The GWSSUS Channel Model

1. Mobile Communications
2. Frequency-Selective Transmission Channels
3. The GWSSUS Channel Model

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Generalized system functions of time variant-systems

Linear time-invariant systems  $\rm (LTI)$  can be completely described with only two system functions,

• the transfer function  $H(f)$  and
• the impulse response  $h(t)$   ⇒   after renaming   $h(\tau)$.

In contrast, four different functions are possible with time-variant systems  $\rm (LTV)$ .  A formal distinction of these functions with regard to time and frequency domain representation by lowercase and uppercase letters is thus excluded.

Therefore a nomenclature change will be made, which can be formalized as follows:

• The four possible system functions are uniformly denoted by  $\boldsymbol{\eta}_{12}$ .
  

• The first subindex is either a  $\boldsymbol{\rm V}$  $($because of German  $\rm V\hspace{-0.05cm}$erzögerung   ⇒   delay time  $\tau)$  or  a  $\boldsymbol{\rm F}$  $($frequency  $f)$.
  

• Either a  $\boldsymbol{\rm Z}$  $($because of German  $\rm Z\hspace{-0.05cm}$eit   ⇒   time  $t)$  or a  $\boldsymbol{\rm D}$  $($Doppler frequency  $f_{\rm D})$  is possible as the second subindex.

Relation between the four system functions

Since, in contrast to line-based transmission, the system functions of mobile communications cannot be described deterministically, but are statistical variables, the corresponding correlation functions must be considered later on. 

In the following, we will refer to these as  $\boldsymbol{\varphi}_{12}$,  and use the same indices as for the system functions  $\boldsymbol{\eta}_{12}$.

• These formalized designations are inscribed in the graphic in blue letters.
• Additionally, the designations used in other chapters or literature are given  (gray letters).  In the other chapters these are also partly used.

${\rm (1)}$   At the top you can see the  »time-variant impulse response«   ${\eta}_{\rm VZ}(\tau,\hspace{0.05cm} t) \equiv h(\tau,\hspace{0.05cm} t)$  in the  "delay–time range".  The associated auto-correlation function  $\rm (ACF)$  is

\[\varphi_{\rm VZ}(\tau_1, t_1, \tau_2, t_2) = {\rm E} \big[ \eta_{\rm VZ}(\tau_1,\hspace{0.05cm} t_1) \cdot \eta_{\rm VZ}^{\star}(\tau_2, t_2) \big]\hspace{0.05cm}. \]

${\rm (2)}$  For the  "frequency–time representation"  you get the  »time-variant transfer function«   ${\eta}_{\rm FZ}(f,\hspace{0.05cm} t) \equiv H(f,\hspace{0.05cm} t)$. 
         The Fourier transform with respect to  $\tau$  is represented in the graph by  ${\rm F}_\tau\hspace{0.05cm}[ \cdot ]$ .  The Fourier integral is written out in full:

\[\eta_{\rm FZ}(f, \hspace{0.05cm} t) = \int_{-\infty}^{+\infty} \eta_{\rm VZ}(\tau,\hspace{0.05cm} t) \cdot {\rm e}^{- {\rm j}\cdot 2 \pi f \tau}\hspace{0.15cm}{\rm d}\tau \hspace{0.05cm}, \hspace{0.3cm} \text{kurz:} \hspace{0.2cm} \eta_{\rm FZ}(f, t) \hspace{0.2cm} \stackrel{f, \hspace{0.05cm} \tau}{\bullet\!\!-\!\!\!-\!\!\!-\!\!\circ} \hspace{0.2cm} \eta_{\rm VZ}(\tau, t) \hspace{0.05cm}.\]

The ACF of this time-variant transfer function is generally:

\[\varphi_{\rm FZ}(f_1, t_1, f_2, t_2) = {\rm E} \big [ \eta_{\rm FZ}(f_1, t_1) \cdot \eta_{\rm FZ}^{\star}(f_2, t_2) \big ]\hspace{0.05cm}.\]

${\rm (3)}$  The  »Scatter–Function«  ${\eta}_{\rm VD}(\tau,\hspace{0.05cm} f_{\rm D}) \equiv s(\tau,\hspace{0.05cm} f_{\rm D})$  corresponding to the left block describes the mobile communications channel in the  "delay–Doppler area".  
         $f_{\rm D}$  describes the  $\text{Doppler frequency}$.   The scatter function results from  ${\eta}_{\rm VZ}(\tau,\hspace{0.05cm} t)$  through Fourier transformation with respect to the second parameter  $t$:

\[ \eta_{\rm VD}(\tau, f_{\rm D}) \hspace{0.2cm} \stackrel{f_{\rm D}, \hspace{0.05cm}t}{\bullet\!\!-\!\!\!-\!\!\!-\!\!\circ} \hspace{0.2cm} \eta_{\rm VZ}(\tau, t)\hspace{0.3cm} \Rightarrow \hspace{0.3cm} \varphi_{\rm VD}(\tau_1, f_{\rm D_1}, \tau_2, f_{\rm D_2}) = {\rm E} \left [ \eta_{\rm VD}(\tau_1, f_{\rm D_1}) \cdot \eta_{\rm VD}^{\star}(\tau_2, f_{\rm D_2}) \right ] \hspace{0.05cm}.\]

${\rm (4)}$  Finally, we consider the so-called  »frequency-variant transfer function«  $\eta_{\rm FD}(f, f_{\rm D})$, i.e. the  "frequency–Doppler representation". 
         According to the graph, this can be reached in two ways:

$$\eta_{\rm FD}(f, f_{\rm D}) \hspace{0.2cm} \stackrel{f_{\rm D}, \hspace{0.05cm}t}{\bullet\!\!-\!\!\!-\!\!\!-\!\!\circ} \hspace{0.2cm} \eta_{\rm FZ}(f, t)\hspace{0.05cm},\hspace{0.5cm}\eta_{\rm FD}(f, f_{\rm D}) \hspace{0.2cm} \stackrel{f, \hspace{0.05cm}\tau}{\bullet\!\!-\!\!\!-\!\!\!-\!\!\circ} \hspace{0.2cm} \eta_{\rm VD}(\tau, f_{\rm D})\hspace{0.05cm}.$$

$\text{Notes:}$ 

• The specified Fourier correlations between the system functions in the graph are illustrated by the outer, dark green arrows and are marked with   ${\rm F}_p\hspace{0.05cm}[\hspace{0.05cm} \cdot \hspace{0.05cm}]$ , 
  the index  $p$  indicates to which parameter  $\tau$,  $f$,  $t$  or  $f_{\rm D}$  does the Fourier transformation refer.

• The inner  (lighter)  arrows indicate the links via the  $\text{inverse Fourier transform}$.   For this we use the notation  ${ {\rm F}_p}^{-1}\hspace{0.05cm}[ \hspace{0.05cm} \cdot \hspace{0.05cm} ]$.

• The applet  "Impulses and Spectra" illustrates the connection between the time and frequency domain, which can be described by formulas using Fourier transformation and Fourier inverse transformation.

Simplifications due to the GWSSUS requirements

The general relationship between the four system functions is very complicated due to non-stationary effects.

Connections between the description functions of the GWSSUS model

Compared to the general model, some limitations have to be made in order to arrive at a suitable model for the mobile communications channel from which relevant statements for practical applications can be derived.

The following definitions lead to the  $\rm GWSSUS$ model 
$( \rm G$aussian  $\rm W$ide  $\rm S$ense  $\rm S$tationary  $\rm U$ncorrelated  $\rm S$cattering$)$:

• The random process of the channel impulse response  $h(\tau,\hspace{0.1cm} t) = {\eta}_{\rm VZ}(\tau,\hspace{0.1cm} t)$  is generally assumed to be complex  (i.e., description in the equivalent low-pass range),  Gaussian  $($identifier  $\rm G)$  and zero-mean  (Rayleigh, not Rice, that means, no line of sight) .
  

• The random process is weakly stationary  ⇒   its characteristics change only slightly with time, and the ACF  $ {\varphi}_{\rm VZ}(\tau_1,\hspace{0.05cm} t_1,\hspace{0.05cm}\tau_2,\hspace{0.05cm} t_2)$  of the time-variant impulse response does not depend on the absolute times  $t_1$  and  $t_2$  but only on the time difference  $\Delta t = t_2 - t_1$.   This is indicated by the identifier  $\rm WSS$    ⇒   $\rm W$ide $\rm S$ense $\rm S$tationary.
  

• The individual echoes due to multipath propagation are uncorrelated, which is expressed by the identifier  $\rm US$   ⇒   $\rm U$ncorrelated $\rm S$cattering.

The mobile communications channel can be described in full according to this graph.  The individual power-spectral densities  (labeled blue)  and the correlation function  (labeled red)  is explained in detail in the following sections.

Auto-correlation function of the time-variant impulse response

We now consider the  $\text{auto-correlation function}$  $\rm (ACF)$  of the time-variant impulse response   ⇒   $h(\tau,\hspace{0.1cm} t) = {\eta}_{\rm VZ}(\tau,\hspace{0.1cm} t)$  more closely.  It shows:

• Based on the  $\rm WSS$ property, the auto-correlation function can be written with  $\Delta t = t_2 - t_1$ :

\[\varphi_{\rm VZ}(\tau_1, t_1, \tau_2, t_2) = \varphi_{\rm VZ}(\tau_1, \tau_2, \Delta t)\hspace{0.05cm}.\]

• Since the echoes were assumed to be independent of each other  $\rm (US$ property$)$, the impulse response can be assumed to be uncorrelated with respect to the delays  $\tau_1$  and  $\tau_2$.  Then:

\[\varphi_{\rm VZ}(\tau_1, \tau_2, \Delta t) = 0 \hspace{0.35cm}{\rm f\ddot{u}r}\hspace{0.35cm} \tau_1 \ne \tau_2\hspace{0.05cm}. \]

• If one now replaces  $\tau_1$  with  $\tau$  and  $\tau_2$  with  $\tau + \Delta \tau$, this auto-correlation function can be represented in the following way:

\[\varphi_{\rm VZ}(\Delta \tau, \Delta t) = \delta(\Delta \tau) \cdot {\it \Phi}_{\rm VZ}(\tau, \Delta t) \hspace{0.05cm}. \]

• Because of the convolution property of the Dirac delta function, the ACF for  $\tau_1 \ne \tau_2$   ⇒   $\Delta \tau \ne 0$ disappears.

• $ {\it \Phi}_{\rm VZ}(\tau, \Delta t) \hspace{0.1cm}$  is the  »delay–time's cross power-spectral density«, which depends on the delay  $\tau \ (= \tau_1 =\tau_2)$  and on the time difference  $\Delta t = t_2 - t_1$ .

$\text{Please note:}$ 

• With this approach, the auto-correlation function  $\varphi_{\rm VZ}(\Delta \tau, \Delta t)$  and the power-spectral density   ${\it \Phi}_{\rm VZ}(\tau, \Delta t) $  are not connected via the Fourier transform as usual, but are linked via a Dirac delta function:

\[\varphi_{\rm VZ}(\Delta \tau, \Delta t) = \delta(\Delta \tau) \cdot {\it \Phi}_{\rm VZ}(\tau, \Delta t) \hspace{0.05cm}. \]

• Not all symmetry properties that follow from the  $\text{Wiener–Khintchine theorem}$  are thus given here.  In particular it is quite possible and even very likely that such a power-spectral density is an odd function.

In the overview in the last section, the  »delay–time's cross power-spectral density«  ${\it \Phi}_{\rm VZ}(\tau, \Delta t) $  can be seen in the top middle.

• Since  $\eta_{\rm VZ}(\tau, t) $,  like any  $\text{impulse response}$,  has the unit  $\rm [1/s]$ , the auto-correlation function has the unit  $\rm [1/s^2]$:

\[\varphi_{\rm VZ}(\Delta \tau, \Delta t) = {\rm E} \left [ \eta_{\rm VZ}(\tau, t) \cdot \eta_{\rm VZ}^{\star}(\tau + \Delta \tau, t + \Delta t) \right ].\]

• But since the Dirac delta function with the time argument   $\delta(\Delta \tau)$ also has the unit  $\rm [1/s]$  the delay–time's cross power-spectral density  ${\it \Phi}_{\rm VZ}(\tau, \Delta t) $  also has the unit $\rm [1/s]$:

\[\varphi_{\rm VZ}(\Delta \tau, \Delta t) = \delta(\Delta \tau) \cdot {\it \Phi}_{\rm VZ}(\tau, \Delta t) \hspace{0.05cm}.\]

Power-spectral density of the time-variant impulse response

Delay's power-spectral density

One obtains the   »delay's power-spectral density«   ${\it \Phi}_{\rm V}(\Delta \tau)$  by setting the second parameter of  ${\it \Phi}_{\rm VZ}(\Delta \tau, \Delta t)$  to  $\Delta t = 0$ .  The graphic on the right shows an exemplary curve.

The delay's power-spectral density is a central quantity for the description of the mobile communications channel.  This has the following characteristics:

• ${\it \Phi}_{\rm V}(\Delta \tau_0)$  is a measure for the "power" of those signal components which are delayed by  $\tau_0$ .  For this purpose, an implicit averaging over all Doppler frequencies  $(f_{\rm D})$  is carried out.
  

• The delay's power-spectral density  ${\it \Phi}_{\rm V}(\Delta \tau)$  has, like  ${\it \Phi}_{\rm VZ}(\Delta \tau, \Delta t)$ , the unit  $\rm [1/s]$.   It characterizes the power distribution over all possible delays  $\tau$.
  

• In the graphic, the power  $ P_0 \approx {\it \Phi}_{\rm V}(\Delta \tau_0)\cdot \Delta \tau$  of those signal components that arrive at the receiver via any path with a delay between  $\tau_0 \pm \Delta \tau/2$. 
  

• Normalizing the power-spectral density  ${\it \Phi}_{\rm V}(\Delta \tau)$  in such a way that the area is  $1$  results in the probability density function  $\rm (PDF)$ of the delay time:

\[{\rm PDF}_{\rm V}(\tau) = \frac{{\it \Phi}_{\rm V}(\tau)}{\int_{0 }^{\infty}{\it \Phi}_{\rm V}(\tau)\hspace{0.15cm}{\rm d}\tau} \hspace{0.05cm}.\]

Note on nomenclature:

• In the book  "Stochastic Signal Theory"  we would have denoted this  $\text{probability density function}$  with $f_\tau(\tau)$.
• To make the relation between  ${\it \Phi}_{\rm V}(\Delta \tau)$  and the PDF clear and to avoid confusion with the frequency $f$  we use the nomenclature given here.
  

$\text{Example 1: Delay models according to COST 207}$

In the 1990s, the European Union founded the working group COST 207 with the aim to provide standardized channel models for cellular mobile communications.  where "COST" stands for  "European Cooperation in Science and Technology".

In this international committee profiles for the delay time  $\tau$  have been developed, based on measurements and valid for different application scenarios.   In the following, four different delay's power-spectral densities are given, where the normalization factor  ${\it \Phi}_0 = {\it \Phi}_{\rm V}(\tau = 0)$  is always used.  The graph shows the PSDs of these profiles in logarithmic representation:

Delay's power-spectral densities according to COST

(1)  profile  $\rm RA$  ("Rural Area"):

\[{\it \Phi}_{\rm V}(\tau)/{\it \Phi}_{\rm 0} = {\rm e}^{ -\tau / \tau_0} \hspace{0.3cm}\text{for}\hspace{0.2cm} 0 < \tau < 0.7\,{\rm µ s}\hspace{0.05cm},\hspace{0.15cm}\tau_0 = 0.109\,{\rm µ s}\hspace{0.05cm}.\]

(2)  profile  $\rm TU$  ("Typical Urban")   ⇒   cities and suburbs:

\[{\it \Phi}_{\rm V}(\tau)/{\it \Phi}_{\rm 0} = {\rm e}^{ -\tau / \tau_0} \hspace{0.3cm}\text{for}\hspace{0.2cm} 0 < \tau < 7\,{\rm µ s}\hspace{0.05cm},\hspace{0.15cm}\tau_0 = 1\,{\rm µ s}\hspace{0.05cm}.\]

(3)  profile  $\rm BU$  ("Bad Urban")   ⇒   unfavourable conditions in cities:

\[{\it \Phi}_{\rm V}(\tau)/{\it \Phi}_{\rm 0} = \left\{ \begin{array}{c} {\rm e}^{ -\tau / \tau_0}\\ 0.5 \cdot {\rm e}^{ (5\,{\rm µ s}-\tau) / \tau_0} \end{array} \right.\quad \begin{array}{*{1}l} \hspace{0.1cm} {\rm for}\hspace{0.3cm} 0 < \tau < 5\,{\rm µ s}\hspace{0.05cm},\hspace{0.15cm}\tau_0 = 1\,{\rm µ s}\hspace{0.05cm}, \\ \hspace{0.1cm} {\rm for}\hspace{0.3cm} 5\,{\rm µ s} < \tau < 10\,{\rm µ s}\hspace{0.05cm},\hspace{0.15cm}\tau_0 = 1\,{\rm µ s} \hspace{0.05cm}. \\ \end{array}\]

(4)  profile  $\rm HT$  ("Hilly Terrain")   ⇒   hilly and mountainous regions:

\[{\it \Phi}_{\rm V}(\tau)/{\it \Phi}_{\rm 0} = \left\{ \begin{array}{c} {\rm e}^{ -\tau / \tau_0}\\ 0.04 \cdot {\rm e}^{ (15\,{\rm µ s}-\tau) / \tau_0} \end{array} \right.\quad \begin{array}{*{1}l} \hspace{-0.25cm} {\rm for}\hspace{0.3cm} 0 < \tau < 2\,{\rm µ s}\hspace{0.05cm},\hspace{0.15cm}\tau_0 = 0.286\,{\rm µ s}\hspace{0.05cm}, \\ \hspace{-0.25cm} {\rm for}\hspace{0.3cm} 15\,{\rm µ s} < \tau < 20\,{\rm µ s}\hspace{0.05cm},\hspace{0.15cm}\tau_0 = 1\,{\rm µ s} \hspace{0.05cm}. \\ \end{array}\]

One can see from the graphics:

• The exponential functions in linear representation now become straight lines.
  

• For logarithmic display, you can read the PSD parameter  $\tau_0$  for  $\rm 10 \cdot lg \ (1/e) = -4.34 \ dB$  as shown in the graph for the  $\rm TU$ profile.

• These four COST profiles are described in the  "Exercise 2.8"  in more detail.

ACF and PSD of the frequency-variant transfer function

The system function   $\eta_{\rm FD}(f, f_{\rm D})$  described in the  "overview in the first section of this chapter"  is also known as the  "Frequency-variant Transfer Function"  where the adjective  "frequency-variant"  refers to the Doppler frequency $f_{\rm D}$.

The associated auto-correlation function  $\rm (ACF)$  is defined as follows:

\[\varphi_{\rm FD}(f_1, f_{\rm D_1}, f_2, f_{\rm D_2}) = {\rm E} \left [ \eta_{\rm FD}(f_1, f_{\rm D_1}) \cdot \eta_{\rm FZ}^{\star}(f_2, f_{\rm D_2}) \right ]\hspace{0.05cm}. \]

By similar considerations as in the  "previous section"  this auto-correlation function can be represented under GWSSUS conditions as follows

\[\varphi_{\rm FD}(\Delta f, \Delta f_{\rm D}) = \delta(\Delta f_{\rm D}) \cdot {\it \Phi}_{\rm FD}(\Delta f, f_{\rm D}) \hspace{0.05cm}.\]

The following applies:

• ${\it \Phi}_{\rm FD}(\Delta f, f_{\rm D})$  is the so-called  »frequency–Doppler's cross power-spectral density«, which is highlighted in the graphic by a yellow background.
  

• The first argument  $\Delta f = f_2 - f_1$  takes into account that ACF and PSD depend only on the frequency difference due to the  "stationarity" .

• The factor  $\delta (\Delta f_{\rm D})$  with  $\Delta f_{\rm D} = f_{\rm D_2} - f_{\rm D_1}$  expresses the  "uncorrelation of the PSD"  with respect to the Doppler shift.
  

• You get from  ${\it \Phi}_{\rm FD}(\Delta f, f_{\rm D})$  to  »Doppler's power-spectral density«   ${\it \Phi}_{\rm D}(f_{\rm D})$  if you set  $\Delta f= 0$. 

• The Doppler's power-spectral density  ${\it \Phi}_{\rm D}(f_{\rm D})$  indicates the power with which individual Doppler frequencies occur.
  

• The  probability density function  $\rm (PDF)$  of the Doppler frequency is obtained from  ${\it \Phi}_{\rm D}(f_{\rm D})$  by suitable normalization.   The PDF has like  ${\it \Phi}_{\rm D}(f_{\rm D})$  the unit  $\rm [1/Hz]$. 

To calculate the Doppler's power-spectral density

\[{\rm PDF}_{\rm D}(f_{\rm D}) = \frac{{\it \Phi}_{\rm D}(f_{\rm D})}{\int_{-\infty }^{+\infty}{\it \Phi}_{\rm D}(f_{\rm D})\hspace{0.15cm}{\rm d}f_{\rm D}} \hspace{0.05cm}.\]

• Often, for example for a vertical monopulse antenna in an isotropically scattered field, the  ${\it \Phi}_{\rm D}(f_{\rm D})$  is given by the  $\text{Jakes spectrum}$ .
  

The frequency–Doppler's cross power-spectral density  ${\it \Phi}_{\rm FD}(\Delta f, f_{\rm D})$  is highlighted in yellow.  The Fourier connections to the neighboring GWSSUS system description functions  ${\it \Phi}_{\rm VD}(\tau, f_{\rm D})$  and  ${\it \varphi}_{\rm FZ}(\Delta f, \Delta t)$ are also marked.

We refer here to the interactive applet  "The Doppler Effect".

ACF and PSD of the delay-Doppler function

The system function shown in the  "overview in the first section of this chapter"  on the left hand side was named  $\eta_{\rm VD}(\tau, f_{\rm D})$ .  

The auto-correlation function  $\rm (ACF)$  of this delay–Doppler function can be written with  $\Delta \tau = \tau_2 - \tau_1$  and  $\Delta f_{\rm D} = f_{\rm D2} - f_{\rm D1}$  taking into account the GWSSUS properties with  $\Delta \tau = \tau_2 - \tau_1$  and  $\Delta f_{\rm D} = f_{\rm D2} - f_{\rm D1}$  as follows

\[\varphi_{\rm VD}(\tau_1, f_{\rm D_1}, \tau_2, f_{\rm D_2}) = \varphi_{\rm VD}(\Delta \tau, \Delta f_{\rm D}) = \delta(\Delta \tau) \cdot {\rm \delta}(\Delta f_{\rm D}) \cdot {\it \Phi}_{\rm VD}(\tau, f_{\rm D}) \hspace{0.05cm}.\]

It should be noted about this equation:

• The first Dirac delta function  $\delta (\Delta \tau)$  takes into account that the delays are uncorrelated  ("Uncorrelated Scattering").

• The second Dirac delta function  $\delta (\Delta f_{\rm D})$  follows from the stationarity  ("Wide Sense Stationary").
  

• The delay–Doppler's power-spectral density  ${\it \Phi}_{\rm VD}(\tau, f_{\rm D})$ , also called  »Scatter Function«  can be calculated from  ${\it \Phi}_{\rm VZ}(\tau, \Delta t)$  or ${\it \Phi}_{\rm FD}(\Delta f, f_{\rm D})$  as follows:

\[{\it \Phi}_{\rm VD}(\tau, f_{\rm D}) ={\rm F}_{\Delta t} \big [ {\it \Phi}_{\rm VZ}(\tau, \Delta t) \big ] = \int_{-\infty}^{+\infty} {\it \Phi}_{\rm VZ}(\tau, \Delta t) \cdot {\rm e}^{- {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2 \pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm D} \hspace{0.05cm}\cdot \hspace{0.05cm}\Delta t}\hspace{0.15cm}{\rm d}\Delta t \hspace{0.05cm},\]

\[{\it \Phi}_{\rm VD}(\tau, f_{\rm D}) = {\rm F}_{f_{\rm D}}^{-1} \big [ {\it \Phi}_{\rm FD}(\Delta f, f_{\rm D}) \big ] = \int_{-\infty}^{+\infty} {\it \Phi}_{\rm FD}(\Delta f, f_{\rm D}) \cdot {\rm e}^{+{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm} 2 \pi \hspace{0.05cm}\cdot \hspace{0.05cm} \tau \hspace{0.05cm}\cdot \hspace{0.05cm} \Delta f}\hspace{0.15cm}{\rm d}\Delta f \hspace{0.05cm}. \]

• Both, the system function  $\eta_{\rm VD}(\tau, f_{\rm D})$  and the derived functions  $\varphi _{\rm VD}(\Delta \tau, \Delta f_{\rm D})$  and  ${\it \Phi}_{\rm VD}(\tau, f_{\rm D})$,  are dimensionless.   For more information on this,  see the specification for  "Exercise 2.6".

• Furthermore, if the GWSSUS requirements are met, the scatter function is equal to the product of the delay's and Doppler's PSDs:

\[{\it \Phi}_{\rm VD}(\tau, f_{\rm D}) = {\it \Phi}_{\rm V}(\tau) \cdot {\it \Phi}_{\rm D}(f_{\rm D})\hspace{0.05cm}.\]

One-dimensional description functions of the GWSSUS model

$\text{Conclusion:}$ 

The figure summarizes the results of this chapter so far.  It should be noted:

(1)   The influence of the delay time  $\tau$  and the Doppler frequency  $f_{\rm D}$  can be separated

• into the blue power-spectral density ${\it \Phi}_{\rm V}(\tau)$
• and the red power-spectral density ${\it \Phi}_{\rm D}(f_{\rm D})$.
  

(2)   The two–dimensional  delay–Doppler's power-spectral density  ${\it \Phi}_{\rm VD}(\tau, f_{\rm D})$  is equal to the product of these two functions.

ACF and PSD of the time-variant transfer function

The following diagram shows all the relationships between the individual power-spectral densities once again in compact form.

Compact summary of all GWSSUS description quantities

This has already been discussed in the last sections:

• the  $\text{delay–time's cross-power-spectral density}$:

$${\it \Phi}_{\rm VZ}(\tau, \Delta t)\hspace{0.55cm}\Rightarrow \hspace{0.3cm}\text{with} \hspace{0.2cm}\Delta t = 0\text{:} \hspace{0.2cm} {\it \Phi}_{\rm V}(\tau),$$

• the  $\text{frequency–Doppler's cross-power-spectral density}$:

$${\it \Phi}_{\rm FD}(\Delta f, f_{\rm D})\hspace{0.3cm}\Rightarrow \hspace{0.3cm}\text{with} \hspace{0.2cm}\Delta f = 0\text{:} \hspace{0.2cm} {\it \Phi}_{\rm D}( f_{\rm D}),$$

• the  $\text{delay–Doppler's cross-power-spectral density}$ :

$${\it \Phi}_{\rm VD}(\tau, f_{\rm D})= {\it \Phi}_{\rm V}(\tau) \cdot {\it \Phi}_{\rm D}(f_{\rm D})\hspace{0.05cm}.$$

The  »frequency–time's correlation function«  (marked yellow in the adjacent graph)  has not yet been considered:

\[\varphi_{\rm FZ}(f_1, t_1, f_2, t_2) = {\rm E} \left [ \eta_{\rm FZ}(f_1, t_1) \cdot \eta_{\rm FZ}^{\star}(f_2, t_2) \right ]\hspace{0.05cm}.\]

Considering again the GWSSUS simplifications and the identity  $\eta_{\rm FZ}(f, \hspace{0.05cm}t) \equiv H(f, \hspace{0.05cm}t)$, the ACF can be also written with  $\Delta f = f_2 - f_1$  and  $\Delta t = t_2 - t_1$  as follows:

\[\varphi_{\rm FZ}(f_1, t_1, f_2, t_2) \hspace{0.3cm} \Rightarrow \hspace{0.3cm}\varphi_{\rm FZ}(\Delta f, \Delta t) = {\rm E} \big [ H(f, t) \cdot H^{\star}(f + \Delta f, t + \Delta t) \big ]\hspace{0.05cm}.\]

It should be noted in this respect:

• You already can see from the name that  $\varphi_{\rm FZ}(\Delta f, \Delta t)$  is a correlation function and not a PSD like the functions  ${\it \Phi}_{\rm VZ}(\tau, \Delta t)$,  ${\it \Phi}_{\rm FD}(\Delta f, f_{\rm D})$  and  ${\it \Phi}_{\rm VD}(\tau, f_{\rm D})$.
  

• The Fourier relationships with the neighboring functions are:

\[{\it \Phi}_{\rm VZ}(\tau, \Delta t) \hspace{0.2cm} \stackrel{\tau, \hspace{0.05cm}\Delta f}{\circ\!\!-\!\!\!-\!\!\!-\!\!\bullet} \hspace{0.2cm} \varphi_{\rm FZ}(\Delta f, \hspace{0.05cm}\Delta t) \hspace{0.2cm} \stackrel{\Delta t,\hspace{0.05cm} f_{\rm D}}{\circ\!\!-\!\!\!-\!\!\!-\!\!\bullet} \hspace{0.2cm} {\it \Phi}_{\rm FD}(\Delta f,\hspace{0.05cm} f_{\rm D}) \hspace{0.05cm}.\]

• If you set the parameters  $\Delta t = 0$  or  $\Delta f = 0$  in this two–dimensional function, the separate correlation functions for the frequency domain or the time domain result:

$$\varphi_{\rm F}(\Delta f) = \varphi_{\rm FZ}(\Delta f, \Delta t = 0) \hspace{0.05cm},\hspace{0.5cm} \varphi_{\rm Z}(\Delta t) = \varphi_{\rm FZ}(\Delta f = 0, \Delta t ) \hspace{0.05cm}.$$

• From the graph it is also clear that these correlation functions correspond to the derived power-spectral densities via the Fourier transform:

\[\varphi_{\rm F}(\Delta f) \hspace{0.2cm} {\bullet\!\!-\!\!\!-\!\!\!-\!\!\circ} \hspace{0.2cm} {\it \Phi}_{\rm V}(\tau)\hspace{0.05cm}, \hspace{0.4cm}\varphi_{\rm Z}(\Delta t) \hspace{0.2cm} {\circ\!\!-\!\!\!-\!\!\!-\!\!\bullet} \hspace{0.2cm} {\it \Phi}_{\rm D}(f_{\rm D})\hspace{0.05cm}.\]

Parameters of the GWSSUS model

According to the results in the last section, the mobile channel is replaced by

• the delay's power-spectral density  ${\it \Phi}_{\rm V}(\tau)$  and
  

• the Doppler's power-spectral density  ${\it \Phi}_{\rm D}(f_{\rm D})$.
  
  

By suitable normalization to the respective area  $1$  the density functions result with respect to the delay time  $\tau$  or the Doppler frequency $f_{\rm D}$.

Characteristic values can be derived from the power-spectral densities or the corresponding correlation functions.  The most important ones are listed here:

$\text{Definition:}$  The  »Multipath Spread«  or  »Time Delay Spread«  $T_{\rm V}$  specifies the spread that a Dirac delta experiences through the channel on statistical average.  
$T_{\rm V}$  is defined as the standard deviation  $(\sigma_{\rm V})$  the random variable  $\tau$:

\[T_{\rm V} = \sigma_{\rm V} = \sqrt{ {\rm E} \big [ \tau^2 \big ] - m_{\rm V}^2} \hspace{0.05cm}.\]

• The mean value  $m_{\rm V} = {\rm E}\big[\tau \big]$  is an  "Average Excess Delay"  for all signal components.

• ${\rm E} \big [ \tau^2 \big ] $  is to be calculated as the root mean square value.

$\text{Definition:}$  The  »Coherence Bandwidth«  $B_{\rm K}$   is the  $\Delta f$  value at which the frequency's correlation function has dropped to half of its value for the first time.

\[\vert \varphi_{\rm F}(\Delta f = B_{\rm K})\vert \stackrel {!}{=} {1}/{2} \cdot \vert \varphi_{\rm F}(\Delta f = 0)\vert \hspace{0.05cm}.\]

• $B_{\rm K}$  is a measure of the minimum frequency difference by which two harmonic oscillations must differ in order to have completely different channel transmission characteristics.

• If the signal bandwidth is  $B_{\rm S} <B_{\rm K}$, then all spectral components are changed in approximately the same way by the channel.
  This means:   Precisely then there is a non-frequency selective fading.

$\text{Example 2:}$  The graphic on the left side shows the delay's power-spectral density  ${\it \Phi}_{\rm V}(\tau)$

Multipath spread and coherence bandwidth

• with  $T_{\rm V} = 1 \ \rm µs$  (red curve),
• with  $T_{\rm V} = 2 \ \rm µs$  (blue curve).

In the right–hand  $\varphi_{\rm F}(\Delta f)$  representation, the coherence bandwidths are drawn in:

• $B_{\rm K} = 276 \ \rm kHz$  (red curve),
• $B_{\rm K} = 138 \ \rm kHz$  (blue curve).

You can see from these numerical values:

• The multipath spread  $T_{\rm V}$,  obtainable from   ${\it \Phi}_{\rm V}(\tau)$ , and the coherence bandwidth  $B_{\rm K}$,  determined by  $\varphi_{\rm F}(\Delta f)$,  stand in a fixed relation to each other:  

$$B_{\rm K} \approx 0.276/T_{\rm V}.$$

• The often  $\text{used approximation}$  $B_{\rm K}\hspace{0.02cm}' \approx 1/T_{\rm V}$  is very inaccurate at exponential  ${\it \Phi}_{\rm V}(\tau)$ .

Let us now consider the time variance characteristics derived from the time correlation function  $\varphi_{\rm Z}(\Delta t)$  or from the Doppler's power-spectral density  ${\it \Phi}_{\rm D}(f_{\rm D})$ :

$\text{Definition:}$  The  »Correlation Time«  $T_{\rm D}$  specifies the average time that must elapse until the channel has completely changed its transmission properties due to the time variance.  Its definition is similar to the definition of the coherence bandwidth:

\[\vert \varphi_{\rm Z}(\Delta t = T_{\rm D})\vert \stackrel {!}{=} {1}/{2} \cdot \vert \varphi_{\rm Z}(\Delta t = 0)\vert \hspace{0.05cm}.\]

$\text{Definition:}$  The  »Doppler Spread«  $B_{\rm D}$  is the average frequency broadening that the individual spectral signal components experience.   The calculation is similar to multipath broadening in that the Doppler spread  $B_{\rm D}$  is calculated as the standard deviation of the random quantity $f_{\rm D}$ :

\[B_{\rm D} = \sigma_{\rm D} = \sqrt{ {\rm E} \left [ f_{\rm D}^2 \right ] - m_{\rm D}^2} \hspace{0.05cm}.\]

• First of all, the Doppler PDF is to be determined from  ${\it \Phi}_{\rm D}(f_{\rm D})$  through area normalization to  $1$. 

• This results in the mean Doppler shift  $m_{\rm D} = {\rm E}[f_{\rm D}]$  and the standard deviation  $\sigma_{\rm D}$.

$\text{Example 3:}$  The diagram is valid for a time–variant channel without direct component.  Shown on the left is the  $\text{Jakes spectrum}$  ${\it \Phi}_{\rm D}(f_{\rm D})$.

Doppler spread and correlation time

The Doppler spread  $B_{\rm D}$  can be determined from this:

\[f_{\rm D,\hspace{0.05cm}max} = 50\,{\rm Hz}\hspace{-0.1cm}: \hspace{-0.1cm}\hspace{0.45cm} B_{\rm D} \approx 35\,{\rm Hz} \hspace{0.05cm},\]

\[f_{\rm D,\hspace{0.05cm}max} = 100\,{\rm Hz}\hspace{-0.1cm}: \hspace{-0.1cm}\hspace{0.2cm} B_{\rm D} \approx 70\,{\rm Hz} \hspace{0.05cm}.\]

The time correlation function  $\varphi_{\rm Z}(\Delta t)$  is sketched on the right, as the Fourier transform of  ${\it \Phi}_{\rm D}(f_{\rm D})$ .

This can be expressed for given boundary conditions with the Bessel function  ${\rm J}_0()$  as:

\[\varphi_{\rm Z}(\Delta t \hspace{-0.05cm} = \hspace{-0.05cm}T_{\rm D}) \hspace{-0.05cm}= \hspace{-0.05cm} {\rm J}_0(2 \pi \hspace{-0.05cm} \cdot \hspace{-0.05cm} f_{\rm D,\hspace{0.1cm}max} \hspace{-0.05cm}\cdot \hspace{-0.05cm}\Delta t ).\]

• The correlation duration of the blue curve is  $T_{\rm D} = 4.84 \ \rm ms$.

• For  $f_{\rm D,\hspace{0.1cm}max} = 100\,{\rm Hz}$  the correlation duration is only half.

• In this case, it generally applies:   $B_{\rm D} \cdot T_{\rm D}\approx 0.17$.

Simulation according to the GWSSUS model

The  »Monte–Carlo method«,  here described for the simulation of a GWSSUS mobile communication channel, is based on work by Rice  [Ric44]  and  Höher [Höh90].

• The two–dimensional impulse response is represented by a sum of  $M$  complex exponential functions.  $M$  can be interpreted as the number of different paths:

\[h(\tau,\ t)= \frac{1}{\sqrt {M}} \cdot \sum_{m=1}^{M} \alpha_m \cdot \delta (t - \tau_m) \cdot {\rm e}^{{\rm j} \hspace{0.05cm} \phi_{m} }\cdot {\rm e}^{ {\rm j} \hspace{0.05cm}2 \pi f_{{\rm D},\hspace{0.05cm} m} t} \hspace{0.05cm}. \]

• First, the delays  $\tau_m$,  the attenuation factors  $\alpha_m$,  the equally distributed phases  $\phi_m$  and the Doppler frequencies  $f_{{\rm D},\hspace{0.1cm} m}$  will be randomly generated according to the GWSSUS specifications.  The base for the random generation of the Doppler frequencies  $f_{{\rm D},\hspace{0.1cm} m}$  is the  $\text{Jakes spectrum}$  ${\it \Phi}_{\rm D}(f_{\rm D})$, which, appropiately normalized, simultaneously indicates the probability density function  $\rm (PDF)$  of the Doppler frequencies.
  

• Because of  ${\it \Phi}_{\rm VD}(\tau, f_{\rm D}) = {\it \Phi}_{\rm V}(\tau) \cdot {\it \Phi}_{\rm D}(f_{\rm D})$  the delay time  $\tau_m$  is independent of the Doppler frequency  $f_{{\rm D},\hspace{0.1cm} m}$  for all  $m$.  This is valid with good approximation for terrestrial land mobile communication.  For the random generation of the parameters  $\alpha_m$  and  $\tau_m$,  which determine the delay–power-spectral density  $ {\it \Phi}_{\rm V}(\tau)$  the following  $\text{COST profiles}$  are available:   $\rm RA$  ("Rural Area"),  $\rm TU$  ("Typical Urban"),  $\rm BU$  ("Bad Urban")  and  $\rm HT$  ("Hilly Terrain").
  

• The greater the number of different paths   $M$  is chosen for the simulation, the better a real impulse response is approximated by the above equation.  However, the higher accuracy of the simulation is at the expense of its duration.  In the literature, favorable values are given for $M$  between  $100$  and  $600$ .
  

Time-variant transfer function  $($the absolute value squared is  simulated$)$

$\text{Example 4:}$  The graphic from  [Hin08]  shows a simulation result:     $20 \cdot \lg \vert H(f, \hspace{0.1cm}t)\vert$  is shown as a 2D plot, where the time-variant transfer function  $H(f, \hspace{0.1cm}t)$  is also referred to as  $\eta_{\rm FZ}(f, \hspace{0.1cm}t)$  in this tutorial.

The simulation is based on the following parameters:

• The time variance results from a movement with  $v = 3 \ \rm km/h$.

• The carrier frequency is  $f_{\rm T} = 2 \ \rm GHz$.
  

• The maximum delay time is  $\tau_{\rm max} \approx 0.4 \ \rm µ s$.

• According to the approximation we obtain the coherence bandwidth  $B_{\rm K}\hspace{0.02cm}' \approx 2.5 \ \rm MHz$.
  

• The maximum Doppler frequency is  $f_\text{D, max} \approx 5.5 \ \rm Hz$.

• The Doppler spread results in  $B_{\rm D} \approx 4 \ \rm Hz$.

Exercises for the chapter

Exercise 2.5: Scatter Function

Exercise 2.5Z: Multi-Path Scenario

Exercise 2.6: Dimensions in GWSSUS

Exercise 2.7: Coherence Bandwidth

Exercise 2.7Z: Coherence Bandwidth of the LTI Two-Path Channel

Exercise 2.8: COST Delay Models

Exercise 2.9: Coherence Time