2013 IEEE International Conference on Ultra-Wideband (ICUWB)
Modified Compressive Sensing Based Receiver for Impulse Radio Communications in UWB Channels Qi Zhang and Jiayin Qin
A. Nallanathan
Department of Electronics and Communications Engineering Sun Yat-Sen University, China Email: {zhqi26, issqjy}@mail.sysu.edu.cn
Department of Electronic Engineering King’s College London, United Kingdom Email:
[email protected]
Abstract—For impulse radio communications, the coherent Rake receiver requires accurate time acquisition and channel estimation whereas the autocorrelation receiver (AcR) requires a wideband analog delay element which has high complexity. In this paper, we propose a modified compressive sensing (CS) based receiver to avoid the stringent time acquisition, channel estimation and complicated wideband analog delay element. Different from the conventional CS algorithms, the proposed receiver modifies the randomly generated base functions according to channel statistics and simplifies the 1 norm minimization as to form the detection template by combining several base functions which have larger correlation coefficients with the received signal. Furthermore, the received signal is demodulated by a symbol rate sampling at the proposed receiver. Computer simulation results show that the proposed receiver outperforms the Rake receiver with 8 fingers and AcR.
I. I NTRODUCTION Impulse radio, which transmits a train of ultra-short pulses in a ultra-wideband (UWB) channel, allows hundreds of multipath components can be resolved and harnessed by the Rake receiver [2]. However, the coherent reception of Rake receiver needs very accurate time acquisition and a large number of Rake fingers to match the multipath components [3]. To avoid the stringent time acquisition, transmitted-reference impulse radios with an autocorrelation receiver (TR-IR/AcR) have been proposed [4]-[6]. The implementation of AcR generally requires a delay element to delay the received pilot pulses to form the correlator template. The delay element, which handles analog signals with several GHz bandwidth, is difficult to build in the low-power integrated fashion [7]-[8]. To avoid the delay element, several alternative implementations of the TR-IR/AcR idea have been proposed. In [8]-[9], the data modulated pulses and pilot pulses are overlapped in time domain but they are separated through frequency shift and code-multiplexed schemes, respectively. The proposed schemes in [8]-[9], although having lower complexity, perform much worse than the AcR. Another method to address the complexity of wideband analog delay element is the digital eigen-based receiver proposed in [10]. In [10], the received pulses are first decomposed into signals in multiple dimensions based on the signal statistics. The signals in the dimensions which have larger eigenvalues are then digitized as in [6]. Since with the digital eigenbased receiver, the sampling rate is significantly reduced, the
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complexity of AcR is reduced accordingly. The key idea of [10] is to find a set of bases to sense the received pulses efficiently. For sparse signals, compressive sensing (CS) is an efficient method to sense and reconstruct the signals [11]. The UWB signals, consisting of extremely short pulse train which has very low duty cycle, may be considered as sparse signals in the time domain. Thus the sparsity of signals may be exploited by applying CS. Application of CS in UWB systems is not new. In [12], Paredes et al employed CS for UWB channel estimation. In [13]-[15], CS was used for direct detection of impulse radio signals without Rake receiver. The above-mentioned schemes in [12]-[15] apply the standard CS algorithms which in general are divided in two steps. The first step is to correlate the unknown signal which is sparse in the basis Ψ with the randomly generated base functions and the second step is the 1 norm minimization of possible projections of the signal onto the basis Ψ [10]. Since the impulse radio is suitable for the low-power and low-complexity applications such as wireless sensor networks, the algorithms proposed in [12]-[15] is still complicated, especially for the minimization step. Inspired by [12]-[15], we propose a modified CS based receiver which is tailored for low complexity impulse radio communications in this paper. In the conventional CS algorithm, the base functions for sensing are generated without considering the UWB channel statistics whereas in the proposed CS algorithm, the randomly generated base functions are modified according to channel statistics. Furthermore, we simplify the 1 norm minimization in CS algorithm as to form the detection template by combining several base functions which have larger correlation coefficients with the received signal. The proposed modified CS based receiver correlates the received signal with the obtained detection template and samples the correlation results at symbol rate to demodulate the signal. The complexities of Rake receiver, AcR and the proposed receiver are compared as follows. The Rake receiver consists of multiple fingers (correlators) where each finger extracts the signal from one of the multipath components [3] whereas the proposed receiver consists of single correlator. Furthermore, each finger of Rake receiver should perform accurate time acquisition and channel estimation, which is searching all the possible positions of pulses and correlating the received pilot signals with the predetermined waveform.
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2013 IEEE International Conference on Ultra-Wideband (ICUWB)
For analog Rake receivers, the received signal cannot be stored. Thus the number of all the possible positions of pulses determines the number of pilot symbols to be transmitted, denoted as Np . The multipath delay spread for a typical UWB channel is tens or hundreds times larger than the pulse duration. Thus to achieve high acquisition accuracy needs the transmission of a large number of pilot symbols. If the received pilot signal can be stored is assumed, the Rake receiver only needs one pilot symbol and the proposed receiver needs one training symbol [12]. For AcR, the possible implementations are as [6]-[7], which require the receivers to sample the received signal at the Nyquist rate. Thus the proposed receiver with the symbol rate sampling scheme has lower complexity. The rest of the paper is organized as follows. Section II describes the system model, including the transmitter, the UWB channel, and the proposed modified CS based receiver. In Section III, bit-error-rate (BER) performance of the proposed receiver is derived. Simulation results are provided and discussed in Section IV. We conclude and summarize our paper in Section V. II. S YSTEM M ODEL A. The Transmitter In this paper, we consider a peer-to-peer impulse radio system in quasi-static UWB environment. In the system, each symbol interval of duration T is divided into Ns frames, each with a duration of Tf . Thus the transmitted signal is described by s −1 N bi ω(t − iT − jTf ) (1) s(t) = i
j=0
where bi ∈ {−1, 1} is the independent and identically distributed data symbol, ω(t) is a causal pulse of duration Tω .
At the receiver, the received signal is expressed as follows r(t) =
i
bi
N s −1
h(t − iT − jTf ) + n(t)
(3)
j=0
where h(t) = ω(t) ⊗ g(t)
(4)
in which ⊗ denotes convolution. In (3), n(t) is lowpass filtered additive white Gaussian noise (AWGN) with two-sided power spectral density No /2. The autocorrelation function of n(t) is E[n(t + τ )n(t)] = No W sinc(W τ )
(5)
where E[•] denotes the expected value of [•] and W (W 1/T ) is the bandwidth of the lowpass filter. The ratio, Eb /No , Ns Tω 2 of the system is defined as N ω (t)dt. 0 o C. The Modified Compressive Sensing (CS) Based Receiver For the CS based receiver, it is important to design a basis Ψ of parameterized waveforms that closely match the received information-carrying signal, which is composed of pulses with different amplitudes and time delays [11]. Thus it is natural designing Ψ to be the collection of different time-delayed versions of the pulse. However, in a dense multipath environment, the received information-carrying signal is distorted by the superposition of randomly arrived multipath components. Therefore, employing the basis Ψ of the collection of timedelayed pulses for CS based receiver may not be optimal. In general, the superposition of randomly arrived pulses can be treated as a multi-dimensional random process [10], which is characterized by its covariance functions. In this paper, the eigenfunctions of above-mentioned covariance functions are considered to form the basis Ψ. The covariance functions of the received pulses are expressed as follows Rk (τ ) = E[h(t + τ + (k − 1)Tc )h(t + (k − 1)Tc )],
B. The Channel Model
0 ≤ τ < Tc , k ∈ {1, 2 · · · , K}
The transmitted signal propagates through a quasi-static dense multipath fading UWB channel. The random channels are generated according to [16], where the clusters and the rays in each cluster form Poisson arrival processes with different, but fixed rates. The amplitude of each ray is modeled as a lognormal distributed random variable. The channel impulse response model can be written, in general, as g(t) =
L
αl δ(t − τl )
(2)
l=1
where L denotes the number of propagation paths, and αl and τl denote, respectively, the amplitude and the delay associated with the lth path. The signals arriving at the receiver are assumed to be perfectly synchronized and the path delays are normalized with τ1 = 0. The perfect synchronization assumption can be fulfilled by wideband acquisition techniques as in [17]-[18]. To preclude intersymbol interference and intrasymbol interference, we set Tf ≥ Tω + Td, where Td is the maximum excess delay of the UWB channel.
(6)
where Tc is the support length of eigenfunctions and K is a positive integer. For convenience, Tc is designed to be Tf . (7) K In this paper, the support length of eigenfunctions, Tc , is chosen to be 0.501 ns. The reason is explained as follows. The objective of analyzing the covariance function of h(t) is to design Ψ closely matching the received information-carrying pulses. Although we design the basis Ψ to be the eigenfunctions of Rk (τ ) instead of the different time-delayed versions of the pulse, the support length of eigenfunctions should be comparable to a pulse duration Tω . In the simulations, we select the pulse ω(t) to be the second derivative of a Gaussian pulse, namely, 2 2 t t ω(t) = 1 − 4π exp −2π (8) τm τm Tc =
where τm = 0.2877 ns, as in [1]. The above pulse has a
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2013 IEEE International Conference on Ultra-Wideband (ICUWB)
duration about 0.6 ns, which is slightly larger than the support length of eigenfunctions, Tc . In this paper, we obtain the covariance functions by using the UWB channel models. The channel impulse response g(t) generated according to UWB channel models is convoluted with the pulse, ω(t), to obtain the channel response, h(t). The covariance functions, Rk (τ ), are estimated by taking ensemble averaging of [h(t + τ + (k − 1)Tc )h(t + (k − 1)Tc )] instead of expectation as that in (6). For a support length Tc of 0.501 ns, the above ensemble average is approximately constant for each t. Let φkp (τ ) denote the eigenfunction which decomposes the covariance function Rk (τ ) such that Tc Rk (τ )φkp (τ )dτ = λkp φkp (τ ) (9)
signals are correlated with the Nt base functions to obtain T bz ψz (t) · r (t + (z − 1)T ) dt, βz = 0
z ∈ {1, 2, · · · , Nt }
where ψz (t) is the normalized base function − 12 T 2 ψz (t) = ψˆz (t) · . ψˆz (t)dt
φ1p (τ ) ≈ φ2p (τ ) ≈ · · · ≈ φKp (τ ), p ∈ {1, 2, 3, · · ·}. (10) The reason may be explained as follows. Unlike the eigenvalues which are determined by the pulse amplitudes, the eigenfunctions are determined by the pulse shape and the pulse arrival rate. Although the channel response h(t) is not a stationary process, the clusters and the rays in each cluster in the channel response have the fixed arrival rates [16]. Thus for different Rk (τ ), the eigenvalues are different while the eigenfunctions are almost the same. To form the detection template, the transmitter transmits Nt training symbols and the receiver correlates the received signal with the Nt base functions in Ψ. It is worth noting that the required Nt training symbols is because the received signal cannot be stored. If we assume that the received signal can be stored, only one training symbol is required to form the detection template at the proposed receiver [12]. In this paper, the Nt base functions are modified according to the average power decay profile, H(t), of the UWB channel where
H(t) = E h2 (t) , t ∈ [0, Tf ). (11) Thus the Nt base function is expressed as follows ψˆz (t) =
N s −1 j=0
H(t − jTf )
K−1
ξz (k)φk1 (t − kTc − jTf ),
(14)
0
t Arrange {|βz |}N z=1 in a decreasing order as
|β1:Nt | > |β2:Nt | > · · · > |βNt :Nt | .
(15)
The detection template is obtained by combining the M base functions with larger absolute correlation coefficients,
0
where λk1 ≥ λk2 ≥ λk3 ≥ ... denote the eigenvalues. The obtained eigenvalues, {λkp }, and eigenfunctions, {φkp (τ )} are applicable to any channel generated by the UWB channel model. After the decomposition of Rk (τ ), it is interesting to find that the eigenfunctions of different Rk (τ ) are almost the same,
(13)
ψd (t) =
M
βz:Nt ψz:Nt (t).
(16)
z=1
The detector implements
> 0; decide bi = +1 Di : ≤ 0; decide bi = −1
(17)
where Di is the decision statistics expressed as follows T r (t + (i − 1)T ) ψd (t)dt. (18) Di = 0
III. P ERFORMANCE A NALYSIS In this section, mathematical formulas for predicting the BER performances of modified CS based receiver for impulse radio communications are derived. By substituting (3) into (18), the decision variable Di can be rewritten as follows D i = b i hd + ni where
hd =
0
and ni =
T
ψd (t) ·
N s −1
(19)
h (t − jTf ) dt
(20)
ψd (t) · n (t + (i − 1)T ) dt.
(21)
j=0
T
0
M Given h(t), {βz:Nt }M z=1 and {ψz:Nt }z=1 , hd is a constant and ni is a Gaussian random variable, T M ni = βz:Nt ψz:Nt (t) · n (t + (i − 1)T ) dt, (22) z=1
k=0
0
(12) which has zero mean and conditional variance of M
No M where {ξz (k) : z = 1, 2, · · · , Nt , k = 0, 1, · · · , K − 1} are the = , {ψ } β 2 . (23) E n2i h(t), {βz:Nt }M z:Nt z=1 z=1 2 z=1 z:Nt generated independent Gaussian random variables with zero mean and unit variance, and φk1 (τ ) is the eigenfunction of Thus, the conditional BER for modified CS based receiver is Rk (τ ) corresponding to the largest eigenvalue λk1 . At the M P r e h(t), {βz:Nt }M proposed modified CS Based Receiver, the received training z=1 , {ψz:Nt }z=1 z ∈ {1, 2, · · · , Nt }
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2013 IEEE International Conference on Ultra-Wideband (ICUWB)
0
0
10
10
−1
−1
10
10
−2
−2
BER
10
BER
10
−3
−3
CS−IR, Nt=30
10
10
−4
TR−IR Rake−1, IR Rake−2, IR Rake−4, IR Rake−8, IR
TR−IR Rake−1, IR Rake−2, IR Rake−4, IR Rake−8, IR
10
15
t
−4
10
20 E /N (dB) b
25
30
10
o
⎞
⎛ Q ⎝
No 2
hd M z=1
15
20 E /N (dB) b
Fig. 1. BER versus Eb /No ; performance comparison of the proposed modified CS based receiver, AcR and Rake receiver for impulse radio communications in UWB channel model CM 2.
=
t
CS−IR, N =100
t
10
CS−IR, N =30
CS−IR, N =100
⎠ 2 βz:N t
(24)
√ ∞ where Q(x) = (1/ 2π) x exp(−y 2 /2)dy. By taking the expectation of (24) with respect to h(t), {βz:Nt }M z=1 , and , we obtain the system BER performance with {ψz:Nt }M z=1 modified CS based receiver as follows
M . (25) P r(e) = E P r e h(t), {βz:Nt }M z=1 , {ψz:Nt }z=1 IV. S IMULATION R ESULTS In this section, we present computer simulated results to validate our designs. In the simulations, the random channels are generated according to [16]. The sampling interval for computer simulations is 0.167 ns. The bandwidth of the lowpass filter is 2.994 GHz. In Fig. 1 and Fig. 2, we compare the BER performances of the proposed modified CS based receiver with the Rake receiver and AcR for impulse radio communications in channel model (CM) 2 and CM 4 UWB channels, respectively. For the impulse radio communications, we choose the number of frames in a symbol duration, Ns , of 8 and the frame durations, Tf , of 60.12 ns in CM 2 UWB channels and 150.30 ns in CM 4 UWB channels, respectively. In the legends of Fig. 1 and Fig. 2, the label “CS-IR” denotes the proposed modified CS based receiver. The label “Rake-N , IR” denotes the impulse radio system with an N -finger selective Rake (SRake) receiver [3], which employs maximum ratio combining. The label “TR-IR” denotes the TR-IR system which employs AcR. For the proposed modified CS based receiver, the parameter K in (7) is 120 in CM 2 channels and 300 in CM 4 channels, respectively, since the support length of eigenfunctions, Tc , is 0.501 ns. The number of base functions to be summed up to form the detection template, M , is selected to be M = 6
25
30
o
Fig. 2. BER versus Eb /No ; performance comparison of the proposed modified CS based receiver, AcR and Rake receiver for impulse radio communications in UWB channel model CM 4.
for the curves with Nt = 30 and M = 12 for the curves with Nt = 100, respectively. At the SRake receiver, the number of pilot symbols, Np , to be transmitted to perform time acquisition and channel estimation is determined by the number of all the possible pulse positions. For CM 2 channels, the multipath delay spread is about 30 ns and for CM 4 channels, the delay spread is about 90 ns. Thus in our simulations, Np is 180 for CM 2 channels and 540 for CM 4 channels, respectively, which is much larger than Nt . For fair comparison, the signal-to-noise ratio (SNR) of the Nt training symbols for the proposed modified CS based receiver and the SNR of the Np pilot symbols for the SRake receiver are the same, which is equal to Eb /No in the simulations. The system model for TR-IR system is similar to that in [4], which is called differential TR-IR. From Fig. 1 and Fig. 2, it is observed that the proposed modified CS based receiver with Nt = 100 outperforms all the Rake receivers and the AcR. In Fig. 3 and Fig. 4, we present the BER performances of the proposed modified CS based receiver with different values of M , which is the number of base functions to be summed up to form the detection template. The Eb /No is selected to be 20 dB and 24 dB for CM 1 and CM 4 UWB channels, respectively. It is found from Fig. 3 and Fig. 4, the BER performance improves with the increase of parameter M . The improvement is gradually slowed down with the increase of M . With the increase of M , the system complexity increases. Thus, the selection of M offers a trade-off between performance and complexity. V. C ONCLUSIONS In this paper, we have proposed a modified CS based receiver for impulse radio communications. The simulation results have shown that the proposed receiver performs better than the Rake receiver with 8 fingers and AcR. Considering stringent time acquisition and channel estimation in UWB
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2013 IEEE International Conference on Ultra-Wideband (ICUWB)
R EFERENCES Nt=30 Nt=50
−2
10
N =100
BER
t
−3
10
5
10
15 M
20
25
30
Fig. 3. BER performances of the proposed modified CS based receiver for various number of base functions to be summed up to form the detection template in UWB channel model CM 2.
−1
10
N =30 t
N =50 t
N =100 t
−2
BER
10
−3
10
−4
10
5
10
15 M
20
25
30
Fig. 4. BER performances of the proposed modified CS based receiver for various number of base functions to be summed up to form the detection template in UWB channel model CM 4.
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channels are challenging tasks, the proposed receiver would be a better choice than the Rake receiver. Furthermore, the proposed receiver avoids the complicated wideband analog delay element which is necessary for the AcR. ACKNOWLEDGMENT This work was supported by the National Natural Science Foundation of China (61173148 and 61202498), the IndustryUniversity-Research Project of Guangdong Province and the Ministry of Education (2011B090400581), the Scientific and Technological Project of Guangzhou City (12C42051578 and 11A11060133).
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