Example Code
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- LabVIEW
Software
Code and Documents
Attachment
Overview
This example presents a method used to recreate a signal from a set of Random sample points. It utilizes a compressive sampling algorithm to construct the signals most important portions. This example documents steps in the experimental process to prove the algorithm being presented.
Description
This Compressive Data Acquisition LabVIEW program presents a signal reconstruction algorithm from a set of single random (scalar) measurements, where the signal is k sparse in some transform domain. Each measurement or observation represents a random projection of the signal onto a single scalar value. By applying Discrete Radon Transform techniques, the original signal can be reconstructed from these random sampled measurements. However, this requires a very large set of measurements which defeats the purpose of compressive data acquisition. To acquire the wideband data below the Nyquist frequency, a rank order filter is applied in the sparse transform domain to extract the k most significant components.
Compressive Sampling
Consider the basic compressive sampling relationship
 | (1) |
| where | y is an M-element vector R is an MxN random binary sampling matrix x is an N-element sampled vector x is k-sparse in some transform domain T 0 < k << N |
The elements of y are given by
 | (2) |
where ri,j is the jth element the ith R random binary sampling basis with probabilities
 | (3) |
 | (4) |
Signal reconstruction
Equations (1) and (2) reveal a standard projection onto a single scalar measurement. Therefore x can be reconstructed from the y projections using
 | (5) |
Convergence
To verify that equation (5) does indeed converge, a simulation was implemented in LabVIEW using
 | (6) |
and letting the systems run in an "infinite" loop. The original signal consisted of the sum of sinusoids. The frequency (sparse) domain results for 7 active components is shown in Fig. 1.
Fig. 1. Signal Convergence in the Sparse Domain
Rank-Order Filter in Sparse Domain
Clearly, we desire to reconstruct x from y from i samples such that i << N. To accelerate the convergence, a k-rank-order filter is applied in the signal's sparse domain. That is
 | (7) |
| where | xHati is the estimate of x at the ith iteration T{•} is a transform mapping to a sparse domain T-1{•} is the transform mapping to the original domain Hk is a k-rank order filter |
The processing loop is terminated when the threshold condition, α, meets
 | (8) |
Compressive Data Acquisition LabVIEW Program
LabVIEW was used to implement the compressive data acquisition algorithm because of its graphical and interactive programming capabilities. The main LabVIEW program is shown in Fig. 2.
Fig. 2. High-level LabVIEW block diagram
The Compressive Sample VI shown in Fig. 3 is simply the dot product of the input Signal and the generated Random Sampling Basis.
Fig. 3. Random sample (measurement) program
The Estimate Signal subroutine implements equation (7) using the Discrete Fourier Transform as the corresponding transform operation T{•}. Analysis of the LabVIEW block diagram in Fig. 4 shows the signal flow representation to obtain an estimate of x.
Fig. 4. LabVIEW block diagram to reconstruct the signal
Fig. 5 shows the results of running the LabVIEW program shown in Fig. 2 using Samples = 4096 (N), %Accuracy = 99 (α), Prob = 0.1 and k = 7.
Fig. 5. LabVIEW front panel results
The simulation results show that for this particular set of parameters, it took 253 iterations (measurements) to reconstruct a wideband signal consisting of k sinusoids with approximate accuracy of 99.39%. Since this simulation is interactive, the iterations required to reconstruct vary. In this simulation we observed iterations in the 200 - 700 range.
Fig. 6 shows a zoomed portion of the signal where the yellow curve is the original signal and the red curve is the reconstructed signal using the rank order filtered convergence algorithm.
Fig. 6. Zoomed in results to show the original signal (yellow) and the signal reconstruction details (red).
Steps to Implement or Execute Code
1. Download and unzip the comp_daq.zip file
2. Open the Compressive Acquisition.lvproj LabVIEW project file
3. Open the Compressive Acquisition Simulation.vi
4. Configure the following controls on the front panel:
- % Accuracy
- Prob
- k
- seed
5. Run the VI
6. See the results on the front panel
Requirements to Run
Software
LabVIEW 2009 or compatible
Hardware
N/A
Additional Information or References
[1] Compressive Sensing Resources, http://www.dsp.ece.rice.edu/cs, Rice University.
Eduardo "Lalo" Pérez, Ph.D. was born in Mexico City, Mexico and obtained a B.S. E.C.E in telecommunications in 1981, M.S. E.C.E in image processing in 1983 and Ph.D. E.C.E. in real-time image processing in 1989 all from The University of Texas at Austin, Austin, TX, USA. Dr. Eduardo "Lalo" Pérez first joined National Instruments in 1985 while completing his Ph.D. and is one of the original members of the LabVIEW development team where he was responsible for designing and implementing the Digital Signal Processing and Data Analysis Libraries, still being used today. Dr. Pérez has over 20 years of industrial experience in the areas of real-time multiprocessor programming for the deployment of audio-visual telecommunications. Lalo has extensive experience in large enterprises - where he successfully deployed global multimedia networks and services for AT&T and Ernst & Young - as well as startups where he forged strategic alliances with Intel, Microsoft, Samsung and Texas Instruments to accelerate adoption of audio-visual services over IP.
**This document has been updated to meet the current required format for the NI Code Exchange.**
Example code from the Example Code Exchange in the NI Community is licensed with the MIT license.
