University:
College
Course:
College Engineering
Academic year:

2023
Views:

387

Pages:

2
Author:

D. Krishnan

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 ) Where 𝛿(𝑡) is unit impulse function. The input signal x(t) is represented as (1) 𝑥(𝑡) = 𝑠𝑏 (𝑡)exp(𝑗2π𝑓𝑐 𝑡) (2) The output, y(t) is the convolution of x(t) and impulse response of the channel. 𝑦(𝑡) = 𝑥(𝑡) ⊗ ℎ(𝑡) (3) ⊗ - represents convolution operator. Substitute equation (1) and (2) in equation (3) 𝑦(𝑡) = 𝑅𝑒{𝑠𝑏 (𝑡)exp(𝑗2π𝑓𝑐 𝑡)} ⊗ ∑𝐿−1 𝑖=0 𝑎𝑖 𝛿(𝑡 − 𝜏𝑖 ) = 𝑅𝑒{∑𝐿−1 𝑖=0 𝑎𝑖 𝑠𝑏 (𝑡 − 𝜏𝑖 )exp(𝑗2π𝑓𝑐 (𝑡 − 𝜏𝑖 ))} (4) (5) After demodulation 𝑦(𝑡) becomes 𝑦𝑏 (𝑡) 𝑦𝑏 (𝑡) = ∑𝐿−1 𝑖=0 𝑎𝑖 𝑠𝑏 (𝑡 − 𝜏𝑖 )exp(−𝑗2π𝑓𝑐 𝜏𝑖 ) (6) i.e., high frequency term, exp(2π𝑓𝑐 𝑡) is eliminated. In equation (6), when 𝜏𝑖 is very small 𝑠𝑏 (𝑡 − 𝜏𝑖 ) ≅ 𝑠𝑏 (𝑡) So 𝑦𝑏 (𝑡) = 𝑠𝑏 (𝑡) ∑𝐿−1 𝑖=0 𝑎𝑖 exp(−𝑗2π𝑓𝑐 𝜏𝑖 ) (7) 𝑦𝑏 (𝑡) = 𝑠𝑏 (𝑡). ℎ (8) Here ℎ is complex fading coefficient. i.e., ℎ = ∑𝐿−1 𝑖=0 𝑎𝑖 exp(−𝑗2π𝑓𝑐 𝜏𝑖 ) , For example, when L= 2, 𝑎0 = 𝑎0 = 1, 𝜏𝑖 = 1 𝑓𝑐 , 1 ℎ = 𝑎0 + 𝑎1 exp(−𝑗2π𝑓𝑐 . 𝑓 ) = 1+ exp(−𝑗2π) , 𝑐 here exp(−𝑗2π) = cos(2π) − 𝑗 sin(2π) = 1 ℎ = 1 + 1 = 2, i.e., constructive interference occurs from complex fading coefficient. Similarly destructive interference occurs when 𝜏𝑖 = 1 2𝑓𝑐 .