Let $mathbf{c}={c_n}_{n{in}mathbb{Z}}in{ell}^1(mathbb{Z})$ and ${f_n}_{n{in}mathbb{Z}}$ be a frame (Riesz basis, respectively) of $L^2(mathbb{R})$. We obtain necessary and sufficient conditions of $mathbf{c}$ under which ${mathbf{c}{ast}_{lambda}f_n}_{n{in}mathbb{Z}}$ becomes a frame (Riesz basis, respectively) of $L^2(mathbb{R})$, where ${lambda}$ > 0 and $(mathbf{c}{ast}_{lambda}f)(t);:=;{sum}_{n{in}mathbb{Z}}c_nf(t-n{lambda})$. When ${mathbf{c}{ast}_{lambda}f_n}_{n{in}mathbb{Z}}$ becomes a frame of $L^2(mathbb{R})$, we present its frame operator and the canonical dual frame in a simple form. Some interesting examples are included.