Mobile radio channel may be modelled as a linear filter with time varying impulse response in continuous time. To show this, consider time variation due to receiver motion and time varying impulse response h(d, t) and x(t), the transmitted signal. The received signal y(d, t) at any position d would be

$ y\left(d,\ t\right)=x\left(t\right)*h\left(d,t\right)=\int_{-\infty{}}^{\infty{}}x\left(\tau{}\right)h\left(d,t-\tau{}\right)d\tau{} $

For Causal System : h(d, t)=0, for t < 0 and

for a stable system( \int_{-\infty{}}^{\infty{}}\left\vert{}h\left(d,t\right)\right\vert{}dt<\infty{} )

Applying Causality condition in the above equation, h(d, t-τ) = 0 for t-τ < 0

τ > t, i.e. the integral limits are changed to

$y\left(d,\ t\right)=\int_{-\infty{}}^tx\left(\tau{}\right)h\left(d,t-\tau{}\right)d\tau{}$

Since the receiver moves along the ground at a constant velocity v, the position of the receiver is d = vt, i.e.,

$y\left(vt,\ t\right)=\int_{-\infty{}}^tx\left(\tau{}\right)h\left(vt,t-\tau{}\right)d\tau{} $

Since v is a constant, y(vt, t) is just a function of t. Therefore the above equation can be expressed as

$y\left(vt,\ t\right)=\int_{-\infty{}}^tx\left(\tau{}\right)h\left(vt,t-\tau{}\right)d\tau{}=x\left(t\right)*h\left(vt,t\right)=x\left(t\right)*d\left(t\right) $

It is useful to discretize the multipath delay axis τ of the impulse response into equal time delay segments called excess delay bins, each bin having a time delay width equal to (τ i+1 - τ i) = ∆τ and τ i = i ∆τ for i is an element of (0,1,2,3...N-1) where N represents the total number of possible equally-spaced multipath components, including the first arriving component. The useful frequency span of the model is 2/∆τ. The model may be used to analyze transmitted RF signals having bandwidth less than 2/∆τ .

If there are N multipaths, maximum excess delay is given by N∆τ .

$\left\{y\left(t\right)=x\left(t\right)*h\left(t,\tau{}\ i\right)\vert{}i=0,1, ...,N-1\right\} $

Bandpass channel impulse response model is

$x\left(t\right)\rightarrow{}h\left(t,\tau{}\right)=Re\{h_b\left(t,\tau{}\right)e^{jwct}\rightarrow{}y\left(t\right)=Re\{r\left(t\right)e^{jwct}\}$

Baseband equivalent channel impulse response model is given by

$\left(t\right)\rightarrow{}\ \frac{1}{2}h_b\left(t,\tau{}\right)\rightarrow{}r\ \left(t\right)=c\left(t\right)*\ \frac{1}{2}h_b(t,\tau{}) $

Average power is

$\frac{\ }{x^2(t)}=\frac{1}{2}\vert{}c\left(t\right){\vert{}}^2 $

The baseband impulse response of a multipath channel can be expressed as

$h_b\left(t,\tau{}\right)=\ \sum_{i=0}^{N-1}\ a_i\left(t,\tau{}\right)\ exp⁡[j\left(2\pi{}f_c{\tau{}}_i\left(t\right)+{\varphi{}}_i\left(t,\tau{}\right)\right)]\delta{}(\tau{}-{\tau{}}_i\left(t\right)) $

where ai(t, τ ) and τ i(t) are the real amplitudes and excess delays, respectively, of the ith multipath component at time t. The phase term 2fc τ i(t) +φi(t, τ ) in the above equation represents the phase shift due to free space propagation of the ith multipath component, plus any additional phase shifts which are encountered in the channel. If the channel impulse response is wide sense stationary over a small-scale time or distance interval, then

$h_b\left(\tau{}\right)=\ \sum_{i=0}^{N-1}\ a_iexp⁡[j\theta{}i]\delta{}(\tau{}-{\tau{}}_i) $

For measuring hb(τ ), we use a probing pulse to approximate (t) i.e.

$p(t)\approx{}\delta{}(t-\tau{}) $

Power delay profile is taken by spatial average of |hb(t, τ)|2 over a local area. The received power delay profile in a local area is given by

$p(\tau{})\approx{}\frac{\ }{k\vert{}h_b\left(t;\tau{}\right){\vert{}}^2} $