Explain an impulse response model of multipath channel.

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teamques10 ★ 69k

The small-scale variations of a mobile radio signal can be directly related to the impulse response of the mobile radio channel. Impulse response is wideband channel characterization and contains all information necessary to simulate or analyze any type of radio transmission through the channel.
A mobile radio channel is modelled as a linear filter with a time varying impulse response, where the time variation is due to receiver motion in space. The impulse response is a useful characterization of the channel as it is used to predict and compare the performance of many different mobile communication systems and transmission bandwidths for a particular mobile channel condition.

Consider the case where time variation is due to receiver motion in space. This is shown in figure below. In the given figure the receiver moves along the ground at some constant velocity ν. For a foxed position d, the channel between the transmitter and the receiver can be modelled as a linear time invariant system. Due to the different multipath waves which have propagation delays which vary over different spatial locations of the receiver, the impulse response of the linear time invariant channel should be a function of the receiver. That is the channel impulse response can be expressed as h (d, t). Let x (t) represents transmitted signal then the received signal y(d, t) at distance d can be expressed as convolution of x (t) and h (d, t).

$y(d, t) = x(t) ⊗ h(d, t)=\int_{-∞}^t x(\tau)h(d, t-τ) dτ ..........(A)$

For a causal system, h (d, t) =0 for t<0, thus equation A reduces to,

$y(d, t)=\int_{-∞}^t x(τ) h(d, t-τ) dτ..........(B)$

Since the receiver moves along the ground at constant velocity ν, the position of the receiver can be expressed as,

$d=νt..........(C)$

Substituting equation C in equation B,

$y(νt, t)=\int_{-∞}^t x(τ) h(νt,t-τ) dτ..........(D)$

Since ν is constant y(νt, t) is just a function of t. Therefore equation D can be expressed as,

$y(νt, t)=\int_{-∞}^t x(τ) \ h(νt, t-τ) \ dτ = x(t) ⊗ h(νt, t0 = x(t) ⊗ h(d, t)..........(E)$

From equation E it is clear that the mobile radio channel can be modelled as a linear time varying channel where the channel with time and distance.

Since ν may be assumed constant over a short time interval, we can let x (t) represents the transmitted bandpass waveform, y (t) the received waveform, and h (t, τ) the impulse response of the time varying multipath radio channel. The impulse response can completely characterize the channel and is a function of both t and τ. The variable t represents the time variations due to motion and τ represents the channel impulse multipath delay for fixed value of t. The signal y (t) can be expressed as a convolution of the transmitted signal x (t) with the channel impulse response.

$y(νt, t)=\int_{-∞}^t \ x(τ) \ h( t-τ) \ dτ = x(t) ⊗ h(t, τ)..........(F)$

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