Abstract: In this dissertation we have explored the behavior of Sagbi Basis under non-central (w.r.t non-commuting variable T) (de)homogenization technique in free associative algebras. Cases of non-central (de)homogenization of Grabner Basis in free associative algebras is discussed in detailed to extract how things work differently when non-central (de)homogenization is applied to Sagbi Basis in free associative algebras