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Vector length 1

A VL of 1 implies that the computation is essentially scalar, and only one complex element can fit in a vector register. An example of such a scenario is when using interleaved double-precision floating-point arithmetic on an SSE2 machine: one 128-bit XMM register is used to store two 64-bit floats that represent the real and imaginary parts of a complex number.

When V L = 1 , the process of generating a program for a hard-coded FFT is as follows:

  1. Elaborate a topological ordering of nodes, where each node represents either a computation at the leaves of the transform, or a computation in the body of the transform (i.e., where smaller sub-transforms are combined into a larger transform);
  2. Write the program header to output, including a list of variables that correspond to registers used by the nodes;
  3. Traverse the list of nodes in order, and for each node, emit a statement that performs the computation represented by the given node. If a node is the last node to use a variable, a statement storing the variable to its corresponding location in memory is also emitted;
  4. Write the program footer to output.

Elaborate

[link] is a function, written in C++, that performs the first task in the process. As mentioned earlier, elaborating a topological ordering of nodes with a depth-first recursive structure is much likeactually computing an FFT with a depth-first recursive program (cf. Listing 3 in Appendix 2 ). [link] lists the nodes contained in the list ` ns ' after elaborating a size-8 transform by invoking elaborate(8, 0, 0, 0) .

  CSplitRadix::elaborate(int N, int ioffset, int offset, int stride) {     if(N > 4) {       elaborate(N/2, ioffset, offset, stride+1);      if(N/4 >= 4) {         elaborate(N/4, ioffset+(1<<stride), offset+(N/2), stride+2);         elaborate(N/4, ioffset-(1<<stride), offset+(3*N/4), stride+2);       }else{        CNodeLoad *n = new CNodeLoad(this, 4, ioffset, stride, 0);         ns.push_back(assign_leaf_registers(n));      }       for(int k=0;k<N/4;k++) {         CNodeBfly *n = new CNodeBfly(this, 4, k, stride);        ns.push_back(assign_body_registers(n,k,N);       }    }else if(N==4) {       CNodeLoad *n = new CNodeLoad(this, 4, ioffset, stride, 1);      ns.push_back(assign_leaf_registers(n));     }else if(N==2) {      CNodeLoad *n = new CNodeLoad(this, 2, ioffset, stride, 1);       ns.push_back(assign_leaf_registers(n));    }   }
Elaborate function for hard-coded conjugate-pair FFT

A transform is divided into sub-transforms with recursive calls at lines 4, 6 and 7, until the base cases of size 2 or size 4 are reached at the leaves of the elaboration. As well as the size-2 and size-4 base cases, which are handled at lines 20-21 and 17-18 (respectively), there is a special case where two size-2 base cases are handled in parallel at lines 9-10. This special case of handling two size-2 base cases as a larger size-4 node ensures that larger transforms are composed of nodes that are homogeneous in size – this is of little utility when emitting V L = 1 code, but it is exploited in "Other vector lengths" where the topological ordering of nodes is vectorized. The second row of [link] is just such a special case, since two size-2 leaf nodes are being computed, and thus the size is listed as 2(x2).

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Source:  OpenStax, Computing the fast fourier transform on simd microprocessors. OpenStax CNX. Jul 15, 2012 Download for free at http://cnx.org/content/col11438/1.2
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