A Review of Indoor Localization Techniques and Wireless Technologies

The acoustic method sounds to be the first localization approach since the Stone Age; nowadays, ultrasound is utilized to incorporate existing mobile devices for localization [33]. Ultrasonic localization is eminent within short ranges due to its low-power penetrating losses through indoor walls, cheap components, and its compatibility with handled devices. On the other hand, [34, 35], localization errors are introduced due to many factors, including multiple reflections from surfaces and synchronization problems between communicating nodes. Ultrasonic localization systems suffer from complex signal processing algorithms [20]. Compared to Ultra-Wideband (UWB) systems, ultrasonic localization systems can tolerate better with low time synchronization [36].

In [37], the authors proposed a localization method based on flight time using a single transceiver; localization median error was 15 cm. The location of ultrasonic nodes should be determined before localization To have accurate results [38].

The technology utilized the build-in microphones in smartphones. The microphones capture the source's sound, which will be used to identify the location with respect to a reference [39, 40]. All smartphones have microphones, while malls, airports, and hospitals have their speakers and microphones, making this technique cost-effective [40, 41]. The transmitted modulated acoustic signal contains information like the time stamp; this will be used to estimate TOF and hence find the target's location using multi-lateration. To avoid annoying people, transmitted power should be low; this requires sophisticated signal processing for detectors to detect the low power transmitted signal.

Infrared (IR) systems are applied in Line of Sight (LOS) scenarios where handsets already have sensors like photodiodes. IR is featured for its simplicity, low weight, compact size, and immunity to interference (unlike RF systems) [42, 43].

However, it suffers from fluorescent light and sunlight interference; also, the hardware of these systems are expensive and costly when performing maintenance [42]. IR systems are composed of IR emitting devices (e.g., LED) and IR sensors (e.g., photodiode). The target wears the IR emitting device; this device emits a signal with a unique identifier; the target's location is identified once the sensors detect the IR signal. The Active Badge is an example of commercial IR based technologies. In [44], pyroelectric infrared (PIR) sensors are used to detect heat radiation changes emitted from people and animals; accuracy is found to be in terms of centimeters; however, they are vulnerable to environmental changes [45].

Visible Light Communications (VLC) can be used in RF sensitive places like hospitals [46]; it has been shown that VLC provides a higher accuracy compared to Wi-Fi systems [47].

With the merge of light-emitting diodes (LED), VLC is widely employed in localization; LED has many advantages, including long-life expectancy, immunity to humidity, low power consumption, and low cost [46]. Also, LED can modulate lightwave signals at high speed [20].

General, light bulbs send coded transmitted light; at the receiver, the sensor compares the detected light with the existing database, which links coded light to the position. And the sensor location inferred.

Localization using VCL can be performed using photodiode based systems which capture the light intensity and image sensor-based systems which can capture light pulses [48]. While photodiode based systems are inexpensive, mobile is equipped with a camera but not photodiode; this makes image sensor-based systems more preferable.

However, both LED and sensors should be in light of sight for accurate localization. AOA is used in visible light systems to ensure precise localization [49]. Light Detection and Ranging localization (LIDAR) provides surrounding obstacles' contour information; with inertial sensors, these systems can provide accurate localization [50].

Frequency modulation (FM) broadcasting can be used for localization for outdoor environments and indoor environments recently [69]. FM operates at 88–108 MHz, which is lower than cellular networks (operates at 0.9 and 1.8 GHz) and Wi-Fi (2.45 GHz). Therefore, FM radio signals are less affected by weather conditions, less sensitive to terrain conditions, and can penetrate walls more comfortable [70]. As it has a larger wavelength (3 m), the interaction with indoor objects and furniture is different from that at Wi-Fi frequencies. Operating at FM frequencies does not interfere with other RF components that operate at 2.4 GHz [71]. Also, FM receivers consume less power.

RSS is used for indoor localization by using the fingerprinting technique; in [72], fingerprinting performance was compared using different machine learning methods, k nearest neighbor (kNN), support vector machine (SVM) classifiers, and Gaussian processes (GP) regression. It was found that the kNN approach achieved the best results. Also, for better accuracy, it was recommended to choose stations with stronger signals. FM performance was compared to Wi-Fi. It was found that Wi-Fi has the best localization performance over large spaces like floors, while FM tends to have the best performance over small places, like rooms.

Cellular networks operate on many frequency bands, including the 0.9 GHz, 1.8, and 2.8 GHz bands. These networks provide better coverage compared to Wi-Fi networks and require no additional infrastructure. Initially, proximity was used for localization, where the mobile location is identified within the cell coverage range; this approach provides extremely low results [73]. Localization is performed usually using RSS, in which the fingerprinting technique is the adopted one. Other researchers consider using TOA, where trilateration is the adopted localization technique. In the case of using RSS fingerprinting, cell stations are regarded as the APs; the accuracy was found to be in the range of 2.5–5.4 m [74].

In [75], localization was performed using RSS fingerprinting using the Global System for Mobile Communications (GSM) received signals. Fingerprints were collected from 29 GSM channels and 6 cells in their work. Localization error was less than 5 m, similar outcomes achieved by [76]. Universal Mobile Telecommunications Service (UMTS) cell for indoor coverage was used for localization by taking fingerprints. In [68], measurements were taken in an office environment. It was found that localization using UMTS small cells is comparable to WLAN in terms of accuracy. Also, Long-Term Evolution (LTE) was utilized for indoor environment, in [77] authors conducted localization using TOA; in their work, the error was less than 8 m for 50% of cases. LTE could be fused with an inertial measurement unit as in [78], where root mean square error (RMSE) was around 3.52 m.

In [79], synthetic aperture navigation (SAN) framework was used to mitigate the effect of multipath signals; a synthetic aperture antenna will capture signals at different time instants. This would be similar to catching the signals from an array. SAN will use the Estimation of Signal Parameters via Rotational Invariance Technique (ESPRIT) algorithm to find DOA to determine the position; in their work, the RMSE of localization was 3.9 m for LTE-SAN compared to 7.19 m for standalone LTE.

In [80], a study on localization using LTE-only and LTE-WLAN fingerprinting was conducted; it was found that LTE-only fingerprinting gives poor results while using LTE-WLAN fingerprinting enhanced the performance by order of 3.5x. Cellular systems can be used to support other existing RF localization systems like RFID [81], Wi-Fi [82‐84],

Wi-Fi is the name of popular wireless networking technology. Wi-Fi operates within the RF bands of 2.5 GHz for IEEE 802.11b, IEEE 802.11 g, and IEEE802.11n, and in 5 GHz for IEEE 802.11a.

A most massive indoor environment such as a university or an office building has already distributed WiFi hotspots that provide whole-building coverage as a network access point. Devices that employ Wi-Fi technology include personal computers, video-game consoles, smartphones, digital cameras, tablet computers, and digital audio players [85, 86].

That the infrastructure cost and user device cost for Wi-Fi can be very low, and Wi-Fi has covered a reception range of about 100 m and has now increased to about 1 km (km). Additionally, Wi-Fi localization based on fingerprinting RSS (Received Signal Strength) [87, 88]. Wi-Fi could be used with other RF localization techniques, like RFID [89]. Wi-Fi covers wider areas than Bluetooth and provides higher throughput; this makes the utilization of Wi-Fi more practical [90]. Examples of commercial Wi-Fi-based positioning systems include RADAR, HORUS, COMPASS, HERECAST, and PlaceLab [88, 91].

ZigBee is a specification based on IEEE 802.15.4 standard. It uses the 868 MHz band in Europe, 915 MHz bands in the USA and Australia, and 2.4 GHz in other regions. ZigBee is used for long-distance transmission between devices in a wireless mesh network. It has low cost, low data transfer rate, short latency time, comparing to WiFi standards. The technology uses the RSS method to estimate the distance between two or more ZigBee sensor devices [92, 93]. Access point (AP) scanning via the WiFi interface leads to high power consumption. To reduce this effect, authors in [94] introduced an energy-efficient indoor localization using ZigBee termed as ZIL, in which the ZigBee interface is used to collect Wi-Fi signals. In [95], a ZigBee localization algorithm based on proximity learning was introduced; the proposed method differs from other traditional triangulation based techniques as it reduces computational time while maintaining accurate positioning.

Bluetooth (IEEE 802.15.1) is intended to enable short-range wireless communication between devices. Bluetooth communicates utilized radio waves with frequencies between 2.402 GHz and 2.480 GHz as Wi-Fi does. If features with cost-effectiveness, low transmission power, battery life, secure and efficient communications, and accessible solutions [96, 97]. The new Bluetooth version termed Bluetooth Low Energy (BLE) can cover a range of 70–100 m and provide 24 Mbps with higher power efficiency [98]. Hence, Bluetooth is not suitable for localization for the large area [66]. In [97], neural networks (NN) are trained using the received signal strength values and their corresponding coordinates in the training phase; once they got trained NN, they can be used to detect user location based on the online RSS measurements. Recently, BLE based localization is utilized in smartphones as iBeacons (Apple) and Eddystone (Google), the smartphone can be used for localization within airports, train stations, big markets, malls and restaurants, where the area map is sent to the smartphone and then localization is performed using the BLE [98].

According to the U.S. Federal Communications Commission (FCC), a UWB signal has absolute bandwidth more than 500 MHz and carrier frequency larger than 2.5 GHz [99]. Using low power consumption in UWB achieves a large bandwidth, high-speed communication, high time resolution, high data rate, and short-wavelength, making UWB stronger against multipath interference and fading. Another useful property of UWB is that it is permitted to occupy low carrier frequencies, where signals can more easily pass through obstacles; UWB is also immune to interference since it has a significantly different spectrum [100]. All those characteristics guarantee UWB a good candidate for indoor wireless positioning. TOA and time difference of arrival (TDOA) have higher accuracy than other localization algorithms because of the high time resolution of the UWB signals, where the multipath effect is minimized. UWB can minimize error to centimetres [68, 101]. In [102], the authors proposed a hybrid localization using Wi-Fi and UWB; this is accomplished by adding UWB beacons to existing Wi-Fi infrastructure. The hybrid method combines the availability of Wi-Fi infrastructure, which will reduce cost and the accuracy of UWB when deploying their algorithm; localization error was limited to 20 cm. Typical UWB systems can localize a limited number of tags; in [103], a UWB localization system called SnapLoc can localize an unlimited number of tags.

In [104], UWB localization system performance was analyzed for LOS and NLOS scenarios. The position was estimated using linearized least square estimation (LLSE), fingerprint estimation (FPE), and weighted centroid estimation (WCE); in their work, it was found that FPE shows the best performance, while LLSE shows the worst.

Radio Frequency Identification (RFID) systems operate on RFID tags' backscattering communication and RFID readers and middleware for processing the signal generated between the tags and the readers [105].

RFID tags are either active, passive, or semi-active. Active tags are equipped with an inbuilt battery embedded in their circuitry. Active RFIDs operate in the ultra-high frequency (UHF) and super high frequency (SHF) range with a detection range of up to 100 m. Thus, active RFID is useful for long-range localization and object tracking [106‐108].

However, active RFID technology is not reliable for sub-meter localization accuracy, and it is not readily available on most portable user devices. Passive tags do not have inbuilt batteries but backscatter the signal received from the base station. Passive RFID is readily used for various applications due to its numerous benefits, including cheap cost, miniaturized size, ease of manufacturing compared to active RFID since it only requires a tag chip and antenna. Passive RFID is useful for sub-meter detections and can detect targets in up to 10 m range [108].

Radio Frequency Identification (RFID) technologies are viral due to their low cost. In such technologies, many reference tags are deployed in advance. Each tag will act as the transmitter, and the Radio Signal Strength Indicator (RSSI) information is measured from the readers around. The target tag's position is then estimated by those reference tags whose RSSI information is closest to the target tag's RSSI information [109, 110].

In [111], authors conducted localization using three types of sensors pairs, including infrared sensor pair, RFID reader and tags, and light-emitting diode LED and light resistor. It was found that localization using RFID outperforms other sensors in terms of accuracy and stability. Table 1 summarizes the characteristics of the localization technologies used nowadays [46, 98, 112‐116].

The distance between AP and mobile is related to the received power/ time of travel; this relationship can be represented in a mathematical expression. For 2D measurements, with two equations, there will be two possible solutions. To have a unique solution, three equations are required; the intersection of these equations will determine the location of the mobile as shown in Fig. 1; if the problem is in 3D (\(x, y,z)\) then four APs are used at least to have a unique solution [123].

Lateration is also used for estimating location through differential measurements (receive signal strength/ time of arrival). Differential measurements are used to reduce the effects of environmental changes. In this case, the transmitted power is unknown (DRSS) or if the time of transmission of the access point is unknown (TDOA), [124– 126]

Generally, if \(m\) APs are collaborating in localization, there will be (\(\frac{m(m-1)}{2}\)) formulated differential equations, among these equations (\(m-1\)) are basic equations, while the rest are redundant, the solution of each basic equation will lie on a hyperbola, the intersection of these hyperbolas gives the mobile's coordinates [127]. Using 3 APs system, there will be two basic equations, while the third equation is a linear combination of the basic ones. Three basic equations are required to have a unique solution, which is achieved by adding another AP, as shown in Fig. 2. Therefore, for 2D localization, 4 APs are required.

For 2D measurements, two angle measurements and single range measurement are required to localize an object as seen in Fig. 3; the range measurement could be the distance between the two arrays. For 3D measurements, two angle measurements, single azimuth measurement, and single range measurement are required [119].

The Bartlett method is considered to be one of the earliest AOA estimation techniques; during the process of making the array manifold, the covariance matrix is implemented; accordingly, the output power of the beamformer is estimated, maximum peaks of \(P(\theta )\) correspond to AOA [136, 137]. where \(y\left[n\right]\) is the array output, \({\varvec{x}}\left[n\right]\) is the array input data vector, \({\varvec{R}}\) is the covariance matrix, and \({\varvec{w}}\) is the weighting vector. For the uniform linear array (ULA) [138], where K is the number of antenna elements:

The drawback of using the Bartlett beamformer is the low resolution as it appears when the impinging signals are very close [138]. The separation between the electrical angles \(\phi \) (equals to \(kdcos(\theta )\)) should be more than (2π/K), adding more elements with suitable design will compensate for these impairments; however, this will add more the cost, size and storage data for calibration [138].

The Capon method uses some of the degrees of freedom to look at the desired signal \({\theta }_{0}\) and use the remaining to suppress the interfering signals [136]; in other words, the algorithm has two constraints, minimize the power from noise and interferer signals \({{\varvec{w}}}^{H}.n[n]\) and keeping the gain of the desired signal constant \({{\varvec{w}}}^{H}. {\varvec{a}}({\widehat{\theta }}_{0})\cong 1\) [136] where \({\varvec{a}}\left({\widehat{\theta }}_{0}\right)\) is the steering vector of the estimated angle of arrival \({\widehat{\theta }}_{0}\). Mathematically array weighting is estimated by [138]:

The weighting vector can be estimated using the Lagrange multiplier method:

Applying in Eq. 4, the total power received is:

This method shows better performance than the Bartlett beamformer as it can distinguish between angles with small separation, and it also suppresses the side lobes levels. One drawback of the Capon method is the expensive calculation for the inverse matrix for large antenna arrays [139]. The Capon beamformer shows acceptable performance in terms of resolution and rejection of interference under the condition that the presumed steering vector of the desired signal is identical to the actual one. However, practically, it is difficult to let the most accurate steering vector similar to the actual one; thus, the performance degrades dramatically [140].

The MUSIC algorithm focus on the eigenvector decomposition of the covariance matrix [141]. The array will scan all possible angles; for an arbitrary steering vector, a projection on signal subspace and projection on noise subspace will be projected. If the steering vector corresponding angle is close to the actual arrival angle, then the steering's projection on the noise subspace will be almost zero [141]. Dividing by zero will give infinite; however, since the steering vector corresponding will not be precisely the actual arrival angle, the division result will be enormous value rather than infinite. where \({{\varvec{U}}}_{n}\) is the noise subspace.

MUSIC is preferred for different reasons, including its ability to measure more than one signal simultaneously [138], in addition to its highly accurate and precise estimation for closed separated signals; however, the algorithm requires the knowledge of the number of incoming signals [138]. MUSIC is sensitive to coherent signals (the same signal is produced from different locations or the multipath of a signal transmitted from a single location); in such a case, \({U}_{n}^{H} a\left(\theta \right)\ne 0\), and hence it become very difficult to detect peaks, this obstacle can be overcome by using spatial smoothing.

Modification on MUSIC algorithm was proposed to overcome the shorting of the algorithm, where \({\varvec{W}}\) is a weighting matrix, for \({\varvec{W}}=I\), this will retrieve the original MUSIC.

If \({\varvec{W}}\) is chosen as: where \({e}_{1}\) the first column of the \(K\times K\) identity matrix, since the weight is of a minimum norm, has its first element equal to unity, and is contained in the noise subspace. This modification is known as the Min-Norm algorithm. Min-Norm shows less bias and, therefore, provides better results than the MUSIC. However, it's applied to ULA only.

It is a polynomial rooting version of MUSIC, which applies only to ULA. The algorithm enhances performance and reduces computational time; it is based on setting up roots of a polynomial, the M-zeros (\({\widehat{z}}_{m}\)), which lies on the unit circle with the largest phases magnitude yields the AOA estimate. where M is the number of arrival signals, and m ranges from 1:M.

The Estimation of Signal Parameters via Rotational Invariance Technique (ESPRIT) is featured with being simple, fast, robust, and power and memory-efficient [142]. The method can detect the direction of multiple incident signals at the antenna array, similar to MUSIC; the ESPRIT uses the eigenvalues to determine DOA and their corresponding delays [143]. The algorithm does search for all possible direction vectors to estimate DOA. Therefore, it reduced computational time and memory usage [144]. The steering matrix can be viewed as [138]:\({{\varvec{A}}}_{2}\) is related to \({{\varvec{A}}}_{1}\) by \(\Phi ,\) which is a diagonal matrix:

The method relies on properties of Eigen decomposition: where \({\varvec{T}}={\varvec{P}}{{\varvec{A}}}^{H}{{\varvec{U}}}_{s}{\left({\Lambda }_{s}-{\sigma }^{2}{\varvec{I}}\right)}^{-1}\) \({{\varvec{U}}}_{2}\) is related to \({{\varvec{U}}}_{1}\) by \(\Psi ,\) which is a diagonal matrix:

Rearranging the equations:

The above equation can be solved using Least-Squares sense (LS-ESPRIT) or by the Total-Least-Squares method (TLS-ESPRIT) [138]. Both \(\Psi \) and \(\Phi \) share the same eigenvalues. The DOA is estimated by estimating the M-eigenvalues (\(\xi \)) of \(\Psi \) [145]: where \(m=1:M\).

It is an asymptotically efficient parametric method that applies to any arbitrary array structure [146]; the weighting provides minimum variance estimates [147]. The algorithm is well-known for its accuracy; its criteria are a multimodal nonlinear multivariate optimization problem, leading to a high computational complexity [148]. Genetic algorithms are used to optimize the WSF and reduce the complexity [149]; it requires the knowledge of the number of directional sources [150].

Recall the covariance matrix \({\varvec{R}}:\)where \({\varvec{A}}\left(\theta \right),{\varvec{P}}\) and \(\Lambda \) are the steering matrix, the statistical expectation of transmitted signal, and diagonal matrix of eigenvalues. By setting \({\varvec{I}}={{\varvec{U}}}_{s}{{\varvec{U}}}_{s}^{H}+{{\varvec{U}}}_{n}{{\varvec{U}}}_{n}^{H}\) and removing the \({\sigma }^{2}{{\varvec{U}}}_{n}{{\varvec{U}}}_{n}^{H}\) from both sides and multiplying the equation by \({{\varvec{U}}}_{s}\) from the right:

Rearranging the equation:

The angles are estimated by solving the following no-linear equation: where \({\varvec{W}}={\left({\widehat{\Lambda }}_{{\varvec{s}}}-{\widehat{{\varvec{\sigma}}}}^{2}{\varvec{I}}\right)}^{2}{\widehat{\Lambda }}_{s}^{-1}\)

The WSF shows better performance when coherent signals exist, rooting version of Weighted Subspace Fitting was introduced for ULA this also referred to as MODE technique,

In [151], a comparison between Min-norm and root-MUSIC was performed; the estimated variance achieved by root-MUSIC is less than or equal to that of the root-min-norm method. In [152], a comparison study between root-WSF, MUSIC, root-MUSIC, ESPRIT, and Capon was conducted using ULA over many scenarios; it was found that Root-WSF has the best and stable performance. In [153], a performance comparison between MUSIC, root-MUSIC and ESPRIT was conducted; it was found that MUSIC has good performance with a low number of array elements and high SNR. While Root-MUSIC needs more elements, ESPRIT is less dependent on the number of array elements and performs better in low SNR scenarios. In [148], a comparison between GA-WSF was compared to MUSIC, it was found that the former outperform the latter in terms of accuracy over different set of SNR.

In [143], ESPRIT and MUSIC's performance was compared using different number antenna elements, different number of snapshots, and different SNR levels; it was found that both techniques provide accurate results. However, MUSIC shows more accurate and stable results. In [154], a comparison between MUSIC, MVDR, Min-norm, ESPRIT DOA algorithms were used in a MIMO-OFDM radar system. It was found that for determining a target, MUSIC shows the best resolution, followed by Min-norm and then ESPRIT. However, if two sources are closely spaced, Min-norm tends to have the best performance, followed by ESPRIT and then MUSIC. MUSIC tends to have the best results than Capon, Bartlett, and Min-norm, as observed by [145].

Cross-correlation is one of the most comprehensive methods used to estimate TOA [134]. In Fig. 4, TOA estimation is demonstrated; once the signal arrived, it matched (correlated) to a known template \(p(t)\) by a match filter MF. The output of the filter is forwarded to a square law device where the sign of the correlated signal is removed, and then the time instant with the maximum peak value represents the time at which the signal arrived first [164].

In multipath propagation adjacent, arrival peaks will have comparable amplitudes to the correct one; therefore, selecting the right peak becomes ambiguous, leading to large errors [164]. This method is preferred to its low complexity; however, it's prone to multipath and noise.

This method is based on the fact that the received signal is a convolution of the transmitted signal and the channel, knowing the channel response will give the signals' arrival times. In the frequency domain, the convolution becomes multiplication. Thus the channel response can be estimated by dividing the received signal by the transmitted signal [133]. The deconvolution problem is defined as if the output of a convolution process \(v(t)\) and the inputs \(s(t)\) are given what would be the other input \(h(t)\) [165]. The Fourier Transform of the channel response \(h\left(t\right)\) is given by: is expressed as [133]: where \(V\left(f\right)\) and \(S\left(f\right)\) are the Fourier Transform of the received signal \(v\left(t\right)\), the transmitted signal \(s\left(t\right)\), and additive white Gaussian noise (AWGN) \(n(t),\) respectively.

By taking the inverse Fourier transform of \(H(f)\) the channel response is estimated and hence TOA. Practically this is done by sweeping the frequency from the lowest frequency (\({f}_{c}-\frac{B}{2}\)) to highest frequency (\({f}_{c}+\frac{B}{2}\)) by a step of \(\Delta f\) [166]. Where \(B\) is the bandwidth, \({f}_{c}\) is the center frequency and \(\Delta f\) is the frequency segment. The range of frequencies is shown in Eq. 22. where \({f}_{0}\) is the lowest frequency and \(n\) is ranged from 1 to \(K\). For every frequency step, different values of \(V\left(f\right), S\left(f\right), H\left(f\right)\) and \(N(f)\) will be observed. A matrix can be generated to represent channel behavior over the bandwidth:

This method performs better than the convolution-based method for resolving the closely separated multipath, but it suffers from the noise enhancement \(\frac{N(f)}{S\left(f\right)}\). Another drawback of this method is the size of memory required for the matrices [133].

In many works in literature, TOA is considered as a special case of channel estimation [164]; in such a case, channel coefficients and time delays are unknowns, the log-likelihood function for the unknown parameters is [133]: where \({T}_{0}\) is the symbol duration. The maximum likelihood estimate is given by [133]:

The algorithm is complicated, requires a huge computational solution, and is not effective in closely spaced multipath [133].

Considering Eq. 23, if \(L\) observations are collected and at each frequency step, and \(M\) signal delays are received at each observation, channel \({\varvec{H}}\) will include information about the frequency segments and time delays. It's possible to view the time delays and the channel behavior on each delay as [166]: where

The nth row of \(U\) times \(A\) represents the channel response at the nth frequency [167].

Substituting the value of H in Eq. 30 into Eq. 29 and multiplying both sides of Eq. 23 by \({{\varvec{S}}}^{-1}\) [166]: where (\({\varvec{Y}}={{\varvec{S}}}^{-1}{\varvec{V}}\)) and (\({\varvec{W}}={{\varvec{S}}}^{-1}{\varvec{N}}\)). The covariance matrix can be taken for Eq. 28 by taking \(L\) observations as in [166]:

Assuming that both signal delays and noise are orthogonal where \({\sigma }^{2}\) is the noise variance.

Using the concept of eigenvalues and eigenvectors, \({\varvec{R}}\) has \(K\) dimensional space which can be decomposed into \(M\) signal delays sub-spaces \({{\varvec{Q}}}_{S}\) and (\(K-M\)) noise sub-spaces \({{\varvec{Q}}}_{N}\) [168], time delays are estimated as:

In Fig. 5, a comparison between MUSIC and IF algorithms is presented, the MUSIC algorithm outperforms the IF algorithm provided by its ability to distinguish closed multipath components. Despite the advantages of the MUSIC algorithm, however, the multipath number is a priori information for the algorithm [133].

The measurements are assumed to be stationary; therefore, the matrix is Hermitian and Toplitz. However, in real-life scenarios, samples taken are finite and small, which will not make the matrix to be Toplitz; the Covariance matrix can be further improved by using the forward–backward covariance matrix (FBCM) [166]: where \({\varvec{J}}\) is an exchange matrix; elements of the matrix are zero except the anti-diagonal where elements are ones.

Hybrid algorithms are used TOA and DOA in [169], where a two-dimensional MUSIC algorithm is used to perform localization in an IR-UWB system. And as in [170], a two-dimensional MUSIC algorithm is used in TOA and DOA estimation using a uniform circular array.

Mobiles can receive power signals until a certain level, where below such a level, the signal will be considered as noise; sensitivity is the term used to express this level. When RSS is below sensitivity, the mobile and AP are no longer in connection, in which the location of the mobile is assumed to be out of the AP coverage; when the RSS level is higher than the threshold, the AP and the mobile are connected, and hence it's within the AP coverage [117, 118]. Proximity gives an idea of whether the mobile is within the coverage area of an AP or not; if the coverage area is large, the information may not be beneficial. However, considering the intersection between APs coverage, this will enhance localization by shrinking the areas of possible locations [195].

In this method, RSS behavior is modeled as a random variable; two stages are performed similarly to the RF-Fingerprinting approach. In the first stage, the study area is divided into J grids, SS measurements are collected from the grid; these data are processed to give a probabilistic distribution for each location's SS behavior. In the second stage, the mobile's RSS from surrounding APs are collected from an unknown location and stored in a vector. Then mobile's location is inferred based on Maximum Likelihood Estimation (MLE) as shown in Eq. 36 [196]: where \(P\left({L}_{j}|{\varvec{s}}{\varvec{s}}\right)\) is the probability that the mobile is located at the location \({L}_{j}\) given that the RSS vector is (\({\varvec{s}}{\varvec{s}}\)). Assuming M APs. The above equation can be expressed as:

where\(P\left({L}_{j}\right)\) is he probability of each grid to be the location where the mobile locates, \(P\left({ss}_{i}|{L}_{j}\right)\) is the probability distribution of RSS from the ith AP at the jth grid, and \(P\left({ss}_{i}\right)\) is the probability for RSS from the ith AP in all grids. Since the mobile location is unknown, then \(P\left({L}_{j}\right)\) is equal.

The algorithm shows a precise analysis of the given data; however, it suffers from extensive labor work [133].

Figure 6 and 7 give comparisons between RSS based algorithm, including the radar algorithm (RF-fingerprinting), GR gridded radar (RF-fingerprinting), ABP (MLE), H1 (MLE), LLS, and NLS (Range based) using WLAN network [133, 197]. As seen in these figures, MLE and RF fingerprinting performance are very similar, while for range-based algorithms, the performance is relatively poor. Also, it can be seen that NLS is more accurate compared to LLS.

In Table 4, a comparison between AOA, TOA, and RSS localization methods is presented; the comparison includes the accuracy, sensitivity to multipath, requiring additional hardware, advantages, and disadvantages [98, 133, 198].