In this paper, we provide the left and right Parseval relations over the quaternions. Moreover, we construct a new infinite family of sequences almost over$Q_{8} $with period$p$by a generalization of Legendre mapping, where$p\equiv 1 \pmod {8}$is prime and the sequences have ideal autocorrelation. The sequences are symmetric when$p\equiv 1 \pmod {16}$, and antisymmetric when$p\equiv 9 \pmod {16} $.