UNDERWATER ACOUSTIC CHANNEL ESTIMATION AND ANALYSIS
Proceedings of the 5th Annual ISC Research Symposium ISCRS 2011 April 7, 2011, Rolla, Missouri
UNDERWATER ACOUSTIC CHANNEL ESTIMATION AND ANALYSIS Jesse Cross Missouri University of Science and Technology Department of Electrical and Computer Engineering ABSTRACT This paper analyzes statistical characteristics of underwater acoustic channels using experimental measurements. Channel impulse responses (CIRs) are estimated in the time domain using a least square method with sliding windows applied to received data. The probability density functions (PDF) of the real part, imaginary part, magnitude, and phase of the CIR are estimated, and the two-sample Kolmogorov-Smirnov test is used to determine how well the magnitude PDF fits the Gamma, Rayleigh, and compound K distributions. The autocorrelation, channel coherence time, and level crossing rate of the channel are also investigated. The experimental results demonstrate that underwater channels often offer poorer communication quality than Rayleigh fading channels. 1. INTRODUCTION Underwater acoustic transmission is an effective means of medium and long range (1-1000 km) underwater wireless communication [1]. It is often more challenging to effectively communicate through an underwater acoustic (UWA) channel than it is through a radio frequency (RF) channel, because UWA channels often exhibit frequency-dependent path loss, excessive multipath delay spread, and fast time variation due to Doppler[1,2]. The use of statistical modeling of RF channels to design transmitters and receivers has been well established in the literature [3,4] since early days of wireless communication. However, there are less statistical studies of UWA channels than there are of RF channels and RF channel models are often borrowed to study UWA channel [1,2]. Past ocean experiments have shown that the UWA channel often exhibits worse performance than a Rayleigh fading channel that is commonly used as the worst case scenario in RF channel modeling [3,4]. Consequently, accurate UWA channel modeling is essential to the evaluation of the channel capacity and to the design of the UWA communication systems. This paper investigate the UWA channel statistics utilizing the data collected in the Reschedule Acoustic Communication Experiment (RACE08) conducted in Narragansett Bay, RI, USA, in March 2008. The probability distribution function (PDF), and autocorrelation, crosscorrelation, and level crossing rate of the baseband Channel Impulse Responses (CIR) are analyzed. A sliding window least squares method with a small sliding step and window width are
used to track the time varying CIRs. The Kolmogorov-Smirnov (KS) test [7] is used to measure the fitness between the experimental PDFs and the PDFs of theoretical distributions. The results show that the experimental PDFs are a close match to the compound K distribution, which can lead to worse performance than Rayleigh fading. Spatial-temporal correlation also has significant impact on the Bit Error Rate (BER) performance of the channels. 2. CHANNEL MODEL, ESTIMATION AND STATISTICS Let be the transmitted passband signal where is the carrier frequency and is the complex baseband equivalent signal. Let be the received passband signal where is the received baseband complex signal. The received baseband complex signal may be represented as
where is the equivalent complex impulse response for the time varying channel. It is assumed there are i multipaths each with its own delay and Doppler shift. Therefore, the channel impulse response may be modeled as
where , , and are the gain, propagation delay, and instantaneous Doppler shift at the i-th path, respectively. Even though the channel is time varying, these parameters will remain constant over a short time period. 2.1. Channel Estimation A time domain sliding window least squares technique is used to estimate the CIR from the baseband transmitted and received data. A sliding window of size is used to estimate the CIR over time as a long data sequence is read The estimated time domain CIR for transmitter m at the pth step is , where is the number of receivers. The CIR for the transmitter/receiver pair, , is , where L is the length of the CIR and s is the sequence of symbols used to perform the estimation. The CIR may be estimated using
1
where is the pseudo inverse, the received MIMO data, and
is where
where . The CIR is re-estimated every seconds where is the slide step of the data window and the symbol period.
is
2.2. Magnitude Distributions The PDF equation for a sequence, X, that has a Rayleigh distribution is
where is a shape parameter. The PDF for a sequence, X, that is gamma distributed is
where is the Euler gamma function and a and b are scalar parameters. If the sequence, X, is compound K distributed, the PDF is
where and are scalar parameters and is the modified Bessel function of the second kind with an order of . If and remains constant, the compound K distribution will from a Rayleigh distribution. As a result, the channel will have lesser quality than a Rayleigh channel as decreases. The first and second moments of the measured data were used to fit the PDF of the measured data to these distributions. The first and second moments of the PDF equations are given in Table 1. Table 1: The first and second moments of the Rayleigh, gamma, and compound K distributions. Rayleigh Gamma Compound K
2.3. Kolmogorov-Smirnov Test The KS test is used to quantify the similarity between two empirical data distributions. No assumptions about the two data distributions are made. The two sample KS test may be used as a goodness of fit test in order to determine whether or not two data sets share the same probability distribution. The null hypothesis is that the distributions of the two datasets do
not differ significantly enough to consider them to follow different distributions. The alternative hypothesis is that the two datasets do not share the same distribution. The KS statistic, , is defined as follows where and are the cumulative distribution functions of the two datasets to be tested and is the supremum of . The alternative hypothesis will be accepted if
where
is a Kolmogorov distributed random variable, , and and are the number of elements in the two datasets. 3. EXPERIMENTAL RESULTS The Reschedule Acoustic Communication Experiment (RACE08) was performed in Narragansett Bay, Rhode Island, on March 2008 by the Woods Hole Oceanographic Institution (WHOI). There were two transmitters suspended four meters above the sea floor and twelve receivers suspended two meters above the sea floor. The water depth varied from nine to fourteen meters. In this paper, only the data collected when the transmitters were 1000 km away from the receivers are considered. The carrier frequency, sampling rate, and bandwidth were 11.5 kHz, 39.0625 kHz, and 3.90625 kHz, respectively. The sequence used to estimate the CIR was 22 seconds long. The sliding window used in the estimation was of size and the interval between each window was . Therefore, the channels were estimated every milliseconds. The estimated channel is 25 symbol periods long. The estimation method described in (3) yielded 2144 estimations of each channel tap for each data packet sent. Figure 1 contains two plots of the estimated CIR versus time for two of the subchannels. Figure 2 shows the PDFs of the real and imaginary components of the experimental data versus fitted Gaussian distributions. It is evident that the Gaussian distribution does not fit well. The Gaussian distribution with the smaller variance fits the peak well but not the tails of the experimental PDF. The converse is true for the Gaussian distribution with the larger variance. Since the real and imaginary components of the Rayleigh distribution follow Gaussian distributions, it is reasonable to conclude that the envelope PDF of the experimental data does not follow a Rayleigh distribution. Figure 3 shows the envelope PDF plotted with fitted Rayleigh, gamma, and compound K distributions. It is visually evident that the experimental PDF does not follow a Rayleigh distribution. The closer fit of the gamma distribution shows that the shape of the experimental PDF is dominated by its gamma component. The KS test is used to quantify the similarity of the distributions. The results of the
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KS test are shown in table 2. At the 5% level of significance ( ), the alternative hypothesis that the two distributions differ significantly is accepted. The distribution parameters were chosen to yield the highest p value from the KS test. The p value is a measure of the extremity of the experimental data if it truly follows the distribution it is tested against. As the p value approaches zero, the alternative hypothesis is accepted with greater confidence.
compared to the Rayleigh distribution is overwhelmingly rejected. The fit of the gamma distribution is much closer than the fit of the Rayleigh distribution; however, the gamma distribution is still rejected. The p value of 0.96 shows that the null hypothesis that the two distributions do not differ enough to be considered different is overwhelming accepted when the experimental PDF is tested against the compound K
Figure 1(a): Estimated CIRs for the subchannel Tx1—Rx1
Figure 2(a): PDF of the Real Part
Figure 1(b): Estimated CIRs for the subchannel Tx2—Rx1 Table 2: KS Test Results Rayleigh Gamma p value Result
0.48 reject
0.16 Reject
Figure 2(b): PDF of the Imaginary Part Compound K 0.96 0.07 accept
As expected, the p values in table 1 increase as the distributions fit the experimental PDF more closely. The extremely small p value of shows that the null hypothesis of the KS test when the experimental PDF is
distribution. It is reasonable to conclude that the experimental PDF follows a compound K distribution and not a Rayleigh or gamma distribution. Figure 4 contains the autocorrelation of the channel taps of the real part, imaginary part, and complex envelope. The autocorrelation appears to decrease exponentially with time. If the coherence threshold is defined to be 0.5, then the according the plot in fig. 4, the coherence time is 0.12 seconds or about symbols. The channel is considered
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invariant during this interval. The cross-correlation between the real and imaginary parts of the CIR is shown in fig. 5. The real and imaginary parts are shown to be uncorrelated.
Figure 5: Crosscorrelation between the real and imaginary parts of the estimated CIR
Figure 3: Experimental PDF versus other Distributions
Figure 6: Level crossing rate Figure 4: Autocorrelation of the estimated CIR taps The measured level crossing rate (LCR) of the channel compared to the theoretical LCR of an equivalent channel that exhibits Rayleigh fading is shown in fig. 6. The theoretical LCR of a Rayleigh channel is given by [8]
where is the maximum Doppler shift and is the level crossing threshold. Except for small values of , the measured LCR is higher than the theoretical Rayleigh LCR. A greater LCR leads to smaller average fade durations but smaller intervals between fades.
4. CONCLUSIONS Many CIRs of the RACE08 UWA channel have been estimated over time. Several characteristics of the channel were then presented. The CIR itself was 25 symbols long. The real and imaginary components of the CIR do not follow Gaussian distributions and that the magnitude PDF of the CIR is not described by a Rayleigh distribution which is often the case for an RF channel. The PDF of the magnitude appears to be dominated by a gamma component and was shown to be described by the compound K distribution. The coherence time of the channel was estimated using the autocorrelation of the channel which appears to decay exponentially. The measured level crossing rate does not correspond to a Rayleigh channel with similar characteristics.
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5. ACKNOWLEDGMENTS This work was supported in part by the National Science Foundation under Grant ECCS-0846486 and the Office of Naval Research under Grants N00014-10-1-0174. The author is grateful for the support provided by the Intelligent Systems Center. 6. REFERENCES [1] M. Stojanovic and J. Preisig, ―Underwater acoustic communication channels: propagation models and statistical characterization,‖ IEEE Commun. Mag., pp.8489, Jan. 2009. [2] A. C. Singer, J. K. Nelson, S. S. Kozat, ―Signal processing for underwater acoustic communications,‖ IEEE Commun. Mag., pp.90-96, 2009. [3] P. A. Bello, ―Characterization of randomly time-variant linear channels,‖ IEEE Trans. Commun. Syst., vol. 11, pp. 360–393, Dec. 1963. [4] C. Xiao, J. Wu, S.-Y. Leong, Y. R. Zheng, and K. B. Letaief, ―A discrete-time model for triply selective MIMO Rayleigh fading channels,‖ IEEE Trans. Wireless Commun., vol. 3, no. 5, pp. 1678–1688, Sept. 2004. [5] W. Yang and T. C. Yang, ―High frequency channel characterization for M-ary frequency-shift keying underwater acoustic communications,‖ J. Acoust. Soc. Am., vol.120, pp.2615-2626, Nov. 2006. [6] W. Yang and T. C. Yang, ―M-ary frequency shift keying communications over an underwater acoustic channel: performance comparison of data with models,‖ J. Acoust. Soc. Am., vol.120, pp.2694-2701, Nov. 2006. [7] F. J. Massey, ―The Kolmogorov-Smirnov test for goodness of fit,‖ J. Am. Stat. Associate, vol.46, pp.68-78, 1951. [8] T. S. Rappaport, Wireless Coomunications: Principle and Practice, second edition, Prentice Hall, 2002.
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UNDERWATER ACOUSTIC CHANNEL ESTIMATION AND ANALYSIS Jesse Cross Missouri University of Science and Technology Department of Electrical and Computer Engineering ABSTRACT This paper analyzes statistical characteristics of underwater acoustic channels using experimental measurements. Channel impulse responses (CIRs) are estimated in the time domain using a least square method with sliding windows applied to received data. The probability density functions (PDF) of the real part, imaginary part, magnitude, and phase of the CIR are estimated, and the two-sample Kolmogorov-Smirnov test is used to determine how well the magnitude PDF fits the Gamma, Rayleigh, and compound K distributions. The autocorrelation, channel coherence time, and level crossing rate of the channel are also investigated. The experimental results demonstrate that underwater channels often offer poorer communication quality than Rayleigh fading channels. 1. INTRODUCTION Underwater acoustic transmission is an effective means of medium and long range (1-1000 km) underwater wireless communication [1]. It is often more challenging to effectively communicate through an underwater acoustic (UWA) channel than it is through a radio frequency (RF) channel, because UWA channels often exhibit frequency-dependent path loss, excessive multipath delay spread, and fast time variation due to Doppler[1,2]. The use of statistical modeling of RF channels to design transmitters and receivers has been well established in the literature [3,4] since early days of wireless communication. However, there are less statistical studies of UWA channels than there are of RF channels and RF channel models are often borrowed to study UWA channel [1,2]. Past ocean experiments have shown that the UWA channel often exhibits worse performance than a Rayleigh fading channel that is commonly used as the worst case scenario in RF channel modeling [3,4]. Consequently, accurate UWA channel modeling is essential to the evaluation of the channel capacity and to the design of the UWA communication systems. This paper investigate the UWA channel statistics utilizing the data collected in the Reschedule Acoustic Communication Experiment (RACE08) conducted in Narragansett Bay, RI, USA, in March 2008. The probability distribution function (PDF), and autocorrelation, crosscorrelation, and level crossing rate of the baseband Channel Impulse Responses (CIR) are analyzed. A sliding window least squares method with a small sliding step and window width are
used to track the time varying CIRs. The Kolmogorov-Smirnov (KS) test [7] is used to measure the fitness between the experimental PDFs and the PDFs of theoretical distributions. The results show that the experimental PDFs are a close match to the compound K distribution, which can lead to worse performance than Rayleigh fading. Spatial-temporal correlation also has significant impact on the Bit Error Rate (BER) performance of the channels. 2. CHANNEL MODEL, ESTIMATION AND STATISTICS Let be the transmitted passband signal where is the carrier frequency and is the complex baseband equivalent signal. Let be the received passband signal where is the received baseband complex signal. The received baseband complex signal may be represented as
where is the equivalent complex impulse response for the time varying channel. It is assumed there are i multipaths each with its own delay and Doppler shift. Therefore, the channel impulse response may be modeled as
where , , and are the gain, propagation delay, and instantaneous Doppler shift at the i-th path, respectively. Even though the channel is time varying, these parameters will remain constant over a short time period. 2.1. Channel Estimation A time domain sliding window least squares technique is used to estimate the CIR from the baseband transmitted and received data. A sliding window of size is used to estimate the CIR over time as a long data sequence is read The estimated time domain CIR for transmitter m at the pth step is , where is the number of receivers. The CIR for the transmitter/receiver pair, , is , where L is the length of the CIR and s is the sequence of symbols used to perform the estimation. The CIR may be estimated using
1
where is the pseudo inverse, the received MIMO data, and
is where
where . The CIR is re-estimated every seconds where is the slide step of the data window and the symbol period.
is
2.2. Magnitude Distributions The PDF equation for a sequence, X, that has a Rayleigh distribution is
where is a shape parameter. The PDF for a sequence, X, that is gamma distributed is
where is the Euler gamma function and a and b are scalar parameters. If the sequence, X, is compound K distributed, the PDF is
where and are scalar parameters and is the modified Bessel function of the second kind with an order of . If and remains constant, the compound K distribution will from a Rayleigh distribution. As a result, the channel will have lesser quality than a Rayleigh channel as decreases. The first and second moments of the measured data were used to fit the PDF of the measured data to these distributions. The first and second moments of the PDF equations are given in Table 1. Table 1: The first and second moments of the Rayleigh, gamma, and compound K distributions. Rayleigh Gamma Compound K
2.3. Kolmogorov-Smirnov Test The KS test is used to quantify the similarity between two empirical data distributions. No assumptions about the two data distributions are made. The two sample KS test may be used as a goodness of fit test in order to determine whether or not two data sets share the same probability distribution. The null hypothesis is that the distributions of the two datasets do
not differ significantly enough to consider them to follow different distributions. The alternative hypothesis is that the two datasets do not share the same distribution. The KS statistic, , is defined as follows where and are the cumulative distribution functions of the two datasets to be tested and is the supremum of . The alternative hypothesis will be accepted if
where
is a Kolmogorov distributed random variable, , and and are the number of elements in the two datasets. 3. EXPERIMENTAL RESULTS The Reschedule Acoustic Communication Experiment (RACE08) was performed in Narragansett Bay, Rhode Island, on March 2008 by the Woods Hole Oceanographic Institution (WHOI). There were two transmitters suspended four meters above the sea floor and twelve receivers suspended two meters above the sea floor. The water depth varied from nine to fourteen meters. In this paper, only the data collected when the transmitters were 1000 km away from the receivers are considered. The carrier frequency, sampling rate, and bandwidth were 11.5 kHz, 39.0625 kHz, and 3.90625 kHz, respectively. The sequence used to estimate the CIR was 22 seconds long. The sliding window used in the estimation was of size and the interval between each window was . Therefore, the channels were estimated every milliseconds. The estimated channel is 25 symbol periods long. The estimation method described in (3) yielded 2144 estimations of each channel tap for each data packet sent. Figure 1 contains two plots of the estimated CIR versus time for two of the subchannels. Figure 2 shows the PDFs of the real and imaginary components of the experimental data versus fitted Gaussian distributions. It is evident that the Gaussian distribution does not fit well. The Gaussian distribution with the smaller variance fits the peak well but not the tails of the experimental PDF. The converse is true for the Gaussian distribution with the larger variance. Since the real and imaginary components of the Rayleigh distribution follow Gaussian distributions, it is reasonable to conclude that the envelope PDF of the experimental data does not follow a Rayleigh distribution. Figure 3 shows the envelope PDF plotted with fitted Rayleigh, gamma, and compound K distributions. It is visually evident that the experimental PDF does not follow a Rayleigh distribution. The closer fit of the gamma distribution shows that the shape of the experimental PDF is dominated by its gamma component. The KS test is used to quantify the similarity of the distributions. The results of the
2
KS test are shown in table 2. At the 5% level of significance ( ), the alternative hypothesis that the two distributions differ significantly is accepted. The distribution parameters were chosen to yield the highest p value from the KS test. The p value is a measure of the extremity of the experimental data if it truly follows the distribution it is tested against. As the p value approaches zero, the alternative hypothesis is accepted with greater confidence.
compared to the Rayleigh distribution is overwhelmingly rejected. The fit of the gamma distribution is much closer than the fit of the Rayleigh distribution; however, the gamma distribution is still rejected. The p value of 0.96 shows that the null hypothesis that the two distributions do not differ enough to be considered different is overwhelming accepted when the experimental PDF is tested against the compound K
Figure 1(a): Estimated CIRs for the subchannel Tx1—Rx1
Figure 2(a): PDF of the Real Part
Figure 1(b): Estimated CIRs for the subchannel Tx2—Rx1 Table 2: KS Test Results Rayleigh Gamma p value Result
0.48 reject
0.16 Reject
Figure 2(b): PDF of the Imaginary Part Compound K 0.96 0.07 accept
As expected, the p values in table 1 increase as the distributions fit the experimental PDF more closely. The extremely small p value of shows that the null hypothesis of the KS test when the experimental PDF is
distribution. It is reasonable to conclude that the experimental PDF follows a compound K distribution and not a Rayleigh or gamma distribution. Figure 4 contains the autocorrelation of the channel taps of the real part, imaginary part, and complex envelope. The autocorrelation appears to decrease exponentially with time. If the coherence threshold is defined to be 0.5, then the according the plot in fig. 4, the coherence time is 0.12 seconds or about symbols. The channel is considered
3
invariant during this interval. The cross-correlation between the real and imaginary parts of the CIR is shown in fig. 5. The real and imaginary parts are shown to be uncorrelated.
Figure 5: Crosscorrelation between the real and imaginary parts of the estimated CIR
Figure 3: Experimental PDF versus other Distributions
Figure 6: Level crossing rate Figure 4: Autocorrelation of the estimated CIR taps The measured level crossing rate (LCR) of the channel compared to the theoretical LCR of an equivalent channel that exhibits Rayleigh fading is shown in fig. 6. The theoretical LCR of a Rayleigh channel is given by [8]
where is the maximum Doppler shift and is the level crossing threshold. Except for small values of , the measured LCR is higher than the theoretical Rayleigh LCR. A greater LCR leads to smaller average fade durations but smaller intervals between fades.
4. CONCLUSIONS Many CIRs of the RACE08 UWA channel have been estimated over time. Several characteristics of the channel were then presented. The CIR itself was 25 symbols long. The real and imaginary components of the CIR do not follow Gaussian distributions and that the magnitude PDF of the CIR is not described by a Rayleigh distribution which is often the case for an RF channel. The PDF of the magnitude appears to be dominated by a gamma component and was shown to be described by the compound K distribution. The coherence time of the channel was estimated using the autocorrelation of the channel which appears to decay exponentially. The measured level crossing rate does not correspond to a Rayleigh channel with similar characteristics.
4
5. ACKNOWLEDGMENTS This work was supported in part by the National Science Foundation under Grant ECCS-0846486 and the Office of Naval Research under Grants N00014-10-1-0174. The author is grateful for the support provided by the Intelligent Systems Center. 6. REFERENCES [1] M. Stojanovic and J. Preisig, ―Underwater acoustic communication channels: propagation models and statistical characterization,‖ IEEE Commun. Mag., pp.8489, Jan. 2009. [2] A. C. Singer, J. K. Nelson, S. S. Kozat, ―Signal processing for underwater acoustic communications,‖ IEEE Commun. Mag., pp.90-96, 2009. [3] P. A. Bello, ―Characterization of randomly time-variant linear channels,‖ IEEE Trans. Commun. Syst., vol. 11, pp. 360–393, Dec. 1963. [4] C. Xiao, J. Wu, S.-Y. Leong, Y. R. Zheng, and K. B. Letaief, ―A discrete-time model for triply selective MIMO Rayleigh fading channels,‖ IEEE Trans. Wireless Commun., vol. 3, no. 5, pp. 1678–1688, Sept. 2004. [5] W. Yang and T. C. Yang, ―High frequency channel characterization for M-ary frequency-shift keying underwater acoustic communications,‖ J. Acoust. Soc. Am., vol.120, pp.2615-2626, Nov. 2006. [6] W. Yang and T. C. Yang, ―M-ary frequency shift keying communications over an underwater acoustic channel: performance comparison of data with models,‖ J. Acoust. Soc. Am., vol.120, pp.2694-2701, Nov. 2006. [7] F. J. Massey, ―The Kolmogorov-Smirnov test for goodness of fit,‖ J. Am. Stat. Associate, vol.46, pp.68-78, 1951. [8] T. S. Rappaport, Wireless Coomunications: Principle and Practice, second edition, Prentice Hall, 2002.
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