How to mix two signals without changing the overall loudness?

Question

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?

Are you asking about mixing (2 inputs, 1 output) or panning (1 input, 2 outputs)? In a sense, they're the opposite of each other. — Dave Tweed, Jan 04 '13 at 18:17
@DaveTweed I want to mix two inputs A and B into one output. — bastibe, Jan 04 '13 at 20:30
In that case, the perception of "loudness" is very dependent on the nature of the signals. — Dave Tweed, Jan 04 '13 at 20:52

score 9 · Accepted Answer · answered Jan 04 '13 at 13:24

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.

How to mix two signals without changing the overall loudness?

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