Similar to existing codes, puncturing and shortening are two general ways to obtain an arbitrary code length and code rate for polar codes. When some of the coded bits are punctured or shortened, it is equivalent to a situation in which the underlying channels of the polar codes are different. Therefore, the quality of bit channels with puncturing or shortening differ from the original qualities, which can greatly affect the construction of polar codes. In this paper, a general construction of polar codes is studied in two aspects: 1) the theoretical foundation of the construction; and 2) the hardware implementation of polar codes encoders. In contrast to the original identical and independent binary-input, memoryless, symmetric (BMS) channels, these underlying BMS channels can be different. For binary erasure channel (BEC) channels, recursive equations can be employed assuming independent BMS channels. For all other channel types, the proposed general construction of polar codes is based on the existing Tal-Vardy's procedure. The symmetric property and the degradation relationship are shown to be preserved under the general setting, rendering the possibility of a modification of Tal-Vardy's procedure. Simulation results clearly show improved error performance with re-ordering using the proposed new procedures. In terms of hardware, a novel pruned folded encoder architecture is proposed which saves the computation for the beginning frozen bits. Implementation results show the pruned encoder achieve 28% throughput improvement.