The partitioning of long or infinite strings of data into shorter sections
or blocks has been used to allow application of the FFT to realize
on-going or continuous convolution Entry 40, Entry 25. These notes
develop the idea of block processing and shows that it is a generalization
of the overlap-add and overlap-save methods Entry 40, Entry 22. They
further generalize the idea to a multidimensional formulation of
convolution Entry 1, Entry 10. Moving in the opposite direction, it is
shown that, rather than partitioning a string of scalars into blocks and
then into blocks of blocks, one can partition a scalar number into blocks
of bits and then include the operation of multiplication in the signal
processing formulation. This is called distributed arithmetic Entry 9
and, since it describes operations at the bit level, is completely
general. These notes try to present a coherent development of these
ideas.
In this section the usual convolution and recursion that implements FIR
and IIR discrete-time filters are reformulated in terms of vectors and
matrices. Because the same data is partitioned and grouped in a variety
of ways, it is important to have a consistent notation in order to be
clear. The nthnth element of a data sequence is expressed h(n)h(n) or, in
some cases to simplify, hnhn. A block or finite length column vector is
denoted h̲nh̲n with nn indicating the nthnth block or
section of a longer vector. A matrix, square or rectangular, is indicated
by an upper case letter such as HH with a subscript if appropriate.
The operation of a finite impulse response (FIR) filter is described by a
finite convolution as
y
(
n
)
=
∑
k
=
0
L
-
1
h
(
k
)
x
(
n
-
k
)
y
(
n
)
=
∑
k
=
0
L
-
1
h
(
k
)
x
(
n
-
k
)
(1)
where x(n)x(n) is causal, h(n)h(n) is causal and of length LL, and the time
index nn goes from zero to infinity or some large value. With a change
of index variables this becomes
y
(
n
)
=
∑
k
=
0
n
h
(
n
-
k
)
x
(
k
)
y
(
n
)
=
∑
k
=
0
n
h
(
n
-
k
)
x
(
k
)
(2)
which can be expressed as a matrix operation by
y
0
y
1
y
2
⋮
=
h
0
0
0
⋯
0
h
1
h
0
0
h
2
h
1
h
0
⋮
⋮
x
0
x
1
x
2
⋮
.
y
0
y
1
y
2
⋮
=
h
0
0
0
⋯
0
h
1
h
0
0
h
2
h
1
h
0
⋮
⋮
x
0
x
1
x
2
⋮
.
(3)
The HH matrix of impulse response values is partitioned into NN by NN
square sub matrices and the XX and YY vectors are partitioned into
length-NN blocks or sections. This is illustrated for N=3N=3 by
H
0
=
h
0
0
0
h
1
h
0
0
h
2
h
1
h
0
H
1
=
h
3
h
2
h
1
h
4
h
3
h
2
h
5
h
4
h
3
etc.
H
0
=
h
0
0
0
h
1
h
0
0
h
2
h
1
h
0
H
1
=
h
3
h
2
h
1
h
4
h
3
h
2
h
5
h
4
h
3
etc.
(4)
x
̲
0
=
x
0
x
1
x
2
x
̲
1
=
x
3
x
4
x
5
y
̲
0
=
y
0
y
1
y
2
etc.
x
̲
0
=
x
0
x
1
x
2
x
̲
1
=
x
3
x
4
x
5
y
̲
0
=
y
0
y
1
y
2
etc.
(5)
Substituting these definitions into (Equation 3) gives
y
̲
0
y
̲
1
y
̲
2
⋮
=
H
0
0
0
⋯
0
H
1
H
0
0
H
2
H
1
H
0
⋮
⋮
x
̲
0
x
̲
1
x
̲
2
⋮
y
̲
0
y
̲
1
y
̲
2
⋮
=
H
0
0
0
⋯
0
H
1
H
0
0
H
2
H
1
H
0
⋮
⋮
x
̲
0
x
̲
1
x
̲
2
⋮
(6)
The general expression for the nthnth output block is
y
̲
n
=
∑
k
=
0
n
H
n
-
k
x
̲
k
y
̲
n
=
∑
k
=
0
n
H
n
-
k
x
̲
k
(7)
which is a vector or block convolution. Since the matrix-vector
multiplication within the block convolution is itself a convolution, (Equation 8)
is a sort of convolution of convolutions and the finite length
matrix-vector multiplication can be carried out using the FFT or other
fast convolution methods.
The equation for one output block can be written as the product
y
̲
2
=
H
2
H
1
H
0
x
̲
0
x
̲
1
x
̲
2
y
̲
2
=
H
2
H
1
H
0
x
̲
0
x
̲
1
x
̲
2
(8)
and the effects of one input block can be written
H
0
H
1
H
2
x
̲
1
=
y
̲
0
y
̲
1
y
̲
2
.
H
0
H
1
H
2
x
̲
1
=
y
̲
0
y
̲
1
y
̲
2
.
(9)
These are generalize statements of overlap save and overlap add
Entry 40, Entry 22. The block length can be longer, shorter, or equal to
the filter length.
Although less well-known, IIR filters can be implemented with block
processing Entry 20, Entry 12, Entry 42, Entry 7, Entry 8. The block form of an IIR
filter is developed in much the same way as for the block convolution
implementation of the FIR filter. The general constant coefficient
difference equation which describes an IIR filter with recursive
coefficients alal, convolution coefficients bkbk, input signal x(n)x(n),
and output signal y(n)y(n) is given by
y
(
n
)
=
∑
l
=
1
N
-
1
a
l
y
n
-
l
+
∑
k
=
0
M
-
1
b
k
x
n
-
k
y
(
n
)
=
∑
l
=
1
N
-
1
a
l
y
n
-
l
+
∑
k
=
0
M
-
1
b
k
x
n
-
k
(10)
using both functional notation and subscripts, depending on which is
easier and clearer. The impulse response h(n)h(n) is
h
(
n
)
=
∑
l
=
1
N
-
1
a
l
h
(
n
-
l
)
+
∑
k
=
0
M
-
1
b
k
δ
(
n
-
k
)
h
(
n
)
=
∑
l
=
1
N
-
1
a
l
h
(
n
-
l
)
+
∑
k
=
0
M
-
1
b
k
δ
(
n
-
k
)
(11)
which can be written in matrix operator form
1
0
0
⋯
0
a
1
1
0
a
2
a
1
1
a
3
a
2
a
1
0
a
3
a
2
⋮
⋮
h
0
h
1
h
2
h
3
h
4
⋮
=
b
0
b
1
b
2
b
3
0
⋮
1
0
0
⋯
0
a
1
1
0
a
2
a
1
1
a
3
a
2
a
1
0
a
3
a
2
⋮
⋮
h
0
h
1
h
2
h
3
h
4
⋮
=
b
0
b
1
b
2
b
3
0
⋮
(12)
In terms of NN by NN submatrices and length-NN blocks, this becomes
A
0
0
0
⋯
0
A
1
A
0
0
0
A
1
A
0
⋮
⋮
h
̲
0
h
̲
1
h
̲
2
⋮
=
b
̲
0
b
̲
1
0
⋮
A
0
0
0
⋯
0
A
1
A
0
0
0
A
1
A
0
⋮
⋮
h
̲
0
h
̲
1
h
̲
2
⋮
=
b
̲
0
b
̲
1
0
⋮
(13)
From this formulation, a block recursive equation can be written that will
generate the impulse response block by block.
A
0
h
̲
n
+
A
1
h
̲
n
-
1
=
0
for
n
≥
2
A
0
h
̲
n
+
A
1
h
̲
n
-
1
=
0
for
n
≥
2
(14)
h
̲
n
=
-
A
0
-
1
A
1
h
̲
n
-
1
=
K
h
̲
n
-
1
for
n
≥
2
h
̲
n
=
-
A
0
-
1
A
1
h
̲
n
-
1
=
K
h
̲
n
-
1
for
n
≥
2
(15)
with initial conditions given by
h
̲
1
=
-
A
0
-
1
A
1
A
0
-
1
b
̲
0
+
A
0
-
1
b
̲
1
h
̲
1
=
-
A
0
-
1
A
1
A
0
-
1
b
̲
0
+
A
0
-
1
b
̲
1
(16)
This can also be written to generate the square partitions of the impulse
response matrix by
H
n
=
K
H
n
-
1
for
n
≥
2
H
n
=
K
H
n
-
1
for
n
≥
2
(17)
with initial conditions given by
H
1
=
K
A
0
-
1
B
0
+
A
0
-
1
B
1
H
1
=
K
A
0
-
1
B
0
+
A
0
-
1
B
1
(18)
ane K=-A0-1A1K=-A0-1A1. This recursively generates square submatrices
of HH similar to those defined in (Equation 4) and (Equation 6) and shows the
basic structure of the dynamic system.
Next, we develop the recursive formulation for a general input as
described by the scalar difference equation (Equation 11) and in matrix operator
form by
1
0
0
⋯
0
a
1
1
0
a
2
a
1
1
a
3
a
2
a
1
0
a
3
a
2
⋮
⋮
y
0
y
1
y
2
y
3
y
4
⋮
=
b
0
0
0
⋯
0
b
1
b
0
0
b
2
b
1
b
0
0