audio - How to mix two signals without changing the overall loudness?

I have two audio signals that I want to mix at various mixing ratios. Initially, I simply went for something like

$y(t) = \alpha \cdot x_1(t) + (1-\alpha) \cdot x_2(t)$
where $\alpha$ is the ratio between zero and one and $x_1$ and $x_2$ are the two signals.

However, I have seen a few panning curves that did not mix signals linearly. They told me that a linear mix would change the overall loudness while mixing.

Is there something to that or is a linear mix the correct way to do it?

Answer

That depends somewhat on the application. In most cases a "constant energy" pan will be best. This can be expressed as $$y(t)=\sqrt{\alpha} \cdot x_{1}(t)+\sqrt{1-\alpha}\cdot x_{2}(t)$$ where $\alpha = .5$ is the point of equal energy.

If you are working with fixed point signals, such as wave files for examples, you may run into clipping problems. That could be a topic for a separate question.