Adaptive Filters and Applications - Digital Signal Processing: Fundamentals and Applications - page 455

to the corrupting noise nðnÞ in the first channel, since both come from the same noise source. Similarly,
the noise signal xðnÞ is not correlated to the desired speech signal sðnÞ .
We assume that the corrupting noise in the first channel is a linear filtered version of the second-
channel noise, since it has a different physical path from the second-channel noise, and the noise source
is time varying, so that we can estimate the corrupting noise nðnÞ using an adaptive filter. The adaptive
filter contains a digital filter with adjustable coefficient(s) and the LMS algorithm to modify the value(s)
of coefficient(s) for filtering each sample. The adaptive filter then produces an estimate of noise yðnÞ ,
which will be subtracted from the corrupted signal dðnÞ¼sðnÞþnðnÞ . When the noise estimate yðnÞ
equals or approximates the noise nðnÞ in the corrupted signal, that is, yðnÞ z nðnÞ , the error signal eðnÞ¼
sðnÞþnðnÞyðnÞ z sðnÞ will approximate the clean speech signal sðnÞ . Hence, the noise is cancelled.
In our illustrative numerical example, the adaptive filter is set to be one-tap FIR filter to simplify
numerical algebra. The filter adjustable coefficient w n is adjusted based on the LMS algorithm (dis-
cussed later in detail) in the following:
w nþ 1 ¼ w n þ 0 : 01 $eðnÞ$xðnÞ
where w n is the coefficient used currently, while w nþ 1 is the coefficient obtained from the LMS
algorithm and will be used for the next coming input sample. The value of 0.01 controls the speed of
the coefficient change. To illustrate the concept of the adaptive filter in Figure 10.2 , the LMS algorithm
has the initial coefficient set to w 0 ¼ 0 : 3 and leads to
yðnÞ¼w n xðnÞ
eðnÞ¼dðnÞyðnÞ
w nþ 1 ¼ w n þ 0 : 01 eðnÞxðnÞ
The corrupted signal is generated by adding noise to a sine wave. The corrupted signal and noise
reference are shown in Figure 10.3 , and their first 16 values are listed in Table 10.1 .
Let us perform adaptive filtering for several samples using the values for the corrupted signal and
reference noise in Table 10.1 . We see that
n ¼ 0 ; yð 0 Þ¼w 0 xð 0 Þ¼ 0 : 3 ð 0 : 5893 Þ¼ 0 : 1768
eð 0 Þ¼dð 0 Þyð 0 Þ¼ 0 : 2947 ð 0 : 1768 Þ¼ 0 : 1179 ¼ sð 0 Þ
w 1 ¼ w 0 þ 0 : 01 eð 0 Þxð 0 Þ¼ 0 : 3 þ 0 : 01 ð 0 : 1179 Þð 0 : 5893 Þ¼ 0 : 3007
n ¼ 1 ; yð 1 Þ¼w 1 xð 1 Þ¼ 0 : 3007 0 : 5893 ¼ 0 : 1772
eð 1 Þ ¼ dð 1 Þ yð 1 Þ ¼ 1 : 0017 0 : 1772 ¼ 0 : 8245 ¼ sð 1 Þ
w 2 ¼ w 1 þ 0 : 01 eð 1 Þxð 1 Þ¼ 0 : 3007 þ 0 : 01 0 : 8245 0 : 5893 ¼ 0 : 3056
n ¼ 2 ; yð 2 Þ¼w 2 xð 2 Þ¼ 0 : 3056 3 : 1654 ¼ 0 : 9673
eð 2 Þ¼dð 2 Þyð 2 Þ¼ 2 : 5827 0 : 9673 ¼ 1 : 6155 ¼ sð 2 Þ
w 3 ¼ w 2 þ 0 : 01 eð 2 Þxð 2 Þ¼ 0 : 3056 þ 0 : 01 1 : 6155 3 : 1654 ¼ 0 : 3567
n ¼ 3 ; /