In this paper, we design a new iterative low-complexity algorithm for computing the Walsh-Hadamard transform (WHT) of an N dimensional signal with a K-sparse WHT. We suppose that N is a power of two and K = O(N^α), scales sub-linearly in N for some α ∈ (0,1). Assuming a random support model for the nonzero transform-domain components, our algorithm reconstructs the WHT of the signal with a sample complexity O(K log_2(N/K)) and a computational complexity O(K log_2(K) log_2(N/K)). Moreover, the algorithm succeeds with a high probability approaching 1 for large dimension N. Our approach is mainly based on the subsampling (aliasing) property of the WHT, where by a carefully designed subsampling of the time-domain signal, a suitable aliasing pattern is induced in the transform domain. We treat the resulting aliasing patterns as parity-check constraints and represent them by a bipartite graph. We analyze the properties of the resulting bipartite graphs and borrow ideas from codes defined over sparse bipartite graphs to formulate the recovery of the nonzero spectral values as a peeling decoding algorithm for a specific sparse-graph code transmitted over a binary erasure channel (BEC). This enables us to use tools from coding theory (belief-propagation analysis) to characterize the asymptotic performance of our algorithm in the very sparse (α ∈ (0,1/3]) and the less sparse (α ∈ (1/3,1)) regime. Comprehensive simulation results are provided to assess the empirical performance of the proposed algorithm.

This is the code for SparseFHT algorithm as presented in the following papers. The code is hosted on github.

[Long] R. Scheibler, S.Haghighatshoar, and M. Vetterli,
*A Fast Hadamard Transform for Signals with Sub-linear Sparsity in the Transform Domain*,
IEEE Trans. Inf. Theory, vol. 61, 2015.

[Short] R. Scheibler, S. Haghighatshoar, and M. Vetterli,
*A Fast Hadamard Transform for Signals with Sub-linear Sparsity*,
Allerton Conference on Communication, Control and Computing, 2013.

Robin Scheibler (email) (homepage)

Please do not hesitate to contact me for help and support! I would be happy to help you run the code.

The code has been tested on Mac OS X 10.7, 10.8, 10.9, and on Ubuntu linux.

Matlab mex wrappers were used for the code generating the figures in the paper. The core of the algorithm is implemented in C.

To reproduce the figures from the paper, type in the following in a matlab shell:

```
cd <path_to_SparseFHT>/matlab/
make_mex_files
make_figures
```

**Note**

- The simulation is fairly time-consuming.
- To speed-up the simulation, the parallel toolbox was used. If you do not have the parallel toolbox, replace the
`parfor`

instructions by`for`

in all the Sim scripts.

**SparseFHT**The fast sparse Hadamard transform algorithm.`[Y, S, U, I] = SparseFHT(X, K, B, C, L, T) Wrapper for the Sparse Fast Hadamard Transform Input arguments: X: input vector (size n) K: the sparsity (and size of y) B: number of buckets C: oversampling factor L: maximum number of iterations of decoder T: Type of algorithm to use ('Random' / 'Deterministic' / 'Optimized') Output arguments: Y: output vector (size K) S: support vector (size K) U: the number of unsatisfied checks (optional) I: the number of loops run (optional)`

**FastHadamard**A straighforward implementation of the conventional fast Hadamard transform. No scaling factor is applied, i.e. applying the algorithm twice will result in the input vector weighted by its length.`[Y] = FastHadamard(X) Input arguments: X: input vector (size has to be a power of two) Output arguments: Y: output vector, the Hadamard transform of X.`

**HadamardBenchmark**Calls a C routine performing a timing comparison of SparseFHT and FastHadamard.`[Tfht Tsfht] = HadamardBenchmark(N, K, B, C, L, R, SEED) Input arguments: N: transform size to investigate (scalar, power of two) K: sparsity parameter, number of non-zero tranform domain coefficients (vector, power of two) B: number of buckets to use in SparseFHT (vector, same size as K, power of two) C: oversampling factor L: maximum number of iterations of decoder R: a length 4 vector containing the following parameters 1. Number of repetitions of one measurement 2. Number of warm-up run 3. Number of iterations for one measurement 4. Maximum magnitude of non-zero components in the sparse signal SEED: A seed for the C random number generator Output arguments: Tfht: The runtime measurement of FastHadamard Tsfht: An array containing the runtime measurement of SparseFHT for every value of K`

The makefile in `./C`

folder will compile all the code as well as a number
of example/test files. This can be used as a basis to reuse the C code directly.

```
cd ../C
make all
```

The libraries included are:

- SparseFHT,
- Fast Hadamard tranform,
- Fast linear algebra in GF2 (boolean matrices and vectors algebra),
- Test files.

2013-2015 (c) Robin Scheibler, Saeid Haghighatshoar, Martin, Vetterli.

This work is licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License. To view a copy of this license, visit http://creativecommons.org/licenses/by-nc-sa/4.0/.

The code is free to reuse for non-commercial and academic purposes. However, please acknowledge its use with the following citation.

```
@article{EPFL-JOUR-204991,
author = {Scheibler, Robin and Haghighatshoar, Saeid and Vetterli, Martin},
title = {A {F}ast {H}adamard {T}ransform for {S}ignals with
{S}ub-linear {S}parsity in the {T}ransform {D}omain},
journal = {IEEE Trans. Inf. Theory}
volume = 61,
year = 2015,
ee = {http://infoscience.epfl.ch/record/204991}
}
```

For any other purposes, please contact the authors.

Comment | Rating | Date |

5 | 2015-06-22 | |

Good stuff | 3 | 2015-11-05 |

2015

Easy to reproduce figures

★★★★☆

★★★★☆

Well-commented code

★★★★★

★★★★★

Robin Scheibler; Saeid Haghighatshoar; Martin Vetterli, "A Fast Hadamard Transform for Signals with Sub-linear Sparsity in the Transform Domain", IEEE Transactions on Information Theory, Vol. 61, No. 4, pp. 2115 - 2132