Consider a linear time-invariant system with impulse response
hn
h
n
.
(Is assumed in these notes that all impulses response
hn
h
n
are real-valued.) (Figure 1)

The block diagram is represented in matrix form as
⋮y0y1y2y3y4y5y6⋮=(
⋱
⋱h0000
⋱h1h000
⋱h2h1h00
⋱h3h2h1h0
0h3h2h1⋱
00h3h2⋱
000h3⋱
⋱
)⋮x0x1x2x3x4x5x6⋮
⋮
y0
y1
y2
y3
y4
y5
y6
⋮
⋱
⋱
h0
000
⋱
h1
h0
00
⋱
h2
h1
h0
0
⋱
h3
h2
h1
h0
0
h3
h2
h1
⋱
00
h3
h2
⋱
000
h3
⋱
⋱
⋮
x0
x1
x2
x3
x4
x5
x6
⋮
where
Q=(
⋱
⋱h0000
⋱h1h000
⋱h2h1h00
⋱h3h2h1h0
0h3h2h1⋱
00h3h2⋱
000h3⋱
⋱
)
Q
⋱
⋱
h0
000
⋱
h1
h0
00
⋱
h2
h1
h0
0
⋱
h3
h2
h1
h0
0
h3
h2
h1
⋱
00
h3
h2
⋱
000
h3
⋱
⋱

The vectors are of infinite length. The matrix has infinitely
many rows and infinitely many columns. We can write convolution
as a matrix-vector multiplication:
yn=hn*xn ⇔y=Qx
⇔
y
n
h
n
x
n
y
Q
x
where QQ is the
convolution matrix. The matrix QQ is also called the
Toeplitz matrix - a Toeplitz matrix is constant
along its diagonals. Note that the impulse response
hnh
n appears in each column of the
matrix QQ.

What is the transpose of the linear time-invariant system with
impulse response
hn
h
n
? Usually, we do not think about the transpose of a
system. However, we can derive the transpose of the system by
using the matrix representation. We just need to look at the
transpose of the matrix QQ. The columns of QQ are the rows of
QT
Q
, so
QT
Q
has the following form:
QT=(
⋱
⋱h3000
⋱h2h300
⋱h1h2h30
⋱h0h1h2h3
0h0h1h2⋱
00h0h1⋱
000h0⋱
⋱
)
Q
⋱
⋱
h3
000
⋱
h2
h3
00
⋱
h1
h2
h3
0
⋱
h0
h1
h2
h3
0
h0
h1
h2
⋱
00
h0
h1
⋱
000
h0
⋱
⋱
We can observe that
QT
Q
is again a convolution matrix (it is constant along the
diagonals). So
QT
Q
represents a linear time-invariant system. The impulse
response of this system can be found by looking at the columns of
QT
Q
,
which we see is the flipped version of
hn
h
n
. Therefore,
QT
Q
represents convolution by
h−n
h
n
. (Figure 2)

y=Qx⇔yn=hn*xn
⇔
y
Q
x
y
n
h
n
x
n
z=QTy⇔zn=h−n*yn
⇔
z
Q
y
z
n
h
n
y
n
Can the filter
hn
h
n
represent an *orthogonal* transformation?
Equivalently, is it possible that
QT
Q
is the inverse of QQ? If
QTQ=I
Q
Q
I
,
then
z=QTy=QTQx=Ix=x
z
Q
y
Q
Q
x
I
x
x
or
zn=xn
z
n
x
n
. But

zn=h−n*yn=h−n*hn*xn
z
n
h
n
y
n
h
n
h
n
x
n

(1)
so

zn=xn
z
n
x
n
only if

h−n*hn=δn
h
n
h
n
δ
n
Note that
h−n*hn=δn⇔H1zHz=1⇔H−eiωHeiω=1⇔|Heiω|2=1
⇔
h
n
h
n
δ
n
H1z
Hz
1
H
ω
H
ω
1
H
ω
2
1
That means Hz Hz must be an
allpass system. * The only orthonormal LTI
systems are allpass systems. * But they provide no
frequency selective filtering - the allpass filter does not
provide a subband decomposition of the signal
xn
x
n
.