Sparse channel estimation for underwater acoustic communication using compressed sensing

1. Proceedings of the 2nd International Conference on Current Trends in Engineering and Management ICCTEM -2014
INTERNATIONAL JOURNAL OF ELECTRONICS AND
17 – 19, July 2014, Mysore, Karnataka, India
COMMUNICATION ENGINEERING TECHNOLOGY (IJECET)
ISSN 0976 – 6464(Print)
ISSN 0976 – 6472(Online)
Volume 5, Issue 8, August (2014), pp. 46-54
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Journal Impact Factor (2014): 7.2836 (Calculated by GISI)
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46
IJECET
© I A E M E
SPARSE CHANNEL ESTIMATION FOR UNDERWATER ACOUSTIC
COMMUNICATION USING COMPRESSED SENSING
Sonia G1, Dr. Mrinal Sarvagya2
1Student, Dept. of Digital Communication and Networking, NMIT, Bangalore
2Professor Dept. of Digital Communication and Networking, NMIT, Bangalore
ABSTRACT
In past decades, there has been a growing interest in the discussion and study of using
underwater acoustic channel as the physical layer for communication systems, ranging from point-to-point
communications to underwater multicarrier modulation networks. This paper includes various
channel estimators that exploit channel sparsity for underwater acoustic system. Compressive
sensing is an emerging field based on the discovery that sparse signals can be reconstructed from
highly incomplete information. For channels with Doppler spread, we adopt a compressed sensing
approach, in form of Orthogonal Matching Pursuit (OMP) and Basis Pursuit (BP) algorithms, and
utilize over complete dictionaries with an increased path delay resolution. Numerical simulation and
experimental data of an OFDM block-by-block receiver are used to evaluate the proposed algorithms
in comparison to the conventional least-squares (LS) channel estimator.
Keywords: Basis Pursuit, Compressed Sensing, Comb Type Pilot Aided Channel Estimation,
Matching Pursuit, Orthogonal Matching Pursuit.
I. INTRODUCTION
The need for underwater wireless communications exists in applications such as remote
control in offshore oil industry, pollution monitoring in environmental systems, collection of
scientific data recorded at ocean-bottom stations, speech transmission between divers, and mapping
of the ocean floor for detection of objects, as well as for the discovery of new resources. Wireless
underwater communications can be established by transmission of acoustic waves.
Underwater acoustic [6](UWA) communication in shallow water is an on-going challenge
due to multipath-induced signal fading and motion induced Doppler shift[7]. Acoustic propagation is
characterized by four major factors: frequency dependent attenuation (e.g., attenuation that increases
with signal frequency), severe time-varying multipath propagation, low speed of sound, and large
Doppler shift. The long channel delay spread leads to significant inter-symbol-interference (ISI) in
single-carrier transmissions. The receiver complexity for channel equalization becomes a major

2. Proceedings of the 2nd International Conference on Current Trends in Engineering and Management ICCTEM -2014
17 – 19, July 2014, Mysore, Karnataka, India
burden when the symbol rate increases. Multicarrier approaches like orthogonal frequency division
multiplexing (OFDM) can equalize the channel at low complexity, but the aforementioned Doppler
effects destroy the orthogonality of the sub-carriers and lead to inter-carrier-interference (ICI). The
combination of large delay spread and significant Doppler effects qualify UWA channels as doubly
(time- and frequency-) spread channels. One known approach to this class of channels is the use of a
basis expansion model (BEM) to reflect the time-varying nature of the UWA channel. Even though
the time-varying nature of channels can be modelled arbitrarily well this way, it also tremendously
increases demands on channel estimation, as the number of unknowns that need to be estimated
increases correspondingly. The only remedy to this challenge is to exploit the fact that UWA
channels are naturally sparse, meaning that most channel energy is concentrated in a few delay
and/or Doppler values.
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II. OFDM
The basic idea underlying OFDM systems is the division of the available frequency spectrum
into several subcarriers. To obtain a high spectral efficiency, the frequency responses of the
subcarriers are overlapping and orthogonal, hence the name OFDM. This orthogonality can be
completely maintained with a small price in a loss in SNR, even though the signal passes through a
time dispersive fading channel, by introducing a cyclic prefix (CP). A block diagram of a baseband
OFDM system is shown in Figure 1.
III. OFDM Pilot-Aided Underwater Acoustic Channel Estimation
Typically, no prior knowledge on the channel is available, and it may vary over time.
Usually, UWA[2][6] channel is one kind of fast time varying channels. Therefore, most practical
multi-carrier UWA communication systems adopt pilot-aided channel estimation technique to track
the fast varying UWA channels.Hence, it needs to be estimated and the estimates updated in a
regular basis. OFDM is a simple way to deal with multipath propagation and overcome problems of
inter-symbol interference (ISI) and inter-carrier interference (ICI). However, OFDM applications in
UWA communications and networks are very scarce. OFDM pilot-aided UWA channel estimation
approaches involve in block-type pilot, comb-type pilot comb-type pilot-aided channel estimation
approach will combine some signal interpolation approaches.
Typically, there are two types of pilots described in Figure2.
• Comb Type Pilot aided
• Block Type Pilot aided

3. Proceedings of the 2nd International Conference on Current Trends in Engineering and Management ICCTEM -2014
17 – 19, July 2014, Mysore, Karnataka, India
48
Figure.2: Types of pilot (a) Comb type, (b) Block type
In the comb-type arrangement, a number of subcarriers are reserved for pilot signals, which
are transmitted continuously. Channel estimation can then be performed uninterruptedly based on
these pilot subcarriers in every symbol. The spacing of pilot subcarrier must be less than the
coherence bandwidth of the channel. Comb-type pilot pattern is suitable for systems operating under
fast-fading channel. Hence comb-type channel estimation is used for UWA channel
In the block-type pilot arrangement, one specific symbol full of pilot subcarriers is
transmitted periodically. The pilot symbol must appear at a frequency tens of times higher than the
Doppler frequency in order to ensure the validity of the channel estimates. In other words, the
interval between two consecutive pilot symbols must be significantly shorter than the channel
coherence time. Consequently, block-type pilot pattern is suitable for systems operating under slow-fading
channels.
The estimation can be based on least square (LS), minimum mean-square error (MMSE).
The LS estimator minimizes the parameter
means the conjugate transpose operation.
It is shown that LS estimator is given by
(k=0,…..,N-1) ……(1)
Without using any knowledge of the statistics of the channels, the LS estimators are calculated with
low complexity, but they suffer from high mean square error.
The MMSE estimator employs second order statistics of the channel conditions to minimize the
mean square error
..….(2)
Where denotes the auto covariance matrix.
The MMSE estimator yields much better performance than LS estimator, especially under low SNR
condition. Major drawback of MMSE estimator is high computational complexity
IV. COMPRESSED SENSING
The Shannon/Nyquist sampling theorem specifies that to avoid losing information when
capturing a signal, one must sample at least two times faster than the signal bandwidth. In many
applications, including digital image and video cameras, the Nyquist rate is so high that too many
samples result, making compression a necessity prior to storage or transmission. In other
applications, including imaging systems (medical scanners and radars) and high-speed analog- to-digital
converters, increasing the sampling rate is very expensive.

4. Proceedings of the 2nd International Conference on Current Trends in Engineering and Management ICCTEM -2014
17 – 19, July 2014, Mysore, Karnataka, India
49
This paper presents a new method to capture and represent compressible signals at a rate
significantly below the Nyquist rate. This method, called compressive sensing[1], employs
nonadaptive linear projections that preserve the structure of the signal; the signal is then
reconstructed from these projections using an optimization process. Compressive sensing address
these inefficiencies by directly acquiring a compressed signal representation without going through
the intermediate stage of acquiring N samples.
Although H has K2 entries, it is defined by Np triplets of (p, bp, p ). Since UWA channels
are sparse, the value of Np is small, hence, it is possible that those Np paths can be identified by
compressed sensing methods based on only a limited number of measurements.
To facilitate implementation, we rewrite z as
……..(3)
If the parameters (bp, p) were available, we could construct the (K × Np)-matrix and solve
for the p using least squares.
A. MATCHING PURSUIT
Matching pursuit is a type of sparse approximation which involves finding the best
matching projections of multidimensional data onto an over-complete dictionary D. The basic idea
is to represent a signal f from Hilbert space H [4]as a weighted sum of functions (called atoms)
taken from D:
………(4)
where n indexes the atoms that have been chosen, and a weighting factor (an amplitude) for each
atom. Given a fixed dictionary, matching pursuit will first find the one atom that has the biggest
inner product with the signal, then subtract the contribution due to that atom, and repeat the process
until the signal is satisfactorily decomposed.
For comparison, consider the Fourier series representation of a signal - this can be described
in the terms given above, where the dictionary is built from sinusoidal basis functions (the smallest
possible complete dictionary). The main disadvantage of Fourier analysis in signal processing is that
it extracts only global features of signals and does not adapt to analysed signals f. By taking an
extremely redundant dictionary we can look in it for functions that best match a signal f. Finding a
representation where most of the coefficients in the sum are close to 0 is desirable for signal coding
and compression.
B. ORTHOGONAL MATCHING PURSUIT
A popular extension of Matching Pursuit (MP) is its orthogonal version: Orthogonal
Matching Pursuit(OMP). The main difference from MP is that after every step, all the coefficients
extracted so far are updated, by computing the orthogonal projection of the signal onto the set of
atoms selected so far. This can lead to better results than standard MP, but requires more
computation.

5. Proceedings of the 2nd International Conference on Current Trends in Engineering and Management ICCTEM -2014
17 – 19, July 2014, Mysore, Karnataka, India
In fact, this algorithm approximates the sparse problem
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…………(5)
with the pseudo-norm (i.e. the number of nonzero elements of )
C. BASIS PURSUIT
Consider the problem of finding the sparsest signal x satisfying a system of linear equations:
Subject to ……..(6)
This problem is known to be combinatorial and NP-hard . A number of approaches to
approximate its solution have been proposed. One of the most well-known approaches is to relax the
zero norm and replace it with the l1-norm:
Subject to …………………..(7)
This approach is often referred to as basis pursuit (BP)
V. RESULTS
0 20 40 60 80 100 120 140
4
2
0
Comb type LSE Channel estimation transmitter
No of subcarriers
Amplitude
Data signal
0 20 40 60 80 100 120 140
1
0
-1
No of subcarriers
Amplitude
Modulated Signal
0 50 100 150
2
0
-2
No of subcarriers
Amplitude
Channel
Figure 3: Comb type LSE channel estimation transmitter-Data signal, Modulated signal, Channel
response
0 20 40 60 80 100 120 140
2
0
-2
Comb type LSE Channel estimation receiver
No of subcarriers
Amplitude
Received signal
0 20 40 60 80 100 120
4
2
0
No of subcarriers
Amplitude
LSE Demodulated Signal
0 20 40 60 80 100 120
4
2
0
No of subcarriers
Amplitude
Sparse Demodulated Signal
Figure 4: Comb type LSE Channel estimation receiver and LSE demodulated signal and Sparse
Demodulated signal

6. Proceedings of the 2nd International Conference on Current Trends in Engineering and Management ICCTEM -2014
17 – 19, July 2014, Mysore, Karnataka, India
51
Fig.3 shows the original data signal modulated using QAM and the pilot symbols inserted in
the channel before transmission using comb type Least Square error estimators .
Fig.4 shows the received signal which is demodulated using LSE Demodulator and Sparse
Demodulator.
From Fig.11 it is seen that the sparse demodulator performs better compared to the LSE
Demodulator.
0 10 20 30 40 50 60
1
0
-1
Matching Pursuit
No Of Samples
Amplitude
Basis
0 50 100 150 200 250 300
2
0
-2
No Of Samples
Amplitude
signal
0 50 100 150 200 250 300
5
0
-5
No Of Samples
Amplitude
recovery
Figure 5: Matching Pursuit
0 50 100 150 200 250 300
8
6
4
2
0
-2
-4
No Of Samples
Amplitude
Compressed Signal using MP
Sparse signal representation
Figure 6: Compressed sampled signal for MP
Fig.5 depicts the compressed sensing algorithm called Matching pursuit .Where the
compressed signal in fig.6 of the original data is recovered using the basis function in the dictionary.
Original
0 50 100 150 200 250 300
2
1
0
-1
2
1
0
-1
-2
No Of Samples
Amplitude
0 50 100 150 200 250 300
-2
No Of Samples
Amplitude
Orthogonal matching pursuit
Recovery
Figure 7: Orthogonal Matching Pursuit Original Signal and recovered signal

7. Proceedings of the 2nd International Conference on Current Trends in Engineering and Management ICCTEM -2014
17 – 19, July 2014, Mysore, Karnataka, India
52
0 50 100 150 200 250 300
4
3
2
1
0
-1
-2
-3
-4
No Of Samples
Amplitude
Compressed Signal using OMP
Sparse signal representation
Figure 8: Compressed sampled signal for OMP
Fig.7 shows the original and recovered signal of compressed data in fig.8 using Orthogonal
matching pursuit algorithm.
0 50 100 150 200 250 300
1
0.5
0
-0.5
-1
No Of Samples
Amplitude
Basis Pursuit
signal
0 50 100 150 200 250 300
1
0.5
0
-0.5
-1
-1.5
No Of Samples
Amplitude
recovery
Figure 9: Basis Pursuit Original signal and recovered signal
0 50 100 150 200 250 300
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
Compressed signal using Basis Pursuit
No Of Samples
Amplitude
Sparse signal representation using BP
Figure 10: Compressed sampled signal for BP
Fig.9 and fig.10 shows the results obtained Basis pursuit algorithm where the original signal
is recovered from the compressed signal

8. Proceedings of the 2nd International Conference on Current Trends in Engineering and Management ICCTEM -2014
17 – 19, July 2014, Mysore, Karnataka, India
53
2 3 4 5 6 7 8 9 10
10
-2
10
-1
10
0
SNR
BER
Comparison of LSE And Sparse channel estimation
Sparse
LSE
Figure 11: Comparision of comb type pilot LSE channel estimation and Sprase channel estimation
0 20 40 60 80 100 120 140 160
0
-2
-4
-6
-8
-10
-12
Comparison of Compressed sensing algorithms
No of iterations
Mean square Error (dB)
MP
OMP
BP
Figure 12: Comparison of compressed sensing algorithms MP, OMP, BP
Fig.12 shows the comparison of all the three compressed sensing algorithms MP, OMP, BP.
From the results it is seen that Basis pursuit performance is better compared to MP and OMP
since its mean square error is almost equal to zero. BP outperform MP and OMP especially for
severe Doppler spread conditions.
VI. CONCLUSION
We considered sparse channel estimation for multicarrier underwater acoustic
communication. Based on the path-based channel model, we linked well-known subspace methods
from the array-processing literature to the channel estimation problem.
Also we employed recent compressed sensing methods, namely Matching Pursuit(MP),
Orthogonal Matching Pursuit (OMP) and Basis Pursuit (BP). Based on the continuous time
characterization of the path delays, we suggested the use of finer delay resolution overcomplete
dictionaries. We also extended the compressed sensing receivers to handle channels with different
Doppler scales on different paths, supplying intercarrier interference (ICI) pattern estimates that can
be used to equalize the ICI. Using extensive numerical simulation and experimental results, we find
that in comparison to the LS receiver the subspace methods show significant performance increase

9. Proceedings of the 2nd International Conference on Current Trends in Engineering and Management ICCTEM -2014
17 – 19, July 2014, Mysore, Karnataka, India
on channels that are sparse, but perform worse if most received energy comes from diffuse
multipath. The compressed sensing algorithms do not suffer this drawback, and benefit significantly
from the increased time resolution using overcomplete dictionaries. When accounting for different
Doppler scales on different paths, BP can effectively handle channels with very large Doppler
spread.
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VII. ACKNOWLEDGEMENT
[1] I would like to acknowledge the contributions from Christian R. Berger, Member, IEEE,
Shengli Zhou, Member, IEEE, James C. Preisig, Member, IEEE, and Peter Willett, Fellow,
IEEE
[2] I would like to Acknowledge the contributions from M.Stojanovic, “Low complexity OFDM
detector for underwater acoustic channels,” IEEE Oceans Conf., Sept. 2006.
VIII. REFERENCES
THESES
[1] Christian R. Berger, Shengli Zhou, James C. Preisig and Peter Willett, “Sparse Channel
Estimation for Multicarrier Underwater Acoustic Communication: From Subspace Methods to
Compressed Sensing” ,IEEE
[2] M.Stojanovic, “Low complexity OFDM detector for underwater acoustic channels,” IEEE
Oceans Conf., Sept. 2006.
[3] C.-J. Wu and D. W. Lin, “Sparse channel estimation for OFDM transmission based on
representative subspace fitting,” in Proc. of Vehicular Technology Conf., Stockholm, Sweden,
May 2005.
JOURNALS
[4] E. Candes and T. Tao, “Near-optimal signal recovery from random projections: Universal
encoding strategies?” IEEE Trans. Inform. Theory, vol. 52, no. 12, pp. 5406–5425, Dec. 2006.
[5] R.Negi and J.Cioffi, “Pilot tone selection for channel estimation in a mobile OFDM system,”
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[6] I. F. Akyildiz, D. Pompili, and T. Melodia, “Challenges for efficient communication in
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[7] T. H. Eggen, A. B. Baggeroer, and J. C. Preisig, “Communication over Doppler spread
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