, for i = Tx, S1, S2, and R, respectively.

Rx

I in which I is

(18)

127

s

HRx,S1VS1 sS1 þ

ϒ ¼

secondary network.

ffiffiffiffiffiffiffiffiffiffiffiffi PS1 d τRx,S1 Rx,S1

ffiffiffiffiffiffiffiffiffiffiffi PR d τRx,R Rx,R

UH Rx

UH Rx

Rx

powers at the terminals Tx, S1, S2, and R are denoted by Pi

s

8 >>>>><

>>>>>:

Rx in PN, is defined by UH

the unitary matrix, σ2

defined by si

s

Interference Alignment in Multi-Input Multi-Output Cognitive Radio-Based Network http://dx.doi.org/10.5772/intechopen.80073 127

Figure 7. System model for interference alignment-based cognitive two-way relay network with primary network and secondary network.

where ϒ is the interference term generated from SN to Rx defined as follows:

3. The effect of CSI quantization on interference alignment in CR

Figure 6. The effect of antenna diversity on the outage probability performance.

126 Cognitive Radio in 4G/5G Wireless Communication Systems

In this section, we investigate a cognitive two-way relaying network composed of a primary network (PN) with one pair of PU and a secondary network (SN) with two source terminals

We consider a MIMO interference network shown in Figure 7, where the transmitter, Tx, and receiver, Rx, are equipped with M1 and N1 antennas in PN, respectively. Each PN transmitter transmits to its corresponding receiver by interfering with the SN nodes, namely, two source terminals (S1 and S2) and a relay terminal. That means Tx transmitter sends messages to its intended receiver Rx, whereas it also causes interference to the unintended receivers in the SN. The SN consists of two source terminals and a relay terminal. We assume that all nodes in SN operate in an AF half-duplex mode with the help of information relaying from each source terminal to R in two phases. All nodes in SN are assumed to have MIMO antennas, and there is no direct transmission between S1 and S2 [34–36]. We consider a scenario where the source terminals and a relay terminal are equipped with NS1 , NS2 , and NR antennas, respectively. In the system model based on IA for cognitive two-way relay network, the received signal at Rx in

networks

and a relay terminal (R).

3.1. System model

PN can be written as

yRx ¼

ffiffiffiffiffiffiffiffiffiffiffiffi PTx d τRx,Tx Rx,Tx

UH Rx

HRx,TxVTx sTx þ ϒ þ n~Rx

, (17)

s

$$\mathcal{Y} = \begin{cases} \sqrt{\frac{P\_{\mathcal{S}\_1}}{d\_{R\_{\mathcal{S}},\mathcal{S}\_1}^{\text{tr}\_{\mathcal{S}}}}} \mathbf{U}\_{\mathcal{R}\_1}^H \mathbf{H}\_{\mathcal{R}\_2,\mathcal{S}\_1} \mathbf{V}\_{\mathcal{S}\_1} \mathbf{s}\_{\mathcal{S}\_1} + \sqrt{\frac{P\_{\mathcal{S}\_2}}{d\_{R\_{\mathcal{S}},\mathcal{S}\_2}^{\text{tr}\_{\mathcal{S}}}}} \mathbf{U}\_{\mathcal{R}\_1}^H \mathbf{H}\_{\mathcal{R}\_2,\mathcal{S}\_2} \mathbf{V}\_{\mathcal{S}\_2} \mathbf{s}\_{\mathcal{S}\_2} & \text{first phase} \\\sqrt{\frac{P\_{\mathcal{R}\_1}}{d\_{R\_{\mathcal{S}},\mathcal{R}}^{\text{tr}\_{\mathcal{S}}}}} \mathbf{U}\_{\mathcal{R}\_2}^H \mathbf{H}\_{\mathcal{R}\_2,\mathcal{R}} \mathbf{V}\_{\mathcal{R}} \mathbf{s}\_{\mathcal{S}\_2} & \text{second phase}. \end{cases} \tag{18}$$

The effective additive white Gaussian noise (AWGN) term with zero mean and unit variance, n~Rx at Rx in PN, is defined by UH Rx nRx, where nRx is the AWGN vector with E nRxnH Rx h i = σ2 Rx I in which I is the unitary matrix, σ2 Rx is the noise variance, and E½ �: is the expectation operator. The transmit powers at the terminals Tx, S1, S2, and R are denoted by Pi , for i = Tx, S1, S2, and R, respectively. Each receive node employs the interference-suppression matrix, Uj, (forj = Rx, R, S1, S2), while each transmit node employs a precoding matrix Vi [37]. The conjugate transpose of the matrix is associated with the Hermitian operator ð Þ: H [38]. The transmit signal vector for the ith user is defined by si . The channel between the ith transmitter and the jth receiver nodes is denoted by Hj,i for both PN and SN. The quantized CSI is passed to the transmitter by the corresponding receiver. Because of limited feedback, the transmitters have imperfect CSI causing certain performance loss. To clarify the effect of CSI quantization error on the performance of interference alignment in underlay cognitive two-way relay networks, we investigate the BER performance, instantaneous capacity, and average PEP of the considered system. Based upon the accuracy parameter, the relation between perfect CSI (rj,i ¼ 0) and imperfect CSI (0 < rj,i ≤ 1) can be given as

$$\mathbf{H}\_{\mathbf{j},i} = \sqrt{1 - \rho\_{\mathbf{j},i}} \hat{\mathbf{H}}\_{\mathbf{j},i} + \sqrt{\rho\_{\mathbf{j},i}} \mathbf{E}\_{\mathbf{j},i} \tag{19}$$

where Hj,i is the real channel matrix and H^ j,i is the estimated channel matrix. The quantization error, Ej,i1mm, can be expressed with the upper bound of 2�Bj,i=ð Þ M1N1�1 , where Bj,i is the CSI exchange amount and M1 and N1 are the numbers of transmit-and-receive antennas, successively [21, 39]. It is assumed that both H^ j,i and Ej,i are independent of Hj,i. Besides, each channel link is also modeled by two additional parameters: the distance between ith transmitter and the jth receiver nodes dj,i and the path loss exponent for the corresponding link, τj,i, regarding for different radio environments, respectively.

channel matrices are constant during the transmission, we ensure that there is no interference from the unintended transmitters and guarantee that received signal achieves f i degrees of freedom [39]. The corresponding signal-to-interference-plus-noise ratio (SINR) for the links

> ∥UH Rx

Ψ þ σn~Rx

∥UH

∥UH S2

∥UH

∥UH S2

PS2 rRx,S2 d τRx,S2 Rx,S2

where Ej,i is the quantization error and k k: is the Euclidean norm. In here, γS2!R and γR!S1 can be found by changing the subscript S1 with S2 of (27) and S2 with S1 of (28). Assuming the channels are reciprocal over SN direct links, thus the channel gains for S1 ! R and R ! S1 and

This section starts by the instantaneous capacity analysis of the proposed system with interference alignment in underlay cognitive two-way relay networks with CSI quantization. We then

The capacity is expressed as the expected value of the mutual information between the transmitting terminal and receiving one. In light of this fact, we consider the method developed in

CRx ¼ log 2 1 þ γTx!Rx

where γTx!Rx is the instantaneous SINR for the corresponding link of Tx ! Rx. On the other hand, end-to-end capacity for the SN, based on the least strong link over two-hop transmis-

H^ Rx,TxVTx∥2

Interference Alignment in Multi-Input Multi-Output Cognitive Radio-Based Network

R H^ R,S1VS1 sS1∥2

H^ S2,RVRsR∥2

ES2,TxVTx∥2 þ σ2

∥UH Rx n~R

n~S2

ERx,S2VS2∥2

, second phase

R ER,TxVTx∥2 þ σ2

2 , (26)

http://dx.doi.org/10.5772/intechopen.80073

, first phase

� �, (30)

, (27)

, (28)

(29)

129

PTx ð Þ 1�rRx,Tx d τRx,Tx Rx,Tx

PS1 1�rR,S1 ð Þ d τR,S1 R,S1

PR 1�rS2 ð Þ ,R d τS2,R S2,R

PTx rR,Tx d τR,Tx R,Tx

PTx rS2,Tx d τS2,Tx S2,Tx

Tx ! Rx, S1 ! R, and R ! S2 can be derived by

γTx!Rx ¼

γS1!R ¼

γR!S2 ¼

ERx,RVR∥2

ERx,S1VS1∥2 þ

Ψ ¼

8 >>>><

>>>>:

3.2. Performance analysis

sion, is denoted as follows:

PS1 rRx,S1 d τRx,S1 Rx,S1

PR rRx,R d τRx,R Rx,R

∥UH Rx

∥UH Rx

S2 ! R and R ! S2 links are identical, respectively.

study the BER and average PEP performance.

[29]; the instantaneous capacity in PN can be expressed as

In the first phase of the transmission (multiple-access phase) in SN, both S1 and S2 transmit their signals simultaneously to the relay terminal, R. Then the received signal at R can be written as

$$\mathbf{y}\_{R} = \sqrt{\frac{\mathbf{P}\_{S\_{1}}}{d\_{R,S\_{1}}^{\pi\_{R,S\_{1}}}}} \mathbf{U}\_{R}^{H} \mathbf{H}\_{R,S\_{1}} \mathbf{V}\_{S\_{1}} \mathbf{s}\_{S\_{1}} + \sqrt{\frac{\mathbf{P}\_{S\_{2}}}{d\_{R\_{x},S\_{2}}^{\pi\_{R\_{x}}}}} \mathbf{U}\_{R}^{H} \mathbf{H}\_{R,S\_{2}} \mathbf{V}\_{S\_{2}} \mathbf{s}\_{S\_{2}} + \sqrt{\frac{\mathbf{P}\_{T\_{x}}}{d\_{R,T\_{x}}^{\pi\_{R}}}} \mathbf{U}\_{R}^{H} \mathbf{H}\_{R,T\_{x}} \mathbf{V}\_{T\_{x}} \mathbf{s}\_{T\_{x}} + \tilde{\mathbf{n}}\_{R} \tag{20}$$

where n~R ¼ UH R nR at the relay terminal in SN is expressed as zero-mean AWGN vector with E nRnH R � � = σ2 RI in which the noise variance at the relay terminal is depicted with σ2 R. Besides, the received signal at S1 and S2 terminals in SN is defined, respectively, as

$$\mathbf{y}\_{S\_1} = \sqrt{\frac{P\_R}{d\_{S\_1,R}^{t\_{S\_1,R}}}} \mathbf{U}\_{S\_1}^H \mathbf{H}\_{S\_1,R} \mathbf{V}\_R \mathbf{s}\_R + \sqrt{\frac{P\_{T\_x}}{d\_{S\_1,T\_x}^{t\_{S\_1,T\_x}}}} \mathbf{U}\_{S\_1}^H \mathbf{H}\_{S\_1,T\_x} \mathbf{V}\_{T\_x} \mathbf{s}\_{T\_x} + \tilde{\mathbf{n}}\_{S\_1, \epsilon} \tag{21}$$

$$\mathbf{y}\_{\text{S}\_{2}} = \sqrt{\frac{P\_{\text{R}}}{d\_{\text{S}\_{2},\text{R}}^{\text{T}\text{s}\_{\text{S}\_{2}}\text{R}}} \mathbf{U}\_{\text{S}\_{2}}^{H} \mathbf{H}\_{\text{S}\_{2},\text{R}} \mathbf{V}\_{\text{R}} \mathbf{s}\_{\text{R}} + \sqrt{\frac{P\_{T\_{x}}}{d\_{\text{S}\_{2},T\_{x}}^{\text{T}\text{s}\_{\text{T}}}} \mathbf{U}\_{\text{S}\_{2}}^{H} \mathbf{H}\_{\text{S}\_{2},T\_{x}} \mathbf{V}\_{T\_{x}} \mathbf{s}\_{T\_{x}} + \tilde{\mathbf{n}}\_{\text{S}\_{2}}.\tag{22}$$

Here, n~S1 and n~S2 are the AWGN vector with E nSknH Sk h i = σ2 Sk I, for k ¼ 1, 2 and the noise variance of σ2 Sk . In addition to that, in the second phase of the signal transmission (broadcast phase), R broadcasts the combined signal yR after multiplying with an ideal amplifying gain, G, which is expressed as

$$\begin{split} & \frac{P\_{S\_1} \left(1 - \rho\_{R, S\_1} \right)}{d\_{R, S\_1}^{\text{tr}\_1, s\_1}} \| \mathbf{U}\_R^H \hat{\mathbf{H}}\_{R, S\_1} \mathbf{V}\_{S\_1} \|^2 + \frac{P\_{S\_2} \left(1 - \rho\_{R, S\_2} \right)}{d\_{R, S\_2}^{\text{tr}\_2, s\_2}} \| \mathbf{U}\_R^H \hat{\mathbf{H}}\_{R, S\_2} \mathbf{V}\_{S\_2} \|^2 \dots \\ & \dots + \frac{P\_{T\_x} \left(1 - \rho\_{R, T\_x} \right)}{d\_{R, T\_x}^{\text{tr}\_x, T\_x}} \| \mathbf{U}\_R^H \hat{\mathbf{H}}\_{R, T\_x} \mathbf{V}\_{T\_x} \|^2 \end{split} \tag{23}$$

where sR ¼ GyR. We assume that both S1 and S2 have knowledge about their own information and can remove back-propagating self-interference from the imposed signals. We also assume that all interference at the receive terminals are perfectly aligned and the following feasible conditions are satisfied for the receive nodes:

$$\mathbf{U}\_{\rangle}^{\text{H}}\mathbf{H}\_{\natural,i}\mathbf{V}\_{i}\mathbf{s}\_{i} = \mathbf{0},\tag{24}$$

$$\text{rank}\left(\mathbf{U}\_{j}^{\text{H}}\mathbf{H}\_{j,i}\mathbf{V}\_{i}\mathbf{s}\_{i}\right) = f\_{i'} \tag{25}$$

where f i is the degree of freedom and rank (.) denotes the rank operation of a matrix. By assuming that the interference is perfectly aligned by the proposed IA algorithm, and the channel matrices are constant during the transmission, we ensure that there is no interference from the unintended transmitters and guarantee that received signal achieves f i degrees of freedom [39]. The corresponding signal-to-interference-plus-noise ratio (SINR) for the links Tx ! Rx, S1 ! R, and R ! S2 can be derived by

$$\gamma\_{T\_x \to R\_x} = \frac{\frac{p\_{T\_x}(1 - \rho\_{R\_{\tilde{x}}, T\_x})}{d\_{R\_{\tilde{x}}, T\_x}} \parallel \mathbf{U}\_{R\_x}^H \hat{\mathbf{H}}\_{R\_x, T\_x} \mathbf{V}\_{T\_x} \parallel^2}{\Psi + \sigma\_{\tilde{n}\_{\tilde{R}\_x}}^{\prime 2\_{\tilde{x}}}} \tag{26}$$

$$\begin{array}{c} \frac{p\_{S\_1} \left(1 - p\_{R\_r S\_1} \right)}{d\_{R\_r S\_1}} \left\| \mathbf{U}\_R^H \hat{\mathbf{H}}\_{R, S\_1} \mathbf{V}\_{S\_1} \mathbf{s}\_{S\_1} \right\|^2 \\\hline \frac{p\_{T\_x} p\_{R\_r T\_x}}{d\_{R\_r T\_x}} \left\| \mathbf{U}\_R^H \mathbf{E}\_{R, T\_x} \mathbf{V}\_{T\_x} \right\|^2 + \sigma\_{\bar{\mathbf{u}}\_k}^2 \end{array} \tag{27}$$

$$\gamma\_{R \to S\_2} = \frac{\frac{p\_\mathcal{R} \left(1 - p\_{S\_2, R}\right)}{d\_{S\_2, R}^{\tau\_{S\_2, R}}} \left\| \mathbf{U}\_{S\_2}^H \hat{\mathbf{H}}\_{S\_2, R} \mathbf{V}\_R \mathbf{s}\_R \right\|^2}{\frac{p\_{T\_x} p\_{S\_2, T\_x}}{d\_{S\_2, T\_x}} \left\| \mathbf{U}\_{S\_2}^H \mathbf{E}\_{S\_2, T\_x} \mathbf{V}\_{T\_x} \right\|^2 + \sigma\_{\hat{u}\_{S\_2}}^2} \tag{28}$$

$$\Psi = \begin{cases} \frac{P\_{\mathcal{S}1} \rho\_{R\_{\mathcal{S}},\mathcal{S}\_{1}}}{d\_{R\_{\mathcal{S}},\mathcal{S}\_{1}}^{\sf R\_{\mathcal{S}},\mathcal{S}\_{1}}} \|\mathbf{U}\_{R\_{\mathcal{S}}}^{H}\mathbf{E}\_{\mathcal{R}\_{\mathcal{S}},\mathcal{S}\_{1}}\mathbf{V}\_{\mathcal{S}\_{1}}\|^{2} + \frac{P\_{\mathcal{S}2} \rho\_{R\_{\mathcal{S}},\mathcal{S}\_{2}}}{d\_{R\_{\mathcal{S}},\mathcal{S}\_{2}}^{\sf R\_{\mathcal{S}}}} \|\mathbf{U}\_{R\_{\mathcal{S}}}^{H}\mathbf{E}\_{\mathcal{R}\_{\mathcal{S}},\mathcal{S}\_{2}}\mathbf{V}\_{\mathcal{S}\_{2}}\|^{2}, & \text{first phase} \\\\ \frac{P\_{R} \rho\_{R\_{\mathcal{S}},R}}{d\_{R\_{\mathcal{S}},R}^{\sf R\_{\mathcal{S}}}} \|\mathbf{U}\_{R\_{\mathcal{S}}}^{H}\mathbf{E}\_{\mathcal{R}\_{\mathcal{S}},\mathcal{R}}\mathbf{V}\_{\mathcal{S}}\|^{2}, & \text{second phase} \end{cases} \tag{29}$$

where Ej,i is the quantization error and k k: is the Euclidean norm. In here, γS2!R and γR!S1 can be found by changing the subscript S1 with S2 of (27) and S2 with S1 of (28). Assuming the channels are reciprocal over SN direct links, thus the channel gains for S1 ! R and R ! S1 and S2 ! R and R ! S2 links are identical, respectively.

#### 3.2. Performance analysis

exchange amount and M1 and N1 are the numbers of transmit-and-receive antennas, successively [21, 39]. It is assumed that both H^ j,i and Ej,i are independent of Hj,i. Besides, each channel link is also modeled by two additional parameters: the distance between ith transmitter and the jth receiver nodes dj,i and the path loss exponent for the corresponding link, τj,i,

In the first phase of the transmission (multiple-access phase) in SN, both S1 and S2 transmit their signals simultaneously to the relay terminal, R. Then the received signal at R can be written as

R HR,S2VS2 sS2 þ

RI in which the noise variance at the relay terminal is depicted with σ2

s

s

phase), R broadcasts the combined signal yR after multiplying with an ideal amplifying gain,

G¼1= ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi PS1 1 � rR,S1

where sR ¼ GyR. We assume that both S1 and S2 have knowledge about their own information and can remove back-propagating self-interference from the imposed signals. We also assume that all interference at the receive terminals are perfectly aligned and the following feasible

> j Hj,iVisi � �

where f i is the degree of freedom and rank (.) denotes the rank operation of a matrix. By assuming that the interference is perfectly aligned by the proposed IA algorithm, and the

R H^ R,S1VS1∥2 þ

R H^ R,TxVTx∥2

R nR at the relay terminal in SN is expressed as zero-mean AWGN vector with

UH S1

UH S2

Sk h i

. In addition to that, in the second phase of the signal transmission (broadcast

PS2 1 � rR,S2 � �

¼ f i

d τR,S2 R,S2

,

= σ2 Sk

∥UH

j Hj,iVisi ¼ 0, (24)

R H^ R,S2VS2∥2

:

, (25)

:

ffiffiffiffiffiffiffiffiffiffiffiffi PTx d τS1,Tx S1,Tx

ffiffiffiffiffiffiffiffiffiffiffiffi PTx d τS2,Tx S2,Tx

ffiffiffiffiffiffiffiffiffiffiffi PTx d τR,Tx R,Tx

UH

HS1,TxVTx sTx þ n~S1

HS2,TxVTx sTx þ n~S2

R HR,TxVTx sTx þ n~R, (20)

R. Besides,

, (21)

: (22)

(23)

I, for k ¼ 1, 2 and the noise

s

ffiffiffiffiffiffiffiffiffiffiffiffi PS2 d τRx,S2 Rx,S2

the received signal at S1 and S2 terminals in SN is defined, respectively, as

HS1,RVRsR þ

HS2,RVRsR þ

UH

s

regarding for different radio environments, respectively.

R HR,S1VS1 sS1 þ

ffiffiffiffiffiffiffiffiffiffi PR d τS1,R S1,R

ffiffiffiffiffiffiffiffiffiffi PR d τS2,R S2,R

Here, n~S1 and n~S2 are the AWGN vector with E nSknH

� �

PTx 1 � rR,Tx � �

d τR,Tx R,Tx ∥UH

∥UH

UH

rank UH

d τR,S1 R,S1 UH S1

UH S2

s

s

yR ¼

E nRnH R � � = σ2

where n~R ¼ UH

variance of σ2

Sk

G, which is expressed as

ffiffiffiffiffiffiffiffiffiffi PS1 d τR,S1 R,S1

UH

128 Cognitive Radio in 4G/5G Wireless Communication Systems

yS1 ¼

yS2 ¼

s

:: þ

conditions are satisfied for the receive nodes:

s

This section starts by the instantaneous capacity analysis of the proposed system with interference alignment in underlay cognitive two-way relay networks with CSI quantization. We then study the BER and average PEP performance.

The capacity is expressed as the expected value of the mutual information between the transmitting terminal and receiving one. In light of this fact, we consider the method developed in [29]; the instantaneous capacity in PN can be expressed as

$$\mathbb{C}\_{\mathbb{R}\_x} = \log\_2 \left( 1 + \gamma\_{T\_x \to \mathbb{R}\_x} \right) \tag{30}$$

where γTx!Rx is the instantaneous SINR for the corresponding link of Tx ! Rx. On the other hand, end-to-end capacity for the SN, based on the least strong link over two-hop transmission, is denoted as follows:

$$\mathcal{C}\_{\mathbb{R}} = \frac{1}{2} \log\_2 \left( 1 + \min \left( \gamma\_{\mathbb{S}\_1 \to \mathbb{R}}, \gamma\_{\mathbb{R} \to \mathbb{S}\_2} \right) \right) + \frac{1}{2} \log\_2 \left( 1 + \min \left( \gamma\_{\mathbb{S}\_2 \to \mathbb{R}}, \gamma\_{\mathbb{R} \to \mathbb{S}\_1} \right) \right). \tag{31}$$

γS1!R and γR!S2 are the instantaneous SINR for the S1 ! R and R ! S2 links, respectively. Average BER for binary phase shift keying (BPSK) modulation can be expressed as

$$\text{BER}\_{\dot{\gamma}} = \mathbb{Q}\left(\sqrt{\mathcal{V}\_{\dot{\gamma}}}\right) \tag{32}$$

where Q xð Þ is the Gaussian Q-function and defined by Q xð Þ¼ 1= ffiffiffiffiffiffi 2π � � p Ð ∞ x e�t 2=2dt [37].

Average pairwise error probability (PEP) can be computed as averaging the Gaussian Qfunction over Rayleigh fading statistics [40], f γTx!Rx ð Þ¼ γ e �γ=γTx!Rx � �=γTx!Rx 1mm, where γTx!Rx ¼ PTx 1 � rRx,Tx � �=d τRx,Tx Rx,Tx σ2 n~Rx

$$\overline{\text{PEP}} = \int\_0^\infty \mathcal{Q}\left(\sqrt{\mathcal{V}\_{\mathcal{T}\_x \to \mathcal{R}\_x}}\right) f\_{\mathcal{V}\_{\mathcal{T}\_x \to \mathcal{R}\_x}}(\boldsymbol{\gamma}) d\boldsymbol{\gamma}.\tag{33}$$

Finally, this integral can be evaluated with the help of Mathematica and average PEP under Rayleigh fading channel can be derived in a closed form as follows:

$$\overline{\text{PEP}} = \frac{1}{2} \left( 1 - \sqrt{\frac{\overline{\mathcal{V}\_{T\_x \to R\_x}}{2 + \overline{\mathcal{V}}\_{T\_x \to R\_x}}}} \right) . \tag{34}$$

Figure 9. Average PEP performance for different amounts of CSI exchange with varying SNR over Rayleigh fading

Interference Alignment in Multi-Input Multi-Output Cognitive Radio-Based Network

http://dx.doi.org/10.5772/intechopen.80073

131

Figure 8. BER performance for different amounts of CSI exchange with varying SNR.

channel in primary network.

#### 3.3. Numerical results

In this section the numerical results are provided with various scenarios to evaluate the performance analysis for IA in underlay cognitive two-way relay networks with CSI quantization. BER performance for direct transmission links of the proposed system is illustrated in Figure 8 over Rayleigh distribution for different amounts of CSI exchange with varying SNR. For convenience, we set dj,i ¼ 3 m and τ ¼ 2:7, and 3�3 MIMO configuration is studied in this figure. Because of the number of interfering links, the quantization error for the Tx ! Rx transmission is greater than the other links (S1 ⇄ R ⇄ S2). Even if the analyzed BER performance of the SN seems better than the PN, it should not be forgotten that SN operates in halfduplex mode. Performance loss in BER due to imperfect CSI (Bj,i ¼ 4, for instance) becomes larger as SNR increases compared to the perfect CSI (for Bj,i ¼ ∞) case.

In Figure 9, the average PEP versus SNR is plotted for dj,i ¼ 3 m and τ ¼ 2:7 over Rayleigh fading channel in PN. It can be noticed from the figure that as SNR increases, average PEP decreases, as expected. To reach the perfect CSI case, we take Bj,i ¼ ∞, and the average PEP performance noticeably enhances. We also consider the case of imperfect CSI (Bj,i ¼ 4) for the comparison purposes in the same figure.

Interference Alignment in Multi-Input Multi-Output Cognitive Radio-Based Network http://dx.doi.org/10.5772/intechopen.80073 131

Figure 8. BER performance for different amounts of CSI exchange with varying SNR.

CR ¼ 1

130 Cognitive Radio in 4G/5G Wireless Communication Systems

γTx!Rx ¼ PTx 1 � rRx,Tx

3.3. Numerical results

� �=d

2 log 2 1 þ min γS1!R; γR!S2

� � � �

where Q xð Þ is the Gaussian Q-function and defined by Q xð Þ¼ 1= ffiffiffiffiffiffi

function over Rayleigh fading statistics [40], f γTx!Rx

τRx,Tx Rx,Tx σ2 n~Rx

PEP ¼

Rayleigh fading channel can be derived in a closed form as follows:

PEP ¼ 1

larger as SNR increases compared to the perfect CSI (for Bj,i ¼ ∞) case.

comparison purposes in the same figure.

ð∞ 0

þ 1

γS1!R and γR!S2 are the instantaneous SINR for the S1 ! R and R ! S2 links, respectively.

BERj ¼ Q ffiffiffiffi

Average pairwise error probability (PEP) can be computed as averaging the Gaussian Q-

Q ffiffiffiffiffiffiffiffiffiffiffiffiffiffi γTx!Rx � � p f γTx!Rx

2 1 �

Finally, this integral can be evaluated with the help of Mathematica and average PEP under

In this section the numerical results are provided with various scenarios to evaluate the performance analysis for IA in underlay cognitive two-way relay networks with CSI quantization. BER performance for direct transmission links of the proposed system is illustrated in Figure 8 over Rayleigh distribution for different amounts of CSI exchange with varying SNR. For convenience, we set dj,i ¼ 3 m and τ ¼ 2:7, and 3�3 MIMO configuration is studied in this figure. Because of the number of interfering links, the quantization error for the Tx ! Rx transmission is greater than the other links (S1 ⇄ R ⇄ S2). Even if the analyzed BER performance of the SN seems better than the PN, it should not be forgotten that SN operates in halfduplex mode. Performance loss in BER due to imperfect CSI (Bj,i ¼ 4, for instance) becomes

In Figure 9, the average PEP versus SNR is plotted for dj,i ¼ 3 m and τ ¼ 2:7 over Rayleigh fading channel in PN. It can be noticed from the figure that as SNR increases, average PEP decreases, as expected. To reach the perfect CSI case, we take Bj,i ¼ ∞, and the average PEP performance noticeably enhances. We also consider the case of imperfect CSI (Bj,i ¼ 4) for the

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi γTx!Rx 2 þ γTx!Rx

! s

γj

Average BER for binary phase shift keying (BPSK) modulation can be expressed as

2 log 2 1 þ min γS2!R; γR!S1

� � � �

� � p (32)

x e�t

ð Þ γ dγ: (33)

: (34)

2=2dt [37].

1mm, where

2π � � p Ð ∞

�γ=γTx!Rx � �=γTx!Rx

ð Þ¼ γ e

: (31)

Figure 9. Average PEP performance for different amounts of CSI exchange with varying SNR over Rayleigh fading channel in primary network.

Figure 10. Capacity vs. SNR of the primary network and secondary network nodes under different CSI scenarios.

Figure 10 examines the capacity analysis with perfect and imperfect CSI for different direct links in PN and SN. The results clearly show that, examining the capacity with perfect CSI, performance improvement becomes larger as the SNR increases.

Figure 11 demonstrates the effects of Bj,i and dj,i parameters on the BER performance for the SN with varying SNR when τ ¼ 2:7 and 3�3 MIMO scheme is used. The results clearly show that for a fixed SNR value, the performance of the considered system increases with the decrease of the dj,i. It can be seen from the same figure that the increase on the amount of CSI exchange Bj,i positively affects the BER performance.

Figure 12 shows the capacity performance of PU in the underlay cognitive two-way relay network over Rayleigh fading channel with varying path loss exponent, τ. The results show a performance improvement while the value of τ decreases. In this plot, Bj,i = 8, dj,i = 3 m, and the 3�3 MIMO scheme are considered. Depending on the environmental conditions for mobile communications, typical τ values, ranging from 1.6 to 5, are used to plot this figure. First, for the line of sight in a building, the environment is considered with the τ values of 1.6 and 1.8. Second, capacity is computed for the free-space environment with τ ¼ 2. Then, the capacity performance is presented with τ values of 2.7 and 3.3 for urban area cellular radio environment. Finally, the shadowed urban cellular radio environment is associated with two different τ values of 3 and 5 to analyze the capacity performance with varying SNR [41].

Figure 12. Capacity changes with SNR for the environmental conditions having different path loss exponents.

Figure 11. BER performance for different amounts of CSI exchange and distances with varying SNR over Rayleigh fading

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channel for secondary network.

Interference Alignment in Multi-Input Multi-Output Cognitive Radio-Based Network http://dx.doi.org/10.5772/intechopen.80073 133

Figure 11. BER performance for different amounts of CSI exchange and distances with varying SNR over Rayleigh fading channel for secondary network.

Figure 10 examines the capacity analysis with perfect and imperfect CSI for different direct links in PN and SN. The results clearly show that, examining the capacity with perfect CSI,

Figure 10. Capacity vs. SNR of the primary network and secondary network nodes under different CSI scenarios.

Figure 11 demonstrates the effects of Bj,i and dj,i parameters on the BER performance for the SN with varying SNR when τ ¼ 2:7 and 3�3 MIMO scheme is used. The results clearly show that for a fixed SNR value, the performance of the considered system increases with the decrease of the dj,i. It can be seen from the same figure that the increase on the amount of CSI

Figure 12 shows the capacity performance of PU in the underlay cognitive two-way relay network over Rayleigh fading channel with varying path loss exponent, τ. The results show a performance improvement while the value of τ decreases. In this plot, Bj,i = 8, dj,i = 3 m, and the 3�3 MIMO scheme are considered. Depending on the environmental conditions for mobile communications, typical τ values, ranging from 1.6 to 5, are used to plot this figure. First, for the line of sight in a building, the environment is considered with the τ values of 1.6 and 1.8. Second, capacity is computed for the free-space environment with τ ¼ 2. Then, the capacity performance is presented with τ values of 2.7 and 3.3 for urban area cellular radio environment. Finally, the shadowed urban cellular radio environment is associated with two different

τ values of 3 and 5 to analyze the capacity performance with varying SNR [41].

performance improvement becomes larger as the SNR increases.

exchange Bj,i positively affects the BER performance.

132 Cognitive Radio in 4G/5G Wireless Communication Systems

Figure 12. Capacity changes with SNR for the environmental conditions having different path loss exponents.

## 4. Conclusion

In this chapter, the system performance of linear interference alignment on the MIMO CR network is investigated under interference leakage. To quantify the performance of the primary system under a certain level of interference leakage, the closed-form outage probability expression is derived for Rayleigh fading channel. In all analyses, the theoretical results closely match with the simulations which confirm the accuracy of the derived expressions.

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In the second part of this work, considering a practical issue, we investigate the performance of interference alignment in underlay cognitive radio network with CSI quantization error over general MIMO interference channel. Amplify-and-forward scheme for two-way relay network under Rayleigh fading is considered. The impact of the CSI exchange amount, the distance between the ith transmitter and the jth receiver nodes, and the path loss exponent on the BER performance, system capacity, and average PEP for the proposed system model are analyzed. We provide the exact closed-form expression for the average PEP in primary network over Rayleigh distribution, while IA algorithm perfectly eliminates the interference. The present performance analysis can be extended to the multiple secondary user pairs, and this approach will be another subject of our future work.

It would be interesting to study on various scenarios, including single-hop, multi-hop, and multi-way networks in future work to analyze the system performance over the recently developed interference alignment algorithms for next-generation 5G wireless communication systems.

## Acknowledgements

The authors wish to express their special thanks to Seda Ustunbas (Wireless Communication Research Laboratory, Istanbul Technical University, Turkey) for useful discussions of this chapter.

## Author details

Arif Basgumus1 , Mustafa Namdar1 , Hakan Alakoca2 , Eylem Erdogan3 and Lutfiye Durak-Ata2 \*

\*Address all correspondence to<EMAIL_ADDRESS>
1 Department of Electrical and Electronics Engineering, Dumlupinar University, Turkey

2 Informatics Institute, Applied Informatics Department, Istanbul Technical University, Turkey

3 Department of Electrical and Electronics Engineering, Istanbul Medeniyet University, Turkey

## References

4. Conclusion

will be another subject of our future work.

134 Cognitive Radio in 4G/5G Wireless Communication Systems

, Mustafa Namdar1

\*Address all correspondence to<EMAIL_ADDRESS>
\*

Acknowledgements

Author details

Arif Basgumus1

Turkey

Turkey

Lutfiye Durak-Ata2

In this chapter, the system performance of linear interference alignment on the MIMO CR network is investigated under interference leakage. To quantify the performance of the primary system under a certain level of interference leakage, the closed-form outage probability expression is derived for Rayleigh fading channel. In all analyses, the theoretical results closely

In the second part of this work, considering a practical issue, we investigate the performance of interference alignment in underlay cognitive radio network with CSI quantization error over general MIMO interference channel. Amplify-and-forward scheme for two-way relay network under Rayleigh fading is considered. The impact of the CSI exchange amount, the distance between the ith transmitter and the jth receiver nodes, and the path loss exponent on the BER performance, system capacity, and average PEP for the proposed system model are analyzed. We provide the exact closed-form expression for the average PEP in primary network over Rayleigh distribution, while IA algorithm perfectly eliminates the interference. The present performance analysis can be extended to the multiple secondary user pairs, and this approach

It would be interesting to study on various scenarios, including single-hop, multi-hop, and multi-way networks in future work to analyze the system performance over the recently developed interference alignment algorithms for next-generation 5G wireless communication systems.

The authors wish to express their special thanks to Seda Ustunbas (Wireless Communication Research Laboratory, Istanbul Technical University, Turkey) for useful discussions of this chapter.

, Hakan Alakoca2

1 Department of Electrical and Electronics Engineering, Dumlupinar University, Turkey 2 Informatics Institute, Applied Informatics Department, Istanbul Technical University,

3 Department of Electrical and Electronics Engineering, Istanbul Medeniyet University,

, Eylem Erdogan3 and

match with the simulations which confirm the accuracy of the derived expressions.

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[38] Jindal N. MIMO broadcast channels with finite-rate feedback. IEEE Transactions on Infor-

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## *Edited by Shahriar Shirvani Moghaddam*

The limitation of the radio spectrum and the rapid growth of communication applications make optimal usage of radio resources essential. Cognitive radio (CR) is an attractive research area for 4G/5G wireless communication systems, which enables unlicensed users to access the spectrum. Delivering higher spectral efficiency, supporting the higher number of users, and achieving higher coverage and throughput are the main advantages of CR-based networks compared to conventional ones. The main goal of this book is to provide highlights of current research topics in the field of CR-based systems. The book consists of six chapters in three sections focusing on primary and secondary users, spectrum sensing, spectrum sharing, CR-based IoT, emulation attack, and interference alignment.

Published in London, UK © 2018 IntechOpen © FeelPic / iStock

Cognitive Radio in 4G/5G Wireless Communication Systems

Cognitive Radio in 4G/5G

Wireless Communication

Systems

*Edited by Shahriar Shirvani Moghaddam*


**6. Conclusions**

It has been shown that the considered critical cases in the stability problem of the distributed chain of logistic equations with delay have an infinite dimension. This leads to the fact that description of their local dynamics is reduced to the study of the nonlocal behavior of boundary value problems of Ginzburg–Landau type solutions. It is known (see, for example, [30]) that the dynamics of such objects can be complicated, and they are characterized by irregular oscillations, multistability phenomena, etc. The dynamic effects essentially depend on the choice of couplings. It has been shown that in a number of cases the solutions are rapidly and slowly oscillating with respect to spatial variable components. The basic results define the structure of problems that are asymptotic with respect to residual solutions to the initial boundary value. The problem of existence, stability, and more complicated asymptotic expansions of exact solutions close to those constructed can be solved, for example, for the case of periodic solutions of normalized equations.

We considered separately the role of the above parameter *θ* = *θ*(*ε*) ∈ [0, 1). We recalled that the dynamic properties of the initial system are determined by the QNF (49), (50), which includes the parameter *θ*. The dynamics of (49), (50) and hence of the boundary value problem (9), (10) may change for different values of this parameter. This is shown in detail in [31]. This implies that an infinite process of forward and reverse bifurcations can occur as *ε* → 0.

Below, we formulate one more conclusion of the general plan. It was shown above that the quasinormal forms that determine the dynamics of the initial boundary value problem are equations of Ginzburg–Landau type. We note that parabolic boundary value problems with one and two spatial variables can act as quasinormal forms depending on the coefficient *σ* of the function *F*(*s*,*ε*) ((12), (13)). The stability of the simplest solutions of these equations is studied in [33]. In particular, it has been established that their stability properties are determined to a large extent by the imaginary components of the diffusion coefficients and of the Lyapunov quantity (coefficients *g* and *q* in (49) and (50)). Numerical analysis of the corresponding criterion makes it possible to formulate the conclusion about the instability of all the simplest solutions of the form · exp(*iωt* + *ikx*). Thus, solution synchronization is a rather rare phenomenon in the considered chains.

It has been demonstrated that the study of the dynamics of logistic equations with delay is reduced to nonlinear dynamics analysis of special families of the parabolic and degenerately parabolic boundary value problems for large values of the coefficient of spatially distributed control. In particular, the phenomenon of hypermultistability is described.

In the study of local dynamics, bifurcation phenomena can be realized in the equilibrium state neighborhood even for asymptotically small delays. Here, the critical case has an infinite dimension in the stability problem. Analogues of the normal form, so-called quasinormal forms, are constructed in this situation, which are universal nonlinear boundary value problems of the parabolic type. Their nonlocal dynamics determine the local behavior of the solutions of the initial boundary value problem.

**Funding:** This work was supported by the Russian Science Foundation (project no. 21-71-30011).

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Conflicts of Interest:** The author declares no conflict of interest. The funder had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.


*3.10. Possible Role of the Expression of CPA4 in Various Infiltrating Immune Cells*

We used the TIMER2 and GEPIA2 databases to further identify the possible role of the expression of CPA4 in various infiltrating immune cells including T cells (general), M1/M2 macrophages, tumor-associated macrophages, B cells, neutrophils, monocytes, NK, CD8+ T cells, and functional DCs as well as T cells such as Th1, Th2, Th9, Th17, Th22, Tfh, exhausted T cells, and Treg. From the results, Th1, T cell exhaustion, and TAM sets marking were greatly connected with the expression of CPA4 in BLCA (Table 2).

**Table 2.** Correlation analysis between CPA4 and markers of immune cells in BLCA patients found in the TIMER2 and GEPIA2.

**Table 2.** *Cont.*

BLCA, Bladder Urothelial Carcinoma; Tfh, Follicular helper T cell; Th, T helper cell; Treg, Regulatory T cell; TAM, tumor-associated macrophage;.None, correlation without adjustment correlation; Purity, correlation adjusted by purity; Tumor, correlation analysis in the tumor tissue of TCGA; Normal, correlation analysis in normal tissue of TCGA; Cor, R value of Spearman's correlation \* *p* < 0.05; \*\* *p* < 0.01; \*\*\* *p* < 0.001.

#### **4. Discussion**

CPA4 (carboxypeptidase A4) is a member of the metallocarboxypeptidase family and is a zinc-containing exopeptidase that catalyzes the release of carboxy-terminal amino acids [34]. In recent years, CPA4 has shown the potential to be a biomarker in the early diagnosis with clinical benefit for certain cancers. Some studies revealed that CPA4 is connected with various cancer cells in its differentiation and growth, including non-smallcell lung cancer and gastric cancer [35,36]. Furthermore, it is reported that CPA4 is located on chromosome 7q32 in a region linked to prostate cancer aggressiveness [11], and Sun suggested that CPA4 is closely associated with colorectal cancer liver metastasis [37]. Although CPA4 expression has been confirmed to have potential significance in multiple types of cancer, no studies have shown the expression level and clinical significance of CPA4 in BLCA. In this study, based on a pan-cancer analysis, we demonstrated that human CPA4 expression levels were highly expressed in 11 types of cancer with corresponding normal tissues (Figure 1), which are consistent with the findings in the previous study reported by Sun and Handa et al. [17,35,38]. We also confirmed that CPA4 is significantly upregulated in BLCA (Figure 1b). Moreover, a previous study has shown that CPA4 expression was detected specifically in the cytoplasm of cancer tissue cells, and in the CPA4-suppressed triple-negative breast cancer (TNBC), viability, and migration were decreased [38]. It can act as a potential biomarker of poor prognosis in TNBC. It is reported that CPA4 might be used as an independent poor prognostic factor in esophageal squamous cell carcinoma [39]. In our study, the results in BLCA are consistent. However, one trial showed that CPA4 is a protective factor in muscle-invasive bladder cancer, contrary to the role of CPA4 in most cancers [40]. A potential reason for the difference is due to updates in the TCGA database and different objects. We studied BLCA, while that study investigated muscle-invasive bladder cancer. We compared the different expression levels of BLCA with age; gender; T, N, and M stage; pathologic stage; subtype; and OS. Surprisingly, we found that higher dead events, higher pathologic stages, and the subtype non-papillary were associated with higher expressions of CPA4 in BLCA, with statistical differences (Figure 2). These findings suggest that CPA4 may be a potential biomarker of poor prognosis in identifying BLCA with poor clinical outcome.

Currently, the function of CPA4 in tumors had not been fully reported. Previous trials suggested that the inhibition of CPA4 could reduce the number of breast cancer cells with stemness properties and may be a potential target for TNBC therapy [41]. The CircCPA4 sponge let-7 regulates the expression of CPA4 and glioma progression [42]. All of these results suggest that CPA4 could be regarded as an emerging target or promising biomarker for cancer therapy. Since the mRNA expression of CPA4 in BLCA was significantly higher than that in normal bladder tissue, we speculated that CPA4 can be regarded as a biomarker to detect BLCA from normal controls. To verify the clinical value of CPA4, an ROC curve analysis was performed to verify the clinical value of CPA4 in the diagnosis of BLCA; our results showed that CPA4 may be a potential diagnostic biomarker between bladder cancer and normal tissues, with an AUC of 0.798 (Figure 2d).

Many studies have shown that CPA4 is a significant biomarker of poor prognosis in lots of cancers and is associated with the upregulation of CPA4 with poor overall survival. In hepatocellular carcinoma, AC10364 inhibited cell proliferation and viability through the abnormal expression of genes including CPA4 associated with tumorigenesis or growth [43]. A paper from Yan suggested that the inhibition of CPA4 might be of great significance for improving early stage non-small cell lung cancer survival after ablation [43]. However, the prognostic value in BLCA of CPA4 has not been investigated. With the increased level of CPA4 related to a higher number of dead events and higher pathologic stages, we speculated that CPA4 is involved in the development of BLCA. In light of the Kaplan–Meier curves, we confirmed that the overexpression of CPA4 is associated with shorter overall survival (OS), disease-specific survival (DSF), and progress-free intervals (PFIs) (Figure 2). Moreover, by univariate and multivariate regression analysis, we found that high CPA4 expression; high pathologic stage; T, N, and M stage; and the subtype papillary were negative predictors for OS in BLCA patients and that CPA4 can be an independent factor correlated with OS (Figure 5). The nomogram more accurately predicted 1-, 3-, and 5-year OS in BLCA patients and could help to screen and determine those high-risk patients (Figure 6).

Through GSEA, CPA4 was found to be involved in epidermal cell differentiation, keratinocyte differentiation, keratinization, etc., indicating CPA4 potentially playing a role in cell metabolism and protein synthesis (Figure 4).

The PPI network of CPA4 protein, which were constructed by STRING, showed the relationship of CPA4 in the progression of BLCA such as LXN, CMA1, SGCE, TPSAB1, etc. (Figure 7). It has been reported that latexin (LXN) can inhibit human CPA4, in which the expression is induced in prostate cancer cells after treatment with histone deacetylase inhibitors [44]. The level of CMA1, a key gene, is significantly correlated with gastric cancer prognosis and infiltration level [45]. SGCE promotes breast cancer stem cell self-renewal, chemoresistance, and metastasis both in vitro and in vivo by stabilizing EGFR [45]. Thus, it is speculated that a high expression of CPA4 may increase the degree of malignancy of tumors through CPA4 interacting proteins, leading to the deterioration of patients' conditions.

Moreover, CPA4 plays a specific role in immune infiltration in bladder cancer. Compellingly, we unraveled that several infiltrating immune cells (Th1cell, Th2 cell, T cell exhaustion, and TAM) were correlated with the expression of CPA4 in bladder cancer using TIMER2 and GEPIA2. Type 1 T helper (Th1) cells produce interferon-gamma [46] (Figure 8, Table 2). The dual inhibition of STAT1 and STAT3 activation downregulates the expression of PD-L1 in cancer cells [47]. T-cell exhaustion is a state of T-cell dysfunction that occurs in many chronic infections and cancers [48]. Scholars have observed that CTLA4 was identified as a crucial negative regulator of the immune system, which transmits an inhibitory signal [49].

There are some limitations in our study. First, basic experiments are needed to verify the results, which were conducted with online public databases. Second, in vivo/vitro experiments are needed to further investigate the potential mechanism of the effect of CPA4 on immune invasion in BLCA.

#### **5. Conclusions**

In conclusion, our study first demonstrated that CPA4 expression increased in BLCA, and univariate and multivariate regression analyses and a nomogram were used to prove that increased CPA4 is correlated with shorter overall survival, which means high risk factors in BLCA patients. Moreover, we illustrated that a high level of CPA4 was positively related to a high pathologic grade; high T, N, and M stages; and the subtype papillary. The immune infiltration in the tumor microenvironment has also been shown to be associated with CPA4. Collectively, this study partially unveiled that CPA4 in BLCA could be regarded

as a potential biomarker for diagnosis and prognosis and may play a special role in immune infiltration.

**Author Contributions:** C.W. designed the original idea, organized the model of TCGA data with relative pictures, and wrote part of the article. Y.Z. was responded for the modification of manuscript and wrote part of the manuscript. Q.X. was responded for the statistics problems and grammar check. M.X. was responsible for the language, bioinformation problems and correcting them. Z.C. and X.Y. was a supporter of this manuscript. Y.H. edited the whole manuscript. All of the authors of this manuscript are responsible for the accuracy, appropriation, and completing any part of this research and agree to open this manuscript. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** All of the data in this manuscript are available and approved by the institution.

**Acknowledgments:** The authors would like to thank all who supported this manuscript as well as provided any useful suggestions and help.

**Conflicts of Interest:** All of the authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript; or in the decision to publish the results.

#### **References**


*A Review of EMG Techniques for Detection of Gait Disorders DOI: http://dx.doi.org/10.5772/intechopen.84403*

*Artificial Intelligence - Applications in Medicine and Biology*

**6. Frequency and time-frequency analysis**

the Vastus Medialis (VM) during walking (**Figure 4**).

methods.

scale.

impairment [19]. In chronic stroke, merging and fractionation of synergies were observed. Merging of muscle synergies results in poor muscle coordination. In children with cerebral palsy, the dimensional space was smaller than it was in the control participants (unimpaired) [42]. However, the modules for cerebral palsy were higher for Duchenne muscular dystrophy (DMD) and typical developing (TD) children [43]. Rodriguez et al. revealed that fewer modules were recruited while walking on treadmill among Parkinson's patients. Thus, the size of dimensional space is crucial for the assessment of gait disorder such as cerebral palsy and Parkinson's [51, 52]. It is also important to properly choose preprocessing before analyzing the synergies as the dimensional space is sensitive to the preprocessing

EMG power spectrum estimation methods can be categorized into parametric and nonparametric techniques. The spectral methods include fast Fourier transform (FFT), multitaper analysis and short-time Fourier transform (STFT) and wavelet transform. The difference between FFT and Wavelet Transformation is that FFT is localized to the frequency domain whereas the latter is localized to time-frequency analysis. Hu [53] recorded cortical and spinal somatosensory evoked potential (CSEP and SSEP), cortical motor evoked potential (CMEP) and spinal cord evoked potential (SCEP). The short time Fourier transformation was applied to the CSEP signal with a Hanning window [53]. The results revealed that the time-frequency analysis is a better marker for spinal injury than time domain analysis. The peak power after spinal injury had lesser energy with more dispersion in time-frequency

The EMG time series signal can be analyzed in the frequency domain for the diagnosis of gait disorders. The frequency spectrum for EMG signals is in range of 0–500 Hz [54]. The FFT algorithm [55] computes the discrete Fourier transform (DFT) of EMG signal more efficiently. The FFT decomposes the EMG signals into periodic sine and cosine waves. We computed the FFT of EMG signal recorded from

*(A) sEMG signal from VM during walking in time domain; (B) frequency domain representation of the signal*

**26**

**Figure 4.**

*using FFT.*

The FFT allows computation of power spectra by squaring of FFT's magnitude [56]. In Parkinson disease, the spectral power of the signal has lower amplitude for the usual tremor than for the unusual tremor, which has peak amplitude of 4–6 Hz during an atypical tremor [15]. The signals associated with nonperiodic tremors are differentiable with FFT [57]. The EMG signal from neuropathic patients with SCI also exhibited distinct power spectrum density and amplitude in comparison to healthy individuals [58]. The application of FFT to the EMG envelope revealed muscle burst discharge in frequency domain ranging from 4 to 7 Hz [15]. Average power spectra computed from fractionated EMG of ALS mice by FFT was significantly higher than the control group. In the ALS group the spectra were skewed towards higher frequency content but single unit recordings revealed the absence of higher motor neuron (MN) frequencies or shortening of MN frequency in ALS mice [32], due to small type firing neurons improperly increasing firing frequency. This phenomenon results in co-contraction thus producing fractionated EMG. Co-contraction in muscles can also be observed in spinal cord injured patients [32]. In a study, EMG signals from lower limbs of dystonic and nondystonic participants while walking were recorded. The non-dystonic participants were also patients suffering from other gait disorders. The power spectral density was computed using FFT with the Welch method of 50% overlap. The median power frequency (MdPF) and total power in low frequency were calculated for each muscle. The results revealed that MdPF for dystonic muscles had shifted to low frequencies and a concurrent increase in total power percentage in low-frequency range was observed [59]. Thus, frequency analysis of EMG signal not only provides us with distinction between normal and abnormal gait behavior but also specific gait abnormalities can be distinguished.


**2. Biological Continuum of Adverse Pregnancy Outcome**

While we divide pregnancy disorders into distinct categories, much of this classification is arbitrary for the purposes of description and study. Delivery at 19 weeks 6 days of gestation is defined as a miscarriage, whereas delivery one day later is a premature birth. In reality, these conditions occur along a continuum. They have common and interrelated risk factors. For example, a woman with a history of a prior unexplained PTB at 28 weeks is at increased risk of spontaneous PTB in a subsequent pregnancy, but is also at increased risk of preeclampsia and FGR in future pregnancies [4]. These disorders also have overlapping biomarkers [12]. Moreover, deficient spiral artery remodeling has been linked with a spectrum of obstetrical syndromes, including pre- eclampsia, FGR, PTB, premature rupture of membranes (PPROM), abortion, and fetal death [4,8,13]. Taken together, these observations suggest that adverse pregnancy events occur along a biological continuum and likely have a common underlying pathophysiology.


**3.5 Evaluation of C-space plots for higher dimensional robot: Example**

Here we will study a specific case for a seven degrees-of-freedom robot, amidst a 2D cluttered environment as shown in fig. 17. The technical parameters of the robot, comprising link-lengths, {li, i = 1,2,…7}and joint limits {i, i = 1,2,…7), are highlighted in Table 2. Circular obstacles are considered for simplicity in computations. The locations of the respective centers and diameters of the obstacles (expressed in suitable units) are presented in Table 3.

Table 2. Technical Facets of the Higher Dimensional Robot Under Consideration

Spatial Path Planning of Static Robots Using Configuration Space Metrics 437

Fig. 18. Four c-space plots for the seven joint revolute robot amidst cluttered workspace of fig. 17 It may be noted that while c-space plot for a 2 d.o.f. robot (working in 2D or 3D task-space) can be composed of irregular non-geometric shapes (refer fig. 7), the same for higher d.o.f. robots are perfectly geometric (refer fig. 18). This is happening because of the incorporation of the concept of 'formidable zones' for higher dimensional robots, wherein we are deliberately allowing the collidable zone to engulf more regions in the c-space. In fact, in most of cases for higher d.o.f. robots, the c-space zones are perfect rectangular in shapes, between the minimum & maximum limits of the participating joint-angles. For example, in fig. 18, (1 vs. 2) c-space slice plot the final rectangle is constituted between 4 vertices, viz.

Based on the formation of c-space maps, collision-free paths are to be ascertained in 2D as well as in 3D. We will analyze the gamut in four functional *quadrants*, which have been

**4. Safe path in configuration space: Logistics & algorithm**

1\_min, 1\_max, 2\_min & 2\_max.

**4.1 Perspective**

Fig. 17. Workspace layout of the seven degrees-of-freedom articulated robot

Table 3. Obstacle Signature, as per Workspace Layout of Figure 16

Now, in this case of seven degrees-of-freedom revolute robot, we will have four different c-space plots, namely, [1 -- 2], [3 -- 4], [5 -- 6] & [6 -- 7]. All of these plots use collision-detection algorithms, described earlier, for each of the obstacles separately, taking into account the concepts of *equivalent circles*. These c-space plots are illustrated in figure 18. A gross estimate reveal that [1 -- 2], [3 -- 4], [5 -- 6] & [6 -- 7] plots occupy a planar area measuring (160x120), (80x95), (40,60) & (60x30) sq. units respectively. It is evident from fig. 18 that although complexity-wise both [1 -- 2] and [3 -- 4] plots are roughly at par, but the former is to be selected as the most significant c-space plot as it is also the largest in size.

Fig. 17. Workspace layout of the seven degrees-of-freedom articulated robot

1 15.88 (x =15, y = 30) 2 13 (x = 12.5, y = 37.5) 3 12.7 (x = 27.5, y = 42.5) 4 19 (x = 35, y = 30) 5 14 (x = 60, y = 15) 6 9.53 (x = 65, y = 50)

Table 3. Obstacle Signature, as per Workspace Layout of Figure 16

also the largest in size.

**Obstacle No. Diameter Location of Centre**

Now, in this case of seven degrees-of-freedom revolute robot, we will have four different c-space plots, namely, [1 -- 2], [3 -- 4], [5 -- 6] & [6 -- 7]. All of these plots use collision-detection algorithms, described earlier, for each of the obstacles separately, taking into account the concepts of *equivalent circles*. These c-space plots are illustrated in figure 18. A gross estimate reveal that [1 -- 2], [3 -- 4], [5 -- 6] & [6 -- 7] plots occupy a planar area measuring (160x120), (80x95), (40,60) & (60x30) sq. units respectively. It is evident from fig. 18 that although complexity-wise both [1 -- 2] and [3 -- 4] plots are roughly at par, but the former is to be selected as the most significant c-space plot as it is

Fig. 18. Four c-space plots for the seven joint revolute robot amidst cluttered workspace of fig. 17

It may be noted that while c-space plot for a 2 d.o.f. robot (working in 2D or 3D task-space) can be composed of irregular non-geometric shapes (refer fig. 7), the same for higher d.o.f. robots are perfectly geometric (refer fig. 18). This is happening because of the incorporation of the concept of 'formidable zones' for higher dimensional robots, wherein we are deliberately allowing the collidable zone to engulf more regions in the c-space. In fact, in most of cases for higher d.o.f. robots, the c-space zones are perfect rectangular in shapes, between the minimum & maximum limits of the participating joint-angles. For example, in fig. 18, (1 vs. 2) c-space slice plot the final rectangle is constituted between 4 vertices, viz. 1\_min, 1\_max, 2\_min & 2\_max.


**2. Pathomechanics of distal femoral fractures**

The most frequent mechanical complication of distal femoral fractures (DFF) is nonunion, reported in up to 19% of patients [3, 4]. Along with nonunion, varus collapse and hardware failure come together. This continues to occur despite advances in fracture fixation implants for this injury, reflecting the need for better understanding of different factors that influence the outcome. Modern implants include those advocated by the AO Foundation decades ago, such as the 95 degree blade-plate and the Dynamic Condylar Screw (DCS) [5–8]. Such devices are seldom used nowadays for acute fracture treatment. Being made of stainless steel, they provided strong fixation and resistance to fatigue, improving dramatically the quality of fixation and ultimately the results. However, their inherent stiffness was still a cause for atrophic nonunions (**Figure 1**). Advances in the knowledge of concepts such as stress and strain highlighted the importance of the mechanical environment in the process of bone healing [9–11]. Application of these concepts by the surgeon is the basis for the construction of biomechanically sound osteosynthesis, in which highly rigid constructs are being replaced by others allowing controlled micromotion, and subsequent stimulus for callus formation.

In more recent years, both anatomical, locked plates (LP), and retrograde interlocking intramedullary nails (RIMN) became the standard for operative management [4, 12]. Both types of fixation show similar rates of bone healing and functional outcome, despite dissimilarities. In some studies, intramedullary devices perform better than plates in terms of axial stiffness, whilst in others side plates were superior [13, 14].  At follow up, nounion and failure in varus occurred. Metaphyseal comminution and a short side plate and working length are considerations to be taken into account to explain the result.*

### *Management of Distal Femoral Fractures DOI: http://dx.doi.org/10.5772/intechopen.110692*

incidence of mechanical failures [4, 15–18]. Although having improved mechanical capabilities compared to previous implants, improper use, or unawareness of the importance of the balance between rigidity, stability and bone healing promotion, can lead to failure (**Figure 2**) [19].

Numerous in vitro reports have been published in recent years, showing improved mechanical capabilities with fixation techniques that combine lateral locking plates and medially placed additional plates, as well as the combination of lateral locking plates and locked intramedullary retrograde nails ("augmented RIMN") [20–25]. These studies report a significant increase in resistance to axial, rotational and bending forces, along with increased stiffness of such constructs, which has led to their application for clinical use, in pursuit of minimizing complications derived from implant failure. Several mechanical stress studies, as well as clinical studies, focus on fractures with bone defects and/or comminution, including fractures in osteoporotic patients, because they have high risk for failure. Ricci et al. published an analysis of the factors that may influence failure of fixation in distal femoral fractures, treated with laterally placed locked plates [26]. Various types of fractures were included: 64%

#### **Figure 2.**

*(a) A3 closed fracture in a 50 year-old patient, treated with anatomical locking plate. AP x-ray at follow up shows atrophic nonunion. Selection of the length of the plate, together with the distribution of screws, may have played a role in the outcome. (b) Same fracture pattern in a 44 year-old patient, with more extended working length and less screw density. Uneventful consolidation has occurred after 6 months.*

of 335 fractures were simple metaphyseal (A1 and A2), or simple articular (C1 or C2). They found that there were factors out of reach from the surgeon, such as severity of the injury (high energy, open fractures), young age (with more severe injuries), comorbidities such as diabetes, and smoking. Plate length, on the other hand, was also a contributing factor: shorter plates were associated with failures more frequently than long plates. The surgeon must therefore balance what is inherent to the fracture and the patient, with the available implants, elaborating a strategy that will ultimately lead to a construct design capable of minimizing the risk of failure.

In our opinion, DFF differ in the pathomechanical analysis that can be made whether they are simple extraarticular, comminuted extraarticular, or intraarticular, as well as if present in young or older (osteopenic) patients. Older patients with poor

#### **Figure 3.**

*Distal extraarticular fracture in a 75 year old patient with concomitant severe ipsilateral hip arthritis and rigidity. Despite absence of comminution, fixation was done with RIMN, augmented with a locking plate. This was decided in consideration of the age of the patient, and the incresed torsional stresses that the construct will withstand throughout the healing process, due to hip stifness.*

#### *Management of Distal Femoral Fractures DOI: http://dx.doi.org/10.5772/intechopen.110692*

bone quality present inherent difficulties in obtaining stable internal fixation. There is data underlying the difficulties to accomplish "limited" weight-bearing in patients of advanced age, so it should be assumed that patients of a certain age will submit the construct to high forces from the early postoperative period [27, 28]. These considerations have decisive influence on the selection of the operative method.

Of note is the fact that advances in implant stability and rigidity may also present concerns about delayed healing. There is the need to balance the overall strength/ stiffness of the construct and the appropriate micromotion that promotes callus formation. With patients of advanced age, however, the benefits of construct stability and endurance may outweigh the achievement of fracture healing, at least initially [28–31]. Postoperative compliance with controlled or "limited" weight bearing is very difficult to achieve in geriatric patients, as mentioned above. Therefore, implant stability and strength is of primary importance (**Figure 3**). These considerations do not apply, nonetheless, in the young patient with similar fractures. In this setting, promotion of bone healing is paramount for long term success and functional restoration. Therefore, careful and judicious application of any fixation method or technique should balance the maintenance of proper reduction while favoring bone healing, the latter by the means of providing a mechanical environment with adequate strain.

As aforementioned, a sound mechanical construct should take into consideration the quality of bone, the general status and age of the patient, and also the type of fracture. We consider it appropriate to analyze distal femoral fractures broadly distinguishing between those with and without articular involvement. Articular involvement has requirements for adequate fixation that differ from extraarticular fractures, whether comminuted or not. Anatomic articular reduction and rigid fixation is a principle that remains unchanged since first advocated by the AO [32]. This requires the use of individual screws for fragment fixation and articular restoration, which is most often followed by implantation of plates and/or intramedullary nails to unite the articular segment to the diaphysis. Within these two main categories, management decisions can be

#### **Figure 4.**

*Flowchart as a proposal for the management of distal femoral fractures. DFR: distal femoral replacement.*

made when we include other components in the analysis (those inherent to the fracture and the patient) pertinent to the treatment plan. This could work as an algorithm for proper selection of surgical approach, implants and design of the construct (**Figure 4**).