N}^{T}{\mathbf{G}}_{N}\left({\widehat{\epsilon}}_{\mathsf{\text{TDA}}\mathsf{\text{EDE}}}\right){\mathbf{1}}_{N}},\phantom{\rule{2em}{0ex}}\end{array}
(31)
where [.]^{+} denotes pseudoinversion and
{\mathbf{G}}_{N}\left({\widehat{\epsilon}}_{\mathsf{\text{TDA}}\mathsf{\text{EDE}}}\right)\triangleq {\mathbf{P}}_{N}\left({\widehat{\epsilon}}_{\mathsf{\text{TDA}}\mathsf{\text{EDE}}}\right){\mathbf{V}}_{N}{\mathbf{V}}_{N}^{H}{\mathbf{P}}_{N}\left({\widehat{\epsilon}}_{\mathsf{\text{TDA}}\mathsf{\text{EDE}}}\right).
(32)
Finally, fine estimate of DCO can be derived from (29) as
\widehat{d}=\frac{\u0109+{\u0109}^{*}{\widehat{\gamma}}_{\mathsf{\text{TDA}}\mathsf{\text{EDE}}}^{*}}{1{\widehat{\gamma}}_{\mathsf{\text{TDA}}\mathsf{\text{EDE}}}{}^{2}}.
(33)
4.7 Summary and discussion
The steps for joint estimation of CFO, DCO, and I/Q imbalance by TDAEDE can be summarized as:

After CP removal, group received samples that belong to the same OFDM block to construct the vector r_{
N
}(m) ≜ [r(0, m),..., r(N  1, m)]^{T};

Cancel the DCO in r_{
N
}(m) by time domain average according to (7) to obtain y_{
N
}(m);

Search for the trail ν that minimizes the smallest eigenvalue of Ω_{2}(ν) according to (19);

Set {\widehat{\epsilon}}_{a,\mathsf{\text{TDAEDE}}} to the ν found during the search and record the eigenvector corresponding to the smallest eigenvalue of {\mathbf{\Omega}}_{2}\left({\widehat{\epsilon}}_{a,\mathsf{\text{TDAEDE}}}\right) in g_{2};

Calculate T_{CFO,TDAEDE} according to its definition, and detect sign ambiguity with (23) and correct it with (24);

Get estimate of I/Q imbalance according to (25) or (26) depending on the sign of T_{CFO,TDAEDE} and then reset it to 0 if it exceeds T_{IQI,TDAEDE} in the validation;

Compensate CFO and I/Q imbalance according to (27) to obtain z_{
N
}(m);

Get fine estimation of DCO according to (31) and (33).
Computation efforts of TDAEDE are mainly determined by the eigendecomposition of Ω_{2}(ν) in the 1D search operation for CFO estimation, as it will execute in each searching step while other operations execute only once. Although computation of {\u1e7c}_{N}^{H}\left(v\right) in TDAEDE is much more timeconsuming than that of {\mathbf{V}}_{N}^{H}{\mathbf{P}}_{N}\left(v\right) in EDE, both of them can be calculated and stored in advance [11] and is therefore not necessary to be computed on the run. What has to be done online for TDAEDE is the eigendecomposition of Ω_{2}(ν). If the wellknown Power Method [25] is employed, eigendecomposition of the 2 × 2 matrix Ω_{2}(ν) is of complexity O(2^{3}) [26, 27], while eigendecomposition of the 3 × 3 matrix Ω_{3}(ν) in EDE is of complexity O(3^{3}) [26, 27]. Therefore, TDAEDE reduces about 1/3 of the computation efforts compared with EDE.