Channel Estimation for PLNC Under Frequency Selective Fading Scenario

Feifei Gao; Chengwen Xing; Gongpu Wang

doi:10.1007/978-3-319-11668-6_4

Channel Estimation for Physical Layer Network Coding Systems pp 35-57 | Cite as

Channel Estimation for PLNC Under Frequency Selective Fading Scenario

Authors
Authors and affiliations

Feifei Gao
Chengwen Xing
Gongpu Wang

Chapter

First Online: 16 September 2014

428 Downloads

Part of the SpringerBriefs in Computer Science book series (BRIEFSCOMPUTER)

Abstract

In this chapter, we consider the channel estimation for PLNC under the more general frequency selective scenario, where orthogonal-frequency-division multiplexing (OFDM) is adopted for data transmission. We propose a two-phase training protocol, which is compatible with the two-phase data transmission, and thus the training block can be embedded into the data frame. Specifically, we design two different types of training methods: (i) block based training, for which we first estimate the cascaded source-relay-source channels, and then recover the individual channels between sources and relay; (ii) pilot-tone (PT) based training, for which we directly estimate the individual channels between sources and relay. Importantly, the identifiability of the channel estimation in both types of the training schemes are fully addressed. Finally, various numerical examples are presented to corroborate our analytical results.

This is a preview of subscription content, to check access.

Appendix 1: Block Training Based Estimation

Proof of Theorem 4.1

From matrix inversion lemma, we know

$$\displaystyle\begin{array}{rcl} & & tr\left ((\mathbf{D}^{H}\mathbf{D})^{-1}\right ) = tr((\mathbf{D}_{ 1}^{H}\mathbf{D}_{ 1} -\mathbf{D}_{1}^{H}\mathbf{D}_{ 2}(\mathbf{D}_{2}^{H}\mathbf{D}_{ 2})^{-1}\mathbf{D}_{ 2}^{H}\mathbf{D}_{ 1})^{-1}) \\ & & \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad + tr((\mathbf{D}_{2}^{H}\mathbf{D}_{ 2} -\mathbf{D}_{2}^{H}\mathbf{D}_{ 1}(\mathbf{D}_{1}^{H}\mathbf{D}_{ 1})^{-1}\mathbf{D}_{ 1}^{H}\mathbf{D}_{ 2})^{-1}) \\ & & = tr((\mathbf{D}_{1}^{H}P_{\mathbf{ D}_{2}}^{\perp }\mathbf{D}_{ 1})^{-1}) + tr((\mathbf{D}_{ 2}^{H}P_{\mathbf{ D}_{1}}^{\perp }\mathbf{D}_{ 2})^{-1}) \geq tr((\mathbf{D}_{ 1}^{H}\mathbf{D}_{ 1})^{-1}) + tr((\mathbf{D}_{ 2}^{H}\mathbf{D}_{ 2})^{-1}),{}\end{array}$$

(4.32)

and the equality holds if and only if $\mathbf{D}_{1}^{H}\mathbf{D}_{2} = {\boldsymbol 0}$. However, we cannot directly claim that the optimal $\mathbf{D}_{1}$ and $\mathbf{D}_{2}$ are orthogonal since $\mathbf{D}_{1}$ and $\mathbf{D}_{2}$ have to follow the circulant structure.

Nonetheless, we first take look at the following optimization:

$$\displaystyle\begin{array}{rcl} \min _{\mathbf{D}_{j}}& & \ tr((\mathbf{D}_{j}^{H}\mathbf{D}_{ j})^{-1}) \\ \mathrm{s.t.}& & \ \|\mathbf{d}_{j}\|^{2} \leq NP_{ j},{}\end{array}$$

(4.33)

for j = 1, 2. Note that (4.33) is the standard optimal channel estimation in single-input single-output (SISO) OFDM [9], whose solutions are

$$\displaystyle{ \vert \tilde{d}_{j,i}\vert ^{2} = P_{ j},\ \ \forall j = 1,2,\ i = 0,\ldots,N - 1. }$$

(4.34)

Meanwhile, the minimum value of $tr((\mathbf{D}_{1}^{H}\mathbf{D}_{1})^{-1})$ is $\frac{2L_{1}-1} {NP_{1}}$ and the minimum value of $tr((\mathbf{D}_{2}^{H}\mathbf{D}_{2})^{-1})$ is $\frac{L_{1}+L_{2}-1} {NP_{2}}$.

Lastly, if we can find a pair of $\mathbf{D}_{1}$ and $\mathbf{D}_{2}$ from (4.34) that satisfies $\mathbf{D}_{1}^{H}\mathbf{D}_{2} = {\boldsymbol 0}$, then these $\mathbf{D}_{1}$ and $\mathbf{D}_{2}$ make the equality in (4.32) hold, and must be the optimal solution to (4.15).

Therefore, the training satisfying both (4.34) and the following equation

$$\displaystyle\begin{array}{rcl} \sum _{i=0}^{N-1}\tilde{d}_{ 1,i}^{{\ast}}\tilde{d}_{ 2,i}e^{-\text{j}2\pi ki/N} = 0,\quad \forall k \in \{-(L_{ 1} + L_{2} - 2),-(L_{1}+L_{2} - 3),\ldots,(2L_{1} - 2)\}& &{}\end{array}$$

(4.35)

is the optimal solution to (4.15).

Proof of Theorem 4.2

The N-point DFT of $\mathbf{b}$ can be expressed as $\tilde{\mathbf{b}} = [\tilde{b}_{0},\tilde{b}_{1},\ldots,\tilde{b}_{N-1}]^{T}$, where $\tilde{b}_{i} = \tilde{h}_{i}^{2}$. Then, only $\tilde{b}_{i}^{1/2} = \tilde{\gamma }_{i}\tilde{h}_{i}$ can be obtained by the rooting operation, where $\tilde{\gamma }_{i} = \pm 1$ contains the channel uncertainty in each carrier. Define $\tilde{\boldsymbol{\gamma }} = [\tilde{\gamma }_{0},\tilde{\gamma }_{1},\ldots,\tilde{\gamma }_{N-1}]^{T}$ and $\tilde{\mathbf{t}} = [\tilde{b}_{0}^{1/2},\tilde{b}_{1}^{1/2},\ldots,\tilde{b}_{N-1}^{1/2}]^{T} = \tilde{\mathbf{h}} \odot \tilde{\boldsymbol{\gamma }}$. Construct an auxiliary vector $\tilde{\boldsymbol{\kappa }} = [\tilde{\kappa }_{0},\tilde{\kappa }_{1},\ldots,\tilde{\kappa }_{N-1}]^{T}$, where $\tilde{\kappa }_{i} \in \{ +1,-1\}$, and define $\tilde{\boldsymbol{\zeta }} = \tilde{\boldsymbol{\gamma }} \odot \tilde{\boldsymbol{\kappa }} = [\tilde{\zeta }_{0},\tilde{\zeta }_{1},\ldots,\tilde{\zeta }_{N-1}]^{T}$, where $\tilde{\zeta }_{i} = \tilde{\gamma }_{i}\tilde{\kappa }_{i}$ belongs to $\{+1,-1\}$. The IDFT of $\tilde{\mathbf{t}} \odot \tilde{\boldsymbol{\kappa }}$ can then be expressed as $\mathbf{t} \otimes \boldsymbol{\kappa } = \mathbf{h} \otimes \boldsymbol{\zeta }$ where $\boldsymbol{\kappa }$ and $\boldsymbol{\zeta }$ are the IDFTs of $\tilde{\boldsymbol{\kappa }}$ and $\tilde{\boldsymbol{\zeta }}$, respectively. Suppose $\boldsymbol{\zeta }$ has length L _ζ. Then, the length of $\mathbf{h}\otimes \boldsymbol{\zeta }$ is $\min \{L_{1} + L_{\zeta } - 1,N\}$.⁶ If L ₁ = N, then the length of $\mathbf{t}$ is N regardless of $\boldsymbol{\zeta }$. However, if L ₁ < N, then the length of $\mathbf{h}\otimes \boldsymbol{\zeta }$ is L ₁ only when L _ζ = 1. In this case, $\boldsymbol{\zeta }$ has only one non-zero value at its first element and is, actually, a scale of the discrete delta function. Therefore, all $\tilde{\zeta }_{i}$’s are equal. Considering the range of $\tilde{\zeta }_{i}$, $\tilde{\boldsymbol{\zeta }}$ can only be $\pm {\boldsymbol 1}$, which corresponds to $\tilde{\boldsymbol{\kappa }} = \pm \tilde{\boldsymbol{\gamma }}$. The so derived $\mathbf{t} \otimes \boldsymbol{\kappa }$ is then $\pm [\mathbf{h},{\boldsymbol 0}_{1\times (N-L_{1})}]^{T}$ whose first L ₁ entries give $\pm \mathbf{h}$.

Proof of Theorem 4.3

After dividing the i th element from $\tilde{\mathbf{c}}$ by $\tilde{b}_{i}^{1/2}$, we obtain $\tilde{\mathbf{z}} = \tilde{\mathbf{c}} \oslash \tilde{\mathbf{t}} = \tilde{\mathbf{g}} \odot \tilde{\boldsymbol{\gamma }}$, where ⊘ denotes the element-wise division. If L ₂ is known and L ₂ < N, from a similar process in Appendix “Proof of Theorem 4.2”, we can obtain $\tilde{\mathbf{g}}$ up to a sign ambiguity by restricting the length of $\mathbf{z}\otimes \boldsymbol{\kappa }$ as L ₂, where $\mathbf{z}$ is the IDFT of $\tilde{\mathbf{z}}$. The effect of $\tilde{\boldsymbol{\kappa }} = \pm \tilde{\boldsymbol{\gamma }}$ can then be removed from $\tilde{\mathbf{z}}$, which results in $\pm \mathbf{g}$.

Proof of Theorem 4.4

From Appendix “Proof of Theorem 4.2”, we know that non-trivial ambiguity happens if there exists a $\boldsymbol{\zeta }$ with length $L_{\zeta } \leq \hat{L}_{1} - L_{1} + 1$ such that the length of $\mathbf{t}\otimes \boldsymbol{\kappa } = \mathbf{h}\otimes \boldsymbol{\zeta }$ is still no greater than $\hat{L}_{1}$.

Define $\bar{\mathbf{F}}_{x}$ as the matrix that contains the last x columns of $\mathbf{F}$. We only need to find the maximum $\hat{L}_{1}$, such that $\bar{\mathbf{F}}_{N-(\hat{L}_{1}-L_{1}+1)}\tilde{\boldsymbol{\zeta }}\neq {\boldsymbol 0}$ for all $\tilde{\boldsymbol{\zeta }}$’s except for $\tilde{\boldsymbol{\zeta }} = \pm {\boldsymbol 1}$. We first take a look into the following lemma.

Lemma 4.1.

For even N, $\bar{\mathbf{F}}_{N/2}^{H}\tilde{\boldsymbol{\zeta }}\neq {\boldsymbol 0}$ for any $\tilde{\boldsymbol{\zeta }}\neq \pm {\boldsymbol 1}$ . For odd N, $\bar{\mathbf{F}}_{(N-1)/2}^{H}\tilde{\boldsymbol{\zeta }}\neq {\boldsymbol 0}$ for any $\tilde{\boldsymbol{\zeta }}\neq \pm {\boldsymbol 1}$ .

Proof.

Bearing in mind that $\tilde{\boldsymbol{\zeta }}$ is a real vector, we know that for even N, if $\bar{\mathbf{F}}_{N/2}^{H}\tilde{\boldsymbol{\zeta }} = {\boldsymbol 0}$, then there is $(\bar{\mathbf{F}}_{N/2}^{{\ast}})^{H}\tilde{\boldsymbol{\zeta }} = {\boldsymbol 0}$. From the structure of $\mathbf{F}$, we know that the i th column of $\bar{\mathbf{F}}_{N/2}^{{\ast}}$ is the $(N/2 - i + 2)$ th column of $\mathbf{F}$. Combining both $\bar{\mathbf{F}}_{N/2}^{H}\tilde{\boldsymbol{\zeta }} = {\boldsymbol 0}$ and $(\bar{\mathbf{F}}_{N/2}^{{\ast}})^{H}\tilde{\boldsymbol{\zeta }} = {\boldsymbol 0}$, we know $\bar{\mathbf{F}}_{N-1}^{H}\boldsymbol{\zeta } = {\boldsymbol 0}$. Therefore, $\boldsymbol{\zeta }$ must lie in the space spanned by the first column of $\mathbf{F}$. From the range of $\tilde{\zeta }_{i}$, we know $\tilde{\boldsymbol{\zeta }}$ can only be $\pm {\boldsymbol 1}$.

For odd N, the i th column of $\bar{\mathbf{F}}_{(N-1)/2}^{{\ast}}$ is the $((N - 1)/2 - i + 2)$ th column of $\mathbf{F}$, and the rest of the proof is the same as above. □

From Lemma 4.1, we know the nontrivial ambiguity does not exist for $\hat{L}_{1} \leq L_{1} + \frac{N} {2} - 1$. In fact, $\hat{L}_{1} = L_{1} + \frac{N} {2} - 1$ is also the maximum allowed value since $\bar{\mathbf{F}}_{N/2-1}^{H}\tilde{\boldsymbol{\zeta }} = {\boldsymbol 0}$ when $\tilde{\boldsymbol{\zeta }}_{i} = [+1,-1,+1,-1,\ldots,+1,-1]^{T}$.

Noting from Theorem 4.2 that $\hat{L}_{1}$ should be smaller than N, we complete the proof.

Proof of Theorem 4.5

For odd N, the maximum $\hat{L}_{1}$ depends on the factorization of N and we do not have a general conclusion, yet. However from Lemma 4.1, we know $\hat{L}_{1} \leq L_{1} + \frac{N-1} {2}$ can guarantee only the sign ambiguity, although it may not be the maximum value.

Algorithm to Extract $\mathbf{h}$ and $\mathbf{g}$

Based on the discussions from Appendix “Proof of Theorem 4.2” to Appendix “Proof of Theorem 4.5”, $\tilde{\boldsymbol{\kappa }} = \pm \tilde{\boldsymbol{\gamma }}$ should be found from

$$\displaystyle\begin{array}{rcl} \|\bar{\mathbf{F}}_{N-\hat{L}_{1}}^{H}(\tilde{\mathbf{t}} \odot \tilde{\boldsymbol{\kappa }})\|^{2} +\| \bar{\mathbf{F}}_{ N-\hat{L}_{2}}^{H}(\tilde{\mathbf{z}} \odot \tilde{\boldsymbol{\kappa }})\|^{2} =\| \bar{\mathbf{F}}_{ N-\hat{L}_{1}}^{H}\tilde{\mathbf{T}}\tilde{\boldsymbol{\kappa }})\|^{2} +\| \bar{\mathbf{F}}_{ N-\hat{L}_{2}}^{H}\tilde{\mathbf{Z}}\tilde{\boldsymbol{\kappa }})\|^{2} = 0,& &{}\end{array}$$

(4.36)

where $\tilde{\mathbf{T}} \triangleq \text{diag}\{\tilde{\mathbf{t}}\}$ and $\tilde{\mathbf{Z}} \triangleq \text{diag}\{\tilde{\mathbf{z}}\}$ are two diagonal matrices.

Due to the noise in practical transmission, the equality in (4.36) cannot hold. Then, $\tilde{\boldsymbol{\kappa }}$ should be found from

$$\displaystyle{ \tilde{\boldsymbol{\kappa }} =\arg \min _{x_{i}\in \{+1,-1\}}\|\boldsymbol{\varXi }\mathbf{x}\|^{2}, }$$

(4.37)

where $\mathbf{x} = [x_{0},x_{1},\ldots,x_{N-1}]^{T}$ is the trial variable and $\boldsymbol{\varXi } \triangleq \left [(\bar{\mathbf{F}}_{N-\hat{L}_{1}}^{H}\tilde{\mathbf{T}})^{T}\right.,\left.(\bar{\mathbf{F}}_{N-\hat{L}_{2}}^{H}\tilde{\mathbf{Z}})^{T}\right ]^{T}$ is a $(2N -\hat{L}_{1} -\hat{L}_{2}) \times N$ matrix. Since the sign ambiguity always exists, we can fix x ₀ = 1. One way is to apply the exhaustive search over 2^N−1 possible candidates. Considering the range of x _i, we can also apply the efficient sphere decoding (SD) algorithm [12, 13] to reduce the complexity.

Appendix 2: PT Based Estimation

Proof of Theorem 4.6

Define $\tilde{\mathbf{h}}^{{\prime}} = [\tilde{h}_{i_{0}},\tilde{h}_{i_{1}},\ldots,\tilde{h}_{i_{(K_{ 1}-1)}}]^{T} = \mathbf{F}_{ K_{1}L_{1}}\mathbf{h}$, where $\mathbf{F}_{K_{1}L_{1}}$ is the matrix that contains the first L ₁ columns and all the (i _k + 1)th rows from $\mathbf{F}$. Without loss of generality, we can assume $i_{k} = kQ + k_{0}$, where $Q = \frac{N} {K_{1}}$ is an integer and $0 \leq k_{0} \leq Q - 1$ is a constant. In this case, $\tilde{\mathbf{h}}^{{\prime}}$ can be equivalently considered as the K ₁-point DFT of $\sqrt{ \frac{N} {K_{1}}} \boldsymbol{\varLambda }\mathbf{h}$, where $\boldsymbol{\varLambda } = \sqrt{\frac{K_{1 } } {N}} \text{diag}\{1,e^{-\text{j}2\pi k_{0}/N},\ldots,e^{-\text{j}2\pi k_{0}(L_{1}-1)/N}\}$.

Once $\tilde{\mathbf{b}}^{{\prime}}\triangleq [\tilde{h}_{i_{0}}^{2},\tilde{h}_{i_{1}}^{2},\ldots,\tilde{h}_{i_{(K_{ 1}-1)}}^{2}]^{T}$ is estimated, from Theorem 4.2, we can obtain $\sqrt{ \frac{ N} {K_{1}}} \boldsymbol{\varLambda }\mathbf{h}$ with sign ambiguity, from which $\mathbf{h}$ can also be determined with sign ambiguity.

Illustration of Conjecture 4.1

For an arbitrary set $\mathcal{L}_{1}$, matrix $\mathbf{F}_{K_{1}L_{1}}$ has full rank because any L ₁ rows from itself form a Vandermonde matrix. If $\tilde{\mathbf{h}}^{{\prime}}$ can be obtained without ambiguity, the channel vector $\mathbf{h}$ can be recovered as $\hat{\mathbf{h}} = \mathbf{F}_{K_{1}L_{1}}^{\dag }\tilde{\mathbf{h}}^{{\prime}}$. Unfortunately, we can only obtain $\tilde{\mathbf{t}}^{{\prime}}\triangleq [\tilde{\gamma }_{i_{0}}\tilde{h}_{i_{0}},\tilde{\gamma }_{i_{1}}\tilde{h}_{i_{1}},\ldots,\tilde{\gamma }_{i_{K_{ 1}-1}}\tilde{h}_{i_{(K_{ 1}-1)}}]^{T}$, where $\tilde{\gamma }_{i_{k}} = +1$ or $\tilde{\gamma }_{i_{k}} = -1$ contains the uncertainty. Define $\tilde{\boldsymbol{\gamma }}^{{\prime}} = [\tilde{\gamma }_{i_{0}},\tilde{\gamma }_{i_{1}},\ldots,\tilde{\gamma }_{i_{(K_{ 1}-1)}}]^{T}$ and construct an auxiliary vector $\tilde{\boldsymbol{\kappa }}^{{\prime}} = [\tilde{\kappa }_{i_{0}},\tilde{\kappa }_{i_{1}},\ldots,\tilde{\kappa }_{i_{(K_{ 1}-1)}}]^{T}$, where $\tilde{\kappa }_{i_{k}} \in \{ +1,-1\}$. Take a look on the following possible estimator of $\mathbf{h}$:

$$\displaystyle{ \hat{\mathbf{h}} = \mathbf{F}_{K_{1}L_{1}}^{\dag }(\tilde{\mathbf{t}}^{{\prime}}\odot \tilde{\boldsymbol{\kappa }}^{{\prime}}) = \mathbf{F}_{ K_{1}L_{1}}^{\dag }(\tilde{\mathbf{h}}^{{\prime}}\odot \tilde{\boldsymbol{\zeta }}^{{\prime}}), }$$

(4.38)

where $\tilde{\boldsymbol{\zeta }}^{{\prime}} = \tilde{\boldsymbol{\gamma }}^{{\prime}}\odot \tilde{\boldsymbol{\kappa }}^{{\prime}} = [\tilde{\zeta }_{i_{0}},\tilde{\zeta }_{i_{1}},\ldots,\tilde{\zeta }_{i_{(K_{ 1}-1)}}]^{T}$ and $\tilde{\zeta }_{i_{k}} = \tilde{\gamma }_{i_{k}}\tilde{\kappa }_{i_{k}} \in \{ +1,-1\}$. If $\tilde{\boldsymbol{\zeta }}^{{\prime}} = \pm {\boldsymbol 1}$, then $\hat{\mathbf{h}} = \pm \mathbf{h}$ and $\mathbf{F}_{K_{1}L_{1}}\hat{\mathbf{h}}$ is still equal to $(\tilde{\mathbf{t}}^{{\prime}}\odot \tilde{\boldsymbol{\kappa }}^{{\prime}})$:

$$\displaystyle{ \mathbf{F}_{K_{1}L_{1}}\mathbf{F}_{K_{1}L_{1}}^{\dag }(\tilde{\mathbf{t}}^{{\prime}}\odot \tilde{\boldsymbol{\kappa }}^{{\prime}}) = (\tilde{\mathbf{t}}^{{\prime}}\odot \tilde{\boldsymbol{\kappa }}^{{\prime}}). }$$

(4.39)

Note that $\mathbf{F}_{K_{1}L_{1}}\mathbf{F}_{K_{1}L_{1}}^{\dag }$ is the projection matrix on the space spanned by $\mathbf{F}_{K_{1}L_{1}}$. For (4.39) to hold, we need $(\tilde{\mathbf{t}}^{{\prime}}\odot \tilde{\boldsymbol{\kappa }}^{{\prime}})$ to stay in the span of $\mathbf{F}_{K_{1}L_{1}}$. Equation (4.39) obviously holds if $\tilde{\boldsymbol{\zeta }}^{{\prime}} = \pm {\boldsymbol 1}$.

Intuitively, since channels are continuous random variable, $\tilde{\mathbf{h}}^{{\prime}}$, who originally stays in the space spanned by $\mathbf{F}_{K_{1}L_{1}}$ in general, does not stay in the same space after taking inner product with $\tilde{\boldsymbol{\zeta }}^{{\prime}}\neq \pm {\boldsymbol 1}$. Moreover, we must require $K_{1}> L_{1}$ since (4.39) always holds for $K_{1} = L_{1}$, which gives $\mathbf{F}_{K_{1}L_{1}}\mathbf{F}_{K_{1}L_{1}}^{\dag } = \mathbf{I}_{K_{1}}$.

Proof of Theorem 4.7

Once $K_{1}> L_{1}$, we can find $\hat{\mathbf{h}} = \pm \mathbf{h}$. Then, $\tilde{h}_{q_{k}}$, $q_{k} \in \mathcal{L}_{2}$ can be determined within sign ambiguity from $\mathbf{F}_{K_{2}L_{1}}\hat{\mathbf{h}}$, where $\mathbf{F}_{K_{2}L_{1}}$ is the matrix that contains all $q_{k} \in \mathcal{L}_{2}$ rows and the first L ₁ columns of $\mathbf{F}$. Then, $\tilde{g}_{q_{k}}$ can be determined with SSA from $\tilde{\mathbf{c}}^{{\prime}}\oslash (\mathbf{F}_{K_{2}L_{1}}\hat{\mathbf{h}})$. Finally, $\hat{\mathbf{g}} = \mathbf{F}_{K_{2}L_{2}}^{\dag }(\tilde{\mathbf{c}}^{{\prime}}\oslash (\mathbf{F}_{K_{2}L_{1}}\hat{\mathbf{h}})) = \pm \mathbf{g}$.

Efficient Algorithm to Estimate $\mathbf{h}$ and $\mathbf{g}$

Based on the discussions in Appendix “Proof of Theorem 4.6” and Appendix “Illustration of Conjecture 4.1”, $\tilde{\boldsymbol{\kappa }}^{{\prime}}$ should be found from

$$\displaystyle\begin{array}{rcl} \|(\mathbf{I} -\mathbf{F}_{K_{1}L_{1}}\mathbf{F}_{K_{1}L_{1}}^{\dag })(\tilde{\mathbf{t}}^{{\prime}}\odot \tilde{\boldsymbol{\kappa }}^{{\prime}})\|^{2} =\| (\mathbf{I} -\mathbf{F}_{ K_{1}L_{1}}\mathbf{F}_{K_{1}L_{1}}^{\dag })\tilde{\mathbf{T}}^{{\prime}}\tilde{\boldsymbol{\kappa }}^{{\prime}}\|^{2} = 0& &{}\end{array}$$

(4.40)

where $\tilde{\mathbf{T}}^{{\prime}}$ is the diagonal matrix with the diagonal elements $\tilde{\mathbf{t}}^{{\prime}}$. When K ₁ PTs from $\mathcal{L}_{1}$ are equispaced, it can be verified that (4.40) is equivalent to checking whether the length of $\mathbf{F}_{K_{1}L_{1}}^{\dag }(\tilde{\mathbf{t}}^{{\prime}}\odot \tilde{\boldsymbol{\kappa }}^{{\prime}})$ is L ₁. Note that, $\tilde{\mathbf{c}}^{{\prime}}$ does not contribute to the determination of $\tilde{\boldsymbol{\kappa }}^{{\prime}}$ here.

In practical transmission, $\tilde{\boldsymbol{\kappa }}^{{\prime}}$ should be found from

$$\displaystyle{ \tilde{\boldsymbol{\kappa }}^{{\prime}} =\arg \min _{ x_{i}\in \{+1,-1\}}\|(\mathbf{I} -\mathbf{F}_{K_{1}L_{1}}\mathbf{F}_{K_{1}L_{1}}^{\dag })\tilde{\mathbf{T}}^{{\prime}}\mathbf{x}\|^{2}. }$$

(4.41)

Again, the SD algorithm [12, 13] can be carried out to reduce the computational complexity.

References

1.
T. Cui, F. Gao, T. Ho, and A. Nallanathan. Distributed space-time coding for two-way wireless relay networks. in Proc. of IEEE ICC, Beijing, China, May 2008, pp. 3888–3892.Google Scholar
2.
T. Cui, T. Ho, and J. Kliewer. Memoryless relay strategies for two-way relay channels: performance analysis and optimization. in Proc. of IEEE ICC, pp. 1139–1143, Beijing, China, May 2008.Google Scholar
3.
R. Zhang, Y.-C. Liang, C. C. Chai, and S. G. Cui. Optimal beamforming for two-way multi-antenna relay channel with analogue network coding. IEEE J. Select. Areas in Commun., 27(5),pp. 699–712, June 2009.Google Scholar
4.
S. Ohno and G. B. Giannakis. Optimal training and redundant precoding for block transmissions with application to wireless OFDM. IEEE Trans. Commun., 50(12), pp. 2113–2123, Dec. 2002.Google Scholar
5.
I. Barhumi, G. Leus, and M. Moonen. Optimal trianing design for MIMO OFDM systems in mobile wireless channels. IEEE Trans. Signal Processing, 51(5), pp. 1615–1624, June 2003.Google Scholar
6.
H. Minn and N. Al-Dhahir. Optimal training signals for MIMO OFDM channel estimation. IEEE Trans. Wireless Commun.,5(6), pp. 158–1168, May 2006.Google Scholar
7.
C. K. Ho, R. Zhang, and Y.-C. Liang. Two-way relaying over OFDM: optimized tone permutation and power allocation. in Proc. of IEEE ICC, pp. 3908–3912, Beijing China, May 2008.Google Scholar
8.
“Wireless LAN medium access control (MAC) and physical layer (PHY) specifications: high speed physical layer in the 5 GHZ band,” IEEE802.11a, 1999.Google Scholar
9.
J. H. Manton. Optimal trianing sequences and pilot tone for OFDM systems. IEEE Commun. Lett., 5(4), pp. 151–153, Apr. 2001.Google Scholar
10.
S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge University Press, 2006.Google Scholar
11.
F. Gao and A. Nallanathan. Blind channel estimation for MIMO OFDM systems via non-redundant linear precoding. IEEE Trans. Signal Processing, 55(2), pp. 784–789, Jan. 2007.Google Scholar
12.
B. Hassibi and H. Vikalo. On the sphere decoding algorithm I: expected complexity. IEEE Trans. Signal Processing, 53(8), pp. 2806–2818, Aug. 2005.Google Scholar
13.
T. Cui and C. Tellambura. An efficient generalized sphere decoder for rank-deficient MIMO systems. IEEE Commun. Lett., 9(5), pp. 423–425, May 2005.Google Scholar

Copyright information

Authors and Affiliations

Feifei Gao
- 1
Chengwen Xing
- 2
Gongpu Wang
- 3

1.School of Information and Science TechnologyTsinghua UniversityBeijingChina
2.School of Information and ElectronicsBeijing Institute of TechnologyBeijingChina
3.School of Computer and Information TechnologyBeijing Jiaotong UniversityBeijingChina

Channel Estimation for PLNC Under Frequency Selective Fading Scenario

Abstract

References

Copyright information

Authors and Affiliations

Personalised recommendations

Cite chapter