资料翻译之二-Statistical Characteristics of Multipath Channels – Scattering Function

第二篇是我之前查资料时看到的，来自于网站，关于无线信道的统计理论(?)的内容，没有完全理解，各位指教。

PS:第一篇翻译没有排好版，看起来非常乱，待我仔细研究下CSDN的排版，有时间再来修改。与之前一样公式编辑太复杂我偷懒省去了，请各位移步原网站查看。

Wireless channel is of time-varying nature in which the parameters randomly change with respect to time. Wireless channel is very harsh when compared to AWGN channel model which is often considered for simulation and modeling. Understanding the various characteristics of a wireless channel and understanding their physical significance is of paramount importance. In these series of articles, I seek to expound various statistical characteristics of a multipath wireless channel by giving more importance to the concept than the mathematical derivation.

 

无线信道由于参数随时间的随机变化而具有时变的特性。相对于常用来模拟和建模的高斯白噪声模型而言，无线信道显得更加的粗糙(harsh)。理解无线信道多变的特性和其本身的物理意义是十分重要的。在接下来的一系列文章中，我会试图以更注重概念而非其数学特性的方式来解释多径无线信道的统计特点。

 

Complex Baseband Mutipath Channel Model:

复变基带多径信道模型

In a multipath channel, multiple copies of a signal travel different paths with different propagation delays τ and are received at the receiver at different phase angles and strengths. These rays add constructively or destructively at the receiver front end, thereby giving rise to rapid fluctuations in the channel. The multipath channel can be viewed as a linear time variant system where the parameters change randomly with respect to time. The channel impulse response is a two dimensional random variable – h(t,τ) that is a function of two parameters –  instantaneous time t and the propagation delay τ. The  channel is expressed as a set of random complex gains at a given time t and the propagation delay τ. The output of the channel y(t) can be expressed as the convolution of the complex channel impulse response h(t,τ) and the input x(t)

?(?)=∫∞0h(?–?,?)?(?−?)??                          (1)

 

在多径信道中，信号的通过不同的路径传播，在经历不同的时延τ后，以不同的相位及强度到达接收端。在接收端，这些(经历不同路径的)信号(rays)会相加或相抵，从而引起信道的剧烈变化(?)。多径信道可以看做一个参数随时间随机变化的线性时变系统。信道冲击响应可以表示为一个具有时刻t和传播时延τ两个随机参数的二维函数h(t,τ)。信道可以被表示为一组由时刻t和时延τ决定的复变增益。信道的输出 y(t)可以表示为复变信道增益h(t,τ)和输入信号的卷积x(t)。如公式(1)。

 

If the complex channel gains are typically drawn from a complex Gaussian distribution, then at any given time t, the absolute value of the impulse response |h(t,τ)| is Rayleigh distributed (if  the mean of the distribution E\left ( h(t,\tau) \right )=0) or Rician distributed (if  the mean of the distribution E[h(t,τ))≠0]. These two scenarios model the presence or absence of a Line of Sight (LOS) path between the transmitter and the receiver.

 

如果上述的复变信道增益是由典型的复变高斯分布演化而来的，则在任意给定时刻t，信道冲击响应的绝对值是符合瑞利分布(如果分布的均值E[h(t,τ))=0)或莱斯分布的(如果分布的均值E[h(t,τ))≠0)。这两种模型分布描述了LOS存在与否时的情景。

 

Here, the values for the channel impulse response are samples of a random process that is defined with respect to time t and the multipath delay τ. That is, for each combination of t and τ, a randomly drawn value is assigned for the channel impulse response. As with any other random process, we can calculate the general autocorrelation function as

?hh(?1,?2;?1,?2)=?[h(?1,?1)h∗(?2,?2)](2)

 

在这里，信道冲击响应的值是根据时刻t和时延对随机过程的采样。也就是说，对于每一对t和，都会对应一个随机的信道冲击响应值。与其他随机过程相似，我们可以通过(2)来计算一般自相关函数。

 

Given the generic autocorrelation function above, following assumptions can be made to restrict the channel model to the following specific set of categories

给出以上的通用的自相关函数，以下假设可以限定信道模型为几种具体的分类。

Wide Sense Stationary channel model广义平稳信道模型

Uncorrelated Scattering channel model不相干散射信道模型

Wide Sense Stationary Uncorrelated Scattering channel model 广义平稳不相干散射信道模型

Wide Sense Stationary (WSS) channel model

In this channel model, the impulse response of the channel is considered Wide Sense Stationary (WSS) , that is the channel impulse response is independent of time t. In other words, the autocorrelation function Rhh(t,τ) is independent of time instant t and it depends on the difference between the time instants Δt=t2−t1 where t1=t and t2=t+Δt. The autocorrelation function for WSS channel model is expressed as

?hh(??;?1,?2)=?[h(?,?1)h∗(?+??,?2)](3)

 

在这种模型下，信道冲击响应被视作一个广义平稳过程，也就是说信道冲击响应与时间是相互独立的。换句话说，自相关函数是独立于t的，并且他取决于时间差。WWS的自相关函数可以表示为(3)。

 

Uncorrelated Scattering  (US) channel model

Here, the individual scattered components arriving at the receiver front end (at different propagation delays) are assumed to be uncorrelated. Thus the autocorrelation function can be expressed as

?hh(?1,?2;?1,?2)=?hh(?1,?2;?1)?(?1−?2)(4)

 

此时，假设(经过不同的传播路径)到达接收端的相互独立的散射信号是互不相关的。因此自相关函数可以表示为(4)。

 

Wide Sense Stationary Uncorrelated Scattering (WSSUS) channel model

The WSSUS channel model combines the aspects of WSS and US channel model that are discussed above. Here, the channel is considered as Wide Sense Stationary and the scattering components arriving at the receiver are assumed to be uncorrelated. Combining both the worlds, the autocorrelation function Rhh(Δt,τ)is

?hh(??,?)=?[h(?,?)h∗(?+??,?)](5)

 

WSSUS信道模型结合了WSS和US模型的特点。此时，视信道为广义平稳的同时散射信号是互不相关的。因此自相关公式为(5)。

 

Scattering Function

The autocorrelation function of the WSSUS channel model can be represented in frequency domain by taking Fourier transform with respect one or both variables – difference in time Δt and the propagation delay τ. Of the two forms, the Fourier transform on the variableΔt gives specific insight to channel properties  in terms of propagation delay τ and the Doppler Frequency f simultaneously. The Fourier transform of the above two-dimensional autocorrelation function on the variable Δt is called scattering function and is given by

?(?,?)=∫∞−∞?hh(??,?)?−?2???????(6)

根据时间差和时延两个参数的其中之一，或者同时考虑着两种参数，WSSUS信道的自相关函数可以通过傅里叶变换转换到频域上。对于两种不同的形式，针对时间差的傅里叶变换可以同时给出在不同的时延和多普勒频率下的信道属性的内在含义。对于之前所述的二维自相关函数对于变量的傅里叶变换被称为散射方程，有公式(6)表示。

 

Fourier transform of relative time Δt is Doppler Frequency. Thus,  the scattering function is a function of two variables – Dopper Frequency f and the propagation delay τ. It gives the average output power of the channel as a function of Doppler Frequency f and the propagation delay τ.

 

对于相对时间的傅里叶变换是多普勒频率。因而，散射方程是多普勒频率和时延的函数。他给出了信道关于多普勒频率和时延的平均输出功率。

 

Two important relationships can be derived from the scattering function – Power Delay Profile (PDP) and Doppler Power Spectrum. Both of them affect the performance of a given wireless channel. Power Delay Profile is a function of propagation delay and the Doppler Power Spectrum is a function of Doppler Frequency.

 

从散射函数中可以派生出两个重要的关系式：能量延时曲线和多普勒能量谱。两者都与给定的无线信道的性能有关。能量时延曲线是传播时延的函数而多普勒能量谱是多普勒频率的函数。

 

Power Delay Profile p(τ) gives the signal intensity received over a multipath channel as a function of propagation delays. It is obtained as the spatial average of the complex baseband channel impulse response as

 

??=?hh0,?=?h?,?27

 

能量时延曲线以传播时延的函数的形式给出了多径传播后的信号密度。它由复变基带信道响应的空间均值表示为(7)。

 

Power Delay Profile can also be obtained from scattering function, by integrating it over the entire frequency range (removing the dependence on Doppler frequency).

 

??=∫∞−∞??,???8

 

PDP也可以有散射函数给出，通过在整个频域上的积分表示为(8)。

 

Similarly, the Doppler Power Spectrum can be obtained by integrating the scattering function over the entire range of propagation delays.

 

?(?)=∫∞−∞?(?,?)??(9)

 

相似的，多普勒能量谱可以通过散射函数在整个传播时延域上的积分表示为(9)。

 

Fourier Transform of Power Delay Profile and Inverse Fourier Transform of Doppler Power Spectrum:

Power Delay Profile is a function of time which can be transformed to frequency domain by taking Fourier Transform. Fourier Transform of Power Delay Profile is called spaced-frequency correlation function. Spaced-frequency correlation function describes the spreading of a signal in frequency domain. This gives rise to the importance channel parameter – Coherence Bandwidth.

 

PDP是时间的函数，可以通过傅里叶变换转换到频域上。PDP的傅里叶变换被称为间隔频率相关函数。这个函数描述了信号在频域上的展开。由此产生了重要的信道参数：相干带宽。

 

Similarly, the Doppler Power Spectrum describes the output power of the multipath channel in frequency domain. The Doppler Power Spectrum when translated to time-domain by means of inverse Fourier transform is called spaced-time correlation function. Spaced-time correlation function describes the correlation between different scattered signals received at two different times as a function of the difference in the received time. It gives rise to the concept of Coherence Time.

 

相似的，多普勒能量谱在频域上描述了多径信道的输出能量。多普勒能量谱通过IFFT的均值转换到时域上后被称为间隔时间相关函数。间隔时间相关函数以接收时间差的函数的形式描述了两个不同时间接收的不同散射信号之间的相关性。由此产生了相干时间的重要概念。