Multitone signals with low crest factor
okay, there is a slightly newer paper by Ivo Mateljan titled Signal selection for the room acoustics measurement.  i ended up reviewing the paper for the 1999 IEEE Mohonk conference, and when it turned up that Ivo was not going to make it to the conference to present the paper, they ask a reviewer to sorta stumble through the issues of the paper.  i ended up talking about what's wrong with MLS more than about Ivo's technique.

if you cannot get a copy of Ivo's paper (it costs money from IEEE), email me and i will send a pdf to you.  in that paper he has a section titled The Low-Crest Multisine which i paste in the text below.  perhaps you can modify it to fit with your selected frequencies.  i dunno.

i still haven't figured out SE's formatting to make the code look like code.  someone else can edit this answer to do that.



Various algorithms were proposed for lowering the multisine crest factor. The following algorithm is slight modification of algorithms proposed in [9] and [10]. It has fast convergence and resulting multisine has good phase randomisation property.

Algorithm: Low-crest multisine (LCMS) generation.


Generate uniform random sequence u of length N.
Transform the sequence to random sign sequence s:

    for i = 1 : N 

        if (u(i) &gt; 0.5) 

            s(i) = 1 

        else 

            s(i) = -1 

    end

Generate the multisine using the phase values of DFT(s).
If the number of iterations is less then 10, clip the time series to 85% of the maximum value, else clip to 90% of the maximum value.
Apply DFT to the clipped time series.
Generate the new multisine using the phase values obtained in the step 5.
Calculate the crest factor. If it is higher than objective value go to the step 4.


Although there is no proof for the convergence of this
algorithm, it has never failed in a more than thousand runs with
different starting seed factor. The number of iterations grows
exponentially when decreasing crest factor. For a multisine
sequence of length 32768 it needs only 4-5 iterations to halve the
crest factor from 4 to 2, but to get the crest factor &lt; 1.5 it needs
several hundreds or more than thousand iterations.

Phase randomisation properties of this signal can be
estimated by testing how it distributes nonlinear distortion in the
estimated PIR. The following numerical experiment is done.
The input sequence of length 4096 drives 9th order bandpass
Butheworth filter (0.1-0.3 fs). The output signal is distorted with
2nd and 3rd order nonlinearities, which are equivalent to 1% of 2nd
harmonic distortion and 0.7% of the 3rd harmonic distortion.

Impulse response is estimated using crosscorrelation and scaled
to maximum value 1. Tables I and II show the rms and peak level
of the impulse response distortion (in percentage of the maximum
value). First 50 values of the impulse response are excluded from distortion calculation to better estimate peak/rms tail distortion. Also, the crest factor of the generated discrete sequence and a crest factor at the filter output are shown.

These results show that low-crest multisine generates the same level and a distribution of distortions as a true random phase multisine, regardless the crest factor value. MLS sequence generates slightly lower distortion rms level (1-1.5dB), but peak/rms ratio is much higher, especially with 2nd order distortion (10dB higher than with multisine excitation). This high peak/rms ratio disqualifies MLS as a reliable excitation signal for echo detection.

For all type of an excitation signal the crest factor at the filter output is almost the same. This is a general conclusion for narrow band systems. The benefit of using signals with lower crest factor is significant only for wideband systems, and when it is necessary to use all of the available measurement dynamic range.

RPMS is the only signal, which do not exhibit a significant crest factor transformation. This fact, constant amplitude and true random phase property fully qualify it as a periodic white noise signal.

[9] Schoukens, J., Pinelton, R., Ven der Ouderaa, E., and Renneboog, J., "Survey of Excitation Signals for FFT Based Signal Analysers", IEEE Trans. Instrumentation and Measurement, vol. 37, September 1988.
[10] Ven der Ouderaa, E., Schoukens, J., and Renneboog, J., "Peak Factor Minimisation of Input and Output Signals of Linear Systems", IEEE Trans. Instrumentation and Measurement, vol. 37, June 1988.