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【MIMO

文章目录

• The Wireless Channel: Propagation and Fading
• • 1.1 Large-Scale Fading
  • • 1.1.1 General Path Loss Model
    • 1.1.2 Okumura/Hata Model
    • 1.1.3 IEEE 802.16d Model
  • 1.2 Small-Scale Fading
  • • 1.2.1 Parameters for Small-Scale Fading
    • • 1. Mean excess delay
      • 2. RMS delay spread
    • 1.2.2 Time-Dispersive vs. Frequency-Dispersive Fading
    • • 1.2.2.1 Fading Due to Time Dispersion: Frequency-Selective Fading Channel
      • 1.2.2.2 Fading Due to Frequency Dispersion: Time-Selective Fading Channel
    • 1.2.3 Statistical Characterization and Generation of Fading Channel
    • • 1.2.3.1 Statistical Characterization of Fading Channel
      • 1.2.3.2 Generation of Fading Channels

The Wireless Channel: Propagation and Fading

Classification of fading channels:

1.1 Large-Scale Fading

1.1.1 General Path Loss Model

1.1.2 Okumura/Hata Model

1.1.3 IEEE 802.16d Model

1.2 Small-Scale Fading

• Small-scale fading: rapid variation of the received signal level in the short term as the user terminal moves a short distance.
• Small-scale fading is attributed to multi-path propagation, mobile speed, speed of surrounding objects, and transmission bandwidth of signal.

1.2.1 Parameters for Small-Scale Fading

• Characteristics of multipath fading channel are often specified by a power delay profile (PDP).
• ‘path’ may also be referred as ‘tap’.

1. Mean excess delay

The mean excess delay τ ‾ \overline\tau τ is given by the first moment of PDP as
τ ‾ = ∑ k a k 2 τ k ∑ k a k 2 = ∑ k τ k P ( τ k ) ∑ k P ( τ k ) \overline{\tau}=\frac{\sum_{k}{a_k^2\tau_k}}{\sum_{k}{a_k^2}}=\frac{\sum_{k}{\tau_kP(\tau_k)}}{\sum_{k}{P(\tau_k)}} τ=∑k​ak2​∑k​ak2​τk​​=∑k​P(τk​)∑k​τk​P(τk​)​
where τ k \tau_k τk​, a k a_k ak​ and P ( τ k ) P(\tau_k) P(τk​) is the channel delay, amplitude and power of the k k kth path, repectively.

2. RMS delay spread

RMS delay spread σ τ \sigma_\tau στ​ is given by the square root of the second central moment of PDP as
σ τ = τ 2 ‾ − ( τ ‾ ) 2 \sigma_\tau=\sqrt{\overline{\tau^2}-(\overline\tau)^2} στ​=τ2−(τ)2 ​
where
τ 2 ‾ = ∑ k a k 2 τ k 2 ∑ k a k 2 = ∑ k τ k 2 P ( τ k ) ∑ k P ( τ k ) \overline{\tau^2}=\frac{\sum_k{a_k^2}\tau_k^2}{\sum_ka_k^2}=\frac{\sum_k{\tau_k^2P(\tau_k)}}{\sum_kP(\tau_k)} τ2=∑k​ak2​∑k​ak2​τk2​​=∑k​P(τk​)∑k​τk2​P(τk​)​
Coherence bandwidth B c B_c Bc​ is generally inversely-proportional to the RMS delay spread, i.e.
B c ≈ 1 σ τ B_c\approx\frac{1}{\sigma_\tau} Bc​≈στ​1​
【About coherence bandwidth: 窄带与宽带】

1.2.2 Time-Dispersive vs. Frequency-Dispersive Fading

Wireless channels can be characterized by two different channel parameters, multipath delay spread and Doppler spread, which cause time dispersion and frequency dispersion, respectively.

1.2.2.1 Fading Due to Time Dispersion: Frequency-Selective Fading Channel

• For the given channel frequency response, frequency selectivity is generally governed by signal bandwidth.
• Due to time dispersion according to multi-paths, channel response varies with frequency.
• Signal bandwidth is narrow: frequency-non-selective fading / flat fading
  • B s ≪ B c B_s \ll B_c Bs​≪Bc​ and T s ≫ σ τ T_s \gg \sigma_\tau Ts​≫στ​
    where B s B_s Bs​ and T s T_s Ts​ are the bandwidth and symbol period of the transmit signal, while B c B_c Bc​ and σ τ \sigma_\tau στ​ is the coherence bandwidth and RMS delay spread.
  • ‘Narrow’ means symbol period T s T_s Ts​ is greater than the delay spread τ \tau τ of the multipath channel h ( t , τ ) h(t,\tau) h(t,τ).
  • The wireless channel maintains a constant (or slowly time-varying) amplitude and linear phase response within a passband.
  • T s T_s Ts​ is greater than τ \tau τ means the current symbol does not affect the subsequent symbol as much over the next symbol period, implying that inter-symbol interference (ISI) is not significant.
• Signal bandwidth is wide: frequency-selective fading
  • B s > B c B_s > B_c Bs​>Bc​ and T s < σ τ T_s < \sigma_\tau Ts​<στ​
  • The channel impulse response has a larger delay spread than a symbol period of the transmit signal, so the multiple-delayed copies of the transmit signal is significantly overlapped with the subsequent symbol, incurring ISI.
  • Frequency-selective fading channel, also referred as wideband channel, since the signal bandwidth is larger than the bandwidth of channel impulse response.
    

1.2.2.2 Fading Due to Frequency Dispersion: Time-Selective Fading Channel

• Depending on the extent of the Doppler spread, the received signal undergoes fast or slow fading.
• Variation in the time domain is related to movement of the transmitter or receiver, which incurs a spread in the frequency domain, known as Doppler shift. f m f_m fm​ is the maximum Doppler shift and B d = 2 f m B_d = 2f_m Bd​=2fm​ is the bandwidth of Doppler spectrum. The coherence time T c T_c Tc​ is inversely proportional to Doppler spread, i.e. T c ≈ 1 / f m T_c \approx 1/f_m Tc​≈1/fm​.
• In a fast fading channel, the coherence time is smaller than the symbol period and thus a channel impulse response quickly varies within the symbol period, i.e.
  T s > T c a n d B s < B d T_s > T_c\quad {\rm and} \quad B_s<B_d Ts​>Tc​andBs​<Bd​
• In a slow fading channel, the channel impulse response varies slowly as compared to variation in the baseband transmit signal, so we can assume that the channel does not change over the duration of one/more symbols (static channel), which implies that the Doppler spread is much smaller than the bandwidth of the baseband transmit signal, i.e.
  T s ≪ T c a n d B s ≫ B d T_s \ll T_c \quad {\rm and}\quad B_s \gg B_d Ts​≪Tc​andBs​≫Bd​

1.2.3 Statistical Characterization and Generation of Fading Channel

1.2.3.1 Statistical Characterization of Fading Channel

• N N N planewaves with arbitrary carrier phases, each coming from an arbitrary direction under the assumption that each planewave has the same average power.

• In the following figure, the planewave arrives from angle θ \theta θ with respect to the direction of terminal movement.
  

• The passband transmit signal is
  x ~ ( t ) = R e [ x ( t ) e j 2 π f c t ] \tilde{x}(t)={\rm Re}\left[x(t)e^{j2\pi f_ct}\right] x~(t)=Re[x(t)ej2πfc​t]
  where x ( t ) x(t) x(t) is the baseband transmit signal.

• Passing through a scattered channel of I I I different propagation paths with different Doppler shifts, the passband received signal can be represented as
  y ~ ( t ) = R e [ ∑ i = 1 I C i e j 2 π ( f c + f i ) ( t − τ i ) x ( t − τ i ) ] = R e [ y ( t ) e j 2 π f c t ] \tilde{y}(t)={\rm Re}\left[\sum_{i=1}^I{C_ie^{j2\pi (f_c+f_i)(t-\tau_i)}x(t-\tau_i)}\right] ={\rm Re}\left[y(t)e^{j2\pi f_ct}\right] y~​(t)=Re[i=1∑I​Ci​ej2π(fc​+fi​)(t−τi​)x(t−τi​)]=Re[y(t)ej2πfc​t]
  where C i C_i Ci​, τ i \tau_i τi​ and f i f_i fi​ denote the channel gain, delay and Doppler shift for the i i ith propagation path, respectively.
  With speed v v v and wavelength λ \lambda λ, the Doppler shift is given as
  f i = f m c o s θ i = v λ c o s θ i f_i=f_m{\rm cos}\theta_i=\frac{v}{\lambda}{\rm cos}\theta_i fi​=fm​cosθi​=λv​cosθi​
  where f m f_m fm​ is the maximum Doppler shift and θ i \theta_i θi​ is the AOA for the i i ith planewave.

• The baseband received signal is
  y ( t ) = ∑ i = 1 I C i e − j ϕ i ( t ) x ( t − τ i ) y(t)=\sum_{i=1}^I{C_ie^{-j\phi_i(t)}x(t-\tau_i)} y(t)=i=1∑I​Ci​e−jϕi​(t)x(t−τi​)
  where ϕ i ( t ) = 2 π { ( f c + f i ) τ i − f i t i } \phi_i(t)=2\pi\{(f_c+f_i)\tau_i-f_it_i\} ϕi​(t)=2π{(fc​+fi​)τi​−fi​ti​}.
  Therefore, the corresponding channel can be modeled as a linear time-varying filter with the following complex baseband impulse response
  h ( t , τ ) = ∑ i = 1 I C i e − j ϕ i ( t ) δ ( t − τ i ) h(t,\tau)=\sum_{i=1}^I{C_ie^{-j\phi_i(t)}\delta(t-\tau_i)} h(t,τ)=i=1∑I​Ci​e−jϕi​(t)δ(t−τi​)

• If the difference is the path delay is much less than the sampling period T s T_s Ts​, then the above equation can be rewrited as
  h ( t , τ ) = h ( t ) δ ( t − τ ^ ) h(t,\tau)=h(t)\delta(t-\hat\tau) h(t,τ)=h(t)δ(t−τ^)
  where h ( t ) = ∑ i = 1 I C i e − j ϕ i ( t ) h(t)=\sum_{i=1}^{I}{C_ie^{-j\phi_i(t)}} h(t)=∑i=1I​Ci​e−jϕi​(t).

• Assuming that x ( t ) = 1 x(t)=1 x(t)=1, the received passband signal is
  y ~ ( t ) = R e [ y ( t ) e j 2 π f c t ] = R e [ { h I ( t ) + j h Q ( t ) } e j 2 π f c t ] = h I ( t ) c o s ( 2 π f c t ) − h Q ( t ) s i n ( 2 π f c t ) \begin{aligned} \tilde{y}(t)&={\rm Re}\left[y(t)e^{j2\pi f_ct}\right]\\ &={\rm Re}\left[\left\{h_I(t)+jh_Q(t)\right\}e^{j2\pi f_ct}\right]\\ &=h_I(t){\rm cos}(2\pi f_ct)-h_Q(t){\rm sin}(2\pi f_ct) \end{aligned} y~​(t)​=Re[y(t)ej2πfc​t]=Re[{hI​(t)+jhQ​(t)}ej2πfc​t]=hI​(t)cos(2πfc​t)−hQ​(t)sin(2πfc​t)​
  where
  h I ( t ) = ∑ i = 1 I C i c o s ϕ i ( t ) , h Q ( t ) = ∑ i = 1 I C i s i n ϕ i ( t ) h_I(t)=\sum_{i=1}^{I}{C_i{\rm cos}\phi_i(t)},\quad h_Q(t)=\sum_{i=1}^{I}{C_i{\rm sin}\phi_i(t)} hI​(t)=i=1∑I​Ci​cosϕi​(t),hQ​(t)=i=1∑I​Ci​sinϕi​(t)

• According to the Central Limit Theorem, h I ( t ) h_I(t) hI​(t) and h Q ( t ) h_Q(t) hQ​(t) can be approximated as Gaussian random variables if I I I is large enough.

  • The amplitude of the received signal y ~ ( t ) = h I 2 ( t ) + h Q 2 ( t ) \tilde{y}(t) = \sqrt{h_I^2(t)+h_Q^2(t)} y~​(t)=hI2​(t)+hQ2​(t) ​ follows the Rayleigh distribution.
  • The power spectrum density (PSD) of the fading process is found by the Fourier transform of the autocorrelation function of y ~ ( t ) \tilde{y}(t) y~​(t)
    S y ~ y ~ ( f ) = { Ω p 4 π f m 1 1 − ( f − f c f m ) 2 , ∣ f − f c ∣ ≤ f m 0 , o t h e r w i s e S_{\tilde{y}\tilde{y}}(f)= \begin{cases} \frac{\Omega_p}{4\pi f_m}\frac{1}{\sqrt{1-\left(\frac{f-f_c}{f_m}\right)^2}},&|f-f_c|\leq f_m\\ 0,&{\rm otherwise} \end{cases} Sy~​y~​​(f)=⎩⎨⎧​4πfm​Ωp​​1−(fm​f−fc​​)2 ​1​,0,​∣f−fc​∣≤fm​otherwise​
    where Ω p = E { h I 2 ( t ) } + E { h Q 2 ( t ) } = ∑ i = 1 I C i 2 \Omega_p=E\left\{h_I^2(t)\right\}+E\left\{h_Q^2(t)\right\}=\sum_{i=1}^{I}C_i^2 Ωp​=E{hI2​(t)}+E{hQ2​(t)}=∑i=1I​Ci2​.
    This PSD is often referred to as the classical Doppler spectrum.
  • If some of the scattering components are much stronger than most of the others, then the amplitude follows the Rician distribution.
    • The strongest scattering component: line-of-sight (LOS) / specular components. All other components: non-line-of-sight (NLOS) / scattering components.
    • The probability density function (PDF) of AoA for all components is
      p ( θ ) = 1 K + 1 p ~ ( θ ) + K K + 1 δ ( θ − θ 0 ) p(\theta)=\frac{1}{K+1}\tilde{p}(\theta)+\frac{K}{K+1}\delta(\theta-\theta_0) p(θ)=K+11​p~​(θ)+K+1K​δ(θ−θ0​)
      where p ~ ( θ ) \tilde p(\theta) p~​(θ) is the PDF of AoA for scattering components, θ 0 \theta_0 θ0​ is the AoA of the specular component, K K K is the Rician factor defined as the ratio of the specular component power c 2 c^2 c2 and the scattering component power 2 σ 2 2\sigma^2 2σ2, i.e.
      K = c 2 2 σ 2 K=\frac{c^2}{2\sigma^2} K=2σ2c2​

1.2.3.2 Generation of Fading Channels

• Any received signal can be considered as the sum of the received signals from an infinite number of scatters. According to the central limit theorem, the received signal can be represented by a Gaussian random variable W 1 + j W 2 W_1+jW_2 W1​+jW2​, where W 1 W_1 W1​ and W 2 W_2 W2​ are the independent and identically-distributed (i.i.d.) Gaussian random variables with zero mean and variance σ 2 \sigma^2 σ2.
• Assume X = W 1 2 + W 2 2 X=\sqrt{W_1^2+W_2^2} X=W12​+W22​ ​ is the amplitude of the complex Gaussian random variable, then X X X is a Rayleigh random variable with the following PDF
  f X ( x ) = x σ 2 e − x 2 2 σ 2 f_X(x)=\frac{x}{\sigma^2}e^{-\frac{x^2}{2\sigma^2}} fX​(x)=σ2x​e−2σ2x2​
  where 2 σ 2 = E { X 2 } 2\sigma^2=E\{X^2\} 2σ2=E{X2}. Furthermore, X 2 X^2 X2 is a chi-square ( χ 2 \chi^2 χ2) random variable (卡方分布).
• In the LOS environment, the amplitude of the received signal can be expressed as X = c + W 1 + j W 2 X=c+W_1+jW_2 X=c+W1​+jW2​ where c c c is the LOS component. X X X is the Rician random variable with PDF
  f X ( x ) = x σ 2 e − x 2 + c 2 2 σ 2 I 0 ( x c σ 2 ) f_X(x)=\frac{x}{\sigma^2}e^{-\frac{x^2+c^2}{2\sigma^2}I_0\left(\frac{xc}{\sigma^2}\right)} fX​(x)=σ2x​e−2σ2x2+c2​I0​(σ2xc​)
  where I 0 ( ⋅ ) I_0(\cdot) I0​(⋅) is the modified zeroth-order Bessel function of the first kind.
  K ∼ − 40 d B K\sim~-40dB K∼ −40dB is Rayleigh fading and K > 15 d B K>15dB K>15dB is Gaussian channel.

【MIMO

文章目录

• The Wireless Channel: Propagation and Fading
• • 1.1 Large-Scale Fading
  • • 1.1.1 General Path Loss Model
    • 1.1.2 Okumura/Hata Model
    • 1.1.3 IEEE 802.16d Model
  • 1.2 Small-Scale Fading
  • • 1.2.1 Parameters for Small-Scale Fading
    • • 1. Mean excess delay
      • 2. RMS delay spread
    • 1.2.2 Time-Dispersive vs. Frequency-Dispersive Fading
    • • 1.2.2.1 Fading Due to Time Dispersion: Frequency-Selective Fading Channel
      • 1.2.2.2 Fading Due to Frequency Dispersion: Time-Selective Fading Channel
    • 1.2.3 Statistical Characterization and Generation of Fading Channel
    • • 1.2.3.1 Statistical Characterization of Fading Channel
      • 1.2.3.2 Generation of Fading Channels

The Wireless Channel: Propagation and Fading

Classification of fading channels:

1.1 Large-Scale Fading

1.1.1 General Path Loss Model

1.1.2 Okumura/Hata Model

1.1.3 IEEE 802.16d Model

1.2 Small-Scale Fading

• Small-scale fading: rapid variation of the received signal level in the short term as the user terminal moves a short distance.
• Small-scale fading is attributed to multi-path propagation, mobile speed, speed of surrounding objects, and transmission bandwidth of signal.

1.2.1 Parameters for Small-Scale Fading

• Characteristics of multipath fading channel are often specified by a power delay profile (PDP).
• ‘path’ may also be referred as ‘tap’.

1. Mean excess delay

The mean excess delay τ ‾ \overline\tau τ is given by the first moment of PDP as
τ ‾ = ∑ k a k 2 τ k ∑ k a k 2 = ∑ k τ k P ( τ k ) ∑ k P ( τ k ) \overline{\tau}=\frac{\sum_{k}{a_k^2\tau_k}}{\sum_{k}{a_k^2}}=\frac{\sum_{k}{\tau_kP(\tau_k)}}{\sum_{k}{P(\tau_k)}} τ=∑k​ak2​∑k​ak2​τk​​=∑k​P(τk​)∑k​τk​P(τk​)​
where τ k \tau_k τk​, a k a_k ak​ and P ( τ k ) P(\tau_k) P(τk​) is the channel delay, amplitude and power of the k k kth path, repectively.

2. RMS delay spread

RMS delay spread σ τ \sigma_\tau στ​ is given by the square root of the second central moment of PDP as
σ τ = τ 2 ‾ − ( τ ‾ ) 2 \sigma_\tau=\sqrt{\overline{\tau^2}-(\overline\tau)^2} στ​=τ2−(τ)2 ​
where
τ 2 ‾ = ∑ k a k 2 τ k 2 ∑ k a k 2 = ∑ k τ k 2 P ( τ k ) ∑ k P ( τ k ) \overline{\tau^2}=\frac{\sum_k{a_k^2}\tau_k^2}{\sum_ka_k^2}=\frac{\sum_k{\tau_k^2P(\tau_k)}}{\sum_kP(\tau_k)} τ2=∑k​ak2​∑k​ak2​τk2​​=∑k​P(τk​)∑k​τk2​P(τk​)​
Coherence bandwidth B c B_c Bc​ is generally inversely-proportional to the RMS delay spread, i.e.
B c ≈ 1 σ τ B_c\approx\frac{1}{\sigma_\tau} Bc​≈στ​1​
【About coherence bandwidth: 窄带与宽带】

1.2.2 Time-Dispersive vs. Frequency-Dispersive Fading

Wireless channels can be characterized by two different channel parameters, multipath delay spread and Doppler spread, which cause time dispersion and frequency dispersion, respectively.

1.2.2.1 Fading Due to Time Dispersion: Frequency-Selective Fading Channel

• For the given channel frequency response, frequency selectivity is generally governed by signal bandwidth.
• Due to time dispersion according to multi-paths, channel response varies with frequency.
• Signal bandwidth is narrow: frequency-non-selective fading / flat fading
  • B s ≪ B c B_s \ll B_c Bs​≪Bc​ and T s ≫ σ τ T_s \gg \sigma_\tau Ts​≫στ​
    where B s B_s Bs​ and T s T_s Ts​ are the bandwidth and symbol period of the transmit signal, while B c B_c Bc​ and σ τ \sigma_\tau στ​ is the coherence bandwidth and RMS delay spread.
  • ‘Narrow’ means symbol period T s T_s Ts​ is greater than the delay spread τ \tau τ of the multipath channel h ( t , τ ) h(t,\tau) h(t,τ).
  • The wireless channel maintains a constant (or slowly time-varying) amplitude and linear phase response within a passband.
  • T s T_s Ts​ is greater than τ \tau τ means the current symbol does not affect the subsequent symbol as much over the next symbol period, implying that inter-symbol interference (ISI) is not significant.
• Signal bandwidth is wide: frequency-selective fading
  • B s > B c B_s > B_c Bs​>Bc​ and T s < σ τ T_s < \sigma_\tau Ts​<στ​
  • The channel impulse response has a larger delay spread than a symbol period of the transmit signal, so the multiple-delayed copies of the transmit signal is significantly overlapped with the subsequent symbol, incurring ISI.
  • Frequency-selective fading channel, also referred as wideband channel, since the signal bandwidth is larger than the bandwidth of channel impulse response.
    

1.2.2.2 Fading Due to Frequency Dispersion: Time-Selective Fading Channel

• Depending on the extent of the Doppler spread, the received signal undergoes fast or slow fading.
• Variation in the time domain is related to movement of the transmitter or receiver, which incurs a spread in the frequency domain, known as Doppler shift. f m f_m fm​ is the maximum Doppler shift and B d = 2 f m B_d = 2f_m Bd​=2fm​ is the bandwidth of Doppler spectrum. The coherence time T c T_c Tc​ is inversely proportional to Doppler spread, i.e. T c ≈ 1 / f m T_c \approx 1/f_m Tc​≈1/fm​.
• In a fast fading channel, the coherence time is smaller than the symbol period and thus a channel impulse response quickly varies within the symbol period, i.e.
  T s > T c a n d B s < B d T_s > T_c\quad {\rm and} \quad B_s<B_d Ts​>Tc​andBs​<Bd​
• In a slow fading channel, the channel impulse response varies slowly as compared to variation in the baseband transmit signal, so we can assume that the channel does not change over the duration of one/more symbols (static channel), which implies that the Doppler spread is much smaller than the bandwidth of the baseband transmit signal, i.e.
  T s ≪ T c a n d B s ≫ B d T_s \ll T_c \quad {\rm and}\quad B_s \gg B_d Ts​≪Tc​andBs​≫Bd​

1.2.3 Statistical Characterization and Generation of Fading Channel

1.2.3.1 Statistical Characterization of Fading Channel

• N N N planewaves with arbitrary carrier phases, each coming from an arbitrary direction under the assumption that each planewave has the same average power.

• In the following figure, the planewave arrives from angle θ \theta θ with respect to the direction of terminal movement.
  

• The passband transmit signal is
  x ~ ( t ) = R e [ x ( t ) e j 2 π f c t ] \tilde{x}(t)={\rm Re}\left[x(t)e^{j2\pi f_ct}\right] x~(t)=Re[x(t)ej2πfc​t]
  where x ( t ) x(t) x(t) is the baseband transmit signal.

• Passing through a scattered channel of I I I different propagation paths with different Doppler shifts, the passband received signal can be represented as
  y ~ ( t ) = R e [ ∑ i = 1 I C i e j 2 π ( f c + f i ) ( t − τ i ) x ( t − τ i ) ] = R e [ y ( t ) e j 2 π f c t ] \tilde{y}(t)={\rm Re}\left[\sum_{i=1}^I{C_ie^{j2\pi (f_c+f_i)(t-\tau_i)}x(t-\tau_i)}\right] ={\rm Re}\left[y(t)e^{j2\pi f_ct}\right] y~​(t)=Re[i=1∑I​Ci​ej2π(fc​+fi​)(t−τi​)x(t−τi​)]=Re[y(t)ej2πfc​t]
  where C i C_i Ci​, τ i \tau_i τi​ and f i f_i fi​ denote the channel gain, delay and Doppler shift for the i i ith propagation path, respectively.
  With speed v v v and wavelength λ \lambda λ, the Doppler shift is given as
  f i = f m c o s θ i = v λ c o s θ i f_i=f_m{\rm cos}\theta_i=\frac{v}{\lambda}{\rm cos}\theta_i fi​=fm​cosθi​=λv​cosθi​
  where f m f_m fm​ is the maximum Doppler shift and θ i \theta_i θi​ is the AOA for the i i ith planewave.

• The baseband received signal is
  y ( t ) = ∑ i = 1 I C i e − j ϕ i ( t ) x ( t − τ i ) y(t)=\sum_{i=1}^I{C_ie^{-j\phi_i(t)}x(t-\tau_i)} y(t)=i=1∑I​Ci​e−jϕi​(t)x(t−τi​)
  where ϕ i ( t ) = 2 π { ( f c + f i ) τ i − f i t i } \phi_i(t)=2\pi\{(f_c+f_i)\tau_i-f_it_i\} ϕi​(t)=2π{(fc​+fi​)τi​−fi​ti​}.
  Therefore, the corresponding channel can be modeled as a linear time-varying filter with the following complex baseband impulse response
  h ( t , τ ) = ∑ i = 1 I C i e − j ϕ i ( t ) δ ( t − τ i ) h(t,\tau)=\sum_{i=1}^I{C_ie^{-j\phi_i(t)}\delta(t-\tau_i)} h(t,τ)=i=1∑I​Ci​e−jϕi​(t)δ(t−τi​)

• If the difference is the path delay is much less than the sampling period T s T_s Ts​, then the above equation can be rewrited as
  h ( t , τ ) = h ( t ) δ ( t − τ ^ ) h(t,\tau)=h(t)\delta(t-\hat\tau) h(t,τ)=h(t)δ(t−τ^)
  where h ( t ) = ∑ i = 1 I C i e − j ϕ i ( t ) h(t)=\sum_{i=1}^{I}{C_ie^{-j\phi_i(t)}} h(t)=∑i=1I​Ci​e−jϕi​(t).

• Assuming that x ( t ) = 1 x(t)=1 x(t)=1, the received passband signal is
  y ~ ( t ) = R e [ y ( t ) e j 2 π f c t ] = R e [ { h I ( t ) + j h Q ( t ) } e j 2 π f c t ] = h I ( t ) c o s ( 2 π f c t ) − h Q ( t ) s i n ( 2 π f c t ) \begin{aligned} \tilde{y}(t)&={\rm Re}\left[y(t)e^{j2\pi f_ct}\right]\\ &={\rm Re}\left[\left\{h_I(t)+jh_Q(t)\right\}e^{j2\pi f_ct}\right]\\ &=h_I(t){\rm cos}(2\pi f_ct)-h_Q(t){\rm sin}(2\pi f_ct) \end{aligned} y~​(t)​=Re[y(t)ej2πfc​t]=Re[{hI​(t)+jhQ​(t)}ej2πfc​t]=hI​(t)cos(2πfc​t)−hQ​(t)sin(2πfc​t)​
  where
  h I ( t ) = ∑ i = 1 I C i c o s ϕ i ( t ) , h Q ( t ) = ∑ i = 1 I C i s i n ϕ i ( t ) h_I(t)=\sum_{i=1}^{I}{C_i{\rm cos}\phi_i(t)},\quad h_Q(t)=\sum_{i=1}^{I}{C_i{\rm sin}\phi_i(t)} hI​(t)=i=1∑I​Ci​cosϕi​(t),hQ​(t)=i=1∑I​Ci​sinϕi​(t)

• According to the Central Limit Theorem, h I ( t ) h_I(t) hI​(t) and h Q ( t ) h_Q(t) hQ​(t) can be approximated as Gaussian random variables if I I I is large enough.

  • The amplitude of the received signal y ~ ( t ) = h I 2 ( t ) + h Q 2 ( t ) \tilde{y}(t) = \sqrt{h_I^2(t)+h_Q^2(t)} y~​(t)=hI2​(t)+hQ2​(t) ​ follows the Rayleigh distribution.
  • The power spectrum density (PSD) of the fading process is found by the Fourier transform of the autocorrelation function of y ~ ( t ) \tilde{y}(t) y~​(t)
    S y ~ y ~ ( f ) = { Ω p 4 π f m 1 1 − ( f − f c f m ) 2 , ∣ f − f c ∣ ≤ f m 0 , o t h e r w i s e S_{\tilde{y}\tilde{y}}(f)= \begin{cases} \frac{\Omega_p}{4\pi f_m}\frac{1}{\sqrt{1-\left(\frac{f-f_c}{f_m}\right)^2}},&|f-f_c|\leq f_m\\ 0,&{\rm otherwise} \end{cases} Sy~​y~​​(f)=⎩⎨⎧​4πfm​Ωp​​1−(fm​f−fc​​)2 ​1​,0,​∣f−fc​∣≤fm​otherwise​
    where Ω p = E { h I 2 ( t ) } + E { h Q 2 ( t ) } = ∑ i = 1 I C i 2 \Omega_p=E\left\{h_I^2(t)\right\}+E\left\{h_Q^2(t)\right\}=\sum_{i=1}^{I}C_i^2 Ωp​=E{hI2​(t)}+E{hQ2​(t)}=∑i=1I​Ci2​.
    This PSD is often referred to as the classical Doppler spectrum.
  • If some of the scattering components are much stronger than most of the others, then the amplitude follows the Rician distribution.
    • The strongest scattering component: line-of-sight (LOS) / specular components. All other components: non-line-of-sight (NLOS) / scattering components.
    • The probability density function (PDF) of AoA for all components is
      p ( θ ) = 1 K + 1 p ~ ( θ ) + K K + 1 δ ( θ − θ 0 ) p(\theta)=\frac{1}{K+1}\tilde{p}(\theta)+\frac{K}{K+1}\delta(\theta-\theta_0) p(θ)=K+11​p~​(θ)+K+1K​δ(θ−θ0​)
      where p ~ ( θ ) \tilde p(\theta) p~​(θ) is the PDF of AoA for scattering components, θ 0 \theta_0 θ0​ is the AoA of the specular component, K K K is the Rician factor defined as the ratio of the specular component power c 2 c^2 c2 and the scattering component power 2 σ 2 2\sigma^2 2σ2, i.e.
      K = c 2 2 σ 2 K=\frac{c^2}{2\sigma^2} K=2σ2c2​

1.2.3.2 Generation of Fading Channels

• Any received signal can be considered as the sum of the received signals from an infinite number of scatters. According to the central limit theorem, the received signal can be represented by a Gaussian random variable W 1 + j W 2 W_1+jW_2 W1​+jW2​, where W 1 W_1 W1​ and W 2 W_2 W2​ are the independent and identically-distributed (i.i.d.) Gaussian random variables with zero mean and variance σ 2 \sigma^2 σ2.
• Assume X = W 1 2 + W 2 2 X=\sqrt{W_1^2+W_2^2} X=W12​+W22​ ​ is the amplitude of the complex Gaussian random variable, then X X X is a Rayleigh random variable with the following PDF
  f X ( x ) = x σ 2 e − x 2 2 σ 2 f_X(x)=\frac{x}{\sigma^2}e^{-\frac{x^2}{2\sigma^2}} fX​(x)=σ2x​e−2σ2x2​
  where 2 σ 2 = E { X 2 } 2\sigma^2=E\{X^2\} 2σ2=E{X2}. Furthermore, X 2 X^2 X2 is a chi-square ( χ 2 \chi^2 χ2) random variable (卡方分布).
• In the LOS environment, the amplitude of the received signal can be expressed as X = c + W 1 + j W 2 X=c+W_1+jW_2 X=c+W1​+jW2​ where c c c is the LOS component. X X X is the Rician random variable with PDF
  f X ( x ) = x σ 2 e − x 2 + c 2 2 σ 2 I 0 ( x c σ 2 ) f_X(x)=\frac{x}{\sigma^2}e^{-\frac{x^2+c^2}{2\sigma^2}I_0\left(\frac{xc}{\sigma^2}\right)} fX​(x)=σ2x​e−2σ2x2+c2​I0​(σ2xc​)
  where I 0 ( ⋅ ) I_0(\cdot) I0​(⋅) is the modified zeroth-order Bessel function of the first kind.
  K ∼ − 40 d B K\sim~-40dB K∼ −40dB is Rayleigh fading and K > 15 d B K>15dB K>15dB is Gaussian channel.

本文标签： MIMO