**7. Conclusions**

It is suggested that digital NCCs be implemented by efficient algorithms and hardware structures for decreasing their high multiplication complexity [35]. With the assist of fast computation for first-order moment, this paper presents an algorithm and a systolic array for fast NCCs that aim to reduce multiplication as much as possible. To do this, the key is to transform the complex inner-product in the NCC into a simple first-order moment according to the statistical properties of the digital inner-product, and then a new NCC formula based on a first-order moment is established in order for eliminating inner-product operations. As a result, by introducing an algorithm without multiplication into the computation of the first-order moment in NCC, we proposed a fast algorithm for NCC with the advantages of simple implementation, less multiplication, no length limitation, and so on. Especially, as the introduced algorithm for first-order moment requests many additions, we also improved it by means of an even-odd relationship to reduce addition complexity and execution time. It is an advantage that the introduced algorithm for the first-order moment can be implemented by systolic structure, so a systolic array composed of latches and adders is designed for implementing fast NCC in parallel. This systolic array is hardware-efficient due to its parallel operation, simple structures and seldom multiplier. This paper analyzes the computational and the hardware complexity for the proposed algorithm and systolic array, and compares them with some existing methods to prove their efficiency. The proposed algorithm and array could also be applied for digital filter and various transforms [36].

There are still many additions in the proposed algorithm and systolic structure. Future studies will focus on further reducing their additions.

**Author Contributions:** The work presented in this paper was completed with collaboration among all authors. Conceptualization, C.P.; Methodology, C.P., Z.L. and X.H.; Formal analysis, Z.L.; Writing—original draft preparation, C.P.; Writing-review and editing, Z.L., X.H., H.L.; Supervision, H.L.; Funding acquisition, X.H. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by the National Natural Science Foundation of China (Grant No. 61801337, 61572012), the Natural Science Foundation of Hubei Province of China (Grant No. 2018CFB661, 2017CFB677).

**Conflicts of Interest:** The authors declare no conflicts of interest.
