Convolution of Complex Signals

The convolution output $y[n]$ between two signals $x[n]$ and $h[n]$ is expressed as:

$$y[n]=\sum _m x[m]h[n-m]$$

Real and imaginary parts

When the two signals are complex, we can write

$$\begin{align*} x[n] &= x_I[n]+jx_Q[n]\\ h[n] &= h_I[n]+jh_Q[n] \end{align*}$$

in which the subscripts $I$ and $Q$ refer to inphase (real) and quadrature (imaginary) parts of the signal.

Now the multiplication in convolution can be split as:

$$\begin{align*} y_I[n]&=\sum _m x_I[m]h_I[n-m] - \sum _m x_Q[m]h_Q[n-m]\\ y_Q[n]&= \sum _m x_Q[m]h_I[n-m] + \sum _m x_I[m]h_Q[n-m] \end{align*}$$

This can also be written as:

$$\begin{align*} y_I[n] &= x_I[n]*h_I[n] - x_Q[n]*h_Q[n]\\ y_Q[n] &= x_Q[n]*h_I[n] + x_I[n]*h_Q[n] \end{align*}$$

This is drawn in the figure below: