Figure 5. Input to the KF x(n)
Figure 6 shows the desired signal (i.e. signal without echo) superimposed on to the Kalman Filter output. As it is clear from this figure, the output of the Kalman filter is not completely matching to the desired non echo signal and therefore the output of the Kalman filter is acting as an input to the LMS adaptive algorithm; result of which is presented in Figure 7.
Figure 6. The desired .wav file  No echo and the KF output
Figure 7 presents the superimposed signal of the desired signal (with no echo) and the output of the LMS adaptive filter. As expected, the combination of the Kalman Filter and the LMS adaptive filter resulted in a very good response in terms of echo cancellation, which is evident from the output signals shown in Figures 7.
Figure 7. Comparison of the desired signal and the hybrid filter output
As well as the graphical illustration of the hybrid filter outputs, numerical comparison of this filter outputs with that of desired has also been considered and presented in Table 1.
Table 1. Numerical comparison of the hybrid filter outputs with that of desired at different times

Amplitude of the input, output and the desired signal at different time

Time (s)

Input to Kalman Filter

Desired Signal

Output of Kalman Filter

Output of LMS adaptive filter

0.4

0.3

0.2

0.07

0.05

1.1

0.35

0.3

0.25

0.23

1.3

0.17

0.12

0.05

0.03

3

0.3

0.25

0.25

0.25

3.25

0.2

0.15

0.15

0.15

Graphical and numerical illustration of the LMS adaptive filter output signal and the desired signal demonstrates that the LMS adaptive filter and the desired signal display similar profile, proving that the echo has been successfully removed.
4. CONCLUSION
In this research, all the existing algorithms of the adaptive filter have been studied, mathematically derived and implemented; out of which, the LMS algorithm was chosen due to the computational simplicity of this adaptive algorithm.
In this paper, a novel approach of using the Kalman Filter and the LMS adaptive algorithm to build a hybrid filter, in an attempt to reduce the effect of echoes over the channel, has been demonstrated. As expected, this merger produced a very good response in terms of echo cancellation. The success of this combination was confirmed by both graphical and numerical analysis as well as the output sound.
The above procedures involved acquisition and processing of both simulated and real data sets (recorded voices). Having considered the examined algorithm and filter, and results obtained, it could be said that this paper has met the specified goals and could act as a stepping stone towards further research in speech enhancement for the communication based applications.
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Vol. 35. Pt.1 2013
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