The impulse response for many filters possesses significant symmetry. This symmetry can generally be exploited to minimize arithmetic requirements and produce area-efficient filter realizations. The following figure shows the impulse response for a 9-tap symmetric FIR filter.
Instead of implementing this filter using the architecture shown in Figure 1, the more efficient signal flow-graph in the following figure can be used. In general, the former approach requires N multiplications and (N-1) additions. In contrast, the architecture in the following figure requires only [N/2] multiplications and approximately N additions. This significant reduction in the computation workload can be exploited to generate efficient filter hardware implementations.
Coefficient symmetry for an even number of terms can be exploited as shown in the following figure.
The following figure shows the impulse response for a negative, or odd, symmetric filter.
This symmetry is exploited in a manner similar to that shown in the preceding figures. In this case, the middle layer of adders are replaced by subtracters, as shown in the following figure.
Filter coefficient symmetry is inferred by the core GUI from the coefficient definition file. You can override this inferred value. When the structure is inferred, the inferred setting is displayed in the Summary page and in the ToolTip for the Coefficient Structure field.