14.11 INTERPOLATOR SCHEDULING

As usual, we employ an affine timing function:

(14.29)

where the row vector s = [s₁ s₂] is the scheduling vector and the column vector p = [n k]^t is any point in the dependence graph. The first component refers to the horizontal axis and the second component refers to the vertical axis. The restrictions on our timing function were discussed in Chapters 10 and 11. We assume that the input data x(n) arrive at consecutive times. Let us study the times associated with the points at the bottom of the graph p = [n 0]^t. Two input samples, x(n) and x(n + 1), arrive at the two points, p₁ = [n 0]^t and p₂ = [nL + 1 0]^t, respectively. Applying the scheduling function in Eq. 14.3, we get

(14.30)

(14.31)

Since the difference t(p₂) − t(p₁) = L, we must have s₁ = 1. A valid scheduling vector that satisfies input data timing must be specified as

(14.32)

The value of s₂ will be determined by our choice of whether we need to pipeline or broadcast the output sample y(n). Choosing s₂ = 0 would result in the broadcast of y(n). Choosing s₂ = ±1 would result in pipelining of y(n). We have three possible valid scheduling functions that we can employ:

(14.33)

(14.34)

(14.35)

Scheduling vector s₁ results in pipelined input x(n) and broadcast output y(n). Scheduling vector s₂ results in broadcast input x(n) and pipelined output y(n). Scheduling vector s₃ results in pipelined input x(n) and pipelined output y(n).

In the following subsections, we explore the possible designs afforded by the above scheduling vectors. We should point out that the advantages of using the approach in this section are reduction in the number of nodes. This comes, however, at the price of processing speed that matches the input data rate and increased task complexity. Later in this chapter, we will explore polyphase designs that do not suffer from these disadvantages.