Important information

If you are thinking of contacting us, please do not e-mail the author to ask for download instructions, installation guidelines, or the toolbox itself. The code itself is well-documented and the package contains a README.txt file providing the essential information about the software. Note that we will NOT help to debug user-generated code that was not included in the provided software package. If, however, you notice a bug in our code, please be so kind to contact the author.

The software package is supplied "as is", without any accompanying support services, maintenance, or future updates. We make no warranties, explicit or implicit, that the software contained in this package is free of error or that it will meet your requirements for any particular application. It should not be relied on for any purpose where incorrect results could result in loss of property, personal injury, liability or whatsoever. If you do use our software for any such purpose, it is at your own risk. The authors disclaim all liability of any kind, either direct or consequential, resulting from your use of these programs.

Minimization of the l-infinity norm finds use in a large number of practical applications, such as vector quantization, approximate nearest neighbor search, peak-to-average power ratio (or crest-factor) reaction, and peak force minimization. The solutions to such minimization problems are called democratic representations (other names that can be found in the literature are anti-sparse, Kashin, or spread representations) and can be obtained by solving specific convex optimization problems. The software package provided here includes three distinct and computationally efficient numerical solvers, namely convex reduction of amplitudes (CRAM), CRAM for Parseval frames (CRAMP), and the fast iterative truncation algorithm (FITRA).

More information on democratic representations and the algorithms provided in the software package can be found in the following papers:

C. Studer, T. Goldstein, W. Yin, and R. G. Baraniuk, "Democratic Representations," submitted to a journal, arXiv preprint: 1401.3420, 2015
C. Studer and Erik G. Larsson, "PAR-Aware Large-Scale Multi-User MIMO-OFDM Downlink," IEEE Journal on Selected Areas in Communications, Vol. 31, No. 2, pp. 303–313, Feb. 2013
C. Studer, W. Yin, and R. G. Baraniuk, "Signal Representations with Minimum l_inf-Norm," Proc. 50th Annual Allerton Conference on Communication, Control, and Computing, Oct. 2012
C. Studer and E. G. Larsson, "PAR-Aware Multi-user Precoder for the Large-Scale MIMO-OFDM Downlink," Proc. IEEE 9th International Symposium on Wireless Communication Systems (ISWCS), Aug. 2012, (invited paper)

Package details

The software package contains three solvers: CRAM, CRAMP, and FITRA, and a simple example script. The example script computes democratic representations for a discrete cosine transform (DCT)-based Parseval frame. The software package is modular so that you can use the three solvers for your own applications. The code is written by C. Studer and T. Goldstein, and is available for free trial, non-commercial research or education purposes, and for non-profit organizations. If you plan on using the code or parts thereof for commercial purposes or if you intend to re-distribute the code or parts thereof, you must contact the author. If you are using our code or parts thereof for your scientific work, you must to provide a reference to this website or at least one of the publications listed above.


The software package has no specific toolbox requirements.


If you agree with the conditions and regulations above, you may download the most recent version here. The zip file (13KB) contains Matlab .m files, as well as a README.txt file describing how to use the example script. Have fun!