mlfllmlzes the output noise power. Since the signal in the output remains constant, minimizing the total output power maximizes the output SNR.
The smallest possible output power is £[e2 ] = £[s2 ] . When this is achievable,
. 2 .
£[(110 - 110 ) ] = o . Therefore, 110 = 110 and e = s . In this case, minimizing output power
causes the output signal to be perfectl y noise free.
3.1.2. Wiener solution to statistical noise cancelling problems
The objective of an optimal unconstrained Wiener solution to certain statistical nOIse cancel1ing problems is to increase performance in SNR using a noise cancelling technique. The analysis is based on the assumption of an infinitely long, two-sided noncausal tapped delay line. However, finite length of fi1ter and causalty are important in practical applications.
Fixed filters are not applicable in noise cancelling because the auto correlation and cross correl ation functions of the primary and reference inputs are generally unknown and often variable with time. On the other hand, adaptive filters are required to 'learn' the statistics initially and to track them if they vary slowly.
For stationary stochastic inputs, the steady-state performance of adaptive filters shows close approximation to that of fixed Wiener filters. Wiener filter theory thus provides a convenient method of mathematically analyzing statistical noise c ancelling problems.
1 ) The Wiener filter theory
For the least mean square filtering problem, Kolmogorov (Kolmogorov, 1 94 1 ) studied the discrete time problem and originally solved it by using the Wold decomposition (Wold, 1 938). On the other hand, Wiener (Wiener, 1 949) studied the continuous time problem and derived the famous Wiener-Hopf integral equation, which requires knowledge of correlation functions of the signal processes of interest.
The equivalent of this integral equation in the d iscrete time case is known as the normal equation. The continuous time solution to the Wiener-Hopf integral equation and the discrete time solution to the normal equation are known collectively as Wiener filters.
Fig. 2-5 shows that for the classic single-input single-output (SISO) Wiener filter with input signal x" ' the output signal y" , and desired response d" , the input and output signals
Chapter 2: Literature review 3. Main approaches
be statistically stationary. The error signal is ell =
d"
- y" . The filter is linear, discrete anddesigned to be optimum in the MMSE sense.
Desired response
n
input
n
H(z)
Yn
- error enFig. 2-5 Single-channel Wiener filter
It i s composed of an infinitely long, two sided tapped delay line.
The discrete autocorrelation function of the input signal X" ' and the cross-correlation
function between
xn
and the desired responsed"
are defined as (2-3) and (2-4) respectively.Rxx (k) == E[xnxn+k ]
RXd (k) == E[xnd,,+k ]
(2-3) (2-4)
The optimal impulse response
h(k)
can then be obtained from the discrete Wiener-Hopf equation (2-5).-
RXd (k)
=Ih(l)Rxx(k
- I) =h(k)
*Rxx(k)
(2-5)1=-00
Thi s form of the Wiener solution i s unconstrained in that the impulse response h(k ) may be causal or noncausal and of finite or infinite duration.
The transfer function of Wiener filter may now be derived as the ratio between the auto power spectral density (PSD) (2-6) of the input signals and the cross PSD (2-7) between the input signal and a desired response are the
z
-transforms ofRxx (k)
andRXd (k)
respectively.-
c;I>xx(z) == IRxx(k)z-k
(2-6)k=-
-c;I>xd (Z) == IRxd (k)z·k
(2-7)k=-oo
The transfer function of the Wiener filter is written as (2-8).
2) The application of Wiener filter theory to adaptive noise cancelling.
An example of a single-channel ANC with correl ated and uncorrelated nOIses In the primary input (2-9) and reference input (2- 1 0) is considered as shown in Fig. 2-6.
dn
= sn+ (ml" +
n,,)
(2-9)(2- 1 0) where
h2n
i s impulse response of channel whose transfer function i sH 2 (z) .
The noises nIl and nIl *h2n
have common origin, and they are correlated with each other, and areuncorrelated with s" . The noises
ml"
andm2"
are uncorrelated with each other, with s" ,1
mln I
Primary inpu�
_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ l I I L _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ J OutputAdapti ve noise canceller
Fig. 2-6 S ingle channel ANC with correlated and uncorrelated noises in the primary and reference inputs (Widrow et aI., 1 975)
If one assumes that the adaptive process has converged and the MMSE solution has been found, then the adaptive filter is equivalent to a Wiener filter.
4>
xx(z)
=4>
mlml(z)
+ 4> nn(z)IH 2 (z)12
4> xd
(Z)
= 4>nn (z)H 2 (Z-I)
Therefore, the Wiener transfer function is
H (z)
= 4> xd(z)
=4>
nn(z)H 2 (Z -I)
4>
xx(Z)
4> m,m,(Z) +
4> nn(Z)IH
2(zf
( 2- 1 1
)
(2- 1 2)(2- 1 3)
Note that this is the two-si ded Wiener filter, i .e., it has a part of its impulse response which is uncausal though this does not affect the arguments which follow.
From the (2- 13),
H (z)
is independent of the primary signal spectrum 4> .IS(z)
and of the primary uncorrelated noi se spectrum 4> m)m)(z) .
Chapter 2: Literature review 3. Main approaches An interesting special case occurs when the additive noise
m2n
in the reference input iszero. Then <l> "'2m2
(z) =
o . Therefore, the optimum transfer function is 1H2(z)
provided of course that
H 2 (z)
i s a minimum phase.(
2-14)
The adaptive filter, as in the balancing of a bridge, causes the noise
nn
to be perfectly nulled at the noise canceller output. However, it shows that the primary uncorrelated noisemln
remains uncancelled.From the above analyses, it shows that the application violates main assumptions to be perfectly applied to a noise cancellation. Therefore, it cannot be cancelled if an uncorrelated noise in primary input exists. In addition, it does not guarantee a stable performance because of a nonminimum phase property.
3.1.3 Effect of signal components in the reference input
In addition to analyses of a noise cancellation, an effect of speech signal components in the reference input is now investigated when it is applied with a speech signal .
It i s based on an application in a real environment, where in a certain circumstance the available reference input to an ANC may contain low-level unwanted signal components including the usual correlated and uncorrel ated noise components, which will cause some cancellation of the primary input signal .
Three main analyses (an unconstrained Wiener filter theory as SNR, si gnal distortion, and noise spectrum at the cancel1er output) have been investigated (Widrow et aI., 1 975).
Fig. 2-7 shows an ANC, where reference input contains signal components and where primary and reference inputs contain additive correlated noises. The z-transforms of auto and cross PSD are expressed as (2- 1 5) and (2-1 6) respectively.
<l> J..<
(z) =
<l> .H(z )IG 2 (z )12
+ <l>nn (z )IH 2 (z )12
<l> xd
(Z) =
<l> ss(Z)G2 (Z -I
) + <l>nn (z)H 2 (Z -I
)(2- 1 5)
I Reference �nput I I I L _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ J Output
Adaptive noise canceller
Fig. 2-7 ANC with signal components in the reference input (Widrow et aI., 1 975)
When the adaptive process has converged, the unconstrained Wiener transfer function of the adaptive filter is
H(z) = <1>..., (z)G2 (z-I ) + <1>nn (z)H2 (z-l )
<1>
SJ(z)IG2 (zf + <1> nn (z)IH 2 (z)12
1 ) Anal ysi s on SNR at the noise canceller output, SNRe
(z)
(2- 1 7)
The transfer function of the propagation path from the signal input to the noise canceller output is 1 -