# Nonlinear estimation for 60GHz millimeter-wave radar system based on Bayesian particle filtering

- Mengwei Sun
^{1}Email author, - Qichao Song
^{2}, - Bin Li
^{1}, - Long Zhao
^{1}and - Chenglin Zhao
^{1}

**2013**:33

https://doi.org/10.1186/1687-1499-2013-33

© Sun et al.; licensee Springer. 2013

**Received: **27 November 2012

**Accepted: **21 January 2013

**Published: **19 February 2013

## Abstract

In the 60GHz millimeter-wave radar communication systems, the nonlinear power amplifier is inevitable. In order to combat this problem, a promising estimation algorithm based on the particle filtering (PF) is presented here. By employing the conception of Bayesian approximation and sequential importance sampling, this appealing Monte Carlo random sampling method can address this complicated statistic estimation problem. In sharp contrast to the classical linear equalization problem, nevertheless, in the considered situation the PF-based method may become invalid due to the hardware nonlinearity and the resulting non-analytical importance function. To remedy this difficulty, based on the linearization technique a novel PF framework is suggested, and we show in particular how to linearize the involved nonlinearity transform in the formulated discrete dynamic state-space modeling (DSM). The merit of this method is that it can efficiently deal with discrete DSMs that are practically nonlinear and non-Gaussian. Experimental simulations verify the superior performance of our presented PF-based detection scheme, which may properly be applied to 60GHz millimeter-wave radar communication systems.

### Keywords

60GHz millimeter-wave communication radar systems PA nonlinearity Particle filtering Taylor series Linearization technique## 1. Introduction

Because of large bandwidth, small size, high-resolution, and all-weather characteristics, 60GHz millimeter-wave radar has widely been used in many fields such as military communication systems and civilian communication. The most common types of 60GHz millimeter-wave radars include anti-collision radar and 60GHz Quadrature Doppler radar [1].

With the growing demand for high-speed wireless transmissions, 60GHz millimeter-wave communication has attracted an increasing amount of interests over the past few years as an effective method of gaining high-speed data rate [2]. In the 60GHz millimeter-wave radar communication, in order to achieve the ultra-high data rate with the regulated transmission bandwidth, high-order modulations have widely been adopted to make efficient use of the spectrum. Two of the most common high-order modulations are QPSK and 16QAM [3, 4].

The absorption from oxygen is most obvious in 60GHz band, and therefore, the significant path-loss may have adverse effects on the transmission quality. To compete against the negative effects, high emission power is usually indispensable in 60GHz millimeter-wave communication. Thus, the resulting high peak-to-average power ratio may pose great challenges to the design of radio components. Unfortunately, it seems that the nonlinear characteristics of power amplifier are practically inevitable for 60GHz millimeter-wave radar devices [5]. As a consequence, it is recognized that there is serious distortion in emission signals which may significant increase the bit error ratio (BER) in receiver. To sum up, improving the performance of signal detections in 60GHz millimeter-wave radar communication in the presence of nonlinear power amplifier remains as a significant challenge and has become one of the major concerns in practice.

Currently, there are two commonly used methods for overcoming the nonlinearity of 60GHz millimeter-wave radar systems. One approach is to simply reduce the radiation power, and apparently the basic idea of this approach is to alleviate the nonlinear distortions directly [6]. An important advantage of this method is that it is easy to implement. Unfortunately, such an output power back-off (OBO) method, however, may directly result in the decline of the signal-to-noise ratio (SNR) in receiver. Another compensation approach can be summarized as a variety of linearization techniques [7, 8]. The deficiency of these techniques is that the complexity is extraordinary and at the same time, the hardware realization is impractical.

In this article, relying on the Bayesian statistical inference and sequential importance sampling (SIS) technique [9, 10], we propose a novel signal detection algorithm for 60GHz millimeter-wave radar communication to effectively address the involved nonlinearity distortion.

The SIS method aims essentially to establish a Monte Carlo (MC) numerical representation of the desired probability distribution which may be too complicated to derive the analytic expression, which primarily consists of a group of discrete particles and the associated weights [11, 12]. Then, these particles and associated weights will be recursively updated, which can then be used to numerically approximate the desired probability distribution [9]. In other words, the discrete particles with their importance weights will provide the Bayesian estimates of the input signal sequence. In this investigation, we apply the SIS method to the signal estimation in 60GHz millimeter-wave radar communication systems in the presence of nonlinear distortion. First, in order to derive the practically feasible importance function (or the related likelihood function) by taking the complex nonlinear transform, we introduce the first-order Taylor expansion and develop a new representative dynamic state-space modeling (DSM) system via the local linearization technique.

Therefore, the involved importance function can conveniently be obtained. And on this basis, the signal detection with nonlinear distortion is realized by resorting to the particle filtering (PF) technique [13]. The experimental simulations are finally provided, which may essentially demonstrate that our suggested method can effectively solve the nonlinear estimation, and hence provide a promising solution to the emerging problem of 60GHz millimeter-wave radar communication systems.

The merits of our proposed algorithm could be summarized into twofold. First, the new method has the characteristic of real-time process. More specifically, the output estimation could be prepared in time as the new observation arrives without the sequential parameter learning. Second, this approach can significantly reduce the complexity and cost of the hardware. For these reasons, the presented detection algorithm could widely be applied in practice.

The remainder of this article is organized as follows. Section 2 provides a model of considered nonlinear 60GHz millimeter-wave radar communication systems, i.e., the received signals are seriously distorted by a power amplifier. In Section 3, we briefly review the PF algorithm and provide computational details for implementing the method in blind estimation. In Section 4, we propose a novel PF method based on the Taylor expansion which is utilized to linearize the nonlinear model. Computer simulations are provided in Section 5. Finally, conclusions are presented in Section 6.

## 2. Nonlinear system model

### 2.1. Nonlinear power amplification

In practice, the power amplification procedure at the transmitter end will inevitably introduce the nonlinear distortion due to hardware imperfections. As a consequence, both amplitude and phase of the output signal in receiver will sharply be distorted, and hence the serious detection error will occur by remarkably degrading the transmission performance. The effects of PA nonlinear include the distortion generated by amplitude modulation–amplitude modulation (AM–AM) conversion and AM–phase modulation (AM–PM) conversion. In this article, the nonlinear PA model regulated by IEEE 802.11ad task group (TG) is adopted [4].

where *V*_{in} and *G*(*V*_{in}) represent the input and output voltage amplitude range in root mean square; *g* denotes the linear gain and we practically set *g* = 4.65; σ_{
s
} is the smoothness factor and *σ*_{
s
} = 0.81; *V*_{sat} is the saturation level and is considered to be 0.58 V.

where *Ψ* (*V*_{
in
}) is the additional phase in degrees. The values of *α*, *β*, *q*_{1}, and *q*_{2} are set to be 2560, 0.114, 2.4, and 2.3, respectively. For the convenience of analysis, we assume a set (*G*, Ψ) = [*g*, *σ*_{
s
}, *V*_{
sat
}, *α*, *β*, *q*_{1}, *q*_{2}] consist of the associative parameters which can denote the AM–AM and AM–PM models.

*V*

_{sat}. We can ignore the amplitude distortion and phase distortion when the amplitude of input is less than 0.1 V. But when the input signal amplitude is greater than 0.1 V, the amplifier will work in nonlinear region; this may result in severe nonlinear distortion, and therefore the performance of the receiver signal will deteriorate seriously.

where *P*_{
sat
} is the output saturation power, *P*_{
in
} is the input power, and *R*_{
backoff
} is the power back-off value in dB. Obviously, the greater back-off value means the smaller the output power, and the smaller the output power means the smaller the nonlinear distortion. More specifically, the nonlinear distortion may partly be alleviated by reducing the input power *P*_{
in
}. Unfortunately, lower transmit power will lead to the lower SNR, this will result in decreasing anti-jamming capability. At the same time, the performance of amplifier devices may greatly be deteriorated. Therefore, the power back-off mechanism can only avoid the negative effects of nonlinear distortion in a certain extent, but fail to effectively improve the system performance under the nonlinear device. It is difficult to be adopted widely in the practical design.

### 2.2. Signal model

The transmitter-end operates as follows. In the beginning, the binary source sequence {*b*_{
i
}} (*i* = 0,1,2,…) is sequentially fed into an *M*-order linear modulator (such as *M*-PSK or *M*-QAM). Subsequently, the modulated data symbols {*x*_{
k
}} (*k* = 0,1,2,…) are passed though a front-end nonlinear PA, and after this process, we can get the emitted symbols {*x*_{
k
}^{†}} whose voltage amplitudes and phases are seriously distorted. Finally, the emission symbols are propagated through single-path channel with AWGN.

*y*

_{ k }} (

*k*= 0,1,2,…) at the receiving end can be given by

where *L* denotes the length of the communication channel impulse response. The complex symbol *x*_{
k
} is a high-order modulated uniform random variable, i.e., *x*_{
k
}∈{*A*}, *k* = 0,1,2,…, and it is independent from the obvious and future symbols. In practice, we set the prior probability of each transmitted symbol equal. And *v*_{
k
} represents the noise value at time *k*, whereas it is an AWGN process with zero mean and variance σ^{2}, i.e., *v*_{
k
} ~ *N*(0, σ^{2}). The coefficient of channel **h** is assumed to be invariant and follow the Gaussian distribution with the mean vector of $\overline{\mathbf{h}}$ and the covariance matrix Σ, i.e., $\mathit{h}\sim N\left(\overline{\mathit{h}},\Sigma \right)$, whereas, it can conveniently be represented by the *L* × 1 vector, **h** = [*h*_{0}, *h*_{1},…,*h*_{L–1}]^{
T
}.

*x*

_{ k }[14]. Correspondingly,

**u**

_{ k }= [0,0,…,

*x*

_{ k }]

^{ T }is an

*L*× 1 vector,

**T**is the

*L*×

*L*state-transition matrix.

Equation (6) is the observation equation, which gives the relationship between the received (or observed) signal *y*_{
k
} and the hidden (or unobserved) state *x*_{
k
}. The nonlinear transform *g*(.) specifies the mapping between the input signal and output signal of the nonlinear power amplifier.

**h**approach single-path. For this reason, under the nonlinear distortion we may simplify the observation model (6) into (7).

Also, in the formulated model above, the propagation channel is assumed to be time-invariant and the gain is normalized to 1.

## 3. PF

### 3.1. Bayesian inference

**h**is known but the transmitted symbols {

*x*

_{0:K}} are unknown. From a Bayesian perspective, the optimal signal detection can be achieved by MAP detection of the transmitted

*x*

_{0:K}, based on the observed sequence

*y*

_{0:K}. The aim of this study is to compute the MAP estimate of state symbols {

*x*

_{0:K}} based on the observed symbols{

*y*

_{0:K}} and the coefficient of channel

**h**. The MAP estimate of the transmitted symbols is given by

*x*

_{0:k}

^{(MAP)}from

*x*

_{0:k−1}

^{(MAP)}when the recent observation

*y*

_{ k }is received. The posterior probability mass function (pmf) is shown as follows [15]

Unfortunately, the involved posterior distribution of interest above is often analytically intractable and hard to derive in practice. More specifically, because of the high-dimensional marginalization integration and the nonlinear process, we have to resort to the approximation method.

### 3.2. Blind estimation using the SIS

As an important expansion of Kalman filtering, the PF method shows great promise to blind nonlinear equalization in 60GHz millimeter-wave communication. With PF, continuous distribution could be approximated by discrete random measures. In the implementation of the PF, SIS plays a crucial role. Relying on the MC method and discrete numerical approximation approach, SIS could deal with many complicated statistic estimations. In this section, we mainly focus on the SIS algorithm and the re-sampling approach which can combat the degeneracy of particles.

*x*

^{(i)}with associated weights ${\tilde{w}}^{\left(i\right)}$ (

*i*= 1,2,…,

*N*). On this basis, the MAP estimation (8) could be drawn and marginal data detection at time

*k*are given by [14]

Here, *N* represents the total number of particles at the same sampling time. *δ* = 1 if **x**_{0:k}*=* **x**_{0:k}^{(i)} and *δ* = 0 otherwise.

We begin SIS algorithm by drawing *N* particles from posterior pmf *p*(*x*_{0:k}|*y*_{0:k},**h**,(*G*,Ψ)). In practice, it is usually intractable to sample directly from the target distribution *p*(*x*_{0:k}|*y*_{0:k},**h**,(*G*,Ψ)), so we design an important distribution π(*x*_{0:k}|*y*_{0:k},**h**,(*G*,Ψ)) from which the particles are more easily to drawn. The design of the important distribution is a critical step in PF, and it varies according to the actual situation. We will present the important distribution adopted in Section 4.

Unfortunately, as has been illustrated by most investigations and noted from (10), the degeneracy of SIS algorithm is usually inevitable, which is referred to the decrease of importance weights over time. Along with a consequence of weight degeneracy, the approximation of posterior probability may seriously deteriorate and even become useless. An efficient approach to alleviate this difficulty is to conduct a re-sampling procedure in the SIS algorithm. The basic idea of such method is to eliminate particles with small normalized importance weight while concentrating upon those particles having larger normalized importance weight.

Thus, in practice, when *N*_{eff} is below a fixed threshold, the re-sampling procedure is used.

## Bayesian signal detection

In this section, we will address the problem of designing the optimal importance function which could be adopted in this nonlinear communication system relying on the local linearization technique. In addition, the implementations of nonlinear equalization are presented.

### 4.1. PF with local linearization

*x*

_{ k }=

*x*

_{ k }

^{*}, we may further get

*x*

_{ k }

^{*}represents the hypothetical signal with known value, which can be drawn from the finite alphabet of modulated symbols. It is conveniently to calculate the optimal importance function π(

*x*

_{ k }|

*x*

_{0:k–1}

^{(i)},

*y*

_{0:k},

**h**,(

*G*,Ψ)) with mean

*m*

_{ * }and covariance

*Σ*

_{*}which could be evaluated for each trajectory using the following formula

*Σ*

_{*}could be calculated by

*m*

_{ * }is given by

Thus, after sampling from the importance function π(*x*_{
k
}|*x*_{
0:k-1
}^{(i)},*y*_{
0:k
},**h**,(*G*,Ψ)), we may get the particles {*x*_{
k
}^{(i)}} at time *k*, and then send these particles to the power amplifier and get new particles {*x*_{
k
}^{(i)}}whose voltage amplitude and phase are distortional.

*p*(

*y*

_{ k }|

*x*

_{ 0:k }

^{(i)},y

_{0:k– 1},

**h**,(

*G*,Ψ))can be expressed for likelihood [9, 15], shown in (18)

where **Σ** is the initial covariance matrix of channel and **x**_{
k
}^{(i)} is an *L* × 1 vector contained by particles at time *k –* 2, *k* – 1, and *k* in the same trajectory *i*. It is important to note that **h** represents the single-path channel, so the value of **h** is approximated to be [1 0 0]^{
T
}.

Then, use Equation (12) to normalize the associated weights into *w*_{
k
}^{(i)}.

Up to now, the discrete random measures {*x*_{
k
}^{(i)}, *w*_{
k
}^{(i)}} have been derived, so the MAP criterion should be carried on to get the estimated symbols.

### 4.2. Implementations

Based on the elaborations above, the Bayesian nonlinear detection algorithm comes in its fullness. At the receiving end, we may simulate the blind signal detection of nonlinear communication systems in the presence of nonlinear distortions in the following four steps:

**for**

*k*= 0:

*K*– 1

- 1
Draw

*N*particles from the importance function π(*x*_{ k }|*x*_{ 0:k-1 }^{(i)},*y*_{ 0:k },**h**,(*G*,Ψ)) ~*N*(*m*_{ * }, Σ_{*}), and feed these particles into the nonlinear power amplifier. - 2
Compute importance weight of each particle according to (18) and (21), then normalize the weight.

- 3
Re-sample in accordance with (13).

- 4
Use the method of MAP estimate (10) to calculate the transmitted symbols.

end for

## 5. Computer simulation

In this article, we focus on a scenario of a time-invariant nonlinear channel with an impulse response with the extended length *L* = 3. We assumed a Gaussian prior for the channel coefficients, $\mathbf{h}\sim N\left(\overline{\mathbf{h}},\Sigma \right)$. The channel mean is assumed to $\overline{\mathbf{h}}=\left[1\phantom{\rule{0.5em}{0ex}}0\phantom{\rule{0.5em}{0ex}}0\right]$ and the covariance matrix is Σ = diag{*δ δ δ*} with *δ* = 10^{–10}. The main reason for this configuration is that it could practically approximate a single-path channel. In our experiment, the transmitted signals {*b*_{
i
}} were modulated to {*x*_{
k
}} by using 16QAM and QPSK schemes, respectively, and the *prior* distribution of {*x*_{
k
}} is assumed known.

^{–2}.

## 6. Conclusions

In this article, we have designed an effective estimation algorithm, which can conveniently be implemented in receiver-end, for nonlinear 60GHz millimeter-wave radar communication systems based on Bayesian inference. This algorithm is implemented by SIS and Taylor expansion. One of the main features of our proposed estimator is that, as a promising solution to address nonlinear estimations, the first-order Taylor expansion is used to approach the involved nonlinear transform, and the associated weight is then recursively computed for each data trajectory in PF. The advantage of this method is its capability to estimate transmitted symbols sequentially and timely without a training sequence, even in the presence of nonlinear distortions due to hardware imperfections. As is shown by the experimental results, the combination of PF with Taylor expansion can effectively combat the nonlinearity distortion in the 60GHz millimeter-wave radar communication systems, which may hence provide a competitive solution to the practical design of 60GHz millimeter-wave communication and other radar communication systems.

## Declarations

### Acknowledgment

This study was supported by the National Natural Science Foundation of China (60972079, 61271180) and the Fundamental Research Funds for the Central Universities (2012RC0103).

## Authors’ Affiliations

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## Copyright

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