Power Delay Profile (PDP) ofers valuable insights into the propagation power fading of Line of Sight (LoS) and Non Line of Sight (NLoS) paths, making it a potential resource for ranging estimation. Despite its potential, the research on the lower bound of ranging error for PDP-based ranging is limited. This paper addresses this gap by introducing the Cramér-Rao bound (CRB) for Power Delay Profile (PDP) based ranging and Received Signal Strength Indicator (RSSI) based ranging, considering a specific indoor channel fading model. Through extensive simulations and analysis, we demonstrate the superiority of PDP-based ranging over RSSI-based ranging. Our findings contribute to a deeper understanding of the potential benefits of PDP-based ranging techniques.
that they solely focused on the first cluster but not other clusters, which typically contains propagation distance information. In contrast, our proposed approach diverges from theirs. We initiate range estimation by utilizing the Power Delay Profile (PDP) of identifiable multiple paths.
Our approach to PDP-based ranging involves utilizing the fading information of LoS and NLoS paths to estimate the LoS distance. The distance-dependent attenuation introduces an upper bound mask on the PDP, further influenced by random efects such as shadowing and reflection/difraction. Therefore, building a statistical model that accounts for propagation distance and PDP is essential. Over the years, various models have been proposed for microwave signal propagation attenuation with respect to (w.r.t.) average power fading [9, 10, 11]. For instance, the Saleh-Valenzuela model [10] presents received signal rays arriving in clusters with independent uniform phases and independent Rayleigh amplitudes decaying exponentially with cluster and ray delays. Additionally, the relationship between average power attenuation and propagation distance has been studied in indoor and outdoor environments [12, 13, 14]. G. Steinböck et al. [13] proposed an indoor model incorporating a delay power spectrum with a primary component (early) following an inverse distance power law (− ) and a reverberant component (tail) decaying exponentially with distance for indoor environments. Based on these models, multi-path fading coeficients are expected to be correlated with propagation distances, with their magnitudes following a Rayleigh distribution whose variance depends on the LoS distance plus NLoS delay distance.
In this paper, we analyze the theoretical performance of the PDP-based ranging approach and the RSSI-based ranging approach using PDP with Saleh-Valenzuela model and the indoor model proposed by G. Steinböck et al, considering random shadowing, reflection/difraction attenuation efects and the discrete nature of multipath. We derive the CRB for PDP-based range estimation via joint parameters estimation and marginalized range estimation for indoor environments. Furthermore, we present a novel CRB for range estimation based on RSSI by analyzing each path’s fading under the selected propagation fading model rather than setting them as fixed numbers. Through simulation, we preliminarily verify that PDP-based ranging has better theoretical performance than RSSI-based ranging. This finding suggests that PDP-based ranging is worth exploring in diferent application scenarios.
This paper is organized as follows: In Section II, we introduce the OFDM model and the selected channel fading model. Sections III and IV delve into the joint parameters estimation CRB and the marginalized ranging estimation CRB for PDP-based ranging, respectively. In Section V, we derive the CRB of classical RSSI-based ranging. Section VI presents the simulation experiments and analysis of the results. Finally, in Section VII, we make a conclusion.
The OFDM model we are considering assumes an OFDM symbol length of , consisting of a Line-of-Sight (LoS) path and Non-Line-of-Sight (NLoS) propagation paths. This model operates with a sampling period of and an OFDM symbol period of 0. One advantage the OFDM model is the elimination of need for precise knowledge of the pulse shape, as it makes = ⎢⎢ ⎣ ⎡[1] 210 . . .
0 = ⎢⎢ ⎣ ⎡ − 2 1 0
[1 ] . . . − 2 0
[ ] · · · . . . · · · · · · . . .
0 ... ⎥⎥ ; 2 0 ⎦ []
⎤ [1 ] [ ] − 2 1 ⎤
... ⎥⎥ , − 2 1 ⎦ use of pilot subcarriers within the pulse shape’s passband.
The received signal vector in the OFDM system can be expressed as: where the received signal vector ∈ × 1 is defined as: = + = + , ⎡[1]⎤
. = ⎢ .. ⎥ ; ⎣ ⎦
[] The matrix ∈ × is filled with pilots and given by: The matrix ∈ (× (+1)) includes pulse shape filtered delayed path responses and is shown as: · · · where and represent the propagation delay and the pulse shape, respectively.
The vector ∈ (+1)× 1 indicates the complex attenuation coeficient (amplitude (+1) and phase ∈ (+1)) and is presented as: ⎡0⎤
. = ⎢⎣ .. ⎥⎦ = ⎢
⎣ ⎡00 ⎤ . .
. ⎥ = (e) = ()e, ⎦ where (* ) represents an operation that converts a vector to a diagonal matrix. The vector ∈ × 1 is a complex Gaussian noise vector, and each element follows a distribution (0, 2 ).
Firstly, we assume that each in is an independent and identically distributed (i.i.d.) random variable drawn from a uniform distribution on the interval [0, 2 ).
Secondly, we consider the matrix to be known and to have been estimated prior to ranging estimation. Additionally, we presume that the estimation error of is negligible, as this paper does not focus on examining its bias impact.
Furthermore, we assume that the multipath scenario includes distinguishable LoS path and
NLoS paths. In this context, each delay between the ℎ NLoS path and the LoS path is
measurable with negligible error. This assumption is grounded in the understanding that the
measurement of delays is considerably more accurate compared to the estimation of path
complex amplitudes.
(
Additionally, we assume that the majority of the system’s subcarriers are within the pulse shape’s passband. In this region, the function ( ) representing the pulse shape is approximately equal to 1. This assumption simplifies the model by considering that most subcarriers experience minimal distortion or attenuation within the passband.
According to the Saleh-Valenzuela model [5], by identifying the first ray of each cluster as the LoS path and the remaining rays as NLoS paths, the probability density function of the fading amplitude for the -th path can be described by a Rayleigh distribution: (| 2 ) = 22 e− 2 , 2 where 2 represents the average power gain of the -th path. It is evident that 2 is associated with the propagation distance of the -th path.
According to the indoor model proposed by G. Steinböck et al. [6], the average power gain can be decomposed into the primary LoS component and the NLoS reverberating component. The gain of the LoS and NLoS paths at a distance is given by:
⎧ () = ⎨ ⎩ 0
0 ︁( ref ︁) ; LoS, ︁( ref ︁) + 0,rev e − ; NLoS, , where 1 represents 0,rev . Furthermore, 2 in (6) is a specific expression of (). Additionally, for each NLoS path distance , it can be represented as: (6) (7) (8) (9) where 0 represents the gain at an arbitrary reference distance ref, 0,rev is the reference gain of the reverberant component, is the reverberation time, is the speed of light, and is the environment path gain exponent.
For localization estimation, we assume that the values of 0 and 0,rev at a reference distance of 1 meter and the value of are known prior information. Therefore, for each path with a distance , the average expected gain 2 can be expressed as: 2 = () =
− 0− + 1e ; ̸= 0, {︃ 0−0 ; where is the delay time from the LoS path to the -th NLoS path, and it is measurable with negligible error as previously hypothesized. 3. Joint Range Estimation CRB for PDP-Based Ranging To model the LoS path distance 0, we apply a Markov chain, disregarding any information about 0 in and . This means that we consider 0 to be independent of the complex attenuation coeficients and the pulse shape filtered delayed path response.
By ignoring the information about 0 in and , we assume that the variations or dynamics of 0 do not directly afect or depend on the complex attenuation coeficients or the pulse shape ifltered delayed path response. Instead, the evolution of 0 is modeled using a Markov chain, where the future values of 0 only depend on its current state and not on its past states.
This simplification allows us to analyze the behavior of 0 using the theory and techniques of Markov chains as
0 → → .
For the joint parameters estimation of = [0, , ], we can express the Fisher Information Matrix (FIM) as below: = ,, − [︁ 2 log (,,|0) ]︁ = ,, ⎣0 ⊺ ⎡00 0 0 0⎤ ⎦ , (10) (11) (12) (13) (14) (15) (16) (17) can be expressed as follows: where ⊺ is matrix transpose operator and the probability density function (pdf) (, , |0) (, , |0) = (|, ) (|0) ().
Within the context of (12), it is possible to represent each pdf as follows: 00 = − 0 = ⊺0 = − 2 log (|0) , 20 2 log (|0) ,
0⊺ (|, ) =
1
2 exp −
=
=0
(|0) = ∏︁ (| 2 (0)) = ∏︁
() = ∏︁ () =
=
=0
︂(
, where () denotes the conjugate transpose. Upon logarithmically processing (12), we obtain: log (, , |0) = log (|, ) + log (|0) + log ().
Through the process of derivation, we can express each element inside (11) as follows: 0 = ⊺0 = − 2 log (, , |0) 0⊺
= 0, = − = − 2 log (|, )
⊺ 2 log (|, ) ⊺ − −
, = ⊺ = − 2 log (|, ) ⊺ ,,00 = 0 2 −0 (+2) + ∑︁ ,,0 = ,,⊺0 = ⎢⎢ 0 − 2
2 −0 21 , · · · The expectations w.r.t. , , and inside (23) can be expressed as follows: 0− − 1 + 1 e 0− + 1e − − ⎞2
⎠ , , √ ︁( ︂( 0+1 + 1 e−
)︂ 1 e− + 0 )︁ 3/2 , · · · ⎥⎥ , ⎤ ⎦ where () is the operation of retaining the diagonal elements while setting all the nondiagonal elements to 0 of the matrix and represents the identity matrix.
Having considered all the factors mentioned earlier, we can now proceed to calculate the joint estimation CRB for the estimation of 0 w.r.t. PDP-based ranging:
Marginalized Range Estimation CRB for PDP-Based Ranging According to (6), the NLoS path complex attenuation coeficients element is an i.i.d. complex zero-mean Gaussian random variable can be expressed as ∈
(+1)× 1 that each + 0 2 −0 (+2) −
2 0 2 2(− 2) (︁ ∑︀ 0
=1 |ℎ0|2 + 21 ︁) − 1 ⎠ (︁ ∑︀ =1 |ℎ|2 + 21 ︁) − 1 ]︂ − 1 . where ∼ (0, ), = ⎢⎣ ...
0 To estimate 0 directly and solely based on ∈ × 1, we can estabilish the pdf of given 2(0) ∈ ℛ(+1)× 1 as follows:
(| 2(0)) = − (det())− 1e− −1, = + 2 , 2(0) = [ 20 · · · 2 ]⊺.
To compute the FIM from the pdf (| 2(0)), which is Gaussian with zero mean and covariance , the FIM can be calculated as follows: 00 = ( 2(00) )⊺ 2(0) 2(0)( 2(0) ).
0 For 2(0) 2(0), its element 2(0) 2(0), can be derived as: {︃ −1 }︃
−1 2 2(0) 2(0), = tr 2 = | −1 |2. where ∈ ℛ(+1)× 1 is a column vector with the -th element being 1 and all other elements being 0. The trace operation, denoted by (· ), computes the sum of the diagonal elements of a matrix. With these definitions, we can compute the FIM as follows:
2(0) 2(0) = ( −1 ) ⊙ ( −1 )* , where ⊙ represents the Hadamard product (element-wise multiplication) and * denotes the conjugate operation. And where ℎ element can be presented as: 2(0)
= 0 [︃ 20 0 · · · 2 ]︃⊺ 0
, 2 = 0 {︃ − 0−0 − 1,
︂[ 2(0) )⊺ 2(0) 2(0)( 0 = ( 0 .
In the case where all data subcarriers can be used and considering the channel model, the RSSI can be measured from the squared Euclidean norm of the magnitude vector . Taking into account the law of large numbers, we can express ‖‖2 as:
‖‖2 = ‖‖2, which represents the sum of squared magnitudes of the individual subcarriers.
Utilizing the squared Euclidean norm, the RSSI measurement provides an aggregate measure of the received signal strength across all the subcarriers, enabling an overall assessment of the signal power.
Since we assume that most of the subcarriers used for transmission are within the passband of the pulse slope where ( ) ≈ 1, ‖‖2 can be derived to: where ‖‖2 = ‖‖2 + = (∑︀=1 2)‖‖2 + 2 ,
‖‖2 = ∑︀=0 ||2 = ∑︀=0(2 + 2).
(0, 22 ).
Using pilots to estimate the channel and perform interference cancellation, the complex attenuation coeficient = + can be decomposed into its real part and imaginary part , both of which are Gaussian random variables. Specifically, we have ∼ (0, 22 ) and ∼
For the LoS path ( = 0) and NLoS paths ( ̸= 0), the variances 2 of the complex attenuation coeficients can be given as follows: • For the LoS path ( = 0): • For the NLoS paths ( ̸= 0): 02 = 0−0 ;
− 2 = 0− + 1e , where 0 is the gain at an arbitrary reference distance , 1 is the reference gain of the reverberant component, is the environment path gain exponent, is the speed of light, and is the reverberation time. The variables 0 and represent the distances of the LoS path and NLoS paths, respectively.
Therefore, the real part and imaginary part of are Gaussian random variables with 2 , where 2 is given by the expressions mentioned above. If we assume that the real variances 2 part and imaginary part of have the same variance 02 for all , then the magnitude squared ‖‖2 follows a Chi-squared distribution with 2( + 1) degrees of freedom. The probability density function (PDF) of the random variable = ||2 given 0 can be expressed as: (2+2)(| 02) =
− 02 02(+1)Γ( + 1) (37) (38) (39) (40) (41) (42) (2+2)(|0) = ︁( (0( 10 ) − 0( 10 ) ︁) (+1) Γ( + 1)
. () = 02( + 1).
Then we can calculate CRB of estimating 0 from ‖‖2 w.r.t. classical RSSI-based ranging as
range from 10 dB to 60 dB, default 20 dB.
ranging from 2.0 to 2.5, default 2.0.
random between 10 and 15. distance of NLOS path (m)
random between 1.10 to 2.00. we can rewrite the pdf of the random variable = ‖‖2 as: where Γ() is Gamma function and is chosen as 1 meter. Replacing 02 by a function of 0,
SNR
In this section, we utilize MATLAB to compute the CRB for PDP-based ranging using joint parameter estimation and marginalized range estimation, as well as classical RSSI-based ranging. Our simulations incorporate the indoor radio propagation model introduced in Section II, which accounts for reverberating efects and path decay. The key parameters employed in the simulations are outlined in Table 1.
To investigate the factors influencing localization error, we focus on two primary factors: the Signal-to-Noise Ratio (SNR) of the channel and the path gain exponent. For each factor, we keep all other parameters at their default values and conduct the simulation to observe the behavior of the CRB.
The obtained performance results are presented in two figures. Figure 1 illustrates the behavior of the square root of the CRB for PDP-based ranging as the SNR increases from 10 dB to 60 dB. As expected, the root of the CRB decreases with higher SNR, indicating improved ranging accuracy due to the higher quality of the received signal. Notably, the performance of PDP-based ranging surpasses that of RSSI-based ranging. Moreover, marginalized range estimation demonstrates superior performance compared to joint estimation. .
In summary, this paper presents a comprehensive analysis of the Cramer-Rao Bound (CRB) for PDP-based positioning using both joint estimation and marginalized range estimation, as well as RSSI-based localization. The derived CRB takes into account the distance-dependent path decay of both Line-of-Sight (LoS) and Non-Line-of-Sight (NLoS) paths. The study investigates the impact of various factors on the CRB, including the Signal-to-Noise Ratio (SNR) of the 3.5
3 2.5 B R 2 C f o t oo1.5 R
1 0.5 0 2 channel and the path gain exponent in diferent environments. The simulation results highlight the superiority of PDP-based localization over RSSI-based localization, with PDP-based ranging achieving better performance across diferent environments. Furthermore, the marginalized range estimation approach demonstrates improved accuracy compared to joint estimation. These findings contribute to the understanding of PDP-based localization and provide insights into its potential for accurate positioning in real-world scenarios.
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