Graph Notation
The Software Synthesis from Dataflow Graphs defines
the following notation in Chapter 2:
- vectors and matrices are indexed in functional notation:
x(3) is the third element of vector x and
M(i,j) represents the value at the ith row
and jth column of the matrix M
- gcd(P) is the greatest common divisor of a finite set
of integers P
- lcm(P) is the least common multiple of a finite set
of integers P
- when gcd(P) = 1, the members of P are co-prime
(a.k.a. relatively prime)
- max(P) and min(P) represent the largest and
smallest numbers of P
- f = a / b is a reduced fraction if a and b
are coprime integers.
Relative Primeness
Relative primeness has meaning for integers, rational numbers, polynomials,
as well as matrices of integers, rational numbers, and polynomials.
In one-dimensional multirate signal processing, one can interchange
the order of upsampling and downsampling if the resampling factors
are relatively prime.
This means that when we are performing a rational sampling rate change
by L / M, we can actually downsample by M first and then upsample by L,
provided that L / M is a reduced fraction.
Putting the downsampling first is far more efficient in terms of
computation.
Another application of relative primeness occurs in multidimensional
multirate signal processing, which is beyond the scope of this class.
One can interchange the order of upsampling and downsampling if the
resampling matrices are relatively prime and commute with respect to
matrix multiplication.
Relative primeness underlies the Smith-McMillan decomposition, which
is key for decoupling systems of differential equations represented
by a matrix of Laplace transforms.
Relative primeness also underlies the Smith form decomposition for integer,
matrices.
The Smith form decomposition is essential for decoupling non-separable
multidimensional signal processing operations.
It was used in the 1970s to solve integer programming problems.
Updated 02/25/99.