The difference between operating a single radar node and an entire net- work is the increased interference caused by collisions, or co-channel interference.
To model this interference, a stochastic model is applied for the interferer geometry.3 The reference node is located at the origin of a plane. The position of the other, interfering, nodes are randomly determined by a
3Stochastic geometry as a tool for research on vehicular networks has previously been proposed in [62], which suggests its suitability in this context.
two-dimensional Poisson Point Process (PPP) with densityλ. Fig. 4.2 illustrates such a scenario.
Choosing a PPP to model the geometry allows the application of results of stochastic geometry to analyze the interference. The main reason PPPs are used, however, is the application OFDM radar networks is intended for, where the nodes are not stationary. This mobility causes a high amount of “spatial randomness”. Such a spatial model has been shown to capture properly these random spatial dynamics affecting the interference [118].
It is important to realise that all nodes represented by this PPP are OFDM transmitters of the same type as the reference node. Because of the homogeneous setup, the results for the reference node are represen- tative for all other nodes as well.
Unlike in the previous chapter, it is no longer possible to neglect the azimuth dependency. As this is a radar system, the azimuthφ of the targets must be estimated as well as the range and Doppler. How the radar system implementation solves this problem is irrelevant for this study; what matters is that the angular resolution results in a receiver directivity which can be expressed as azimuth-dependant antenna gain G(φ). The transmitters are assumed to be omnidirectional so they can communicate with all other nodes. Section 4.3.1 discusses this in greater detail.
For the radar processing methods from Section 3.3 to still be applicable, the total interference must be WGN. Conditioning on a certain spatial configuration, assumeI interferers, withFIx,ibeing the transmit frame of the i-th interferer. The noise matrix Z now does not only contain the receiver noise, but also energy from the interfering transmit symbols.
By assuming synchronous interference (see Section 4.3), the total noise matrix can be analyzed element-wise:
(Ztotal)k,l=Zk,l+
I−1X
i=0
pbi(FIx,i)k,l
(FTx)k,lejϕi. (4.10) Note that the (FIx)k,lare zero for interferers which use a different channel from the reference node, assuming perfect orthogonality.
On top of the receiver noise,Ztotalnow contains a sum of complex values with random amplitude bi and phase ϕi; the latter can be modelled
as uniformly distributed within [0,2π). The former is modelled by an exponential path loss,
bi=gi β
rαi G(φi), (4.11)
where ri is the distance to the origin, φi the azimuth and α the path loss exponent. β is a constant attenuation factor which is assumed to fulfill
β=PTx
c20
(4π)2fC2 (4.12)
in correspondence with free space path loss. gi is an optional random small-scale power attenuation parameter with distribution functionFg(g) (as the fading of the nodes is identically distributed, the distribution itself
does not depend oni). Section 4.4 discusses the cases where gi= 1 (i.e.
no fading), or i.i.d. exponentially distributed with unit mean (Rayleigh fading), but ifFg(g) is known, other types of fading can be analyzed in the same fashion. This fading parameter covers multi-path propagation of the interference signals.
In the special case where the modulation has constant amplitude (e.g.
as in PSK) and the amplitude is Rayleigh distributed, Ztotal is a sum of i.i.d. random variables, and therefore is normal distributed. For the more general case where thebifollow any distribution (the definition of the path loss (4.11) states that thebi depend on the distance of the in- terferers to the reference node and are thusnot identically distributed), the central limit theorem is considered, which suggests that a small num- ber of summands (ten to twelve) suffice for Ztotal to be approximately Gaussian [106, Chap. 2].
For the rest of this chapter, the index “total” shall be omitted and Z is used to describe the compound noise and interference with total two-sided noise power σN2 + ˜Y, where ˜Y denotes the random variable4 representing the total interference power.
4.3.1 Influence of directivity
As stated above, the nodes transmit omnidirectionally, in order to allow broadcasting. The radar processing unit however must have a way to de-
4In the following, all the random variables related to the PPP are typeset without serifs.
0◦ φ0
90◦
180◦
270◦ 0.0
0.5 1.0
(a)Gcone(φ)
0◦ φ0
90◦
180◦
270◦ 0.0
0.5 1.0
(b)Gsinc(φ) Figure 4.3: Gain functions Gcone(φ) andGsinc(φ)
tect the azimuth of an object, and therefore have an azimuth-dependant receiver gain. As a consequence, an interferer causes less interference when its transmission originates from a different angle than the refer- ence target (see also Fig. 4.2).
This must be factored in when calculating the total interference power, note thatG(φ) is already part of (4.11).
The actual shape ofG(φ) is highly dependent on the specific implementa- tion. If applicable, an approximation ofG(φ) by a simple representation can help to obtain manageable analytic results.
Two directivity functions used in the following are the cone shape, Gcone(φ) =1|φ|<φ
0, −π
2 ≤φ < π
2 (4.13)
and a sinc shape,
Gsinc(φ) = sinc2 φ
φ0
, −π
2 ≤φ < π
2. (4.14)
Both are defined by a beam widthφ0and are depicted in Fig. 4.3. It must be emphasized that these directivity functions are crude approximations of realistic beam shapes, but are useful for analytical derivations.