Multidimensional Dataflow Graphs - Signal Processing Systems - page 1126

Thus actor A has to produce a number of times (3,1) tokens, and actor B has
to consume a number of times (1,3) tokens. The equations can be solved to yield
( r A , 1 , r A , 2 )=( 1 , 3 ) ,and ( r B , 1 , r B , 2 )=( 3 , 1 ) . In other words, actor A has to produce

( 3 , tokens.

These equations are independent of initial tokens

3 ) tokens, and actor B has to consume ( 3 , 3 )

in the AB arc's buffer.
Different interpretations of these initial tokens are possible. Figure 2 illustrates one
of them. The buffer contains initial tokens in two columns of a single row, that is a
delay of

( x , y )

. A possible schedule is shown at the right in the figure: A, B, 2(A),
2(B), labeled as (1, 2, 3, 4, and 6). Note that the state returns to its initial value (1,2).
An alternative interpretation of multidimensional initial tokens is discussed in
[ 13 , 14 ] .

( 1 , 2 )

3 Arbitrary Sampling

Two canonical actors that play a fundamental role in MD signal processing are the
compressor (or downsampler) and the expander (or upsampler) [ 13 , 14 ] . For the sake
of pictorial representation, we will confine to the 2D case. Consider a 2D “signal”
s

2 , defined on a rectangular subspace 0

K 2 .
This bounded subspace is represented in Fig. 3 as a back-ground grid, with K 1 =7,
and K 2 = 6. Notice that although s

( k ) , k =( k 1 , k 2 ) ∈ R ≤ k 1 ≤ K 1 ∧ 0 ≤ k 2 ≤

is a continuous function, we do represent its
supporting domain as a grid of k-points that can be arbitrary dense, see Fig. 3 . The
signal s

( k )

( k ) can be sampled to yield the discrete signal s ( n )= s ( k = V n ) ,where V is

2 is such that the sample points k

a2 × 2 non-singular sampling matrix ,and n ∈ Z =

V n fall within the rectangle )

in the example). In fact, the set of n-points that satisfy this condition are the integral
points in the parallelogram

K defined by the matrix K = diag ( K 1 , K 2 ) ( diag ( 7 , 6

V − 1 K . Thus, VQ Q defined by the matrix Q = = K and

2 2 . The matrix Q is called the support matrix .

n ∈Q∩ Z ↔ k = V n ∈K ∩ Z

A typical sampling matrix is

v 1 , 1

0

V 1 =

0 v 2 , 2

2

= ∈ Q ∩ Z

which yields a rectangular lattice of sample points k V 1 n , n (black 1

= , =

dots). With v 1 , 1 2 v 2 , 2 3, the support matrix for this lattice is

3

50
02

.

Q 1 = .

A second typical sampling matrix is

v 1 , 1 =

v 1 v 1 , 2 = v 2

V 2 =

v 2 , 1 = v 1 v 2 , 2 = − v 2