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Based on fast finite-impulse response (FIR) algorithms (FFAs), this paper proposes new parallel FIR filter structures, which are beneficial to symmetric coefficients in terms of the hardware cost, under the condition that the number of taps is a multiple of 2 or 3. The proposed parallel FIR structures exploit the inherent nature of symmetric coefficients reducing half the number of multipliers in subfilter section at the expense of additional adders in preprocessing and postprocessing blocks. Exchanging multipliers with adders is advantageous because adders weigh less than multipliers in terms of silicon area; in addition, the overhead from the additional adders in preprocessing and postprocessing blocks stay fixed and do not increase along with the length of the FIR filter, whereas the number of reduced multipliers increases along with the length of the FIR filter. For example, for a four-parallel 72-tap filter, the proposed structure saves 27 multipliers at the expense of 11 adders, whereas for a four-parallel 576-tap filter, the proposed structure saves 216 multipliers at the expense of 11 adders still. Overall, the proposed parallel FIR structures can lead to significant hardware savings for symmetric convolutions from the existing FFA parallel FIR filter, especially when the length of the filter is large.