M.4.2 TRP Integration for Constant Step Size Grid Type

38.521-23GPPNRPart 2: Range 2 StandaloneRadio transmission and receptionRelease 17TSUser Equipment (UE) conformance specification

Different approaches to perform the TRP integration from the respective EIRP measurements are outlined in the next sub clauses for the constant step size grid type.

M.4.2.1 TRP Integration using Weights

In many engineering disciplines, the integral of a function needs to be solved using numerical integration techniques, commonly referred to as “quadrature”. Here, the approximation of the integral of a function is usually stated as a weighted sum of function values at specified points within the domain of integration. The derivation from the closed surface TRP integral

to the classical discretized summation equation used for OTA

The weights for this integral are based on the sin⋅ weights. More accurate implementations are based on the Clenshaw-Curtis quadrature integral approximation based on an expansion of the integrand in terms of Chebyshev polynomials. This implementation does not ignore the measurement points at the poles (=0^o and 180^o) where sin = 0. The discretized TRP can be expressed as

which the sin⋅ weights replaced by a weight function W( and extends the sum over I to include the poles There is no simple closed-form expression for the Clenshaw-Curtis weights; however, a numerical straightforward approach is available, i.e.,

with

and

The Clenshaw-Curtis weights are compared to the classical sin ⋅ weights in Tables M.4.2.1-1 and M.4.2.1-2 for two different numbers of latitudes. The TRP measurement grid consists of N+1 latitudes and M longitudes with

and

Table M.4.2.1-1: Samples and weights for the classical sin ⋅ weighting and Clenshaw-Curtis quadratures with 12 latitudes (=16.4^o)

Classical sin		Clenshaw-Curtis
 [deg]	Weights	 [deg]	Weights
0	0	0	0.008
16.4	0.08	16.4	0.079
32.7	0.154	32.7	0.155
49.1	0.216	49.1	0.216
65.5	0.26	65.5	0.26
81.8	0.283	81.8	0.283
98.2	0.283	98.2	0.283
114.6	0.26	114.6	0.26
130.9	0.216	130.9	0.216
147.3	0.154	147.3	0.155
163.6	0.08	163.6	0.079
180	0	180	0.008

Table M.4.2.1-2: Samples and weights for the classical sin ⋅ weighting and Clenshaw-Curtis quadratures with 13 latitudes (=15^o)

Classical sin		Clenshaw-Curtis
 [deg]	Weights	 [deg]	Weights
0	0	0	0.007
15	0.0678	15	0.0661
30	0.1309	30	0.1315
45	0.1851	45	0.1848
60	0.2267	60	0.227
75	0.2529	75	0.2527
90	0.2618	90	0.262
105	0.2529	105	0.2527
120	0.2267	120	0.227
135	0.1851	135	0.1848
150	0.1309	150	0.1315
165	0.0678	165	0.0661
180	0	180	0.007