Parameter estimation using the sliding-correlator’s output for wideband propagation channels
- Xuefeng Yin^{1}Email author,
- Cen Ling^{1},
- Myung-Don Kim^{2} and
- Hyun Kyu Chung^{2}
https://doi.org/10.1186/s13638-015-0400-8
© Yin et al. 2015
Received: 4 November 2014
Accepted: 28 May 2015
Published: 11 June 2015
Abstract
In this contribution, a high-resolution parameter estimation algorithm is derived based on the Space-Alternating Generalized Expectation-maximization (SAGE) principle for extracting multipath parameters from the output of sliding correlator (SC). The SC allows calculating channel impulse responses with a sampling rate less than that required by Nyquist criterion, and hence is widely used in real-time wideband (e.g., >500 MHz) channel sounding for the fifth generation wireless communication scenarios. However, since the sounding signal needs to be sent repetitively, the SC-based solution is unacceptable for time-variant channel measurements. The algorithm proposed here estimates multipath parameters by using a parametric model of both low- and high-frequency components of the SC’s output. The latter was considered as distortions and discarded in the conventional SC-based channel sounding. The new algorithm allows estimating path parameters with less repetitions of transmitting the sounding signal and still exhibits higher estimation accuracy than the conventional method. Simulations are conducted and illustrate the root mean square estimation errors and the resolution capability of the proposed algorithm with respect to the bandwidth and the length of the SC’s output. These studies pave the way for measuring time-variant wideband propagation channels using SC-based solutions.
Keywords
1 Introduction
Measurement-based channel models are important for verifying the performance of wireless communication systems in realistic propagation scenarios [1, 2]. Geometry-based stochastic channel models, such as the WINNER spatial channel models [3], IMT-Advanced models [4], and COST2100 multiple-input multiple-output (MIMO) models [5], have been proposed in various standards and widely used to generate single- and multi-link channel realizations at the carrier frequency up to 6 GHz with a bandwidth up to 100 MHz.
Recently, researches on the fifth generation (5G) wireless communications have been paid a lot of attention. The European 7th framework project “Mobile and wireless communications Enablers for the Twenty-twenty Information Society (METIS)” announced a white paper which describes the typical applications and propagation environments considered in 5G [6]. According to the definition by the METIS project, the candidate frequency bands for 5G applications range from 0.45 to 85 GHz, and the bandwidth is from 0.5 up to 2 GHz [6]. At present, the shortage of measurement-based channel models for these frequencies, particularly in the millimeter (mm)-wave bands hinders both the progress of 5G standardization and the designing of 5G-based communication systems and networks. Characterization of mm-wave channel with bandwidth beyond 0.5 GHz for various types of applications and environments began to attract much research attentions recently.
Data acquisition for wideband channels is usually performed by using the equipment such as oscilloscope, spectrum analyzer, and vector network analyzer. The latter two kinds of equipment usually do not have the capability of recording the complex time-domain signals, and thus not suitable for investigating the wideband channel characteristics extracted from multipath parameters. For sampling the mm-wave signals, the oscilloscope devices are required to have sampling rate up to 100 GHz, which is not easy to achieve. Furthermore, due to the small storage in the oscilloscope devices, measurement of wideband channel becomes very time-consuming. A solution tackling these problems is to down-sample the received wideband signals and store the data at a low speed which allows transferring data in real-time from local memory to external disk. Then by using a so-called sliding correlation (SC) technique, a time-dilated approximate channel impulse response (CIR) can be calculated by low-pass filtering (LPF) the received data if the sounding signals can be sent repetitively. The LPF in the receiver is applied to remove the distortion components which have higher frequencies [7]. It has been shown in [8] that pre-filtering techniques can also be applied in the transmitter side to achieve the same objective. Due to the benefits of low complexity in the receiver design and acceptable costs, the SC-based data acquisition has been widely adopted [9–12].
However, the SC-based data acquisition has two problems. First, the higher-frequency components in the SC’s output considered as distortions still carry information of channel parameters and, thus, should be exploited to improve the accuracy of parameter estimation. A drawback resulted when higher-frequency components are considered is that the time-dilated approximation of CIR is unavailable, and as a consequence, conventional peak-searching estimation methods adopted in the SC-based channel estimation are inapplicable. Second, the time-dilated CIR generated by the conventional SC requires the sounding signal being sent repetitively. The number of the repetitions, also called as sliding factor, is usually in the 10^{3} order of magnitude or even higher [7]. In the cases where channels are time-variant, the CIR may not be calculated within the channel coherence time. As a consequence, the mobile to mobile (M2M) channel measurements cannot be conducted by using the SC-based solution. Recently, a Space-Alternating Generalized Expectation-maximization (SAGE) estimation approach was introduced in [13] which is derived based on a parametric model characterizing the SC’s output, allowing the estimation of multipath parameters by using higher-frequency components. However, this solution still relies on the SC’s output obtained by sending the sounding signals many times. No thorough investigation has been carried out so far for the feasibility of accurate parameter estimation based on the SC’s outputs without sending the sounding signals repetitively.
In this contribution, the SAGE algorithm originally derived in [13] based on a parametric model for both low- and high-frequency components of SC’s output is elaborated. Its performances in estimating multipath parameters are investigated extensively by using simulations. It shows that without discarding the higher-frequency components of SC’s output, the estimation accuracy, particularly for delay parameters, can be improved substantially. In addition, another benefit of this novel estimation algorithm not found previously is discovered; that is, the estimation of path parameters, including Doppler frequency, can be performed by using only a fraction of the SC’s output. Hence, the overall observation span can be kept less than channel coherence time in time-variant cases, and characterizing time-variant channels through SC-based measurements, which cannot be performed before, becomes feasible. Simulations are carried out to compare the performance of the proposed algorithm with the conventional method, and investigate the impact of selecting different bandwidth of LPF and the length of the SC’s output on the RMSEEs, resolution capability of the algorithm.
Explanation of adopted symbols
Explanation | |
---|---|
Symbol | |
u(t) | Transmitted pseudo-noise (PN) binary chip sequence |
L | Number of chips in the sequence of u(t) |
V _{0} | Magnitude of the chips in u(t) |
a _{ i } | Values of the chips in u(t) with i being chip index. |
a _{ i }∈[−1,1] | |
f _{ c } | Chip rate of the sequence u(t) |
u ^{′}(t) | PN sequence correlated with received signals |
f c′ | Chip rate of u ^{′}(t) |
γ | “Sliding factor” of the sliding correlator. γ=f _{ c }/(f _{ c }−f c′) |
a _{ u }(τ) | Auto-correlation function of u(t) in the delay domain τ |
a _{ u }(τ/γ) | Time-dilated version of a _{ u }(τ) |
B | Bandwidth of a low-pass-filter (LPF) applied to sliding |
correlator’s output | |
h(τ) | Channel impulse response in the delay domain |
\(\hat {h}(\tau)\) | Estimate of h(τ) |
\(\hat {(\cdot)}\) | Estimate of given argument |
n | Half of the width of the LPF normalized by f _{ c }/γ |
B _{ n } | Width of the LPF B _{ n }=[−n f _{ c }/γ,n f _{ c }/γ] |
N | Number of frequency components of the signals |
r(f) | Baseband representation of received signals in the |
frequency domain | |
M | Total number of propagation paths in a channel |
α _{ ℓ } | Complex attenuation coefficient of the ℓth path |
τ _{ ℓ } | Delay of the ℓth path |
ν _{ ℓ } | Doppler frequency of the ℓth path |
n(f) | White Gaussian noise represented in the frequency |
domain | |
w(f) | White Gaussian noise with spectral height equal to N _{0} |
y(f) | Output of sliding correlator |
s(f) | The signal component of the output of sliding correlator |
n ^{′}(f) | The noise component in the output of sliding correlator |
p(f;τ _{ ℓ },ν _{ ℓ }) | Convolution results between u(f) distorted by a channel |
and the clean sequence u ^{′}(f) | |
y | Concatenated received signals in frequencies, i.e. |
y=[y(f);f∈(f _{1},…,f _{ N })] | |
Θ | The parameters of propagation paths in a channel |
\(\hat {\boldsymbol {\Theta }}^{[0]}\) | Initial estimates of Θ |
\(\hat {\boldsymbol {\Theta }}^{[i]}\) | Estimates of Θ obtained in the ith SAGE iteration |
\(\hat {\boldsymbol {\Theta }}_{\text {SAGE}}\) | The estimates of Θ obtained when the SAGE algorithm |
converges. | |
x _{ ℓ }(f) | Admissible hidden data defined in the SAGE algorithm |
Λ(θ _{ ℓ }) | Likelihood function of the parameters θ _{ ℓ } |
\(\hat {x}^{[i]}_{\ell }(f)\) | Estimated admissible hidden data in the E-step of the ith |
SAGE iteration | |
W | Diagonal matrix with its diagonal elements equal to |
E[|n ^{′}(f)|^{2}],f=f _{1},…,f _{ N }. | |
η(τ,ν) | Objective function maximized in the M-step |
ϱ | Fraction of total length of SC’s output |
T | The time duration of the CIR |
Notations | |
(·)^{∗} | Complex conjugate operation |
(·)^{T} | Transpose operation |
(·)^{H} | Hermitian transpose operation |
∗ | Convolution operation |
E[·] | Expectation operation |
\(\arg \min \limits _{a}\) | Minimization operation with respect to a |
\(\arg \max \limits _{a}\) | Maximization operation with respect to a |
(W)^{−1} | Inverse operation of matrix W |
2 Signal model
As elaborated in [9, 10] and [7], the SC performs a specific cross-correlation operation, e.g., between a pseudo-noise (PN) random sequence u(t) with chip rate f _{ c } and another sequence u ^{′}(t) with chip rate f c′. According to [7], both sequences contain exactly the same chips, and the chip rates are related as \(f_{c}'=\frac {\gamma -1}{\gamma }f_{c}\), where γ is called sliding factor. By sample-wise multiplying these two sequences in the time domain for multiple cycles which start with linearly increasing time-offsets and summing the products over individual cycles of u ^{′}(t), a time-dilated approximate a _{ u }(τ/γ) of the autocorrelation function a _{ u }(τ)=E[u(t)u ^{∗}(t−τ)] can be calculated by low-pass-filtering the SC’s output with bandwidth B=[−f _{ c }/γ,f _{ c }/γ].
In the channel sounding cases, the received signal is the convolution of the transmitted sequence u(t) with the CIR h(τ), the output of the SC after the LPF with bandwidth B, is the time-dilated approximate \(\hat {h}\left (\tau /\gamma \right)\) of the CIR. Here, \(\hat {h}\left (\tau /\gamma \right)\) is a time-dilated version of \(\hat {h}\left (\tau \right) = h\left (\tau \right)\ast a_{u}\left (\tau \right)\) with ∗ denoting the convolution operation. It is well-accepted that only \(\hat {h}(\frac {\tau }{\gamma })\) obtained with the LPF bandwidth B can be used to estimate the channel parameters [7, 14]. Whether the components obtained with larger B, e.g., B _{ n }=[−n f _{ c }/γ,n f _{ c }/γ], n>1, are applicable for estimating the characteristics of h(τ) is necessary to investigate. It is worth mentioning that the time-dilated CIRs with bandwidth of B _{ n } can be obtained by averaging the temporal output of a SC received within the time of L/(f c′n).
The parameters Θ=[α _{ ℓ },ν _{ ℓ },τ _{ ℓ };ℓ=1,…,M] in (4) are unknown and need to be estimated. For simplicity, let us assume that the data y=[y(f);f∈(f _{1},…,f _{ N })] with N being the total number of frequency bins, obtained within the duration long enough for generating one observation of \(\hat {h}(\frac {\tau }{\gamma })\), is available. Estimation of Θ needs to be carried out given y. Notice that this assumption is realistic in the case where the channel coherence time is so short that the SC cannot generate multiple consecutive CIRs for a stationary channel. However, the parameter estimation algorithm derived in the later part of the paper can be easily extended to the case where multiple CIRs are available.
3 Parameter estimation
The maximum likelihood estimate (MLE) of Θ can be derived based on the signal models (4) to (8). However, obtaining the MLE of Θ requires solving a 4M-dimensional optimization problem. The computational complexity involved prohibits any practical implementation. In the following, we present the SAGE algorithm which can iteratively update the subsets of Θ and output the approximate of the MLE of Θ when the estimation process converges [15, 16].
with \(\hat {\alpha }_{\ell '}^{[i]}=\int \alpha _{\ell '} \delta (\alpha _{\ell '} - \hat {\alpha }_{\ell '}^{[i]})\mathrm {d}\alpha _{\ell '}\) and \(\hat {\tau }_{\ell '}^{[i]}\), \(\hat {\nu }_{\ell '}^{[i]}\) obtained similarly.
For notational convenience, \(\boldsymbol {\hat {x}}^{[i]}\) is used in the sequel to represent \(\boldsymbol {\hat {x}}^{[i]}=\left [\hat {x}^{[i]}\left (\,f\right); f\in \left [f_{1},\dots,f_{N}\right ]\right ]\).
The amplitude estimate \(\hat {\alpha }_{\ell }^{[i+1]}\) can be calculated by using (14).
When the convergence is achieved, e.g., the parameter estimates do not change as the iteration continues, the estimation procedure is suspended, and the parameter estimates obtained in the current iteration are considered to be the final result.
3.1 Discussion of the complexity of algorithm implementation and its influence on the SC device
The complexity of the proposed SAGE algorithm increases along with the number of paths to estimate, the total number of iterations, and the values of B _{ n }, ϱ which determine the number of data samples to be considered when calculating the objective function (16). Reducing the algorithm complexity can be performed in different ways. For example, instead of estimating a large number of multipath components, we may determine an appropriate model order in advance by applying the Akaike Information Criterion [19], and furthermore, the complexity involved in the parameter estimate updating procedure can be reduced by using advanced searching methods [20, 21].
When being implemented in reality, the proposed estimation method requires the SC’s outputs that are the results of filtering the original received sequence by using a LPF of bandwidth B _{ n }, n>1. It can be shown that the relationship B _{ n }≤f c′ is maintained if n≤γ−1 is selected. Therefore, enlarging the bandwidth of the SC’s output up to B _{ n } with n≤γ−1 would not introduce any additional requirement on increasing the sampling rate at the input of the SC. In such a case, the original complexity in the SC device is not influenced by B _{ n }, 1≤n≤γ−1. The studies in the sequel are conducted under the constraint 1≤n≤γ−1.
4 Simulation study
Simulation studies are conducted to evaluate the performance of the proposed algorithm under the influence of different B _{ n } settings and the fraction ϱ of SC’s output being considered. These two parameters determine how the SC’s output is selected and applied for channel estimation. With larger B _{ n }, more high-frequency components can be involved in parameter estimation, which may improve the estimation accuracy. In the conventional SC-based channel estimation, ϱ=1 is usually adopted. In the proposed algorithm, ϱ<1 can be selected, which allows reducing the observation time required in each snapshot, and consequently, measurements of a time-variant channel can be performed within channel coherence time. The impact of B _{ n } and ϱ on the performance of the proposed algorithm is of importance for understanding the effectiveness of the algorithm. Therefore, we select B _{ n } and ϱ as parameters in the simulation studies.
Parameter setting of the simulations
Values | |
---|---|
Parameters for SC configuration | |
Type of PN sequence | m-sequence |
Chip rate of PN sequence u(t) | 500 MHz |
Chip rate of PN sequence u ^{′}(t) | 494.62 MHz |
Sliding factor γ | 93 |
Normalized sliding factor γ/L | 3 |
Number L of chips in u(t) | 31 |
Over-sampling rate | 2 |
Repetition times of u(t) in Tx | 93 |
Repetition times of u ^{′}(t) in Rx | 92 |
Time duration T for complete IR | 5.766 μs |
Bandwidth B _{ n } of LPF | \(B_{n}=[-n\frac {f_{c}}{\gamma },n\frac {f_{c}}{\gamma }]\), n=1,2,… |
Fraction ϱ of the SC’s output | ϱ=1/m, m=1,2,… |
Parameters of synthetic paths | |
Delay | Unif. dist. within [20,70] ns |
Doppler frequency | Unif. dist. within [−1×10^{5},1 |
×10^{5}] Hz^{a} | |
Complex attenuation coefficient | Circularly symmetric Gaussian |
random variables with | |
magnitude within the range | |
of \([1\cdot 10^{-6}, \sqrt {2}\cdot 10^{-6}]\) | |
Noise’s properties in synthetic channels | |
Noise spectral height | max([|α _{ ℓ }|;ℓ=1,…,M])· |
10^{−SNR/20} | |
with SNR represented in dB and | |
max([|α _{ ℓ }|;ℓ=1,…,M]) being | |
the maximum of the path | |
magnitudes |
The performance of the proposed SAGE algorithm can be investigated from two perspectives, i.e., the RMSEEs in the case where paths are well-separated, and the resolution capability in separating multipath in the case where paths are closely spaced. Sections 4.1, 4.2, and 4.3 are dedicated to the investigation of the RMSEE behavior of the SAGE algorithm in the case with well-separated paths, and Section 4.4 to the resolution capability of the algorithm. When a channel consists of well-separated paths, the received signals of multipath components are mutually orthogonal, and the behavior of the SAGE algorithm can be represented by that of maximum-likelihood estimation (MLE) method in the single-path scenarios [16]. Therefore, the performance of MLE in single-path scenarios are studied in Sections 4.1, 4.2, and 4.3.
4.1 SC’s output with B _{ n } and ϱ as parameters
4.2 Objective functions of delay and Doppler frequency with B _{ n } and ϱ as parameters
4.2.1 η(τ;ν=ν ^{′}) versus B _{ n }
4.2.2 η(ν;τ=τ ^{′}) versus B _{ n }
4.2.3 η(τ;ν=ν ^{′}) versus ϱ
4.2.4 η(ν;τ=τ ^{′}) versus ϱ
4.2.5 η(τ,ν) versus B _{ n }
Notice that by using the SC’s output that allows generating one CIR, the Doppler frequency estimation resolution is very low due to the short observation duration T. By using the simulation settings in Table 2, the total observation span is calculated to be T=L γ/f _{ c }=31·93·2·10^{−9}=5.77 μs. Thus, the Doppler frequency estimation resolution is 1/(2T)=43 KHz. Since the differences of Doppler frequencies of paths are empirically much less than 43 KHz, it is important to jointly estimate the delay and Doppler frequency in order to resolve the paths in the delay domain. Furthermore, due to the low Doppler frequency estimation resolution, observations with high SNRs are always preferable in order to obtain less estimation errors. Our simulation results here show that the SNR should be kept beyond 10 and 40 dB in order to obtain RMSEE (ν) less than 10 Hz when the LPF bandwidth B _{ n } is set with n≥5 and n≤3, respectively.
4.3 RMSEEs of delay and Doppler frequency
4.3.1 RMSEE (τ) versus B _{ n }
4.3.2 RMSEE (τ) versus ϱ
4.3.3 RMSEE (ν) versus B _{ n }
4.3.4 RMSEE (ν) versus ϱ
4.3.5 RMSEE (τ) versus B _{ n } and ϱ
4.3.6 RMSEE (ν) versus B _{ n } and ϱ
4.4 The SAGE performance in two-path scenarios
4.4.1 RMSEE (τ) and RMSEE (ν) versus B _{ n }
The performance of the derived SAGE algorithm is evaluated in a two-path scenario, where the parameters of the paths are (τ _{1},ν _{1},α _{1})=(22 ns,−40 Hz,3) and (τ _{2},ν _{2},α _{2})=(28 ns,40 Hz,1), respectively. The noise components are added with N _{0}=max(|α _{1}|^{2},|α _{2}|^{2})10^{−ζ/10} where max(a,b) returns the maximum of the given arguments a and b, ζ denotes the SNR in dB. To limit the simulation times, the SAGE algorithm was executed to estimate the parameters of two paths within maximum 5 iterations. Since the true paths are set with different magnitudes, the path estimated by the SAGE algorithm with larger magnitude is considered to the estimate of the first path, and the other estimated path is the estimate of the second path, i.e., the weaker path.
4.4.2 RMSEE (τ) and RMSEE (ν) versus ϱ
4.4.3 Performance of the SAGE algorithm in a multipath propagation scenario
5 Conclusions
In this contribution, a parametric generic model was proposed to describe the output of the sliding correlator (SC) which is usually utilized to calculate the time-dilated wideband propagation channel impulse responses (CIRs). Based on the model proposed, a Space-Alternating Generalized Expectation-maximization (SAGE) algorithm was derived for extracting the delays, Doppler frequencies, and complex attenuations of multipath from the SC’s output that contains only one observation of time-dilated CIR. Simulation results have shown that the conventional constraint that only the low-frequency component in the SC’s output is applicable for channel estimation is unnecessary when the proposed estimation method is used. Furthermore, more high-frequency components considered, the higher the estimation accuracy can be achieved. Compared with the conventional approach which estimates the channel based on the time-dilated CIR, the proposed method is applicable not only for estimating the multipath’s Doppler frequencies but also returns more accurate estimates than the conventional method, e.g., the delay estimation errors are at least one order of magnitude less than those obtained by using the conventional method. Simulation results also demonstrated that the root mean squared estimation errors (RMSEEs) can be reduced by enlarging the bandwidth of a low-pass-filter (LPF) applied in the SC. When only a part of the SC’s output is available, the parameters can still be estimated provided the bandwidth of the LPF is no less than three times of the transmitted signal bandwidth divided by the sliding factor. In cases where only fractions of SC’s output is considered for estimation, the RMSEEs increase along with the data amount due to the improved output signal to noise ratio and the enhanced resolution capability particularly in the Doppler frequency domain. These results revealed the potential of applying the proposed high-resolution method in the SC-based parameter estimation for mm-wave wideband channel characterization.
6 Appendix 1: Derivation of (14)
By taking the complex conjugates of both sides in (23), (14) is finally obtained.
7 Appendix 2: Derivation of (16)
From (26) it is obvious that minimization of L(θ _{ ℓ }) with respect to θ _{ ℓ } is equivalent with maximization of an objective function η(τ _{ ℓ },ν _{ ℓ }) defined as shown in (16).
Declarations
Acknowledgements
This work was supported by Institute for Information & communications Technology Promotion (IITP) grant funded by the Korea government (MSIP) [B0101-15-222, Development of core technologies to improve spectral efficiency for mobile big-bang], the general project of national Natural Science Foundation of China (NSFC) (Grant No. 61471268), the national NSFC key program (Grant No. 61331009), and the international cooperation project “System design and demo-construction for cooperative networks of high-efficiency 4G wireless communications in urban hot-spot environments” granted by the Science and Technology Commission of Shanghai Municipality, China.
Authors’ Affiliations
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