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# Statistics Distribution of Higher-Order Nonlinear Residual Errors

## Computer Simulations

### 4.5 Statistics Distribution of Higher-Order Nonlinear Residual Errors

In Section 4.4.1, we discuss the ERLE for six methods and the “slope” method in Section 3.3.1 has been used in AEC1/slope1, AEC1/AES3/slope1, and AEC1/AES3/slope2. In order to analyze the three methods, we will observe statistical distributions. If the residual error is linearly than which order echo, the residual error is closer than estimate value. The straight line is estimated nonlinear residual error

and the point is real nonlinear residual error . ˆnl

Y Ynl

(1) speech signal + real system:

First, we input speech signal and real system that is real loudspeaker and real room impulse response.

0 0.5 1 1.5 2 2.5 3 3.5 0

0.5 1 1.5 2 2.5

AEC1/slope1(speech+real)

Amplitude in Echo Replica |Y(m)|

Amplitude in Residual Echo |E3+(m)|

Fig. 4.11 Statistic distribution of AEC1/slope1 in speech signal + real system

0 0.5 1 1.5 2 2.5 3

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

AEC13/AES3/slope1(real+speech)

Amplitude in Echo Replica |Y(m)|

Amplitude in Residual Echo |E5+(m)|

Fig. 4.12 Statistic distribution of AEC13/AES3/slope1 in speech signal + real system

0 1 2 3 4 5 6 7 8 x 10-3 0

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

AEC13/AES3/slope3(real+speech)

Amplitude in Echo Replica |Y3(m)|

Amplitude in Residual Echo |E5+(m)|

Fig. 4.13 Statistic distribution of AEC13/AES3/slope3 in speech signal + real system

The point is closer than straight line in Fig. 4.13 than Fig. 4.11 and Fig. 4.12.

(2) WGN signal + polynomial system:

Second, we input white gauss noise signal and the pseudo system includes two parts that are polynomial function in Fig. 3.10 and exponential function in Fig.

4.2.

0 5 10 15 20 25 30 35 0

2 4 6 8 10 12

AEC1/slope1(WGN+polynomial)

Amplitude in Echo Replica |Y(m)|

Amplitude in Residual Echo |E3+(m)|

Fig. 4.14 Statistic distribution of AEC1/slope1 in WGN signal + polynomial system

0 5 10 15 20 25 30 35

0 2 4 6 8 10 12

AEC13/AES3/slope1(WGN+polynomial)

Amplitude in Echo Replica |Y(m)|

Amplitude in Residual Echo |E5+(m)|

Fig. 4.15 Statistic distribution of AEC13/AES3/slope1 in WGN signal + polynomial system

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0

2 4 6 8 10 12

AEC13/AES3/slope3(WGN+polynomial)

Amplitude in Echo Replica |Y3(m)|

Amplitude in Residual Echo |E5+(m)|

Fig. 4.16 Statistic distribution of AEC13/AES3/slope3 in WGN signal + polynomial

The point is closer than straight line in Fig. 4.16 than Fig. 4.14 and Fig. 4.15.

(3) speech signal + polynomial system:

Third, we input speech signal and the pseudo system includes two parts that are polynomial function in Fig. 3.10 and exponential function in Fig. 4.2.

0 5 10 15 20 25 30 35 40 45 50 0

5 10 15 20 25 30

AEC1/slope1(speech+polynomial)

Amplitude in Echo Replica |Y(m)|

Amplitude in Residual Echo |E3+(m)|

Fig. 4.17 Statistic distribution of AEC1/slope1 in speech signal + polynomial system

0 5 10 15 20 25 30 35

0 5 10 15 20 25 30 35 40 45 50

AEC13/AES3/slope1(speech+polynomial)

Amplitude in Echo Replica |Y(m)|

Amplitude in Residual Echo |E5+(m)|

Fig. 4.18 Statistic distribution of AEC13/AES3/slope1 in speech signal + polynomial system

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0

5 10 15 20 25 30 35

AEC13/AES3/slope3(speech+polynomial)

Amplitude in Echo Replica |Y3(m)|

Amplitude in Residual Echo |E5+(m)|

Fig. 4.19 Statistic distribution of AEC13/AES3/slope3 in speech signal + polynomial system

The point is closer than straight line in Fig. 4.19 than Fig. 4.17 and Fig. 4.18.

(4) WGN signal + real system:

Fourth, we input white gauss noise signal and real system that is real loudspeaker and real room impulse response.

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0

0.05 0.1 0.15 0.2 0.25 0.3 0.35

AEC1/slope1(WGN+real)

Amplitude in Echo Replica |Y(m)|

Amplitude in Residual Echo |E3+(m)|

Fig. 4.20 Statistic distribution of AEC1/slope1 in WGN signal + real system

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

AEC13/AES3/slope1(WGN+real)

Amplitude in Echo Replica |Y(m)|

Amplitude in Residual Echo |E5+(m)|

Fig. 4.21 Statistic distribution of AEC13/AES3/slope1 in WGN signal + real

system

0 0.5 1 1.5

x 10-3 0

0.05 0.1 0.15 0.2 0.25 0.3 0.35

AEC13/AES3/slope3(WGN+real)

Amplitude in Echo Replica |Y3(m)|

Amplitude in Residual Echo |E5+(m)|

Fig. 4.22 Statistic distribution of AEC13/AES3/slope1 in WGN signal + real system

The point is closer than straight line in Fig. 4.22 than Fig. 4.20 and Fig. 4.21.

From the above statistical distribution, we can find roughly the AEC13/AES3/slope3 is more linear than the AEC13/AES3/slope1 and AEC1/slope1.

In order to accurately find which one can suppress nonlinear residual error more, we define correlation mismatch factorε which is variance of estimate nonlinear residual error and real nonlinear residual error. If the correlation mismatch factorε is less, the method is more linear.

real estimate

Enl Enl

ε = − (4.5.1)

ˆ

### )

estimate n

avg

Enl = m Y⋅ (4.5.2)

Table 4.3 Correlation mismatch factor ε

AEC1/slope1 AEC1/AES3/slope1 AEC1/AES3/slope3

Speech & Real 5.4690e-004 5.0143e-004 5.0079e-004

WGN & Polynomial 0.0121 0.0123 0.0058

Speech & polynomial 3.8030e-004 3.1465e-004 3.1465e-004 WGN & Real 3.4303e-005 3.3021e-005 2.3639e-005

The correlation mismatch factor ε of AEC1/AES/slope3 is less than AEC1/slope1 and AEC/AES/slope1 and AEC1/AES/slope3 can suppress more nonlinear residual error than the others. Table 4.3 can prove the ERLE of AEC1/AES/slope3 is the best than the others in Section 4.4.1.

### Chapter 5 Conclusion

In this thesis, we introduce different structures for estimating the nonlinear residual error and near-end speech signal. However, the method using the linear echo to estimate the nonlinear residual error in Section 3.3.1 is not accurate. The method using the NAEC to estimate the nonlinear residual error needs large computational operation in Section 3.3.2. We integrate the two methods to propose the new method that can estimate the nonlinear residual error more accurate and save the calculating operation. From the simulation results in Fig. 4.8-11, we know the new method is better than the others methods. Besides, the statistics distribution and correlation mismatch factor can prove the proposed method better than the others. We provide the input signal of WGN or speech signal and the system of polynomial or real loudspeaker for the simulation, and the new method are better than the others methods.

In addition, we introduce the other suppression structure which is NR+AEC structure. In low SNR condition, the NR+AEC structure is better than AEC+NR structure. The first NR suppresses the large noise, so the noise has less affect on AEC.

The AEC can estimate the echo path accurately. In high SNR condition, the NR+AEC structure is worse than AEC+NR structure. This is because the NR+AEC structure causes the nonlinear disturbance by the NR. So, AEC can’t estimate the accurate echo path. We also extend the NAES to other nonlinear adaptive filter which is Volterra model. Volterra model can estimate nonlinear memory residual error that the loudspeaker is time-variant. So, the performance is better than Hammerstein model that can estimate memoryless residual error only.

The simulation result shows the Volterra model is better than Hammerstein model for steady input signal, but the input signal is real speech model for communication that the performance is worse than Hammerstein model. This is because the convergence speed of Volterra structure is too slow. When the length of input signal period isn’t enough to adaptive the nonlinear channel, the performance will come down.

Using higher-order nonlinear echoes as bases to estimate nonlinear residua error is not only can estimate lower-order nonlinear echoes accurately, but also estimate high-order nonlinear echoes roughly.

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