3. Works related to sparse signal processing-based multipath mitigation

Sparse Signal Processing otherwise known as compressive Sensing (CS), is a classical Signal Processing technique efficiently acquiring and reconstructing a signal completely from reduced number of measurements, by exploiting its compressibility. CS has become a very interesting research area in recent years due to its theoretical and practical utility to capture a wide range of signals at a rate significantly lower than the Nyquist rate representing signal with lesser number of coefficients.

Optimal demodulation and decoding in wireless communication systems often requires accurate knowledge of the channel impulse response. Typically, this is accomplished by searching the channel with a known training sequence and linearly processing the channel with sparse impulse response. On the other hand, conventional linear channel estimation schemes, such as the least-squares method, fail to take advantage of on the anticipated sparsity of the channels. In contrast, it is observed that a CS channel estimate obtained as a solution significantly outperforms a least-squares based channel estimate in terms of the mean squared error (MSE) when it comes to learning sparse (or approximately sparse) channels.

This section highlights some of the most prominent state-of-the-art techniques, which have gained a lot of interest in the research community. The Critical review of literature indicates that exhaustive research has been done by several researchers to develop techniques to improve the performance of software GPS receivers under multipath environment. Researchers have concentrated the methods based on compressive sensing implemented in software based GPS receivers for accurate undisturbed reception and positioning.

algorithms such as Least Mean Squares (LMS) and various Recursive Least Squares (RLS) are considered to mitigate the error [12]. The estimated multipath error for a typical signal

In a simulated multipath environment, the reflection geometry is used in combination with a special GPS antenna arrangement to detect and track multipath. In the highly non stationary environment, Researchers also used Kalman Filter, particle filters and multiple differential GPS receivers to remove multipath errors in final positioning [13]. Code multipath is calibrated and estimated using spherical harmonics in static applications, similarly for kinematic applications, the multipath error mitigation is carried out by Mozaviet et al. [14] using wavelet transform. The estimation of frequency components of multipath error signal using spectral analysis and its effective mitigation using time varying digital filters are designed by Yedukondalu et al. [11]. The four types of filters, namely, Butterworth, Type I and II Chebyshev and Elliptic filters, are examined for mitigation of multipath and their performance are compared. It is observed that by applying digital filters of different cut-off frequencies over the spectrum of the multipath, one can significantly reduce the multipath errors. It was found that Butterworth

3. Works related to sparse signal processing-based multipath mitigation

Sparse Signal Processing otherwise known as compressive Sensing (CS), is a classical Signal Processing technique efficiently acquiring and reconstructing a signal completely from reduced number of measurements, by exploiting its compressibility. CS has become a very interesting research area in recent years due to its theoretical and practical utility to capture a wide range of signals at a rate significantly lower than the Nyquist rate representing signal

Optimal demodulation and decoding in wireless communication systems often requires accurate knowledge of the channel impulse response. Typically, this is accomplished by searching the channel with a known training sequence and linearly processing the channel with sparse impulse response. On the other hand, conventional linear channel estimation schemes, such as the least-squares method, fail to take advantage of on the anticipated sparsity of the channels. In contrast, it is observed that a CS channel estimate obtained as a solution significantly outperforms a least-squares based channel estimate in terms of the mean squared error (MSE)

This section highlights some of the most prominent state-of-the-art techniques, which have gained a lot of interest in the research community. The Critical review of literature indicates that exhaustive research has been done by several researchers to develop techniques to improve the performance of software GPS receivers under multipath environment. Researchers have

when it comes to learning sparse (or approximately sparse) channels.

is 0.8 and 2.1 m on L1 and L2 carriers, respectively.

2.4. Other filtering methods

10 Multifunctional Operation and Application of GPS

filter reduced the error most effectively.

with lesser number of coefficients.

Dragunas and Borre et al. [19] proposed the sparse deconvolution based Projection onto Convex Sets (POCS) method which is used to mitigate the multipath in indoor environments. The author compared the several multipath mitigation techniques suitable for the indoor environments. By using the proposed method the author chooses one of the secondary paths as LOS signal. In this method, the author achieves better resolution than the conventional methods. An extension to this work, Dragunas et al. [20] presented a modified Projection onto Convex Sets (POCS) that optimizes the Coarse/Acquisition codes employed in Global Positioning Systems. The author deals with the problem of joint LOS code delay and carrier phase estimation of GPS signals in a multipath environment. The modified POCS algorithm acts as the most resistant in closely-spaced multipath static channels both when LOS code delay and carrier phase estimation are concerned. Another sparse based modified iterative Projection onto convex sets (POCS) method proposed by Negin Sokhandan and Ali Broudman [21] is used to reduce the multipath error in harsh environment. The algorithm estimates the channel impulse response (CIR) and removes the spurious noise peaks at each iteration. This method is carried out to estimate the LOS time of arrival from the position of its first non-zero element that passes a certain threshold. The modified POCS algorithm correctly estimates the code delay and carrier phase for GPS signals with few iterations. Hence, faster performance has been achieved when compared to conventional POCS.

Kumar and Lau et al. [22] implemented the deconvolution approach for the code phase and carrier phase estimation. The deconvolution approach shows that it is very different from POCS approach where each path can be estimated. The deconvolution approach can accurately estimate the Line of Sight (LOS) signal. Initially the channel impulse response is computed and by getting the deconvolution filter coefficients, multipath can be removed by convolving the measurements with deconvolution filter coefficients and the code and carrier phase can be estimated and finally the LOS is found.

The novel sparse reconstruction method for mitigating the multipath induced code delay estimation has been implemented by Fei and Liao et al. [23] in GPS receivers. The author exploited to enhance the direct signal without affecting the accuracy of the GPS code delay estimates. The coherent accumulation of received GPS signals and by transforming it into frequency domain and parameters of multipath signals are estimated by sparse reconstruction algorithm. The author estimates the code delay without affecting the accuracy of the GPS by sparse reconstruction method. Tian and Li et al. [24] proposed a novel method based on nonnegative matrix factorization (NMF) spectral unmixing for land seismic additive random noise attenuation. In this method, the noisy seismic signal is first decomposed into a collection of intrinsic mode functions (IMFs) instead of being directly processed. Then, a sparse NMF is used to unmix the STFT spectrum of each IMF. By separating the sub-spectrums by the inverse STFT, the sub-signals can be easily acquired. Finally, the desired signal is reconstructed from the sub-signals by K-means clustering algorithm. Bostan and Kamilov et al. [25] proposed a novel statistically-based discretization paradigm and derive a class of maximum a posterior (MAP) estimators for solving ill-conditioned linear inverse problems. It proposes the theory of sparse stochastic processes, which specifies the continuous –domain signals as solutions of linear stochastic differential equations. It provides the algorithm that handles the nonconvex problems and by applying it to the reconstruction algorithm and finally compares the performance of estimators, associated with the models of increasing sparsity.

Broumandan and Lin et al. [15] established a way to enhance the performance of GNSS time arrival estimation techniques in multipath environments by determining the multipath channel estimation using equivalent discrete-time linear time-invariant system method which is modeled as a Moving Average system. It modeled the multipath channel as a sparse channel by describing the number of parameters of the channel is less than the number of unknowns in the Moving Average model. The author compares the performance of the sparse estimation with the Cramer-Rao Lower Bound (CRLB) of the parameter estimation problem and the least square estimate. It provides the better sparse signal recovery method to estimate the channel impulse response.

#### 3.1. Sparse signal deconvolution

Sparse de-convolution finds variety of application in accurate estimation of multipath channels with sparse impulse response of a channel is calculated by degradation version of convolution matrix. After down conversion to baseband, the signal from all the satellites can be represented in complex baseband representation as.

$$\mathbf{v}(t) = \sum\_{s=0}^{S-1} \alpha\_s \cdot \mathbf{c}\_s \left(\mathbf{t} - \tau\right) \mathbf{e}^{j2\pi \left(\mathbf{f}\_s \ast \mathbf{f}\_d\right) \mathbf{t}} \tag{3}$$

where α(s) is the channel attenuation from the sth satellite to the receiver, τ is the time delay or code phase of the C/A code and fd is the Doppler frequency for the sth satellite.

We assume that the observed GPS signal y from a multipath channel can be written as

$$y = H\mathbf{x} + n\tag{4}$$

To find a signal x, so that Hx is very similar to y, i.e., here it is needed to find a signal x which is consistent with the observed data y. For D (y, Hx), the mean square error can be calculated as

Review on Sparse-Based Multipath Estimation and Mitigation: Intense Solution to Counteract the Effects…

D yð Þ¼ ; Hx k k <sup>y</sup> � Hx <sup>2</sup>

The squared error between y and Hx is minimized by finding the norm difference of D (y, Hx) that will give a signal x, which is as consistent with y as possible, according to the square error

convolution filter be {1,-1, 1,-1….M} and signal be of length M. Convolution sum will have

100 ⋯ 0 �11 0 ⋯ 0 0 �1 1 ⋯ ⋮ ⋮ 0 �11 0 0 ⋮ ⋯ �1 1 0 0 … 0 �1

which is definitely invertible. Even if H were invertible, it may be very ill-conditioned, in which case, this solution amplifies the noise, sometimes to such an extent that the solution is useless. The role of the regularization term R(x) is exactly to address this problem. The regularizer R(x) should be chosen so as to penalize the undesirable/unwanted behavior in x.

By assuming that the GPS signal of interest after acquisition x, is known to be sparse. i.e., x has relatively few non-zero values, i.e., x consists of a few impulses and is otherwise zero. In this case, the R(x) may be defined to be the number of non-zero values of x. R(x) is not a convex function of x, which is not differentiable then the objective function J(x) is very difficult to minimize and therefore J(x) will have many local minima. To minimize J(x), it is better to choose J(x) to be a convex function of x that measures sparsity, but which is also convex. For this reason, when x is known to be sparse, the regularization function R(x) is often chosen to be the L1-norm. Hence, the approach is to estimate x from y by minimizing the objective function,

> J xð Þ¼ <sup>y</sup> � Hx � � � � 2

The requirement for development of fast algorithm is to minimize the equation and its related functions. This is carried out by another significant algorithm called iterated soft-thresholding algorithm (ISTA), also referred as Thresholded Landweber (TL) algorithm. ISTA is a combination of the Landweber algorithm and soft-thresholding. To minimize J(x), consider first the

criterion. To minimize D (y, Hx) by setting x = H�<sup>1</sup>

3.1.1. L1-norm regularized linear inverse problem

3.1.2. Soft-thresholding algorithm (ISTA)

minimization of the simpler objective function

length equal to N + M-1. So H in this case will have N � M-1dimension

H ¼

<sup>2</sup> (6)

http://dx.doi.org/10.5772/intechopen.76521

13

y; however, H may not be invertible. Let

<sup>2</sup> þ λk kx <sup>1</sup> (7)

where x is the signal of interest which is to be estimated, n is additive noise, and H is a matrix representing the degradation process. The estimation of actual GPS signal x from the faded version y can be treated as a linear inverse problem. An appropriate objective function, J(x) has been formulated to solve linear inverse problems and to find the signal x, by minimizing J(x).

Generally, the chosen objective function is the sum of two terms:

$$J(\mathbf{x}) = D(y, H\_x) + \lambda R(\mathbf{x}) \tag{5}$$

where.

D(y, Hx) measures the discrepancy between y and x.

R(x)—Regularization term (or penalty function).

λ—Regularization parameter (positive value).

To find a signal x, so that Hx is very similar to y, i.e., here it is needed to find a signal x which is consistent with the observed data y. For D (y, Hx), the mean square error can be calculated as

$$D(y, H\_x) = \left\| y - H\_x \right\|^2\_2 \tag{6}$$

The squared error between y and Hx is minimized by finding the norm difference of D (y, Hx) that will give a signal x, which is as consistent with y as possible, according to the square error criterion. To minimize D (y, Hx) by setting x = H�<sup>1</sup> y; however, H may not be invertible. Let convolution filter be {1,-1, 1,-1….M} and signal be of length M. Convolution sum will have length equal to N + M-1. So H in this case will have N � M-1dimension

$$\mathbf{H} = \begin{bmatrix} 1 & 0 & 0 & \cdots & 0 \\ -1 & 1 & 0 & \cdots & 0 \\ 0 & -1 & 1 & \cdots & \vdots \\ \vdots & 0 & -1 & 1 & 0 \\ 0 & \vdots & \cdots & -1 & 1 \\ 0 & 0 & \cdots & 0 & -1 \end{bmatrix}$$

which is definitely invertible. Even if H were invertible, it may be very ill-conditioned, in which case, this solution amplifies the noise, sometimes to such an extent that the solution is useless. The role of the regularization term R(x) is exactly to address this problem. The regularizer R(x) should be chosen so as to penalize the undesirable/unwanted behavior in x.

#### 3.1.1. L1-norm regularized linear inverse problem

sparse stochastic processes, which specifies the continuous –domain signals as solutions of linear stochastic differential equations. It provides the algorithm that handles the nonconvex problems and by applying it to the reconstruction algorithm and finally compares the perfor-

Broumandan and Lin et al. [15] established a way to enhance the performance of GNSS time arrival estimation techniques in multipath environments by determining the multipath channel estimation using equivalent discrete-time linear time-invariant system method which is modeled as a Moving Average system. It modeled the multipath channel as a sparse channel by describing the number of parameters of the channel is less than the number of unknowns in the Moving Average model. The author compares the performance of the sparse estimation with the Cramer-Rao Lower Bound (CRLB) of the parameter estimation problem and the least square estimate. It provides the better sparse signal recovery method to estimate the channel

Sparse de-convolution finds variety of application in accurate estimation of multipath channels with sparse impulse response of a channel is calculated by degradation version of convolution matrix. After down conversion to baseband, the signal from all the satellites can be represented

where α(s) is the channel attenuation from the sth satellite to the receiver, τ is the time delay or

where x is the signal of interest which is to be estimated, n is additive noise, and H is a matrix representing the degradation process. The estimation of actual GPS signal x from the faded version y can be treated as a linear inverse problem. An appropriate objective function, J(x) has been formulated to solve linear inverse problems and to find the signal x, by minimizing J(x).

y ¼ Hx þ n (4)

J xð Þ¼ D yð Þþ ; Hx λR xð Þ (5)

code phase of the C/A code and fd is the Doppler frequency for the sth satellite.

Generally, the chosen objective function is the sum of two terms:

D(y, Hx) measures the discrepancy between y and x.

R(x)—Regularization term (or penalty function).

λ—Regularization parameter (positive value).

We assume that the observed GPS signal y from a multipath channel can be written as

ð3Þ

mance of estimators, associated with the models of increasing sparsity.

impulse response.

where.

3.1. Sparse signal deconvolution

12 Multifunctional Operation and Application of GPS

in complex baseband representation as.

By assuming that the GPS signal of interest after acquisition x, is known to be sparse. i.e., x has relatively few non-zero values, i.e., x consists of a few impulses and is otherwise zero. In this case, the R(x) may be defined to be the number of non-zero values of x. R(x) is not a convex function of x, which is not differentiable then the objective function J(x) is very difficult to minimize and therefore J(x) will have many local minima. To minimize J(x), it is better to choose J(x) to be a convex function of x that measures sparsity, but which is also convex. For this reason, when x is known to be sparse, the regularization function R(x) is often chosen to be the L1-norm. Hence, the approach is to estimate x from y by minimizing the objective function,

$$\mathbf{J}(\mathbf{x}) = \left\| \mathbf{y} - \mathbf{H}\mathbf{x} \right\|\_2^2 + \lambda \left\| \mathbf{x} \right\|\_1 \tag{7}$$

#### 3.1.2. Soft-thresholding algorithm (ISTA)

The requirement for development of fast algorithm is to minimize the equation and its related functions. This is carried out by another significant algorithm called iterated soft-thresholding algorithm (ISTA), also referred as Thresholded Landweber (TL) algorithm. ISTA is a combination of the Landweber algorithm and soft-thresholding. To minimize J(x), consider first the minimization of the simpler objective function

$$J(\mathbf{x}) = \|\mathbf{y} - H\mathbf{x}\|\_2^2 = (\mathbf{y} - H\mathbf{x})^T(\mathbf{y} - H\mathbf{x})\tag{8}$$

$$J(\mathbf{x}) = y^T y - 2y^T H \mathbf{x} + \mathbf{x}^T H^T H \mathbf{x} \tag{9}$$

Note that the quadratic term in Eq. (12) is simply x<sup>T</sup>

<sup>G</sup>kð Þ¼� <sup>x</sup> <sup>2</sup>HTy � <sup>2</sup> <sup>α</sup><sup>I</sup> � <sup>H</sup>TH xk <sup>þ</sup> <sup>2</sup>αx, Setting, <sup>∂</sup>

1 α

Hence, by using MM procedure to obtain x value at each iteration is given by Landweber

1 α

In this simulation, four multipath components are considered with time varying amplitude and the phase. Initially the GPS signal needs to be framed in the form of sparse signal. This can be done in the acquisition stage only, the sparse representation of this signal easily decomposed in the form of basis function and the coefficient term. Then one can easily reconstruct the sparse coefficient of minimum number of non-zero coefficient by random by l1 minimization. The code and carrier tracking loop of the software GPS receiver has to be synchronized if and only if the lock is achieved. Due to multipath error, the code loop error may be varied more than 1 chip delay and the carrier loop (Costas) is also intercoupled with this, so error may be introduced in the carrier tracking loop also hence, both the tracking errors

The objective function (J) and the 2000 samples of the recovered GPS signal after ISTA algorithm is plotted in Figures 7 and 8 respectively. The recovered signal is further given to the acquisition stage to find the visible satellites (SVN's) and allocate those SVN to initiate the tracking stage. The code and carrier tracking error is observed after recovering the GPS signal using MM method. The significant improvement in carrier tracking is achieved within 50 msec

x ¼ xk þ

xkþ<sup>1</sup> ¼ xk þ

should be carefully minimized to certain extent to achieve the lock.

minimize Gk(x) more easily

update equation as

Figure 7. Objective function.

∂ ∂x

4. Results and discussion

x instead of x<sup>T</sup>

Review on Sparse-Based Multipath Estimation and Mitigation: Intense Solution to Counteract the Effects…

HT

Gkð Þ¼ x 0

http://dx.doi.org/10.5772/intechopen.76521

∂x

<sup>H</sup><sup>T</sup>ð Þ <sup>y</sup> � Hxk (14)

<sup>H</sup><sup>T</sup>ð Þ <sup>y</sup> � Hxk (15)

Hx. Therefore, we can

15

Because J(x) in Eq. (8) is differentiable and convex, thus one can obtain its minimizer by setting the derivative with respect to x to zero. The derivative of J(x) is given by

$$\frac{\partial}{\partial \mathbf{x}} \mathbf{J}(\mathbf{x}) = -2\mathbf{H}^{\mathrm{T}} + 2\mathbf{H}^{\mathrm{T}} \mathbf{H} \mathbf{x}$$

Setting the derivative to zero gives a system of linear equations,

$$\frac{\partial}{\partial \mathbf{x}} \mathbf{J}(\mathbf{x}) = 0 \text{ implies } (H^T H)\mathbf{x} = H^T \mathbf{y}.$$

So the minimizer of J(x) in Eq. (9) is given by

$$\mathbf{x} = \left(\boldsymbol{H}^T \boldsymbol{H}\right)^{-1} \boldsymbol{H}^T \boldsymbol{y} \tag{10}$$

#### 3.1.3. Majorization-minimization (MM) approach

However, it is not able to solve these equations easily. Since GPS data is a very long, then H will be very large matrix and solving the system of equations may require huge memory and computation time. Moreover, the matrix H<sup>T</sup> H is not invertible, or ill-conditioned. By using the Majorization-minimization (MM) approach to minimize J(x) in Eq. (10), solving a system of linear equations can be avoided. At each iteration k of the MM approach, a function Gk(x) that coincides with J(x) at xk has been found. A majorizer Gk(x) has introduced that can be minimized more easily without solving a system of Eqs.

A function Gk(x) that majorizes J(x) by adding a non-negative function to J(x),

$$\mathbf{G}\_k(\mathbf{x}) = f(\mathbf{x}) + \text{Non}-\text{negative function of } \mathbf{x} \tag{11}$$

When Gk(x) coincides with J(x) at x = xk, the non-negative function added to J(x) should be equal to zero at xk then Gk(x) to be

$$\mathbf{G}\_{k}(\mathbf{x}) = f(\mathbf{x}) + (\mathbf{x} - \mathbf{x})^{T} \left( aI - HH^{T} \right) (\mathbf{x} - \mathbf{x}\_{k}) \tag{12}$$

The function which is added to J(x) is clearly zero at xk so that Gk(x) equals to J(xk) as required. To ensure the function added to J(x) is non-negative, for all x, the scalar parameter α must be chosen to be equal to or greater than the maximum eigenvalue of H<sup>T</sup> H, i.e., α≥ max eig (H<sup>T</sup> H). Then the matrix αI- H<sup>T</sup> H is a positive semi-definite matrix, meaning that v<sup>T</sup> (αI- HT H) v ≥ 0. Now, using MM procedure, to obtain xk + 1, function Gk(x) is minimized. Expanding Gk(x) in (12) gives

$$\mathbf{G}\_{k}(\mathbf{x}) = \mathbf{y}^{T}\mathbf{y} - 2\mathbf{y}^{T}H\mathbf{x} + \mathbf{x}^{T}H\mathbf{x} + \left(\mathbf{x} - \mathbf{x}\_{k}\right)^{T}\left(aI - H^{T}H\right)\left(\mathbf{x} - \mathbf{x}\_{k}\right) \tag{13}$$

Note that the quadratic term in Eq. (12) is simply x<sup>T</sup> x instead of x<sup>T</sup> HT Hx. Therefore, we can minimize Gk(x) more easily

$$\frac{\partial}{\partial \mathbf{x}} \mathbf{G}\_k(\mathbf{x}) = -2H^T y - 2\{\alpha I - H^T H\} \mathbf{x}\_k + 2\alpha \mathbf{x}, \text{Setting } \frac{\partial}{\partial \mathbf{x}} \mathbf{G}\_k(\mathbf{x}) = \mathbf{0}$$

$$\mathbf{x} = \mathbf{x}\_k + \frac{1}{\alpha} H^T (y - H \mathbf{x}\_k) \tag{14}$$

Hence, by using MM procedure to obtain x value at each iteration is given by Landweber update equation as

$$\mathbf{x}\_{k+1} = \mathbf{x}\_k + \frac{1}{\alpha} H^T (\mathbf{y} - H\mathbf{x}\_k) \tag{15}$$

#### 4. Results and discussion

J xð Þ¼ k k <sup>y</sup> � Hx <sup>2</sup>

the derivative with respect to x to zero. The derivative of J(x) is given by

∂ ∂x

Setting the derivative to zero gives a system of linear equations,

∂ ∂x

So the minimizer of J(x) in Eq. (9) is given by

14 Multifunctional Operation and Application of GPS

3.1.3. Majorization-minimization (MM) approach

mized more easily without solving a system of Eqs.

equal to zero at xk then Gk(x) to be

Then the matrix αI- H<sup>T</sup>

(12) gives

Because J(x) in Eq. (8) is differentiable and convex, thus one can obtain its minimizer by setting

J xð Þ¼ -2H<sup>T</sup> <sup>þ</sup> 2HTHx

J xð Þ¼ 0 implies <sup>H</sup>TH <sup>x</sup> <sup>¼</sup> <sup>H</sup>Ty:

<sup>x</sup> <sup>¼</sup> <sup>H</sup>TH �<sup>1</sup>

However, it is not able to solve these equations easily. Since GPS data is a very long, then H will be very large matrix and solving the system of equations may require huge memory and computation time. Moreover, the matrix H<sup>T</sup> H is not invertible, or ill-conditioned. By using the Majorization-minimization (MM) approach to minimize J(x) in Eq. (10), solving a system of linear equations can be avoided. At each iteration k of the MM approach, a function Gk(x) that coincides with J(x) at xk has been found. A majorizer Gk(x) has introduced that can be mini-

When Gk(x) coincides with J(x) at x = xk, the non-negative function added to J(x) should be

T

The function which is added to J(x) is clearly zero at xk so that Gk(x) equals to J(xk) as required. To ensure the function added to J(x) is non-negative, for all x, the scalar parameter α must be

Now, using MM procedure, to obtain xk + 1, function Gk(x) is minimized. Expanding Gk(x) in

H is a positive semi-definite matrix, meaning that v<sup>T</sup> (αI- HT

T

Gkð Þ¼ x J xð Þþ Non � negative function of x (11)

A function Gk(x) that majorizes J(x) by adding a non-negative function to J(x),

Gkð Þ¼ x J xð Þþ ð Þ x � x

chosen to be equal to or greater than the maximum eigenvalue of H<sup>T</sup>

Gkð Þ¼ <sup>x</sup> yTy � <sup>2</sup>yTHx <sup>þ</sup> xTHx <sup>þ</sup> ð Þ <sup>x</sup> � xk

<sup>2</sup> <sup>¼</sup> ð Þ <sup>y</sup> � Hx <sup>T</sup>ð Þ <sup>y</sup> � Hx (8)

HTy (10)

<sup>α</sup><sup>I</sup> � HH<sup>T</sup> ð Þ <sup>x</sup> � xk (12)

H, i.e., α≥ max eig (H<sup>T</sup>

<sup>α</sup><sup>I</sup> � <sup>H</sup>TH ð Þ <sup>x</sup> � xk (13)

H).

H) v ≥ 0.

J xð Þ¼ <sup>y</sup>Ty � <sup>2</sup>yTHx <sup>þ</sup> xTHTHx (9)

In this simulation, four multipath components are considered with time varying amplitude and the phase. Initially the GPS signal needs to be framed in the form of sparse signal. This can be done in the acquisition stage only, the sparse representation of this signal easily decomposed in the form of basis function and the coefficient term. Then one can easily reconstruct the sparse coefficient of minimum number of non-zero coefficient by random by l1 minimization. The code and carrier tracking loop of the software GPS receiver has to be synchronized if and only if the lock is achieved. Due to multipath error, the code loop error may be varied more than 1 chip delay and the carrier loop (Costas) is also intercoupled with this, so error may be introduced in the carrier tracking loop also hence, both the tracking errors should be carefully minimized to certain extent to achieve the lock.

The objective function (J) and the 2000 samples of the recovered GPS signal after ISTA algorithm is plotted in Figures 7 and 8 respectively. The recovered signal is further given to the acquisition stage to find the visible satellites (SVN's) and allocate those SVN to initiate the tracking stage. The code and carrier tracking error is observed after recovering the GPS signal using MM method. The significant improvement in carrier tracking is achieved within 50 msec

Figure 7. Objective function.

period where as in the case of code tracking, error is settled down quickly within 100 msec of GPS data as shown in Figures 9 and 10. The lock has been achieved with 0.5 chips spacing of early, late and prompt code replicas, hence the navigation data can easily be demodulated and

Review on Sparse-Based Multipath Estimation and Mitigation: Intense Solution to Counteract the Effects…

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17

In this Chapter, various techniques have been discussed to nullify the effect of the multipath, we have provided an in depth review of existing multipath mitigation techniques. These techniques were classified in categories according to the involved process before and after correlation with the C/A code. Compressive sensing is a promoting tool for the next generation communication systems. However, it still faces a number of challenges in the real time implementation. In multipath applications, compressive sensing exploits the GPS signal need to be converted to a sparse equivalent structure then the channel impulse response of the filter is determined from the convolution matrix. For reconstruction, the challenge resides in how to separate the LOS signal from composite signals in multi-channel environments, where the channel powers and behaviors evolve over time. A comparison of several compressive techniques was given and discussed. The sparse recovery of the signal is obtained from

Ganapathy Arul Elango\*, B. Senthil Kumar, Ch.V.M.S.N. Pavan Kumar and C. Venkatramanan

Department of Electronics and Communication Engineering, Sree Vidyanikethan Engineering

[1] Michel SB, Van Dierendonck AJ. GPS receiver architecture and measurements. Proceed-

[2] Borre K, Akos DM, Bertelsen N, Rinder P, Jensen SH. A software defined GPS and Galileo receiver. New York: Birkhäuser Bostonin, Springer Science & Business Media; 2007. ISBN-

[3] Wildemeersch M. Fortuny-Guasch J. Radio Frequency Interference Impact Assessment on Global Navigation Satellite Systems. Security Technology Assessment Unit, EC Joint

the pseudorange is calculated for each satellite.

unconstrained optimization algorithms.

ings of the IEEE. 1999;87:48-64

Research Centre. In: EUR 24242 EN. 2010

10 978–0–8176-4390-4

\*Address all correspondence to: arulelango2012@gmail.com

5. Conclusion

Author details

College, Tirupathi, India

References

Figure 8. Recovered GPS signal through ISTA algorithm.

Figure 9. Carrier loop tracking error for SVN-12.

Figure 10. Code loop tracking error for SVN-12.

period where as in the case of code tracking, error is settled down quickly within 100 msec of GPS data as shown in Figures 9 and 10. The lock has been achieved with 0.5 chips spacing of early, late and prompt code replicas, hence the navigation data can easily be demodulated and the pseudorange is calculated for each satellite.
