Derivation of complex channel coefficient of multipath channel

Wireless multi path communication and analysis of complex channel coefficient




In wireless system, multiple components of the signal will reach at the receiver through different
path. It includes direct line of sight path, scattered path, diffracted path and reflected path. The
signal used for detection will be recovered from the total signal received.
In the receiver, due to multipath propagation, multiple copies of same transmitted signal will reach.
It will have two issues
1. Each signal will have different power or magnitude
2. Compared to line of sight signal, other signal will have delay.
Let us consider the ith path between the transmitter and receiver, 𝑎𝑖 represents the amplitude and
𝜏𝑖 represents the delay.
Consider a system with L number of paths, the following figure represents the power delay profile
of the system. The x-axis represents the delay and y axis represents the received signal power.




Let the channel is modeled as a linear time invariant (LTI) channel. Then the impulse response is
ℎ(𝑡) = ∑𝐿−1
𝑖=0 𝑎𝑖 𝛿(𝑡 − 𝜏𝑖 ) (1)
Where 𝛿(𝑡) is unit impulse function.
The input signal x(t) is represented as