α_K(f) = k₁ + k₂ * f ^k₃ *l
where

• f is the frequency in Mhz and
• l the length in metres

d=0.35 mm:: k₁=7.9 dB/km, k₂=15.1 dB/km, k₃=0.62
d=0.40 mm: k₁=5.1 dB/km, k₂=14.3 dB/km,k₃=0.59
d=0.50 mm: k₁=4.4 dB/km,k₂=10.8 dB/km,k₃=0.60
d=0.60 mm: k₁=3.8 dB/km,k₂=9.2 dB/km,k₃=0.61.

From these numerical values the following follows.
The attenuation factor α_K(f) and the attenuation function α_K(f) = α(f) * l depend significantly on the pipe diameter.
The cables laid since 1994 with d=0.35 mm and d=0.5 mm have a 10% greater attenuation factor than the older lines with d=0.4 mm or d=0.6 mm. This smaller diameter, which is based on the manufacturing and installation costs, significantly reduces the range l_max of the transmission systems used on these lines, so that in the worst case scenario expensive intermediate regenerators have to be used.

The current transmission methods for copper lines prove only a relatively narrow frequency band, for example 120 kHz with ISDN and ˜1100 kHz with DSL. For f=1MHz the attenuation factor of a 0.4 mm cable is around 20 dB/km, so that even with a cable length of l = 4 km the attenuation does not exceed 80 dB.

The k-parameters of the attenuation factor α_I(f) can be converted into corresponding α-parameters, α_I(f): .

α_I(f) = k₁ + k₂ * f ^k₃
α_II(f) = α₀ + α₁ * f + α₂ * √f

₀∫^{_B[ α_I(f) - α_II(f)]2df is a minimum}

Clearly It is obvious that α₀ = k₁. The parameters α₁ and α₂ are dependent on the underlying bandwidth B and are:

α₁ = 15 * (

B / f0

)^{k₃ - 1} *

k₃ - 0.5 / (k₃ + 1.5)(k₃ + 2)

*

k₂ / f₀

α₂ = 10 * (

B / f0

)^{k₃ - 0.5} *

1 - k₃ / (k₃ + 1.5)(k₃ + 2)

*

k₂ / √ f₀