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Ahern and Rudin have constructed an example of a totally real embedding of $S^3$ in $\mathbb{C}^3$ explicitly. As a generalization of their example, we give an explicit example of a CR regular embedding of $S^{4n-1}$ in $\mathbb{C}^{2n+1}$. Consequently, we show that the odd dimensional sphere $S^{2m-1}$ with $m>1$ admits a CR regular embedding in $\mathbb{C}^{m+1}$ if and only if $m$ is even.

CR regular embeddings of 𝑆⁴ⁿ⁻¹ in ℂ²ⁿ⁺¹