The Least Squares Error (LSE) estimation method can be used to
estimate the system h[m] by minimizing the squared error between
estimation and detection.
Figure 4. Discrete-Time LTI System
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The generalized equation for the output of the system
shown in Figure 3 is
, (2)
and
this can be re-written in the matrix form.
,
(3)
,
(4)
The data matrix D has a Toeplitz
structure, which has constant diagonal entries. The system output error, e, can be calculated with the
equation,
.
(5)
where y is the desired output of the system.
This equation is used to estimate the squared error using equation,
.
(6)
Substituting (4) into (6), the squared error S can be expanded to
.
(7)
Since the squared error S is a scalar component, equation (7) can be differentiated with respect to h.
(8)
Then the filter component h can be solved in terms of input data matrix D and desired output y as
.
(9)
However, in the above case, when
estimating h, if y is not known, it is best to assume
that 
= y[n], which will minimize the error from
(6), along with the error for the estimated filter h, 
.
Thus equation (9) can be rewritten for as
.
(10)