The symmetric pairs technique (see Symmetry Exploitation in Digital Interpolators/Decimators [Ref 9] ) is used to exploit coefficient symmetry when implementing an Interpolation filter in the Systolic Multiply-Accumulator architecture. When P polyphase subfilters are generated from symmetric filter coefficients, not all the subfilters contain a set of coefficients that are themselves symmetric. The symmetric pairs technique observes that adding and subtracting two corresponding non-symmetric phases produces two new phases containing symmetric coefficients.
The following example demonstrates this technique for a 15-tap interpolate by 3 filter. The filter coefficients, a, b, c, d, e, f, g, h, g, f, e, d, c, b, a produce the following subfilters:
h 0 = a, d, g, f, c
h 1 = b, e, h, e, b
h 2 = c, f, g, d, a
Subfilters h 0 and h 2 are not symmetric. Applying the symmetric pairs technique produces the following subfilters:
h 0 = a+c, d+f, d,g, f+d, c+a
h 1 = b, e, h, e, b
h 2 = c-a, f-d, g-g, d-f, a-c
Now both h 0 and h 2 are symmetric with h 2 being negative symmetric. The filter can now be implemented utilizing symmetry, giving the associated resource savings. The output from subfilters h 0 and h 2 must be added and subtracted and then scaled by a factor of 0.5 to produce the original filter output. This Figure shows the resulting structure.
Note: For some configurations an extra DSP Slice is required to implement the recombination of the phases.
Note: When interpolating by 2 with an odd number of symmetric coefficients, this technique is not required as the resulting polyphase subfilters are symmetric.