# Convolution of real with complex signal

Let $$x[n]$$ be real signal and $$y[n]=\exp(j3\pi n)$$ be a complex signal

Would the convolution between those two signals be $$x[n] * \Re(y[n]) + jx[n]*\Im(y[n])$$?

\begin{align*}x[n]*y[n] &= (x*y)[n]\\ \\ &= \sum_{m=-\infty}^{\infty} x[m]y[n-m] \\ \\ &= \sum_{m=-\infty}^{\infty} x[m]\Re(y[n-m])+ jx[m]\Im(y[n-m]) \\ \\ &= \left[\sum_{m=-\infty}^{\infty} x[m]\Re(y[n-m])\right] + \left[\sum_{m=-\infty}^{\infty} jx[m]\Im(y[n-m])\right] \\ \\ &= x[n]*\Re(y[n])+jx[n]*\Im(y[n])\\ \end{align*}
\begin{align} x[n] \star y[n] & = x[n] \star \left( \Re \{ y[n] \} + j ~~ \Im \{ y[n] \} \right) \\ &= x[n] \star \Re \{ y[n] \} + j ~~ x[n] \star \Im \{ y[n] \} \end{align}