Computing real signal with minimum absolute values from even magnitude spectrum

I want to derive a real audio signal from an arbitrary even magnitude spectrum.

The phase spectrum affects the values of the signal in the time domain; for example, a phase of 0 for all frequencies amounts to summing cosines in the time domain, which will result in a large peak at t = 0. I want to minimize the absolute values of the signal in the time domain without scaling the magnitude spectrum

I want this so that the audio signal can be multiplied by the largest scalar value possible without clipping. Is there a way to compute the phase spectrum that will do that? If there is a stupidly simply solution, I apologize in advance.

• i presume you want to minimize the max absolute value (which is the $L_1$ norm) while keeping the power (which is the $L_2$ norm) constant. otherwise you can minimize your absolute values by just turning the volume down. – robert bristow-johnson Oct 21 '17 at 4:44
• I'm aiming for a signal that can be as loud as possible without clipping. – BatWannaBe Oct 21 '17 at 4:46
• by "loud as possible", do you mean simply as much power as possible? do you want to consider A-weighting or similar in your definition of how loud a signal is? – robert bristow-johnson Oct 21 '17 at 6:30
• I meant just power – BatWannaBe Oct 21 '17 at 6:51
• May I also ask about the application you have? There's a lot of research on OFDM PAPR reduction that applies very much here, but we might really be finding a solution for the problem you state that doesn't actually help with what you want to do in the end. – Marcus Müller Oct 21 '17 at 10:05