Part A | 9.5
224 Part A Fundamentals
For the derivation of the above it is reminded that
e D x y :
Also, using the orthogonality condition (9.135), the following important result can be derived
S ye .!/ D H.!/S we .!/
SweD0
) S ye .!/ D 0 D S ey .!/ :
It is noted here that
R ee ../ D F
1
fS ee .!/g D
C1 Z
1
S ee .!/e
i!! d! ) he; ei
D R ee .0/ D
C1 Z
1
S ee .!/ d! :
An alternative expression for the minimum error PSD
can be derived on the basis of the following
S yy .!/ D jH.!/j
2 S ww .!/ D
jS xw .!/j
2
S 2
ww .!/
S ww .!/
D
jS xw .!/j
2
S ww .!/
:
Then
S ee .!/ D S xx .!/ S yy .!/
D
S xx .!/S ww .!/ jS xw .!/j
2
S ww .!/
:
(9.139)
Further treatment of (9.137) is possible since the additive measurement noise is uncorrelated to the unknown
signal x.t/ at the instrument input. In this case,
S xw .!/ D S xu .!/ C S xv .!/
D G
.!/S xx .!/ because S xv .!/ D 0 :
(9.140)
Also
S ww .!/ D S uu .!/ C S vv .!/
D jG.!/j
2 S xx .!/ C S vv .!/
since u D v C w :
(9.141)
Therefore, the Wiener filter in (9.137) becomes as follows
H.!/ D
G
.!/S xx .!/
jG.!/j 2 S xx .!/ C S vv .!/
:
(9.142)
The above allows us to obtain an estimate of waveform x.t/ appearing at the instrument input port. As
can be seen, to achieve this the signal’s (waveform)
statistics need to be known in advance, i. e., the signal PSD S xx .!/ or autocorrelation R xx ../, as well as
the dynamics of the measurement instrument, i. e., its
transfer function G.!/ or its impulse response g.t/;
last, but not least, prior information of the additive
noise’s statistics, ie, its PSD S vv .!/ or autocorrelation R vv ../.
Two important specific subcases of (9.137) are
given below:
1. Zero-forcing equalization [9.1, 5]:
If S vv .!/ S xx .!/
H.!/ D
1
G.!/
:
(9.143)
Evidently, in this subcase the statistics of either the
unknown input signal or the additive noise need to
be known in advance. Furthermore, one can easily
verify that, if the inverse of the instrument’s transfer function G.!/ does not introduce unstable poles,
estimation error e vanishes.
2. Matched filter [9.1, 5]:
If jG.!/j
2 S xx .!/ S vv .!/
H.!/ D G
.!/
S xx .!/
S vv .!/
:
(9.144)
In this subcase the major impairment toward obtaining an estimate of x.t/ is the additive noise rather
than the instrument distortion, as was the case previously. Moreover note that if S xx .!/ < S vv .!/, the
instrument’s output w .t/ will be noise-like, and yet
by use of the Wiener filter it is possible to extract
the unknown information signal x.t/.
Matched filter estimator design is widely used in
communications engineering. At least as a first approximation, transfer function G.!/ models the telecommunication channel that introduces attenuation monotonically increasing with distance from the transmitter
source. In a typical system, both the information signal
x.t/, as well as disturbance noise v .t/ can be considered as white noise, i. e., signals with the following
property
S nn .!/ D N 0 ; 8! , R nn ../ D N 0 •../; 8 : (9.145)
As can be seen, the case of a white noise signal demonstrates practically no predictability, since the value of
the signal at any time instant is entirely uncorrelated
224 Part A Fundamentals
For the derivation of the above it is reminded that
e D x y :
Also, using the orthogonality condition (9.135), the following important result can be derived
S ye .!/ D H.!/S we .!/
SweD0
) S ye .!/ D 0 D S ey .!/ :
It is noted here that
R ee ../ D F
1
fS ee .!/g D
C1 Z
1
S ee .!/e
i!! d! ) he; ei
D R ee .0/ D
C1 Z
1
S ee .!/ d! :
An alternative expression for the minimum error PSD
can be derived on the basis of the following
S yy .!/ D jH.!/j
2 S ww .!/ D
jS xw .!/j
2
S 2
ww .!/
S ww .!/
D
jS xw .!/j
2
S ww .!/
:
Then
S ee .!/ D S xx .!/ S yy .!/
D
S xx .!/S ww .!/ jS xw .!/j
2
S ww .!/
:
(9.139)
Further treatment of (9.137) is possible since the additive measurement noise is uncorrelated to the unknown
signal x.t/ at the instrument input. In this case,
S xw .!/ D S xu .!/ C S xv .!/
D G
.!/S xx .!/ because S xv .!/ D 0 :
(9.140)
Also
S ww .!/ D S uu .!/ C S vv .!/
D jG.!/j
2 S xx .!/ C S vv .!/
since u D v C w :
(9.141)
Therefore, the Wiener filter in (9.137) becomes as follows
H.!/ D
G
.!/S xx .!/
jG.!/j 2 S xx .!/ C S vv .!/
:
(9.142)
The above allows us to obtain an estimate of waveform x.t/ appearing at the instrument input port. As
can be seen, to achieve this the signal’s (waveform)
statistics need to be known in advance, i. e., the signal PSD S xx .!/ or autocorrelation R xx ../, as well as
the dynamics of the measurement instrument, i. e., its
transfer function G.!/ or its impulse response g.t/;
last, but not least, prior information of the additive
noise’s statistics, ie, its PSD S vv .!/ or autocorrelation R vv ../.
Two important specific subcases of (9.137) are
given below:
1. Zero-forcing equalization [9.1, 5]:
If S vv .!/ S xx .!/
H.!/ D
1
G.!/
:
(9.143)
Evidently, in this subcase the statistics of either the
unknown input signal or the additive noise need to
be known in advance. Furthermore, one can easily
verify that, if the inverse of the instrument’s transfer function G.!/ does not introduce unstable poles,
estimation error e vanishes.
2. Matched filter [9.1, 5]:
If jG.!/j
2 S xx .!/ S vv .!/
H.!/ D G
.!/
S xx .!/
S vv .!/
:
(9.144)
In this subcase the major impairment toward obtaining an estimate of x.t/ is the additive noise rather
than the instrument distortion, as was the case previously. Moreover note that if S xx .!/ < S vv .!/, the
instrument’s output w .t/ will be noise-like, and yet
by use of the Wiener filter it is possible to extract
the unknown information signal x.t/.
Matched filter estimator design is widely used in
communications engineering. At least as a first approximation, transfer function G.!/ models the telecommunication channel that introduces attenuation monotonically increasing with distance from the transmitter
source. In a typical system, both the information signal
x.t/, as well as disturbance noise v .t/ can be considered as white noise, i. e., signals with the following
property
S nn .!/ D N 0 ; 8! , R nn ../ D N 0 •../; 8 : (9.145)
As can be seen, the case of a white noise signal demonstrates practically no predictability, since the value of
the signal at any time instant is entirely uncorrelated