Mean Square Error Estimation of an MA(2) Signal

​
MA(2) parameter
b
1
0.5
MA(2) parameter
b
2
-0.5
MA(2) noise variance
0.5
observation noise variance
0.1
This Demonstration displays an MA(2) desired signal
x(k)
that is corrupted by white Gaussian noise, its mean square error (MSE) estimate

x
(k)
and the resulting simulated MSE (using 50,000 samples). The optimum tap weights
h
i
are also displayed. Use the sliders to vary the parameters of the MA(2) signal model for
x(k)
,
b
1
,
b
2
and
2
σ
w
, as well as the Gaussian observation noise variance
2
σ
n
. For comparison, the performance of an equal weighted moving average filter is shown.

Details

The desired signal is an MA(2) process—that is,
x(k)=w(k)+
b
1
w(k-1)+
b
2
w(k-2)
, where
w(k)
is white Gaussian noise with zero mean and variance
2
σ
w
.
The observed signal is
y(k)=x(k)+n(k)
, where
n(k)
is white Gaussian observation noise with zero mean and variance
2
σ
n
.
A three-tap finite impulse response (FIR) filter MSE estimate of
x(k)
is

x
(k)=
h
0
y(k)+
h
1
y(k-1)+
h
2
y(k-2)
.
The optimum tap weights
h
i
minimize
E(x(k)-)
2

x
(k)

.
Let the optimum tap weights filter be defined as
h=(
h
0
,
h
1
,
h
2
)
;
R
y
is the autocorrelation of
y(k)
and
r
xy
is the cross-correlation between
x(k)
and
y(k)
.
Then the optimum filter is found from
h=
-1
R
y
r
xy
, and the resulting MSE is
R
x
(0)-h
r
xy
.
The initial values of
b
1
=1
,
b
2
=1
,
2
σ
w
=1/4
and
2
σ
n
=1/4
produce results that match Example 7.14[1, pp. 415–416].
A new desired MA(2) signal
x(k)
is created each time a system parameter is changed.

References

[1] K. S. Shanmugan and A. M. Breipohl, Random Signals: Detection, Estimation and Data Analysis, New York: Wiley, 1991.

Permanent Citation

Victor S. Frost, University, Kansas
​
​"Mean Square Error Estimation of an MA(2) Signal"​
​http://demonstrations.wolfram.com/MeanSquareErrorEstimationOfAnMA2Signal/​
​Wolfram Demonstrations Project​
​Published: September 26, 2025