IEEE Signal Processing - July 2018 - 130

Appendix 2
Theorem 2
The impulse response h [n] of a discrete-time linear timeinvariant (LTI) system described by (18) can be determined
as follows:
h [n] = P (E ) [w d [n] u [n]],

(A9)

where w d [0] = 0 and w d [n] is the solution to the following
homogeneous difference equation:
Q (E ) w d [n] = 0 for n $ 1,

(A10)

subject to the following auxiliary conditions:
w d [1] = g = w d [N - 1] = 0, and w d [N ] = 1 / a N.
(A11)
Proof
The terms on the left side of (23) are rewritten as follows:
w d [n] u [n] =
=
E [w d [n] u [n]] =
=
=

w d [n] (u [n - 1] + d [n])
w d [n] u [n - 1] + w d [0] d [n]
w d [n + 1] u [n + 1]
w d [n + 1] (u [n - 1] + d [n] + d [n + 1])
[Ew d [n]] u [n - 1] + w d [1] d [n]
+ w d [0] d [n + 1]
E 2 [w d [n] u [n]] = w d [n + 2] u [n + 2]
= w d [n + 2] (u [n - 1] + d [n] + d [n + 1]
+ d [n + 2])
= [E 2 w d [n]] u [n - 1] + w d [2] d [n]
+ w d [1] d [n + 1] + w d [0] d [n + 2]
h
E N [w d [n] u [n]] = [E N w d [n]] u [n - 1]
N

+

/ w d [N - r] d [n + r] .

r= 0

(A12)

By setting w d [0] = 0 and substituting all the auxiliary conditions given in (A11) into (A12), all the terms in (A12) are simplified as follows:
w d [n] u [n] = w d [n] u [n - 1]
E [w d [n] u [n]] = [Ew d [n]] u [n - 1]
E 2 [w d [n] u [n]] = [E 2 w d [n]] u [n - 1]
h
E N [w d [n] u [n]] = [E N w d [n]] u [n - 1] + 1 d [n] . (A13)
aN

130

Substituting all the terms in (A13) into (23), we obtain the following equation:
[Q (E ) w d [n]] u [n - 1] + d [n] = d [n]
or
[Q (E ) w d [n]] u [n - 1] = 0,
which is satisfied if
Q (E ) w d [n] = 0 for n $ 1.

(A14)

As in the continuous-time case, w d [n] is the solution to the
homogeneous difference equation (A14) subject to the auxiliary conditions given in (A11).
■
The solution given in Theorem 2, (A9), is very general and
can be used for any causal or noncausal system. In the following corollary, it is shown how the general solution (A9)
can be simplified for three possible cases: N > M, N = M,
and N < M.
Corollary 2
The unit impulse response h [n] of a discrete-time LTI system
S specified by (18) can be obtained for the cases N > M,
N = M, and N < M as follows:
* Case N > M:
h [n] = [P (E ) w d [n]] u [n - 1]

(A15)

* Case N = M:
h [n] =

bN
d [n] + [P (E ) w d [n]] u [n - 1]
aN

(A16)

* Case N < M:
h [ n] =

M-N

/

r=1

b r d [n + r] + b 0 d [n] + [P (E ) w d [n]] u [n - 1],

(A17)

where
M-N-r

br =

/

l=0

b N + r + l w d [N + l ],

r = 0, 1, f, M - N.
(A18)

The intermediate signal w d [n] is the solution to the homogeneous equation (A10) subject to the auxiliary conditions given
in (A11).
Proof
Essentially as for Theorem 2, all the terms on the right side
of (A9) can be simplified by setting w d [0] = 0 and using
the auxiliary conditions given in (A11).
■

IEEE Signal Processing Magazine

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July 2018

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