# related to random nature of wireless channel

I am reading An Approximate BER Analysis for Ambient Backscatter Communication Systems With Tag Selection wherein it is mentioned that "when the distance between two nodes is very small and line of sight is maintained then a random wireless channel can be assumed to behave as a constant".

I am trying to understand how the above statement is true ? Is there any formula for it?

• "I'm reading a paper": So cite that paper properly, context never hurts! Aug 11, 2021 at 9:30
• well do you understand how wave propagation works? Aug 11, 2021 at 9:30
• "An Approximate BER Analysis for Ambient Backscatter Communication Systems With Tag Selection" This is the paper wherein the assumption about constant channel is given in Remark 3.
– paru
Aug 11, 2021 at 10:45
• thanks, but this throws us back at my other question: do you understand how the channel your signal perceives and the physics of wave propagation relate? Aug 11, 2021 at 10:52
• Thanks for your quick response....Yes I know the physics behind wave propogation. An EM wave when moves away from source then its strength decreases i.e., its strength is inversely proportional to the distance.... But not getting how the channel and distance are related.
– paru
Aug 11, 2021 at 10:55

A VERY simple model of a a relatively narrow band propagation can be built as the sum of the direct contribution and the reflections of the environment as treating all of this as spherical waves. The transfer function of the channel would look like

$$H(\omega) = \frac{r_{ref}}{r_d} e^{-j k \cdot r_d}+ \sum_n R_n\frac{r_{ref}}{r_k} e^{-j k \cdot r_d}$$

Where $$r_ref$$ is a suitable reference distance, $$r_d$$ the direct distance between sender and receiver, $$r_n$$ the total travel path length of reflection path n, $$R_n$$ the cumulative reflection coefficient of path n and $$k = \omega/c$$ the wave number.

If the length of any path changes then the phase term $$e^-kr$$ changes as well and hence the entire transfer function changes. The amplitude of each path does also change with $$r$$ but the effect is typically a lot smaller.

Reflections have a tendency to change frequently since the stuff the wave bounces of moves: door opens, people walk around, etc.

If the distance between transmitter and receiver is small, then direct path is a lot shorter than the reflections and we can assume $$1/r_d >> 1/r_n$$ and hence the direct path dominates and the reflections can be ignored. In this case the transfer function remains constant as long as neither transmitter or receiver move.

That isn't always the case: For example if the transmitter have strong directivity and the receiver is in a null or notch of the transmitter's directivity pattern, then the direct path can be attenuated quite a bit and the reflections matter again.