TEQ calculation

Figure 3.1: System model proposed in the thesis.

Let x_n be the training symbol of transmission system and r_n be the received signal at the transmitter. the dash line block represents TEQ will be added into the system after calculation of the method. Since the work is done on the VDSL system at training stage, we will have a introduction about the signal. VDSL transceiver uses a Frequency Division Duplex (FDD) to separate upstream and downstream transmission. In the trial standard [10], the frequency plan consists of two upstream bands are denoted by 1U, 2U and two downstream bands are denoted by 1D, 2D. Two upstream bands and two downstream bands as shown in Fig. 3.2. The values of the splitting frequencies f_i are given in Table. 3.1.

Table 3.1: VDSL band separating frequency

Separating Frequencies f₀ f₁ f₂ f₃ f₄ f₅ (MHz) 0.25 0.138 3.75 5.2 8.5 12

The optional band between 25 kHz and 138 kHz is to be negotiated during

Figure 3.2: VDSL band allocation.

FDD schemes, each tone is used for either downstream or upstream, but not simultaneously. For example, in upstream application, zero are padded in the downstream tones. In this case, the downstream tones are referred to as the null tones. Similarly, in downstream application, the upstream tones are referred to as the null tones. In the training VDSL symbol, some of the data tones are reserved for pilots and the others used for transmitting Special Operation Channel (SOC) message. In the training stage, even tones are reserved and constellation point of 00 is transmitted on even tones. A Special Operation Channel (SOC) message which carries one byte of information is transmitted in every DMT training symbol. The bit mapping of training symbol is given in the Tab. 3.2 .

Table 3.2: Training symbol bit mapping Tone index Constellation point

Even 00

1, 11, 21,· · · ,10n+1,· · · SOC message bits 0,1 3, 13, 23,· · · ,10n+3,· · · SOC message bits 2,3 5, 15, 25,· · · ,10n+5,· · · SOC message bits 4,5 7, 17, 27,· · · ,10n+7,· · · SOC message bits 4,5 9, 19, 29,· · · ,10n+9,· · · 00

The selected constellation points shall be pseudo-random rotated by 0, π/2, π or 3π/2, and the sequence is reset at every DMT symbol. Once the message bit is generated, it is mapped into 4QAM constellation and the encode the constel-lation point in to the complex vector z size (M/2), half of the DFT size(M/2) is

considered to be integer since we use FFT process and the input is the power of 2). Before the Modulation by the IDFT, we must map z into a double sized s in order to generate a complex-to-real IDFT, thus the vector s is in the form,

s_i = z_i, i = 0,· · · , (M/2 − 1) (3.1) s_i = conj(z_M−i), i = M/2,· · · , (M − 1) (3.2) where conj(·) denotes the conjugate of the complex value, and si, z_i denotes the ith element of s, z. Thus we could have a real vector after the IDFT block.

We summarize the properties of VDSL training symbol, there are Null tones in frequency domain, and there are constant part of signal, this enable us to get rid of the noise and knowing where part of the ISI are in frequency domain.

When the channel order is smaller than the length of the cyclic prefix, we know there is no IBI (inter-block interference) after removing guard samples (cyclic removal). In the absence of channel noise, the outputs of the DFT matrix at the receiving end are the scaled versions of the transmitter inputs. The scalars are the M -point DFT of the channel impulse response. In this case, the null tones will be nothing but channel noise. However, if the channel order is larger than the length of cyclic prefix, there will be IBI even after removing guard samples.

The output of the null tones now has not only channel noise but also interference from the data tones of the previous block due to IBI (assuming channel order is smaller than N , the length of one block). We observe that the outputs of the null tones will be small if TEQ has effectively shortened the channel.

In this thesis, we propose to a semi-blind TEQ design method for VDSL systems by minimizing the ISI present in the null tones. The design does not require the channel impulse response. To be more specific, suppose the number of data tones is M_d and the number of null tones is M_n, where M = M_d+ M_n. The numbers M_d, M_n are determined by the spectral plan. Considering the i-th output block, we collect the outputs of the data tones and the outputs of the null tones respectively in vectors di and ni, for i = 1, 2,· · · , B, where B denotes the number of received output block available for equalizer design. The dimensions

of d_i and n_i are respectively M_d and M_n. We compute the averaged vectors,

d = 1 B

B i=1

d_i,

n = 1 B

B i=1

n_i.

We note that the averaged null tone vector n is mostly interference from data tones as averaging remove most noise. Moreover the interference comes mainly from pilot tones because the symbols in message-bearing tones are different from block to block. Two objective functions will used here.

φ₁ = n^†n. (3.3)

φ₂ = d^†d

n^†n. (3.4)

In the first case, we will optimize the TEQ to minimize interference in null tones, characterized by φ₁. In the second case, we will find TEQ to maximize the ratio of data tone energy over the null tone energy. In both cases, the TEQ is constrained to have unit energy, i.e.,

i=0|t(i)|² = 1.

Remarks. Notice that our method is semi-blind. Namely, the receiver knows the training symbol contains pilots tones but it knows neither the channel impulse response nor the spectrum of channel noise. We can compare our method to the blind channel equalization method in [9]. The close form equalizer solution requires the channel impulse response and the second order statistics of channel noise in [9].

In what follows, we will see that the objective functions in (3.3) and (3.4) can be formulated as quadratic terms of the TEQ coefficients and the problem can be solved elegantly by computing eigen vectors of appropriately defined positive definite matrices. Suppose the TEQ has order T ,

T (z) = T

i=0

t(i)z⁻ⁱ.

The output of the TEQ can be written as

x(n) = T

=0

t()r(n− ).

Let the i-th intput vector of the DFT matrix be

x_i =

Then x_k can be written in terms of TEQ coefficients as

x_i = i-th data tone vector d_i can be expressed as

d_i = W₁x_i,

where W₁ is an M_d× M submatrix of the M × M DFT matrix W, obtained by removing the rows that correspond to the null tones. Similarly, we can express the null tone vector n_i as

n_i = W₂x_i,

where where W₂is an M_n×M submatrix of the M ×M DFT matrix W, obtained by removing the rows that correspond to the data tones. Using (3.5), we can write d_i and n_i respectively as

di = W₁Rit, ni = W₂Rit.

Using these expressions, we have

where

R = 1 B

B i=1

R_i. Therefore, we have

d^†d = t^†RW^†₁W₁R

  

A

t = t^†At,

n^†n = t^†RW^†₂W₂R

  

B

t = t^†Bt,

where A and B are square matrices of size (T + 1). Also, both matrices are positive definite. The function functions given in (3.3) and (3.4) become

φ₁= t^†Bt, φ₂ = t^†At

t^†Bt.

Now both objective functions are written as quadratic forms of the TEQ coeffi-cients. The energy constraint 

i=0|t(i)|² = 1 becomes t^†t = 1.

Optimal solutions

• Objection function φ1. The problem of We can use Rayleigh’s principle to minimize φ₁ subject to the constraint t^†t = 1. The optimal t is the eigen vector corresponding to the smallest eigen value of B.

• Objection function φ2. We can use two methods to find the optimal t that maximize φ₂ subject to the constraint t^†t = 1.

– Method 1: As B is positive definite, we can write decompose B as B = C^−†C⁻¹. Then φ₂ can written as the ratio φ₂ = _t†^t_C^†At_†Ct. Let u = C⁻¹t, then t = Cu and

φ₂ = u^†C^†ACu u^†u .

Using Rayleigh’s principle, φ₂ can be maximized by choosing u to be the eigen vector corresponding to the largest eigen value of C^†AC.

– Method 2: The design problem can be stated as “maximize d^†d.

Solving 3.6 leads to a TEQ that satisfies the generalized eigenvector problem

At = λBt (3.7)

The solution for t is the eigenvector corresponding to the largest gen-eralized eigenvalue of (B⁻¹A).

Remarks on method 1 and method 2. The same TEQ solution could be found by both methods, however method 2 has a computation advan-tage to method 1, since method 2 only needs a matrix inversion to form the appropriate defined positive definite matrix instead of one cholosky decom-position, matrix inverse, and two matrix multiplication of method 1. The computation effort is reduced.

3.1 Design Procedure

In our proposed TEQ design, we need to compute M × (T + 1) matrix R and (T + 1)× (T + 1) matrix A = R^†W^†₁W₁R, B = R^†W₂^†W₁R. These matrices can be computed efficiently as detailed below.

Efficient computation of R. Observe that the entries of R are drawn from the (M + T )× 1 vector r given by

Therefore the matrix R can be obtained by simply computing the average received vector r.

Efficient computation of A and B. Notice that A = E^†₁E₁, B = E^†₂E₂, where

E₁ = W₁R, E₂ = W₂R.

and E₁, E₂ can be formed from collecting the rows of WR, where E₁ corresponds to data tones and E₂ corresponds to the null tones. We could also observe the column of R is just a data shifting with two different samples from the previous column, we gave a example of fast calculating E₁.

E₁ = (e₀ e₁ · · · eT) R = (r₀ r₁ · · · rT)

where e_i,r_i represents the ith column of E₁,R. the first column e₀ could be calculated as

e₀ = W₁r₀

which is the DFT of r₀ and collects the elements corresponds to the data tones.

The next column is a data shift and two different samples from the previous one which we could observe from



thus we could have the relation of where ⊗ denotes component wise production. and w^−kl denotes the (k, l) com-ponent of DFT matrix. We can summarize the design procedure for TEQ opti-mization using φ₂ as follows. The design using φ₁ is similar.

1. Collect received signal and compute the average received vector r given in (3.8).

2. Obtain A and B by first computing E₁ = W₁R and E₂ = W₂R using DFT, and then computing A = E^†₁E₁ and B = E^†₂E₂.

3. Compute B⁻¹A.

4. Obtain the optimal t by computing the eigen vector corresponding to the largest eigen value of B⁻¹A.

The choice of ∆. It is the effective channel that we are dealing with, thus we are choosing the best SIR window delay of a equalized channel that is not revealed. However, from the nature of the effective channel it’s main impulse is still near the peak of the original channel, and we place the center of SIR window where the effective channel main impulse is and most of the power of effective channel is inside the SIR window. The delay can be choose as

 = gp− L

2 (3.10)

where gp is the group delay of original channel and denoted as the estimation of main peak of original channel.

Chapter 4