Simulation Results

In this section, we use several examples to demonstrate the performance of the proposed method. The channel normalized root-mean-square error (NRMSE) is defined as

NRMSE = 1 estimate of channel impulse response matrix H after removing the unitary matrix ambiguity by the least squares method [21]. I = 100 is the number of Monte Carlo runs. The input source symbols are independent and identically distributed (i.i.d.) QPSK signals. The channel noise is temporally and spatially white Gaussian. The signal-to-noise ratio (SNR) at the output is defined as SNR = signal component of the received signal (see Figure 2.1).

1) Simulation 1 – optimal selection of precoding sequences In this simulation, we use the following model

H(z) =

to demonstrate the effect of different precoding sequences on the performance of the pro-posed method. In experiment 1, the first sequence is chosen as {0.767 1.07 1.07 1.07}, which satisfies (2.28) and (2.29). The second and third sequences are chosen based on

(2.30) for P = 4 and τ = 0.5878 with the two possible peak positions: m = 0 and m = 3.

By computation, the corresponding µ for the three cases are 40.0, 4.66 and 22.1, respec-tively. Thus m = 0 is the optimal selection. Figure 2.2 shows that for SNR=10 dB, there are about 5∼7 dB and 5∼9 dB difference in NRMSE between the optimal one and two others.

In experiment 2, we use the precoding sequences that satisfy (2.30) with m = 0, but with different τ to test the effect of τ on the identification performance. Figure 2.3 shows that for each sequence, when the number of samples (for each transmitter) is fixed at 1000, the NRMSE decreases as SNR increases and is roughly constant for SNR≥ 20 dB. A possible explanation is that for sufficiently large SNR, the NRMSE is contributed mainly by numerical error rather than by channel noise. Figure 2.3 also shows that the identification performs better for smaller τ , which is consistent with the conclusion at the end of Section 2.3.1.

2) Simulation 2 – channel order overestimation

In this simulation, we use the following channel model H(z) = dB, and 1000 samples (for each transmitter) for simulation. The precoding sequences are chosen as (2.30) with m = 0 and τ = 0.2, 0.4, 0.6, and 0.8. Figure 2.4 shows the NRMSE increases with increasing channel order overestimation. We see the proposed method is quite robust to channel order overestimation when τ is small. For example, with τ = 0.4, when ( ˆL− L) increases from 0 to 3, the NRMSE increases from -25.5dB to -21dB, which is still a low value.

3) Simulation 3 – a 3-input 2-output channel

In this simulation, we use the 3-input 2-output model H(z) = to illustrate the performance of the proposed method for channel with more transmitters than receivers. Note that H is full column rank, but the channel is not irreducible [21]

because H(0) is not full rank, and it is not column reduced [21] either because H(2) is not full rank. In experiment 1, the precoding sequences (P = 4) are given as in (2.30) with m = 0 and m = 3, respectively. Figure 2.5 shows that the NRMSE decreases as the number of data samples increases for SNR=10 dB. As expected, m = 0 case (the optimal selection) is better than m = 3 case.

In experiment 2, we use the precoding sequences that satisfy (2.30) with m = 0, but with different τ to test the effect of τ on the identification performance. Figure 2.6 shows that for each sequence, when the number of samples (for each transmitter) is fixed at 1000, the NRMSE decreases as SNR increases and is roughly constant for SNR ≥ 25 dB due to numerical error. Figure 2.6 also shows the identification performs better for smaller τ .

4) Simulation 4 – channel equalization performance

In this simulation, we use the channel model given in (2.34) to demonstrate the perfor-mance of the proposed method for channel equalization. We use the precoding sequences that satisfy (2.30) with m = 0, but with different τ to test the effect of τ on the equalization performance. For simplicity, we use the minimum mean square error (MMSE) equalizer.

The equalizer is a 17-tap Wiener filter with 12-tap reconstruction delay whose jth output ˆ

u_j(k) is an estimate of u_j(k) for j = 1, 2,· · · , K. Since the precoding scheme is applied at the transmitter, we need to multiply ˆu_j(k) by the corresponding p(k)⁻¹ to obtain an estimate of s_j(k) for j = 1, 2,· · · , K. The number of samples is 1200. We first identify the channel using the first 400 samples and then do equalization. To obtain smoother curves, we use I = 300 as the number of Monte Carlo runs rather than 100.

Figure 2.7 shows that under low SNR, the proposed method performs better when τ is large; however, under high SNR, the proposed method performs better when τ is low. A possible explanation is as follows.

Channel estimates become more accurate as τ becomes smaller, but the gains p(k)⁻¹ =

√1

τ, k = 1, 2,· · · , P −1 become larger and result in larger noise amplification at the receiver.

Both channel estimation error and channel noise contribute to the (maximum likelihood) detection performance, i.e., the symbol error rate. In the low SNR region, the detrimental effect of noise amplification outweighs the benefit of small estimation error; whereas in the high SNR region, accurate channel estimation weighs more than the noise amplification effect. Hence we choose a small τ when SNR is high and a large τ when SNR is low.

5) Simulation 5 – Comparisons with other methods

In this simulation, we generate 100 2-input 4-output random channels with order L = 2;

each element in the channel impulse response matrix is a complex circular Gaussian random

variable with unit variance. We compare the proposed method with a generalized space time block codes (GSTBC)[23] based method. Both methods require periodic precoding sequences. For the proposed method, the precoding sequence is chosen as {1.500 0.767 0.767 0.767}; whereas the entries in the precoding sequence for the GSTBC method is chosen as random entries with modulus 1 for each random channel simulation [23]. The performance of the proposed method is also compared with a linear prediction (LP)[2, chap. 6] based method, and an outer product decomposition algorithm (OPDA)[20]. Both methods do not require a periodic precoder. MMSE equalizers are used for the proposed method, LP method, and OPDA method. For the GSTBC method, we use the customized equalizer proposed in [23]. Figure 2.8(a) shows that when the number of samples is 1200 (for each transmitter), the identification performance of the proposed method is better than those of the other three methods excepting the GSTBC method for SNR ≥ 13 dB.

However, Figure 2.8(b) shows the equalization performance of the proposed method is only better than those of the LP and OPDA methods and worse than the GSTBC method.

The inconsistency of the channel estimation and equalization performance of the proposed method and the GSTBC method for SNR≤ 13 dB may be due to the different precoding sequences and equalizers used. Figure 2.9 shows that when the number of samples is 200 (for each transmitter), the identification and equalization performance of the proposed method is better than that of the GSTBC method for SNR≤ 15 dB. Figure 2.9 shows that when the number of samples is small, the proposed method has better performance than the GSTBC method under low SNR.

100 200 300 400 500 600 700 800 900 1000

−22

−20

−18

−16

−14

−12

−10

−8

−6

Number of Samples

Channel NRMSE(dB)

m=0 m=3

non−optimal p(n)

Figure 2.2. Channel NRMSE versus number of samples

0 5 10 15 20 25 30 35 40

−35

−30

−25

−20

−15

−10

SNR(dB)

channel NRMSE(dB)

τ=0.2 τ=0.4 τ=0.6 τ=0.8

Figure 2.3. Channel NRMSE versus output SNR

0 1 2 3 4 5 6

−30

−28

−26

−24

−22

−20

−18

−16

−14

−12

−10

overestimated channel order

channel NRMSE(dB)

τ=0.2 τ=0.4 τ=0.6 τ=0.8

Figure 2.4. Channel NRMSE versus ( ˆL− L)

100 200 300 400 500 600 700 800 900 1000

−22

−21

−20

−19

−18

−17

−16

−15

−14

−13

−12

number of samples

channel NRMSE(dB)

m=0 m=3

Figure 2.5. 3-input 2-output model: channel NRMSE versus number of samples

0 5 10 15 20 25 30 35 40

−30

−28

−26

−24

−22

−20

−18

−16

−14

−12

SNR(dB)

channel NRMSE(dB)

τ=0.2 τ=0.4 τ=0.6 τ=0.8

Figure 2.6. 3-input 2-output model: channel NRMSE versus output SNR

0 5 10 15 20 25 30 35 40

10⁻⁴ 10⁻³ 10⁻² 10⁻¹ 10⁰

SNR(dB)

symbol error rate

τ=0.2 τ=0.4 τ=0.6 τ=0.8

Figure 2.7. Symbol error rate versus output SNR

0 5 10 15 20

(a) Channel NRMSE versus output SNR

0 5 10 15 20

(b) Symbol error rate versus output SNR

Figure 2.8. Comparison of NRMSE and symbol error rate, number of input samples = 1200

(a) Channel NRMSE versus output SNR

0 2 4 6 8 10 12 14 16 18 20

(b) Symbol error rate versus output SNR

Figure 2.9. Comparison of NRMSE and symbol error rate, number of input samples = 200

Chapter 3

Identification of MIMO Single Carrier Zero Padding Channels

In this chapter, we propose a blind identification method based on periodic precod-ing for another transmission systems, sprecod-ingle carrier with zero paddprecod-ing block transmission systems. The method uses periodic precoding on the source signal before transmission.

The estimation of the channel impulse response matrix consists of two steps: (1) obtain the channel product matrix by solving a lower-triangular linear system and (2) obtain the channel impulse response matrix by computing the positive eigenvalues and eigenvectors of a Hermitian matrix formed from the channel product matrix. The method is applicable to MIMO channels with more transmitters or more receivers. A sufficient condition for identi-fiability is simply that the channel impulse response matrix is full column rank. The design of the precoding sequence which minimizes the noise effect in covariance matrix estimation is proposed and the effect of the optimal precoding sequence on channel equalization is discussed. Simulations are used to demonstrate the performance of the method.

3.1 System Model and Formulation

Consider the K-input J -output discrete time SC-ZP block transmission baseband model shown in Figure 3.1. At the transmitter, the kth input signal v_k(n) is first multiplied by a positive P -periodic sequence, p(n) ∈ R, to obtain sk(n) = p(n)v_k(n), where p(n+P ) = p(n),

∀ n. Then sk(n) is passed through a serial-to-parallel block whose output is

¯s_k(i) = [s_k(iM ) s_k(iM + 1) · · · sk(iM + M − 1)]^T. (3.1)

-N^? _{- S/P}

Figure 3.1. An MIMO SC-ZP block transmission baseband model with periodic precoding

Then ¯s_k(i) is passed through a zero padding prefilter F₁ = [I_M 0^T_P×M]^T ∈ R^{(M +P )×M} whose transmitted through the MIMO FIR channel. At the receiver, the jth received signal is x_j(n) = t_j(n) + w_j(n), where t_j(n) is the signal component at the output and w_j(n) is the channel noise seen at the jth receiver. If we define x(n) = [x₁(n) x₂(n) · · · xJ(n)]^T ∈ C^J, then x(n) can be written as

x(n) = impulse response from the kth transmitter to the jth receiver, and L = max_j,k{Ljk} is the order of the MIMO channel. We assume that H(L) 6= 0J×K. Group the sequence of x(n) as ¯x(i) = [x(iN )^T x(iN + 1)^T · · · x(iN + N − 1)^T]^T ∈ C^{J N}, and define ¯u(i) ∈ C^KN and

¯

w(i) ∈ C^{J N} similarly as ¯x(i), we have

¯

x(i) = H0u(i) + H¯ 1¯u(i− 1) + ¯w(i), (3.4) where H0 is a J N×KN block lower-triangular Toeplitz matrix with the first block column being [H(0)^T H(1)^T · · · H(L)^T 0^T_J×K· · · 0^TJ×K]^T ∈ C^{J N×K}, and H1 is a J N × KN block upper-triangular Toeplitz matrix with the first block row being [0J×K· · · 0J×K H(L) H(L− 1)· · · H(1)] ∈ C^J×KN. We assume that the receivers are synchronized with the transmit-ters. In addition, the following assumptions are made throughout this chapter.

(B1) The source signal v(n) = [v₁(n) v₂(n) · · · vK(n)]^T ∈ C^K is a zero mean white se-quence with E[v(m)v(n)^∗] = δ(m− n)IK ∈ R^K×K, where δ(·) is the Kronecker delta function. The noise is white zero mean with E[w(m)w(n)^∗] = δ(m− n)σw²I_J ∈ R^J×J. In addition, the source signal is uncorrelated with the noise w(n), i.e., E[v(m)w(n)^∗] = 0_K×J, ∀ m, n.

(B2) An upper bound ˆL of the channel order L is known, P = ˆL + 1, and M > P is a multiple of P .

(B3) The channel impulse response matrix H = [H(0)^T H(1)^T · · · H(L)^T]^T is full column rank, i.e., rank(H) = K.

In the next section, we derive an algorithm for blind identification of the MIMO channel impulse response matrix H using second-order statistics of the received data.

3.2 Blind Channel Identification

In this section, we derive the proposed method under assumptions (B1), (B2), and (B3). We discuss an optimal design of the precoding sequence, which takes into account the noise effect in the estimation of covariance matrix of the received data, so as to increase the accuracy in the computation of the channel product matrix HH^∗ and thus reduce the channel estimation error. With the proposed optimal precoding sequence, the computation of HH^∗ becomes particularly simple. Taking eigen-decomposition of HH^∗, we obtain the channel impulse response matrix H up to a unitary matrix ambiguity.

3.2.1 The Identification Method

We first derive the proposed method for the case where the channel order L is known with P = L + 1, there are more receivers, i.e., J ≥ K, and the noise is absent. The cases of channel order overestimation and more transmitters than receivers (i.e., K > J ) are given at the end of this sub-section. The effects of noise and optimal design of the precoding sequence are discussed in Section 3.2.2.

From (3.4), we know that only the last L block columns of H₁ are non-zero and zeros are padded in the last P block rows of ¯u(i− 1) and ¯u(i) (see (3.2)). Hence the product H₁u(i¯ − 1) equals the zero vector and (3.4) can be written as follows (noiseless case):

¯

Define S ∈ R^{J (L+1)×J(L+1)} as the matrix whose first block sub-diagonal entries are all I_J (i.e., S(J + 1 : J (L + 1), 1 : J L) = I_{J L}), and all remaining entries are zero. Rewrite (3.7) as x_f(i) = [p(0)H p(1)SH · · · p(L)S^LH]v_f(i) = H_pv_f(i). (3.8)

Taking expectation of x_f(i)x_f(i)^∗, we get the covariance matrix From [37, p.414], we know that the general matrix equationP_p

j=1AjXBj = C can be equiv-alently expressed as a matrix-vector equation form, hPp

j=1B^T_j ⊗ Aj

i

vec(X) = vec(C), where vec(·) is the vec-function which stacks up columns of a matrix. Hence the matrix equation (3.10) can be written in the following vector form:

vec(R_f) = vec Here G is a block Toeplitz lower-triangular matrix shown as follows:

G = blocks. Since G is square, the solution to (3.11) is

vec(HH^∗) = G⁻¹vec(R_f) (3.13)

provided p(0) 6= 0. We use the solution obtained in (3.13) to form a Hermitian matrix Q = HH^∗. Then under the assumption (B3), we can obtain the channel impulse response matrix, up to a unitary matrix ambiguity, by choosing the K largest eigenvalues and the associated eigenvectors of Q, like the way at the end of Section 2.2.1.

Remark 1: So far we have assumed that the channel order L is known. If only an upper bound ˆL≥ L is available, then following the same process given in this sub-section, we ob-tain vec(H\ovH^∗_ov) = [PL^ˆ

k=0Sk⊗ Sk]⁻¹vec(Rf) where Hov = [H^T 0| {z }· · · 0

Lˆ−L blocks

]^T ∈ C^{J ( ˆL+1)×K}. Then we can also obtain Q = HovH^∗_ov. Note that the last ( ˆL−L) block columns and block rows of Q are zero. Then similar to the discussion in Section 2.2.2, we can also identify the channel impulse response matrix.

Remark 2: The proposed method can apply to the case of more transmitters than re-ceivers. Please see the discussion in Section 2.2.3.

3.2.2 Optimal Design of the Precoding Sequence

When the noise is present, the covariance matrix R_f contains the contribution of noise.

Thus (3.9) becomes

Rf = E[xf(i)xf(i)^∗] = HpH^∗_p+ σ_w²IF, (3.14) where F = J (L + 1). In this case, (3.11) becomes

vec(R_f) = G· vec(HH^∗) + σ_w²vec(I_F). (3.15) From (3.13), the approximate solution of vec(HH^∗) is

vec(HH\^∗) = G⁻¹vec(R_f). (3.16)

It follows from (3.16) and (3.15) that

vec(HH\^∗) = vec(HH^∗) + σ²_wG⁻¹· vec(IF)

| {z }

z

= vec(HH^∗) + σ_w²z. (3.17)

The vector z = [z₁z₂ · · · zF²]^T in (3.17) is the solution of Gz = vec(I_F). Since the matrix G is completely determined by the precoding sequence p(n), we seek to choose p(n) so that kzk²2 is minimized. To this end, we need to analyze the relations between z and p(n). By expanding the matrix equation Gz = vec(I_F), we find that



and z_j = 0 for all other indices j. We write (3.18) as the following matrix equation.

 relations between z and p(n), is reduced to (3.19), and minimization of kzk²2 is equivalent to minimization ofkmk²2, which is a nonlinear function of g0, g1,· · · , gL. Then the problem

is to minimizekmk²2 by choosing g₀, g₁,· · · , gL, subject to suitable constraints. Specifically, we formulate the problem as

Minimize_{g0,g1··· ,gL}kmk²₂ subject to

g_n ≥ τ > 0, ∀ 0 ≤ n ≤ L (3.20)

1 L + 1

XL n=0

gn= 1 . (3.21)

Roughly, constraint (3.20) requires that at each instant, the power gain (g_n= p(n)²) is no less than τ with 0 < τ < 1; constraint (3.21) normalizes the power gain of the precoding sequence of each transmitter to 1.

It is easy to show that for L = 1, the problem has a unique global minimizer given by g₀ = 2− τ and g1 = τ . For general L ≥ 2 case, the standard Kuhn-Tucker conditions [38] of the nonlinear minimization problem do not seem to yield easily a unique analytical solution. However, the problem can be easily solved numerically (for fixed L and τ ), say using the Matlab Optimization Toolbox. Extensive numerically solutions, with different L, τ , and initial guess, have indicated that a global minimizer exists and is given by

g0 = L + 1− Lτ, g1 = g2 =· · · = gL = τ. (3.22)

In the following, we show that the solution (3.22) is also the global minimizer of an upper bound of kmk²2. We know kmk²2 = kG⁻¹s yk²2 ≤ kG⁻¹s k²2 · kyk²2 = (L + 1)kG⁻¹s k²2, wherekG⁻¹_s k2 is the 2-induced norm of G⁻¹_s . Since G_sis triangular and Toeplitz, it follows from [32] that for any fixed integer L≥ 1,

kG⁻¹s k²2 ≤ 1

(α + 2)²β²[(α + 1)^2(L+1)+ 2(L + 1)(α + 2)− 1] , f(α, β), (3.23) where α = max_{i=1,2,··· ,L}|^g_gⁱ₀| and β = |g0|. Hence we know kmk²2 ≤ (L + 1)f(α, β). Since for any α > 0 and β > 0, ^{∂f (α,β)}_∂α > 0 (see Appendix C) and ^{∂f (α,β)}_∂β =−²_βf (α, β) < 0 , we know for any fixed β > 0, f (α, β) is an increasing function of α, and for any fixed α > 0, f (α, β) is a decreasing function of β. Hence to minimize f (α, β), we should choose α as small as possible and choose β as large as possible subject to β ≤ L + 1 − Lτ and α ≥ _L+1^τ_−Lτ. It follows that (3.22) is a global minimizer of the upper bound (L + 1)f (α, β).

Since g_n = p(n)² and p(n) > 0, the optimal precoding sequence is p(n) =

( √L + 1− Lτ , n = 0

√τ , 1≤ n ≤ L . (3.24)

We consider next the effect of τ on channel identification. From (3.19) and [30, 31], we know m = G⁻¹_s y, where G⁻¹_s is a lower-triangular Toeplitz matrix with [¯g0g¯1 · · · ¯gL]^T ∈

R^L+1 as its first column, and For the optimal solution in (3.22), the corresponding ¯gn in (3.25) can be expressed as follows: The following proposition shows that kmk²₂ is a continuous and strictly increasing function of τ on (0, 1). In other words, for 0 < τ < 1, kmk²₂ decreases as τ decreases, and thus as τ decreases, the noise effect in the estimation of the covariance matrix R_f is reduced and hence identification performance improves.

Proof : See Appendix D.

3.2.3 Computation of G

⁻¹₀

With the precoding sequence p(n) chosen as (3.24), the matrix G in (3.12) becomes

G₀ =

where a = L+1−Lτ, and b = τ. The inverse of G0can be obtained by forward substitutions as G⁻¹₀ vec(Rf) in (3.16) is thus quite easy to compute once the optimal precoding sequence is given.

3.2.4 Identification Algorithm

So far, we have proposed a new method to identify the MIMO channels for the single carrier zero padding block transmission system using optimal designed periodic precoding which minimize the noise effect in the estimation of the covariance matrix R_f. With zero padding, the computation of the channel product matrix HH^∗ becomes particularly simple, since it amounts to solving a lower-triangular linear system. The channel impulse response matrix H is then computed, up to a unitary matrix ambiguity, from the channel product matrix via an eigen-decomposition. We summarize the proposed method as the following algorithm:

1) Select the optimal precoding sequence p(n) given by (3.24), and form G⁻¹₀ as in (3.29).

2) Collect the received data as ¯x(i) and pick up the first (L + 1) block entries of ¯x(i) as x_f(i). Then estimate the covariance matrix R_f via the time average

Rˆf = 1 S

XS i=1

x_f(i)x_f(i)^∗, (3.30)

where S is the number of data block.

3) Computevec(HH\^∗) = G⁻¹₀ vec( ˆR_f) to obtain the elements of HH^∗.

4) Form the matrix Q = HH^∗ and obtain the channel impulse response matrix by comput-ing the K largest eigenvalues and the associated eigenvectors of Q.

在文檔中 多輸入多輸出有限脈衝系統的盲判別與對等化(III) (頁 20-36)