**2. System model and notation**

2 Advances in Discrete Time Systems

*bit*

QAM input

of interest.

*stream* S / P IFFT

serial bits in the frequency-domain.

**Figure 1.** Block Diagram of a Discrete Multitone System.

P / S + **CP**

*x (n)* CIR *h*

the channel with impulse response is generally corrupted by the additive white Gaussian noise and near-end crosstalk. The received signal is also equalised by PTEQs without TEQ concerned. The per-tone equalisation structure is based on transferring the TEQ-operations into the frequency-domain after FFT demodulation, which results in a multitap PTEQ for each tone separately. Then, the parallel of received symbols are eventually converted into

> AWGN+NEXT **n**k

It is a well known issue that in DMT theory, there is no overlapping between tones due to orthogonality derived from the discrete Fourier transformation among them. In practice, frequency-selective fading channel generally destroys such orthogonal structure leading to information interfering from adjacent tones as commonly known as ICI. In such case, information supposedly belonging to a particular tone generally smear into adjacent tones and leave some residual energy in them. The idea of a mixed-tone PTEQ for DMT-based system has been proposed in [9]. By recovering adaptively the knowledge of residual interfering signal energy from adjacent tones, the mixed-tone exponentially weighted least squares criterion can be shown to offer an improved signal to noise ratio (SNR) of the tone

In order to improve the convergence properties, the orthogonal gradient adaptive (OGA) has been presented by introducing orthogonal projection to the filtered gradient adaptive (FGA) algorithm. When the forgetting-factor is optimised sample by sample whereas a fixed forgetting-factor is used for FGA algorithm [10]. A normalised version of the OGA (NOGA) algorithm that has been introduced with the mixed-tone cost function and fixed step-size presented in [11]. With the purpose of the good tracking behaviour and recovering to a steady-state, it is necessary to let the step-size automatically track the change of system. Consequently, the concept of low complexity adaptive step-size approach based on the FGA

algorithm is introduced for the per-tone equalisation in DMT-based systems in [12].

step-size parameter relating to the mean convergence factor.

In this chapter, the focus is therefore to present low complexity orthogonal gradient-based algorithms for PTEQ based on the adaptive step-size approaches related to the mixed-tone criterion. The convergence behaviour and stability analysis of proposed algorithms will be investigated based on the mixed-tone weight-estimated errors. The convergence analysis of mechanisms will be carried out the steady-state and mean-square expressions of adaptive

*y (n)* S / P + **CP**

sliding FFT

**PTEQs**

P / S QAM output

*bit stream* In this section, the basic structure of the DMT transceiver is illustrated in Fig. 1. We describe that the data model and notation based on an FIR model of the DMT transmission channel is presented as [7]

$$\mathbf{y} = \mathbf{H} \cdot \mathbf{X} + \mathbf{n} \quad \text{(}$$

$$\underbrace{\begin{bmatrix} y\_{k,l+\Delta} \\ \vdots \\ y\_{k,N-l+\Delta} \end{bmatrix}}\_{\mathbf{Y}\_{k\bar{l}+\Delta N-1+\Delta}} = \underbrace{\begin{bmatrix} \left\| \begin{bmatrix} \bar{\mathbf{h}}^{T} \end{bmatrix} \mathbf{0} \right\| \cdot \cdots \cdot \left| \begin{bmatrix} \mathbf{0} \\ \mathbf{0}\_{2} \end{bmatrix} \mathbf{0} \right| \\\ \mathbf{0}\_{1} \cdot \mathbf{0} \left[ \begin{bmatrix} \mathbf{0}\_{2} \end{bmatrix} \right] \mathbf{0} \end{bmatrix} \cdot \begin{bmatrix} \mathcal{P}\_{\nu} & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & \mathcal{P}\_{\nu} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & \mathcal{P}\_{\nu} \end{bmatrix} \cdot \begin{bmatrix} \mathcal{Z}\_{N} & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & \mathcal{Z}\_{N} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & \mathcal{Z}\_{N} \end{bmatrix}}\_{\mathbf{H}}.$$

$$\underbrace{\begin{bmatrix} \mathbf{x}\_{k-1,N} \\ \mathbf{x}\_{k,N} \\ \mathbf{x}\_{k+1,N} \end{bmatrix}}\_{\mathbf{X}\_{k-1,k+1} \times \mathbf{M}} + \underbrace{\begin{bmatrix} \eta\_{k,l+\Delta} \\ \vdots \\ \eta\_{k,N-l+\Delta} \end{bmatrix}}\_{\mathbf{M}\_{l+\Delta N-l-1+\Delta}}.\tag{1}$$

where *l* denotes as the first considered sample of the *k*-th received DMT-symbol. This depends on the number of tap of equaliser (T) and the synchronisation delay (∆). The vector **y***k*,*i*:*<sup>j</sup>* of received samples *i* to *j* of *k*-th DMT-symbol is as **y***k*,*i*:*<sup>j</sup>* = [*yk*,*<sup>i</sup>* ··· *yk*,*j*] *<sup>T</sup>*. A sequence of the *N* × 1 **x***k*,*<sup>N</sup>* transmitted symbol vector is as **x***k*,*<sup>N</sup>* = [*xk*,0 ··· *xk*,*N*−1] *<sup>T</sup>*. The size *N* is of inverse discrete Fourier transform (IDFT) and DFT. The parameter *ν* denotes as the length of cyclic prefix. The matrices **0**(1) and **0**(2) are also the zero matrices of size (*N* − *l*) × (*N* − *L* + 2*ν* + ∆ + *l*) and (*N* − *l*) × (*N* + *ν* − ∆). The vector **h**¯ is the **h** channel impulse responce (CIR) vector in reverse order. The (*N* + *ν*) × *N* matrix P*<sup>ν</sup>* is denoted by

$$\mathcal{P}\_{\boldsymbol{\nu}} = \begin{bmatrix} \mathbf{0}\_{\boldsymbol{\nu} \times (N-\boldsymbol{\nu})} & \mathbf{I}\_{\boldsymbol{\nu}}\\ \hline & \mathbf{I}\_{N} \end{bmatrix} \boldsymbol{\nu}$$

which adds the cyclic prefix. The I*<sup>N</sup>* is *N* × *N* IDFT matrix and modulates the input symbols. The η*k*,*l*+∆:*N*−1+<sup>∆</sup> is a vector with additive white Gaussian noise (AWGN) and near-end cross-talk (NEXT).

Some notation will be used throughout this chapter as follows: *E*{·} is the expectation operator and diag(·) is a diagonal matrix operator. The operators (·)*T*, (·)*H*, (·)<sup>∗</sup> denote as the transpose, Hermitian and complex conjugate operators, respectively. The parameter *k* is the DMT symbol index and **I***<sup>a</sup>* is an *a* × *a* identity matrix. A tilde over the variable indicates the frequency-domain. The vectors are in bold lowercase and matrices are in bold uppercase.

#### **3. Per-tone equalisation**

In this section, we show the concept of per-tone equaliser (PTEQ). We refer the readers to [7] for more details. The per-tone equalisation structure is based on transferring the TEQ-operations into the frequency-domain after DFT demodulation, which results in a T-tap PTEQ for each tone separately. For each tone *i* (*i* = 1... *n*), the TEQ-operations are shown as follows [7]

$$\vec{d}\_{\rm nl} = \overbrace{\vec{z}\_{\rm n}}^{\text{1-top PTEQ}} \cdot \overbrace{\text{row}\_{\rm n}(\mathcal{F}\_{\rm N}) \cdot (\mathbf{Y} \cdot \mathbf{w})}^{\text{1DFT}} \tag{2}$$

10.5772/52158

141

http://dx.doi.org/10.5772/52158

*<sup>m</sup>***p**ˆ *<sup>m</sup>* are illustrated in two dimentional

**Figure 2.** The tap-weight estimated PTEQ vector **p**ˆ *<sup>m</sup>* and its orthogonal projection Π<sup>⊥</sup>

1 y

y*n*

 

*<sup>J</sup>*(*k*) = <sup>1</sup> 2

*m* for *m* ∈ *M*. The number of the adjacent tones *M* is of tone of interest.

y*m*

1 y*m*

2 y*m*

subspace *S*<sup>2</sup> [9].

tones, where *M* = 3. [11]

as

 

 ? <sup>1</sup>  ? ?  1

Adaptive Step-Size Orthogonal Gradient-Based Per-Tone Equalisation in Discrete Multitone Systems

pˆ *m m* A 3

*mx*

*m* [

*<sup>λ</sup>k*−*<sup>i</sup> <sup>m</sup>* {*ξm*(*i*)} 2, (6)

pˆ *m*

<sup>1</sup> pˆ *<sup>m</sup>*

<sup>2</sup> pˆ *<sup>m</sup>*

**Figure 3.** Block structure of the proposed mixed-tone PTEQ **p**ˆ *<sup>m</sup>* for *m* ∈ *M* with the use of combining estimates of *M*-adjacent

A mixed-tone exponentially weighted least squares cost function to be minimised is defined

*k* ∑ *i*=1

where *λ<sup>m</sup>* is the forgetting-factor and *ξm*(*k*) is the mixed-tone weight-estimated error at tone

*M* ∑ *m*=1

1 1 pˆ *m m* A 3

2 2 pˆ *m m* A 3

$$=\text{row}\_{\boldsymbol{\mathfrak{u}}}(\underbrace{\boldsymbol{\mathcal{F}}\_{\boldsymbol{N}}\cdot\mathbf{Y}}\_{\text{T DFTs}})\cdot\underbrace{\mathbf{w}\cdot\boldsymbol{\tilde{z}}\_{\boldsymbol{\mathfrak{u}}}}\_{\text{T}-\text{tap PTEQ }\mathbf{v}\_{\boldsymbol{\mathfrak{u}}}}\tag{3}$$

where ˜*dn* is the output after frequency-domain equalisation for tone *n*. The *z*˜*<sup>n</sup>* is the (complex) one-tap PTEQ for tone *n*. The parameter **w** is of (real) T-tap TEQ and F*<sup>N</sup>* is an *N* × *N* DFT matrix [7]. Note that **Y** is an *N* × T Toeplitz matrix of received signal samples as vector **y** in (1). From (3), the T DFT-operations are cheaply calculated by means of a sliding DFT. It is demonstrated in [7] that every T-tap PTEQ **v***<sup>n</sup>* exists a T-tap PTEQ **p**˜ *<sup>n</sup>* which consists of only one DFT and T − 1 real difference terms as its input.

The PTEQ output *<sup>x</sup>*ˆ*k*,*<sup>n</sup>* can be specified as follows

$$\mathbf{x}\_{k,n} = \tilde{\mathbf{p}}\_n^H \cdot \underbrace{\begin{bmatrix} \mathbf{I}\_{\mathbf{T}-1} \begin{bmatrix} \mathbf{0} & \mathbf{\cdot} \mathbf{I}\_{\mathbf{T}-1} \\ \mathbf{0} & \mathbf{\cdot} \mathbf{\cdot} \mathbf{\cdot} \end{bmatrix} \cdot \mathbf{y}}\_{\mathbf{E}\_n} \cdot \mathbf{y} \tag{4}$$

$$
\tilde{\mathbf{p}} = \tilde{\mathbf{p}}\_n^H \cdot \tilde{\mathbf{y}}\_{k,n} \,\prime \tag{5}
$$

where **p**˜ *<sup>n</sup>* is the T-tap complex-valued PTEQ vector for tone *n*. The **F***<sup>n</sup>* is a (T − 1) × (*N* + T − 1) matrix [7]. The F*N*(*n*, :) is the *nth* row of F*N*. By using the sliding DFT, the first block row of matrix **F***n* in (4) extracts the difference terms, while the last row corresponds to the usual DFT operation as detailed in [7] and [13]. The vector **y** is of channel output samples as described in (1). The ˜**y***k*,*<sup>n</sup>* is the sliding DFT output for tone *<sup>n</sup>* at each symbol *<sup>k</sup>*.

**F***n*

#### **4. A mixed-tone cost function**

In this section, we describe a mixed-tone cost function by means of the orthogonal projection matrix. The idea of using orthogonal projection of adjacent equalisers to include the information of interfering tones has been presented firstly in [9]. The illustration of the vector **p**ˆ *<sup>m</sup>* and its orthogonal projection as well as **x**˜*<sup>m</sup>* for 2-dimensional subspace *S*<sup>2</sup> is shown in Fig. 2. The error vector **e**<sup>⊥</sup> *m* associated with the orthogonal projection of vectors **p**ˆ *<sup>m</sup>* and Π<sup>⊥</sup> *<sup>m</sup>***p**ˆ *<sup>m</sup>*, where Π<sup>⊥</sup> *<sup>m</sup>* denotes as the orthogonal projection matrix of **p**ˆ *<sup>m</sup>*(*k*), will be presented in the update of the vector **p**ˆ *<sup>m</sup>*(*k*) where *k* �= *m*. Therefore, the mixed-tone cost function derived as the sum of weight-estimated errors is optimised in order to achieve the solutions for frequency-domain equalisation. It is designed to work in conjunction with the complex-valued frequency-domain equalisation structure.

4 Advances in Discrete Time Systems

follows [7]

TEQ-operations into the frequency-domain after DFT demodulation, which results in a T-tap PTEQ for each tone separately. For each tone *i* (*i* = 1... *n*), the TEQ-operations are shown as

> 1 DFT

· **w** · *z*˜*<sup>n</sup>* T−tap PTEQ **v***<sup>n</sup>*

(F*N*) · (**Y** · **w**) , (2)

, (3)

·**y** , (4)

˜*dn* =

The PTEQ output *<sup>x</sup>*ˆ*k*,*<sup>n</sup>* can be specified as follows

**4. A mixed-tone cost function**

shown in Fig. 2. The error vector **e**<sup>⊥</sup>

*<sup>m</sup>***p**ˆ *<sup>m</sup>*, where Π<sup>⊥</sup>

complex-valued frequency-domain equalisation structure.

**p**ˆ *<sup>m</sup>* and Π<sup>⊥</sup>

*<sup>x</sup>*ˆ*k*,*<sup>n</sup>* <sup>=</sup> **<sup>p</sup>**˜ *<sup>H</sup> n* · 

= **<sup>p</sup>**˜ *<sup>H</sup>*

1-tap PTEQ *z*˜*<sup>n</sup>* ·row*<sup>n</sup>*

= row*n* (F*N* · **Y**)

which consists of only one DFT and T − 1 real difference terms as its input.

 T DFTs

where ˜*dn* is the output after frequency-domain equalisation for tone *n*. The *z*˜*<sup>n</sup>* is the (complex) one-tap PTEQ for tone *n*. The parameter **w** is of (real) T-tap TEQ and F*<sup>N</sup>* is an *N* × *N* DFT matrix [7]. Note that **Y** is an *N* × T Toeplitz matrix of received signal samples as vector **y** in (1). From (3), the T DFT-operations are cheaply calculated by means of a sliding DFT. It is demonstrated in [7] that every T-tap PTEQ **v***<sup>n</sup>* exists a T-tap PTEQ **p**˜ *<sup>n</sup>*

> **I**T−<sup>1</sup> **0 -I**T−<sup>1</sup> **0** F*N*(*n*, :)

 **F***n*

where **p**˜ *<sup>n</sup>* is the T-tap complex-valued PTEQ vector for tone *n*. The **F***<sup>n</sup>* is a (T − 1) × (*N* + T − 1) matrix [7]. The F*N*(*n*, :) is the *nth* row of F*N*. By using the sliding DFT, the first block row of matrix **F***n* in (4) extracts the difference terms, while the last row corresponds to the usual DFT operation as detailed in [7] and [13]. The vector **y** is of channel output samples as

In this section, we describe a mixed-tone cost function by means of the orthogonal projection matrix. The idea of using orthogonal projection of adjacent equalisers to include the information of interfering tones has been presented firstly in [9]. The illustration of the vector **p**ˆ *<sup>m</sup>* and its orthogonal projection as well as **x**˜*<sup>m</sup>* for 2-dimensional subspace *S*<sup>2</sup> is

presented in the update of the vector **p**ˆ *<sup>m</sup>*(*k*) where *k* �= *m*. Therefore, the mixed-tone cost function derived as the sum of weight-estimated errors is optimised in order to achieve the solutions for frequency-domain equalisation. It is designed to work in conjunction with the

described in (1). The ˜**y***k*,*<sup>n</sup>* is the sliding DFT output for tone *<sup>n</sup>* at each symbol *<sup>k</sup>*.

*<sup>n</sup>* · **y**˜ *<sup>k</sup>*,*<sup>n</sup>* , (5)

*m* associated with the orthogonal projection of vectors

*<sup>m</sup>* denotes as the orthogonal projection matrix of **p**ˆ *<sup>m</sup>*(*k*), will be

**Figure 2.** The tap-weight estimated PTEQ vector **p**ˆ *<sup>m</sup>* and its orthogonal projection Π<sup>⊥</sup> *<sup>m</sup>***p**ˆ *<sup>m</sup>* are illustrated in two dimentional subspace *S*<sup>2</sup> [9].

**Figure 3.** Block structure of the proposed mixed-tone PTEQ **p**ˆ *<sup>m</sup>* for *m* ∈ *M* with the use of combining estimates of *M*-adjacent tones, where *M* = 3. [11]

A mixed-tone exponentially weighted least squares cost function to be minimised is defined as

$$J(k) = \frac{1}{2} \sum\_{m=1}^{M} \sum\_{i=1}^{k} \lambda\_m^{k-i} \{\xi\_m(i)\} \,^2 \prime \tag{6}$$

where *λ<sup>m</sup>* is the forgetting-factor and *ξm*(*k*) is the mixed-tone weight-estimated error at tone *m* for *m* ∈ *M*. The number of the adjacent tones *M* is of tone of interest.

$$\begin{aligned} \boldsymbol{\upleft}\boldsymbol{\upzeta}\_{m}(i) &= \boldsymbol{\uppi}\_{m}(i) - \boldsymbol{\uphat{\mathbf{p}}}\_{m}^{H}(k)\boldsymbol{\uphat{\mathbf{y}}}\_{m}(i) - \left(\boldsymbol{\Pi}\_{l}^{\perp}(k)\boldsymbol{\uphat{\mathbf{p}}}\_{l}(k)\right)^{H}\boldsymbol{\uphat{\mathbf{y}}}\_{l}(i) \\ &- \left(\boldsymbol{\upPi}\_{l+1}^{\perp}(k)\boldsymbol{\uphat{\mathbf{p}}}\_{l+1}(k)\right)^{H}\boldsymbol{\uphat{\mathbf{y}}}\_{l+1}(i) \\ &- \dots - \left(\boldsymbol{\upPi}\_{L}^{\perp}(k)\boldsymbol{\uphat{\mathbf{p}}}\_{L}(k)\right)^{H}\boldsymbol{\uphat{\mathbf{y}}}\_{L}(i) \end{aligned} \tag{7}$$
 
$$\text{for } m \neq l, \ L \le M - 1. \tag{7}$$

where the parameter *<sup>x</sup>*˜*m*(*k*) is the *<sup>k</sup>th* transmitted DMT-symbol on tone *<sup>m</sup>*. The vector **<sup>p</sup>**<sup>ˆ</sup> *<sup>m</sup>*(*k*) is of complex-valued T-tap PTEQ for tone *m*. The vector ˜**y***m*(*k*) is the DFT output for tone *m* at symbol *k*.

The orthogonal projection matrix Π<sup>⊥</sup> *<sup>l</sup>* (*k*) which is the matrix difference determined by the tap-weight estimated vector **p**ˆ*l*(*k*) as [14]

$$\begin{split} \Pi\_{l}^{\perp}(k) &= \mathbf{\tilde{I}} - \hat{\Pi}\_{l}(k) \\ &= \mathbf{\tilde{I}} - \mathbf{\hat{p}}\_{l}(k) \left[ \mathbf{\hat{p}}\_{l}^{H}(k) \, \mathbf{\hat{p}}\_{l}(k) \right]^{-1} \, \mathbf{\hat{p}}\_{l}^{H}(k) \end{split} \tag{8}$$

10.5772/52158

143

http://dx.doi.org/10.5772/52158

**5.1. A Mixed-Tone Normalised Orthogonal Gradient Adaptive (MT-NOGA)**

The orthogonal gradient adaptive (OGA) algorithm is formulated from the FGA algorithm [10] by introducing an orthogonal constraint between the present and previous direction vectors [17]. This OGA algorithm employs the optimised forgetting-factor on a sample-by-sample basis, so that the direction vector is orthogonal to the previous direction

Adaptive Step-Size Orthogonal Gradient-Based Per-Tone Equalisation in Discrete Multitone Systems

We then demonstrate the derivation of the mixed-tone normalised orthogonal gradient adaptive (MT-NOGA) algorithm for PTEQ in DMT-based systems. With this mixed-tone criterion in Section 4, the tap-weight estimate vector **p**ˆ *<sup>m</sup>*(*k*) at symbol *k* for *m* ∈ *M* is given

where *µm*(*k*) is the step-size parameter and **d***m*(*k*) is the T × 1 direction vector.

**<sup>g</sup>***m*(*k*) = −∇**p**<sup>ˆ</sup> *<sup>m</sup>*(*k*)*J*(*k*)

<sup>=</sup> <sup>−</sup>*ξm*(*k*) *∂ξm*(*k*)

*L* ∑ *l*=1 Π⊥

where *ξm*(*k*) is the *a priori* mixed-tone weight-estimated error at symbol *k* for *m* ∈ *M* as

**<sup>g</sup>***m*(*k*) = *<sup>λ</sup>m*(*k*)**g***m*(*<sup>k</sup>* <sup>−</sup> <sup>1</sup>) + **<sup>y</sup>**˜ *<sup>m</sup>*(*k*)*ξ*<sup>∗</sup>

A procedure of an orthogonal gradient adaptive (OGA) algorithm to determine *λm*(*k*) has been described in [17] by projecting the gradient vector **g***m*(*k*) onto the previous direction

**d***m*(*k*) = *λm*(*k*) **d***m*(*k* − 1) + **g***m*(*k*)

where **g***m*(*k*) is the negative gradient of cost function *J*(*k*) in (6) and *λm*(*k*) is the

By differentiating *J*(*k*) in (6) with respect to **p**ˆ *<sup>m</sup>*(*k*), we then get the gradient vector **g***m*(*k*) as

*<sup>∂</sup>***p**<sup>ˆ</sup> *<sup>m</sup>*(*k*) <sup>=</sup> **<sup>y</sup>**˜ *<sup>m</sup>*(*k*) *<sup>ξ</sup>*<sup>∗</sup>

*H*

*<sup>l</sup>* (*k*)**p**ˆ*l*(*k*)

*<sup>m</sup>*(*k*) is the complex conjugate of the mixed-tone estimated error at symbol *k* for

The direction vector **d***m*(*k*) can be obtained recursively as

*<sup>m</sup>* (*k* − 1)**y**˜ *<sup>m</sup>*(*k*) −

We introduce the updating gradient vector **g***m*(*k*) by

**p**ˆ *<sup>m</sup>*(*k*) = **p**ˆ *<sup>m</sup>*(*k* − 1) + *µm*(*k*) **d***m*(*k*) , (10)

= *<sup>λ</sup>m*(*k*) **<sup>d</sup>***m*(*<sup>k</sup>* − <sup>1</sup>) − ∇**p**<sup>ˆ</sup> *<sup>m</sup>*(*k*)*J*(*k*) , (11)

*<sup>m</sup>*(*k*) . (12)

**y**˜*l*(*k*) , for *m* �= *l* , *L* ≤ *M* − 1. (13)

*<sup>m</sup>*(*k*) , (14)

**algorithm**

vector.

adaptively as

forgetting-factor at symbol *k*.

*ξm*(*k*) = *x*˜*m*(*k*) − **p**ˆ *<sup>H</sup>*

*m* ∈ *M* as given in (13).

where *ξ*<sup>∗</sup>

where ˜ **I** denotes as an identity matrix and Πˆ *<sup>l</sup>*(*k*) is the projection matrix onto the space spanned by the tap-weight vector **p**ˆ*l*(*k*). We note that the orthogonal projection matrix Π<sup>⊥</sup> *<sup>l</sup>* (*k*) is mentioned by the vector **p**ˆ*l*(*k*) for *l* �= *m*.

With the definition for this cost function, the *mth*-term on the right hand side of (9) represents as the estimated mixed-tone error of the symbol *<sup>k</sup>* due to the *<sup>m</sup>th*-tone of equaliser **<sup>p</sup>**<sup>ˆ</sup> *<sup>m</sup>*(*k*) for *m* ∈ *M* as depicted in Fig. 3.

$$\mathfrak{F}\_{\mathfrak{M}}(i) = \mathfrak{X}\_{\mathfrak{M}}(i) - \mathfrak{p}\_{\mathfrak{m}}^{H}(k)\mathfrak{y}\_{\mathfrak{m}}(i) - \sum\_{l=1}^{L} \left(\Pi\_{l}^{\perp}(k)\mathfrak{p}\_{l}(k)\right)^{H}\mathfrak{y}\_{l}(i) \text{ , for } m \neq l \text{ , } L \le M - 1. \tag{9}$$
