Abstract

A new technique, which can compensate for the lack of channel bandwidth in an optical wireless orthogonal frequency division multiplexing (OFDM) link based on a light emitting diode (LED), is proposed. It uses an adaptive sampling and an inverse discrete cosine transform in order to convert an OFDM signal into a sparse waveform so that not only is the important data obtained efficiently but the redundancy one is removed. In compressive sensing (CS), a sparse signal that is sampled below the Nyquist/Shannon limit can be reconstructed successively with enough measurement. This means that the CS technique can increase the data rate of visible light communication (VLC) systems based on LEDs. It is observed that the data rate of the proposed CS-based VLC-OFDM link can be made 1.7 times greater than a conventional VLC-OFDM link (from 30.72 Mb/s to 51.2 Mb/s). We see that the error vector magnitude (EVM) of the quadrature phase shift keying (QPSK) symbol is 31% (FEC limit: EVM of 32%) at a compression ratio of 40%.

© 2014 Optical Society of America

1. Introduction

A visible light is defined as the electromagnetic radiation whose wavelengths lie in the range that can be detected and interpreted by the human brain. The rapid development in visible light emitting diodes (LEDs) has enabled researchers to use them in many applications that require illumination, such as displays, and signal devices. A visible light communication (VLC) using white LEDs, where the LEDs are used for both illumination and wireless transmission, has been proposed to be used for multimedia mobile services such as indoor positioning, broadband wireless multimedia services (over 1 Gb/s), and inter-vehicular communication [1].

One of the technical requirements for developing these kinds of services in a network-based VLC is to have a broadband wireless channel bandwidth so that access users would transmit flexibly multimedia data irrespective of its bandwidth. However, it was very difficult to support this because the commercial LED devices have a low 3-dB physical bandwidth (<10 MHz). To complement and overcome the drawbacks of the conventional approach, several techniques have been proposed to improve the bandwidth efficiency [26]. Gruber et al. reported that the modulation bandwidth of visible LEDs increased from 3 to 20 MHz by removing the yellow phosphor that has a slow time constant with the help of a blue filter [2]. This technique reduces the intensity of the received light because of the insertion of the blue filter. As a result, the wireless-transmission distance reduces, while the received channel noise is also attenuated. Therefore, it is important to optimize the relation between the received optical power and channel noise in order to minimize the noise figure of VLC system using the blue filter. Using an electronic equalizer is also an efficient technique to improve the channel-bandwidth of the wireless channel in a VLC network because the electronic equalizer compensates the signal components with a small frequency response [3]. However, each equalizer requires a different driving circuit because each white LED has its own distinct frequency response. The cost of a VLC module also increases because it takes longer to produce them in a commercial scale when these optical and electrical devices are added. Kottke et al. proposed a 1.25-Gb/s wireless optical transmission technique that uses discrete multi-tones (DMTs) and RGB-LEDs [5]. However, an RGB-LED costs more than a white LED, which discourages commercialization. Moreover, their wireless-transmission distance (10 cm) is too short to use in wireless multimedia services such as Wi-Fi and Bluetooth.

In this paper, a signal compression and reconstruction scheme based on compressive sensing (CS) is proposed. It improves the channel bandwidth of an optical wireless orthogonal frequency-division multiplexing (OFDM) link. The CS scheme shows that perfect reconstruction of a signal can be attained with enough CS measurements of a sparse signal [7]. A sparse signal is one that is sampled significantly below the Nyquist/Shannon limit. Figure 1 shows a wireless optical link using the proposed technique. Here, an OFDM/quadrature phase-shift keying (QPSK) signal is compressed using adaptive sampling and inverse discrete cosine transform (IDCT) at the VLC transmitter (Tx), and then transmitted wirelessly to the VLC receiver (Rx). The compressed waveform is reconstructed at the VLC Rx using L1-minimization. As shown in Fig. 1, a sparse matrix is used for adaptive sampling, while the IDCT is computed using a domain transformer. To the best of our knowledge, this is the first work to transmit and reconstruct compressed data using a compressive sensing technique in VLC systems.

 figure: Fig. 1

Fig. 1 Compressed optical OFDM signal transmission and reconstruction in a VLC system using adaptive sampling with IDCT domain and L1-minimization.

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The rest of the paper is organized as follows: In Section II, the basic CS theory is explained briefly in order to enhance the understanding of the proposed technique. An adaptive sampling technique that uses IDCT is described. The technique generates a compressed OFDM/QPSK signal with high sparsity. We also describe the L1-minimization method, which is used to reconstruct the compressed waveform at the VLC-Rx. The experimental setup to verify the compression and reconstruction of the OFDM/QPSK signal is described in Section III. The experimental results are presented and discussed in Section IV. Finally, the conclusions are made in Section V.

2. Our proposed approach

2.1 Compressive sensing theory

The CS-based signal compression and reconstruction approach has generated a tremendous amount of excitement in the signal processing community, because it enables a potentially large reduction (under Nyquist/Shannon limit) in the sampling and computation cost. Many recent techniques for data reconstruction exploit the fact that the signal may be compressible in a prescribed basis. In other words, wired and wireless data can be reconstructed accurately (or even exactly) with a small number of sparse measurements. This approach is similar to the data- acquisition technique of CS.

An original or uncompressed signal can be represented as a column vector by vectorization of the original signal. In the framework of CS, the unknown vectorxm, can be exactly recovered or well approximated from n compressed measurements (n<m) [8]. The measurements can be written in the vector form asv=Φxn . This can be done with certain conditions on the n×m measurement matrixΦ, and an m×m basis matrixΨ=[ψ1|ψ2||ψm], where {ψi}i=1m is a basis in m. Here, x is a linear combination of{ψ1,,ψm}, as written in the matrix form in Eq. (1).

x=Ψα
Only k of the m coefficient entries in α is nonzero, withk<<m. This is why Φn×m is called a sparse matrix. To be more precise, the CS technique proves that a sparse vector x can be recovered from v=Φx by using the equationx=Ψ1α. The solution αm to the problem can be written as shown in Eq. (2).
α=argminαmα1subjecttoν=ΦΨα
whereα1=i=1m|αi| for α=(α1,,αm) with certain conditions on Θ=ΦΨ (such as the restricted isometric properties or incoherence). Instead of directly estimating the vectorx, the linear relationship between xand vis used. Under the assumption that the vector x is sparse in a given basis matrix Θ, the measurement v uniquely determinesx; as long as n is sufficiently large [9, 10]. For an exact reconstruction from the compressed measurements, many suboptimal methods such as basis pursuit and basis marching are proposed [10, 11].

2.2 Adaptive sampling of the OFDM/QPSK signal

An adaptive sampling technique is proposed to reduce the complexity and to increase the data rate of OFDM transmission. As shown in Fig. 2, first the samples are extracted from the original OFDM/QPSK samples at regular intervals in order to reduce the errors that can be generated by extracting the samples from only certain parts of its scope (vreg'). Second, samples are extracted at the local minimum and maximum points to track the rapid movements in the complex OFDM signal (vmin_max'). This is done to capture the analogue characteristics of OFDM signal precisely. All these samples (v'), which are composed of the samples at regular intervals (vreg') and the local maxima and minima samples(vmin_max'), are converted into the sparse domain using an inverse discrete cosine transform (IDCT), because it guarantees sparsity in the complicated OFDM signal.

 figure: Fig. 2

Fig. 2 Process of sparse signal generation using adaptive sampling and IDCT.

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2.3 Reconstruction of OFDM/QPSK signal using L1-minimization

Figure 3 shows a method to reconstruct the noisy OFDM/QPSK signal from the given measurement (v). It is based on L1-minimzation. Here, x^is reconstructed in the receiver from the given measurement by an iterative process that reconstructs the shape of the original OFDM/QPSK signal using Eq. (3).

x^=Ψα+ε
whereαis sparse, Ψm×n,and εis a relatively small measurement error. We iteratively learn Ψfrom the received measurement by updating the projection matrices and the sparse matrices as shown in Eq. (4). Here, ε<<1is the tolerance. Using projection Ψ and α in the problem, we obtain:
minΨ,αα1subjecttoΦxΦΨα2ε
We apply the procedure iteratively until all the values of the original signal are recovered from the given measurement.

 figure: Fig. 3

Fig. 3 L1-minimization-based original signal reconstruction procedure.

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The signal x^ is the reconstructed OFDM/QPSK signal. The procedure is iterated until the recovered signal meets our requirement. In this approach, it is assumed that the measured sample data from the original OFDM/QPSK signal can be well represented by a linear combination of a few intensity values. By iteratively determining the projections, the matrix can be reconstructed using random projections and the noise effect is reduced during the reconstruction procedure. The shape of the reconstructed signal can be optimized and improved by repeating the reconstruction procedure.

3. Experiments

3.1 Experimental setup for wireless optical link based on compressive sensing

Figure 4 shows the experimental setup of the proposed scheme. The setup was used to wirelessly transmit and reconstruct a compressed OFDM/QPSK signal. To evaluate the performance of the proposed technique, the difference in the signal quality between a compressed OFDM/QPSK signal and the original one was measured. The original OFDM/QPSK signal was generated and recovered using offline-processing-1 block, while the compressed OFDM/QPSK signal was handled by the offline-processing-2 block. First, a QPSK-encoded OFDM signal was generated by MATLAB®. The size of fast Fourier transform (FFT) was chosen as 2048 to optimize the signal quality of the proposed scheme. One frame containing the training sequence was inserted into every 1000 OFDM symbols as the preamble. The size of cyclic prefix (CP) was 224 for all 1200 OFDM subcarriers. After generating the OFDM/QPSK waveform using offline digital processing, it is converted to an analog waveform by an arbitrary waveform generator (AWG: Tektronix 7122C) that sampled at 31.25 Msample/s. The spectrum of the converted analog OFDM/QPSK signal waveform ranged from dc to 15.56 MHz. Before loading the OFDM/QPSK waveform on the AWG, the generated OFDM/QPSK signal was separated into an in-phase (I) and a quadrature-phase (Q) channel with real values in the digital domain. All the data of the Q-channel, which had only imaginary values, were transformed to purely real values using an imaginary-to-real numerical converter in offline processing. Then, both the I and Q channel data streams with real values were loaded to the AWG so that they would be transmitted separately.

 figure: Fig. 4

Fig. 4 Experimental setup for reconstruction of the compressed OFDM/QPSK signal in an optical wireless link based on a white LED.

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In the second offline processing, the OFDM/QPSK signal was resampled by adaptive sampling and was then transformed into a sparse waveform using IDCT based on the CS technique. After the CS process, the compressed OFDM/QPSK signal was generated.

Then, the original OFDM/QPSK signal and the compressed one from the AWG were equalized and amplified by a 1st order equalizer and a low noise amplifier (LNA: 25 dB gain, 30 MHz 3-dB bandwidth). This is done to obtain enough bandwidth for the white LED to modulate the OFDM/QPSK signal, which is sampled at 31.25 Msample/s. Figure 5 shows the frequency response of white LEDs with a 1st order equalizer. A filled square line shows the response with the 1st order equalizer and an LNA, while the open circle line shows the response with only a white LED. As shown in Fig. 5, it was observed that the 3-dB channel response of VLC-transmitter (Tx) was 25 MHz for the 1st order equalizer with LNA. It shows that the light from a white LED can be modulated by the OFDM/QPSK signal with the help of a 1st order equalizer.

 figure: Fig. 5

Fig. 5 Channel response of VLC-Tx, filled square line: 1st order equalizer with LNA, open circle line: only white LED.

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After being equalized, the two kinds of OFDM/QPSK signals were combined with a dc source (bias current: 300 mA) using a bias-T and then the diffused light of the white LED was modulated. The white LED was operated in the quasilinear region by controlling the bias current as well as the electrical output power of AWG.

The modulated light of the white LED was transmitted wirelessly to a VLC-Rx. It was placed one meter away from the VLC-Tx and aligned to have a direct line-of-sight (LOS) between the Tx and the Rx. The VLC-Rx was made of a dichroic optical bandpass filter (Thorlab, FD2B, FWHM bandwidth: 50 nm), which eliminated the phosphorescent component from the white LED, a biconvex glass lens (Thorlab, LB1723, focal length: 60 mm), and an APD module. The APD module (Hamamatsu C5331) consisted of an avalanche photodiode and an integrated TIA (transimpedance amplifier, 3-dB bandwidth of 100 MHz), which amplified the low-level photocurrent generated by the APD. The illuminance of 120 lux was measured in front of VLC-Rx at 1-m wireless transmission length. The original OFDM/QPSK waveform was sampled by a real-time oscilloscope (DPO: Tektronix 7200B) and the OFDM symbols were recovered by combining the I and Q channels after synchronization. Next, the compressed waveform uncompressed into an OFDM/QPSK signal using L1-minimization, and its OFDM symbol was recovered by the same process as the original OFDM/QPSK signal.

3.2 Experimental results

Figure 6 shows how the OFDM/QPSK signal can be changed into a sparse waveform using the proposed scheme. Figure 6(a) shows waveforms before (upper waveform) and after (lower waveform) the usage of adaptive sampling at a compression ratio of 30%. The compression ratio is defined in Eq. (5).

Compressionratio(%)=(1LengthofcompresseddataLengthoforiginaldata)×100
The lower waveform of Fig. 6(b) shows the waveform of OFDM/QPSK signal transformed by Inverse discrete cosine transform (IDCT) after adaptive sampling. The inset of Fig. 6(b) shows is an enlargement of a 1-µs window of the waveform that is transformed by IDCT. As shown in Fig. 6(b), it was observed that the transformed waveform was changed in to a sparse one with a high sparsity using the IDCT. After IDCT, the transformed OFDM/QPSK signal was wirelessly transmitted to the VLC-Rx, and then reconstructed using L1-minimization.

 figure: Fig. 6

Fig. 6 Sparse waveform generation using adaptive sampling and IDCT, (a) upper waveform: original OFDM/QPSK signal, lower waveform: adaptive sampled OFDM/QPSK signal, (b) upper waveform: result of OFDM/QPSK signal transformed by IDCT, lower waveform: result of adaptive sampled one transformed by IDCT.

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Figure 7 shows the variation of EVM as well as the effective data rate against the compression ratio. Each constellation of QPSK was measured at three compression ratios (0%, 30%, and 50%). The effective data rate is the improved data rate after compression. The filled square line shows the EVM of the QPSK symbols, while the open square line corresponds to its effective data rate. The EVM of the QPSK symbols was 32% at the forward error correction (FEC) limit (bit error rate of 10−3). As shown in Fig. 7, at the EVM of 32%, it was observed that the original OFDM/QPSK signal could be compressed by up to 40%. In other words, it was seen that the data rate of the QPSK symbols could be increased from 30.72 Mb/s to 51.2 Mb/s by using the proposed technique. In addition, it was seen that the EVM values of QPSK increased as the compression ratio increased. It was because both the sparsity of the OFDM/QPSK samples and the number of OFDM/QPSK samples for the perfect reconstruction were not sufficient.

 figure: Fig. 7

Fig. 7 The variation of EVM as well as the effective data rate against the compression ratio. The dashed line indicates the FEC limit (EVM of 32% = BER of 10−3).

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Figure 8 shows the curves for the signal to noise ratio (SNR) and BER in each modulation format as the compression ratio increases. The filled square line shows the SNR curves of the QPSK signal, and the open square line shows the results for 16-QAM. The filled inverse triangle line show the BER of QPSK symbols, and the open inverse triangle line shows the BER of 16-QAM symbols. Figure 8 represents that SNR is dependent on the compression rate, but our proposed method does not rapidly decrease the SNR even though the compression rate is over Shannon/Nyquist sampling rate. It was reported that a minimum SNR of 17 dB and 10 dB, respectively, were required so that both the 16-QAM and QPSK symbols could be transmitted successfully with the help of FEC (BER of 10−3) [12,13]. As shown in Fig. 8, it was found that the compression ratio of 10% could be obtained at an SNR of 17 dB in case of 16-QAM. This means that an SNR greater than 10 dB is required to obtain the same compression ratio (40%) as a QPSK signal in an error free optical wireless transmission. In case of high order modulation format more than 16-QAM, it is expected that the effective data rate will be reduced due to the decrease in compression ratio as the level of modulation format grow higher because the minimum SNR, which is required for error free transmission, also increased at high order modulation more than 16-QAM.

 figure: Fig. 8

Fig. 8 Variation of SNR and BER against compression ratio according to the modulation format (QPSK and 16-QAM), filled squares: QPSK symbols, open squares: 16QAM symbols, filled inverse triangles: BER of QPSK symbols, open inverse triangles: BER of 16-QAM symbols.

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It is necessary to observe the impact of CS on a coded OFDM system in order to find if a FEC can correct the errors of data successively in spite of the SNR loss of CS. For the coded OFDM system, a rate 1/2 convolutional code was used in the experimental setup. The change of SNR was measured before and after using the convolutional code.

Figure 9 shows the feed-forward convolutional encoder diagram. The encoder constraint length was 7. The first output in the encoder diagram is the modulo-2 sum of the rightmost and the four leftmost elements in the diagram’s array of input values. The seven-digit binary number 1111001 captures this information, and is equivalent to the octal number 171. It thus becomes the first entry of the code generator matrix. Here, each triplet of bits uses the leftmost bit as the most significant bit. The second output corresponds to the binary number 1011011, which is equivalent to the octal number 133.

 figure: Fig. 9

Fig. 9 Used a rate 1/2 feed-forward convolutional encoder diagram.

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Figure 10 shows how the SNR of QPSK signal, which is coded by the used convolutional code, changes as the compression ratio increases. The upper figure shows each SNR change in case of two kinds of OFDM systems (uncoded OFDM and coded OFDM). The lower figure shows the difference of SNR between uncoded OFDM system and coded one. As shown in Fig. 10, the 3-dB reduction of SNR was observed in case of using the convolutional code comparing to the uncoded OFDM system. It is caused by the following reason. The length of coded OFDM symbol, which is compressed by the proposed adaptive sampling, increases because the length of coded OFDM symbol is longer than that of uncoded OFDM symbol. Therefore, the amount of error grows bigger due to the adaptive sampling comparing to the uncoded OFDM system. The number of error, which is generated during the reconstruction of coded OFDM symbol according to the L1-minimization, also increases. As a result, these kinds of errors reduce the SNR of QPSK signal more than that of uncoded OFDM system.”

 figure: Fig. 10

Fig. 10 SNR change of reconstructed QPSK signal against compression ratio before and after the convolutional code.

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4. Conclusion

A new technique for white-LED-based optical wireless transmission was proposed to improve the channel capacity. It involved using adaptive sampling with IDCT and L1-minimization based on the CS technique. It was observed that a compression ratio of 40% could be obtained in case of QPSK symbols, showing that error-free optical wireless transmission (FEC limit: BER of 10−3) is possible using the proposed technique. Moreover, in case of 16-QAM, a compression ratio of 10% was measured. It means that the data rate of QPSK symbols can be greater than that of the original sample by 1.7 times (from 30.72 Mb/s to 51.2 Mb/s) with the help of CS. These experimental results tell us that the proposed technique can solve the problem of insufficient bandwidth resulting from supporting multiple broadband multimedia services in many transmission systems as well as in white-LED-based optical wireless links.

Acknowledgments

S.M. Yoon and Y.-Y. Won were supported by the ICT R&D program of MSIP/IITP, Republic of Korea [14-823-04-006]. Y.-Y. Won was also supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT & Future Planning [2012R1A1A1012531]. Won was also supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT & Future Planning [NRF-2014R1A1A1002890].

References and links

1. A. Jovicic, J. Li, and T. Richardson, “Visible light communication: opportunities, challenges and the path to market,” IEEE Commun. Mag. 51(12), 26–32 (2013). [CrossRef]  

2. J. Gruber, S. C. J. Lee, K.-D. Langer, T. Koonen, and J. W. Walewski, “Wireless high-speed data transmission with phosphorescent white-light LEDs,” in Proceedings of 33th European Conference and Exhibition on Optical Communication (Berlin, 2007), pp. 1–2.

3. H. L. Minh, D. O’Brien, G. Faulkner, L. Zeng, K. Lee, D. Jung, and Y. Oh, “High-speed visible light communications using multiple resonant equalization,” IEEE Photon. Technol. Lett. 20(14), 1243–1245 (2008). [CrossRef]  

4. H. L. Minh, D. O’Brien, G. Faulkner, L. Zeng, K. Lee, D. Jung, Y. Oh, and E. T. Won, “100-Mb/s NRZ visible light communications using a postequalized white LED,” IEEE Photon. Technol. Lett. 21(15), 1063–1065 (2009). [CrossRef]  

5. C. Kottke, J. Hilt, K. Habel, J. Vucic, and K.-D. Langer, “1.25 Gbit/s visible light WDM link based on DMT modulation of a single RGB LED luminary,” in Proceedings of 38th European Conference and Exhibition on Optical Communication (Amsterdam, 2012), paper We.3.B.4. [CrossRef]  

6. J. Vucic, C. Kottke, S. Nerreter, K.-D. Langer, and J. W. Walewski, “513 Mbit/s visible light communications link based on DMT-modulation of a white LED,” J. Lightwave Technol. 28(24), 3512–3518 (2010).

7. D. L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory 52(4), 1289–1306 (2006). [CrossRef]  

8. D. L. Donoho and M. Elad, “Optimally sparse representation in general (nonorthogonal) dictionaries via ll minimization,” Proc. Natl. Acad. Sci. U.S.A. 100(5), 2197–2202 (2003). [CrossRef]   [PubMed]  

9. E. Candès and T. Tao, “Decoding by linear programming,” IEEE Trans. Inf. Theory 51(12), 4203–4215 (2005). [CrossRef]  

10. S. S. Chen, D. L. Donoho, and M. A. Saunders, “Atomic decomposition by basis pursuit,” SIAM Rev. 43(1), 129–159 (2001). [CrossRef]  

11. J. A. Tropp, “Greed is good: Algorithmic results for sparse approximation,” IEEE Trans. Inf. Theory 50(10), 2231–2242 (2004). [CrossRef]  

12. R. A. Shafik, S. Rahman, and R. Islam, “On the extended relationships among EVM, BER, and SNR as performance metrics,” in Proceedings of 4th International Conference on Electrical and Computer Engineering (Dhaka, 2006), pp. 408–411. [CrossRef]  

13. Application note, “8 hints for making and interpreting EVM measurements” (Agilent Technologies, 2005), cp.literature.agilent.com/litweb/pdf/5989–3144EN.pdf.

References

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  1. A. Jovicic, J. Li, and T. Richardson, “Visible light communication: opportunities, challenges and the path to market,” IEEE Commun. Mag. 51(12), 26–32 (2013).
    [Crossref]
  2. J. Gruber, S. C. J. Lee, K.-D. Langer, T. Koonen, and J. W. Walewski, “Wireless high-speed data transmission with phosphorescent white-light LEDs,” in Proceedings of 33th European Conference and Exhibition on Optical Communication (Berlin, 2007), pp. 1–2.
  3. H. L. Minh, D. O’Brien, G. Faulkner, L. Zeng, K. Lee, D. Jung, and Y. Oh, “High-speed visible light communications using multiple resonant equalization,” IEEE Photon. Technol. Lett. 20(14), 1243–1245 (2008).
    [Crossref]
  4. H. L. Minh, D. O’Brien, G. Faulkner, L. Zeng, K. Lee, D. Jung, Y. Oh, and E. T. Won, “100-Mb/s NRZ visible light communications using a postequalized white LED,” IEEE Photon. Technol. Lett. 21(15), 1063–1065 (2009).
    [Crossref]
  5. C. Kottke, J. Hilt, K. Habel, J. Vucic, and K.-D. Langer, “1.25 Gbit/s visible light WDM link based on DMT modulation of a single RGB LED luminary,” in Proceedings of 38th European Conference and Exhibition on Optical Communication (Amsterdam, 2012), paper We.3.B.4.
    [Crossref]
  6. J. Vucic, C. Kottke, S. Nerreter, K.-D. Langer, and J. W. Walewski, “513 Mbit/s visible light communications link based on DMT-modulation of a white LED,” J. Lightwave Technol. 28(24), 3512–3518 (2010).
  7. D. L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory 52(4), 1289–1306 (2006).
    [Crossref]
  8. D. L. Donoho and M. Elad, “Optimally sparse representation in general (nonorthogonal) dictionaries via ll minimization,” Proc. Natl. Acad. Sci. U.S.A. 100(5), 2197–2202 (2003).
    [Crossref] [PubMed]
  9. E. Candès and T. Tao, “Decoding by linear programming,” IEEE Trans. Inf. Theory 51(12), 4203–4215 (2005).
    [Crossref]
  10. S. S. Chen, D. L. Donoho, and M. A. Saunders, “Atomic decomposition by basis pursuit,” SIAM Rev. 43(1), 129–159 (2001).
    [Crossref]
  11. J. A. Tropp, “Greed is good: Algorithmic results for sparse approximation,” IEEE Trans. Inf. Theory 50(10), 2231–2242 (2004).
    [Crossref]
  12. R. A. Shafik, S. Rahman, and R. Islam, “On the extended relationships among EVM, BER, and SNR as performance metrics,” in Proceedings of 4th International Conference on Electrical and Computer Engineering (Dhaka, 2006), pp. 408–411.
    [Crossref]
  13. Application note, “8 hints for making and interpreting EVM measurements” (Agilent Technologies, 2005), cp.literature.agilent.com/litweb/pdf/5989–3144EN.pdf .

2013 (1)

A. Jovicic, J. Li, and T. Richardson, “Visible light communication: opportunities, challenges and the path to market,” IEEE Commun. Mag. 51(12), 26–32 (2013).
[Crossref]

2010 (1)

2009 (1)

H. L. Minh, D. O’Brien, G. Faulkner, L. Zeng, K. Lee, D. Jung, Y. Oh, and E. T. Won, “100-Mb/s NRZ visible light communications using a postequalized white LED,” IEEE Photon. Technol. Lett. 21(15), 1063–1065 (2009).
[Crossref]

2008 (1)

H. L. Minh, D. O’Brien, G. Faulkner, L. Zeng, K. Lee, D. Jung, and Y. Oh, “High-speed visible light communications using multiple resonant equalization,” IEEE Photon. Technol. Lett. 20(14), 1243–1245 (2008).
[Crossref]

2006 (1)

D. L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory 52(4), 1289–1306 (2006).
[Crossref]

2005 (1)

E. Candès and T. Tao, “Decoding by linear programming,” IEEE Trans. Inf. Theory 51(12), 4203–4215 (2005).
[Crossref]

2004 (1)

J. A. Tropp, “Greed is good: Algorithmic results for sparse approximation,” IEEE Trans. Inf. Theory 50(10), 2231–2242 (2004).
[Crossref]

2003 (1)

D. L. Donoho and M. Elad, “Optimally sparse representation in general (nonorthogonal) dictionaries via ll minimization,” Proc. Natl. Acad. Sci. U.S.A. 100(5), 2197–2202 (2003).
[Crossref] [PubMed]

2001 (1)

S. S. Chen, D. L. Donoho, and M. A. Saunders, “Atomic decomposition by basis pursuit,” SIAM Rev. 43(1), 129–159 (2001).
[Crossref]

Candès, E.

E. Candès and T. Tao, “Decoding by linear programming,” IEEE Trans. Inf. Theory 51(12), 4203–4215 (2005).
[Crossref]

Chen, S. S.

S. S. Chen, D. L. Donoho, and M. A. Saunders, “Atomic decomposition by basis pursuit,” SIAM Rev. 43(1), 129–159 (2001).
[Crossref]

Donoho, D. L.

D. L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory 52(4), 1289–1306 (2006).
[Crossref]

D. L. Donoho and M. Elad, “Optimally sparse representation in general (nonorthogonal) dictionaries via ll minimization,” Proc. Natl. Acad. Sci. U.S.A. 100(5), 2197–2202 (2003).
[Crossref] [PubMed]

S. S. Chen, D. L. Donoho, and M. A. Saunders, “Atomic decomposition by basis pursuit,” SIAM Rev. 43(1), 129–159 (2001).
[Crossref]

Elad, M.

D. L. Donoho and M. Elad, “Optimally sparse representation in general (nonorthogonal) dictionaries via ll minimization,” Proc. Natl. Acad. Sci. U.S.A. 100(5), 2197–2202 (2003).
[Crossref] [PubMed]

Faulkner, G.

H. L. Minh, D. O’Brien, G. Faulkner, L. Zeng, K. Lee, D. Jung, Y. Oh, and E. T. Won, “100-Mb/s NRZ visible light communications using a postequalized white LED,” IEEE Photon. Technol. Lett. 21(15), 1063–1065 (2009).
[Crossref]

H. L. Minh, D. O’Brien, G. Faulkner, L. Zeng, K. Lee, D. Jung, and Y. Oh, “High-speed visible light communications using multiple resonant equalization,” IEEE Photon. Technol. Lett. 20(14), 1243–1245 (2008).
[Crossref]

Gruber, J.

J. Gruber, S. C. J. Lee, K.-D. Langer, T. Koonen, and J. W. Walewski, “Wireless high-speed data transmission with phosphorescent white-light LEDs,” in Proceedings of 33th European Conference and Exhibition on Optical Communication (Berlin, 2007), pp. 1–2.

Jovicic, A.

A. Jovicic, J. Li, and T. Richardson, “Visible light communication: opportunities, challenges and the path to market,” IEEE Commun. Mag. 51(12), 26–32 (2013).
[Crossref]

Jung, D.

H. L. Minh, D. O’Brien, G. Faulkner, L. Zeng, K. Lee, D. Jung, Y. Oh, and E. T. Won, “100-Mb/s NRZ visible light communications using a postequalized white LED,” IEEE Photon. Technol. Lett. 21(15), 1063–1065 (2009).
[Crossref]

H. L. Minh, D. O’Brien, G. Faulkner, L. Zeng, K. Lee, D. Jung, and Y. Oh, “High-speed visible light communications using multiple resonant equalization,” IEEE Photon. Technol. Lett. 20(14), 1243–1245 (2008).
[Crossref]

Koonen, T.

J. Gruber, S. C. J. Lee, K.-D. Langer, T. Koonen, and J. W. Walewski, “Wireless high-speed data transmission with phosphorescent white-light LEDs,” in Proceedings of 33th European Conference and Exhibition on Optical Communication (Berlin, 2007), pp. 1–2.

Kottke, C.

Langer, K.-D.

J. Vucic, C. Kottke, S. Nerreter, K.-D. Langer, and J. W. Walewski, “513 Mbit/s visible light communications link based on DMT-modulation of a white LED,” J. Lightwave Technol. 28(24), 3512–3518 (2010).

J. Gruber, S. C. J. Lee, K.-D. Langer, T. Koonen, and J. W. Walewski, “Wireless high-speed data transmission with phosphorescent white-light LEDs,” in Proceedings of 33th European Conference and Exhibition on Optical Communication (Berlin, 2007), pp. 1–2.

Lee, K.

H. L. Minh, D. O’Brien, G. Faulkner, L. Zeng, K. Lee, D. Jung, Y. Oh, and E. T. Won, “100-Mb/s NRZ visible light communications using a postequalized white LED,” IEEE Photon. Technol. Lett. 21(15), 1063–1065 (2009).
[Crossref]

H. L. Minh, D. O’Brien, G. Faulkner, L. Zeng, K. Lee, D. Jung, and Y. Oh, “High-speed visible light communications using multiple resonant equalization,” IEEE Photon. Technol. Lett. 20(14), 1243–1245 (2008).
[Crossref]

Lee, S. C. J.

J. Gruber, S. C. J. Lee, K.-D. Langer, T. Koonen, and J. W. Walewski, “Wireless high-speed data transmission with phosphorescent white-light LEDs,” in Proceedings of 33th European Conference and Exhibition on Optical Communication (Berlin, 2007), pp. 1–2.

Li, J.

A. Jovicic, J. Li, and T. Richardson, “Visible light communication: opportunities, challenges and the path to market,” IEEE Commun. Mag. 51(12), 26–32 (2013).
[Crossref]

Minh, H. L.

H. L. Minh, D. O’Brien, G. Faulkner, L. Zeng, K. Lee, D. Jung, Y. Oh, and E. T. Won, “100-Mb/s NRZ visible light communications using a postequalized white LED,” IEEE Photon. Technol. Lett. 21(15), 1063–1065 (2009).
[Crossref]

H. L. Minh, D. O’Brien, G. Faulkner, L. Zeng, K. Lee, D. Jung, and Y. Oh, “High-speed visible light communications using multiple resonant equalization,” IEEE Photon. Technol. Lett. 20(14), 1243–1245 (2008).
[Crossref]

Nerreter, S.

O’Brien, D.

H. L. Minh, D. O’Brien, G. Faulkner, L. Zeng, K. Lee, D. Jung, Y. Oh, and E. T. Won, “100-Mb/s NRZ visible light communications using a postequalized white LED,” IEEE Photon. Technol. Lett. 21(15), 1063–1065 (2009).
[Crossref]

H. L. Minh, D. O’Brien, G. Faulkner, L. Zeng, K. Lee, D. Jung, and Y. Oh, “High-speed visible light communications using multiple resonant equalization,” IEEE Photon. Technol. Lett. 20(14), 1243–1245 (2008).
[Crossref]

Oh, Y.

H. L. Minh, D. O’Brien, G. Faulkner, L. Zeng, K. Lee, D. Jung, Y. Oh, and E. T. Won, “100-Mb/s NRZ visible light communications using a postequalized white LED,” IEEE Photon. Technol. Lett. 21(15), 1063–1065 (2009).
[Crossref]

H. L. Minh, D. O’Brien, G. Faulkner, L. Zeng, K. Lee, D. Jung, and Y. Oh, “High-speed visible light communications using multiple resonant equalization,” IEEE Photon. Technol. Lett. 20(14), 1243–1245 (2008).
[Crossref]

Richardson, T.

A. Jovicic, J. Li, and T. Richardson, “Visible light communication: opportunities, challenges and the path to market,” IEEE Commun. Mag. 51(12), 26–32 (2013).
[Crossref]

Saunders, M. A.

S. S. Chen, D. L. Donoho, and M. A. Saunders, “Atomic decomposition by basis pursuit,” SIAM Rev. 43(1), 129–159 (2001).
[Crossref]

Tao, T.

E. Candès and T. Tao, “Decoding by linear programming,” IEEE Trans. Inf. Theory 51(12), 4203–4215 (2005).
[Crossref]

Tropp, J. A.

J. A. Tropp, “Greed is good: Algorithmic results for sparse approximation,” IEEE Trans. Inf. Theory 50(10), 2231–2242 (2004).
[Crossref]

Vucic, J.

Walewski, J. W.

J. Vucic, C. Kottke, S. Nerreter, K.-D. Langer, and J. W. Walewski, “513 Mbit/s visible light communications link based on DMT-modulation of a white LED,” J. Lightwave Technol. 28(24), 3512–3518 (2010).

J. Gruber, S. C. J. Lee, K.-D. Langer, T. Koonen, and J. W. Walewski, “Wireless high-speed data transmission with phosphorescent white-light LEDs,” in Proceedings of 33th European Conference and Exhibition on Optical Communication (Berlin, 2007), pp. 1–2.

Won, E. T.

H. L. Minh, D. O’Brien, G. Faulkner, L. Zeng, K. Lee, D. Jung, Y. Oh, and E. T. Won, “100-Mb/s NRZ visible light communications using a postequalized white LED,” IEEE Photon. Technol. Lett. 21(15), 1063–1065 (2009).
[Crossref]

Zeng, L.

H. L. Minh, D. O’Brien, G. Faulkner, L. Zeng, K. Lee, D. Jung, Y. Oh, and E. T. Won, “100-Mb/s NRZ visible light communications using a postequalized white LED,” IEEE Photon. Technol. Lett. 21(15), 1063–1065 (2009).
[Crossref]

H. L. Minh, D. O’Brien, G. Faulkner, L. Zeng, K. Lee, D. Jung, and Y. Oh, “High-speed visible light communications using multiple resonant equalization,” IEEE Photon. Technol. Lett. 20(14), 1243–1245 (2008).
[Crossref]

IEEE Commun. Mag. (1)

A. Jovicic, J. Li, and T. Richardson, “Visible light communication: opportunities, challenges and the path to market,” IEEE Commun. Mag. 51(12), 26–32 (2013).
[Crossref]

IEEE Photon. Technol. Lett. (2)

H. L. Minh, D. O’Brien, G. Faulkner, L. Zeng, K. Lee, D. Jung, and Y. Oh, “High-speed visible light communications using multiple resonant equalization,” IEEE Photon. Technol. Lett. 20(14), 1243–1245 (2008).
[Crossref]

H. L. Minh, D. O’Brien, G. Faulkner, L. Zeng, K. Lee, D. Jung, Y. Oh, and E. T. Won, “100-Mb/s NRZ visible light communications using a postequalized white LED,” IEEE Photon. Technol. Lett. 21(15), 1063–1065 (2009).
[Crossref]

IEEE Trans. Inf. Theory (3)

D. L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory 52(4), 1289–1306 (2006).
[Crossref]

E. Candès and T. Tao, “Decoding by linear programming,” IEEE Trans. Inf. Theory 51(12), 4203–4215 (2005).
[Crossref]

J. A. Tropp, “Greed is good: Algorithmic results for sparse approximation,” IEEE Trans. Inf. Theory 50(10), 2231–2242 (2004).
[Crossref]

J. Lightwave Technol. (1)

Proc. Natl. Acad. Sci. U.S.A. (1)

D. L. Donoho and M. Elad, “Optimally sparse representation in general (nonorthogonal) dictionaries via ll minimization,” Proc. Natl. Acad. Sci. U.S.A. 100(5), 2197–2202 (2003).
[Crossref] [PubMed]

SIAM Rev. (1)

S. S. Chen, D. L. Donoho, and M. A. Saunders, “Atomic decomposition by basis pursuit,” SIAM Rev. 43(1), 129–159 (2001).
[Crossref]

Other (4)

C. Kottke, J. Hilt, K. Habel, J. Vucic, and K.-D. Langer, “1.25 Gbit/s visible light WDM link based on DMT modulation of a single RGB LED luminary,” in Proceedings of 38th European Conference and Exhibition on Optical Communication (Amsterdam, 2012), paper We.3.B.4.
[Crossref]

J. Gruber, S. C. J. Lee, K.-D. Langer, T. Koonen, and J. W. Walewski, “Wireless high-speed data transmission with phosphorescent white-light LEDs,” in Proceedings of 33th European Conference and Exhibition on Optical Communication (Berlin, 2007), pp. 1–2.

R. A. Shafik, S. Rahman, and R. Islam, “On the extended relationships among EVM, BER, and SNR as performance metrics,” in Proceedings of 4th International Conference on Electrical and Computer Engineering (Dhaka, 2006), pp. 408–411.
[Crossref]

Application note, “8 hints for making and interpreting EVM measurements” (Agilent Technologies, 2005), cp.literature.agilent.com/litweb/pdf/5989–3144EN.pdf .

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Figures (10)

Fig. 1
Fig. 1 Compressed optical OFDM signal transmission and reconstruction in a VLC system using adaptive sampling with IDCT domain and L1-minimization.
Fig. 2
Fig. 2 Process of sparse signal generation using adaptive sampling and IDCT.
Fig. 3
Fig. 3 L1-minimization-based original signal reconstruction procedure.
Fig. 4
Fig. 4 Experimental setup for reconstruction of the compressed OFDM/QPSK signal in an optical wireless link based on a white LED.
Fig. 5
Fig. 5 Channel response of VLC-Tx, filled square line: 1st order equalizer with LNA, open circle line: only white LED.
Fig. 6
Fig. 6 Sparse waveform generation using adaptive sampling and IDCT, (a) upper waveform: original OFDM/QPSK signal, lower waveform: adaptive sampled OFDM/QPSK signal, (b) upper waveform: result of OFDM/QPSK signal transformed by IDCT, lower waveform: result of adaptive sampled one transformed by IDCT.
Fig. 7
Fig. 7 The variation of EVM as well as the effective data rate against the compression ratio. The dashed line indicates the FEC limit (EVM of 32% = BER of 10−3).
Fig. 8
Fig. 8 Variation of SNR and BER against compression ratio according to the modulation format (QPSK and 16-QAM), filled squares: QPSK symbols, open squares: 16QAM symbols, filled inverse triangles: BER of QPSK symbols, open inverse triangles: BER of 16-QAM symbols.
Fig. 9
Fig. 9 Used a rate 1/2 feed-forward convolutional encoder diagram.
Fig. 10
Fig. 10 SNR change of reconstructed QPSK signal against compression ratio before and after the convolutional code.

Equations (5)

Equations on this page are rendered with MathJax. Learn more.

x=Ψα
α= argmin α m α 1 subject to ν=ΦΨα
x ^ =Ψα+ε
min Ψ,α α 1 subject to ΦxΦΨα 2 ε
Compression ratio (%)=( 1 Length of compressed data Length of original data )×100

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