Tapped-delay line filters (basically FIR filters) are best to simulate multiple echoes originating from same source. Hence they can be used to model multipath scenarios.

## Choose frequency-flat or frequency selective:

If the TDL filter has only one tap (\(N=1\)), it simulates a ** frequency-flat fading** channel. In contrast, if \(N>1\), then it simulates a

*channel.*

**frequency-selective fading**Each path in the multipath fading model is associated with a corresponding attenuation factor (\(a_n\)) and the path-delay (\(\tau_n\)). In continuous time, the complex path attenuation suffered by a signal on a given path is given by

$$ \tilde{a_n}(t) = a_n(t) exp\left[ -j 2 \pi f_c \tau_n(t) \right] $$

The complex channel response is given by

$$ h(t,\tau) = \sum_{n=0}^{N-1} \tilde{a}_n(t) \delta(t-\tau_n(t)) $$

In the equation above, the attenuation and path delay vary with time. This simulates a * time-variant complex channel*.

If we were to simulate a scenario where there is absolutely no movements or other changes in the transmission channel, the channel can remain fairly time invariant. Thus the * time-invariant complex channel* becomes

$$ h(t) = \sum_{n=0}^{N-1} \tilde{a}_n \delta(t-\tau_n) $$

Usually, the pair (\(a_n,\tau_n\)) is described as a * Power Delay Profile (PDP)* plot. A sample power delay profile plot for a fixed, discrete, three ray model with its corresponding implementation using a tapped-delay line filter is shown in the following figure

## Choose underlying distribution:

The next level of modeling involves, introduction of randomness in the above mentioned model there by rendering the channel response time variant. If the path attenuations are typically drawn from a complex Gaussian random variable, then at any given time \(t\), the absolute value of the impulse response \(\left |h(t,\tau)\right |\) is

- Rayleigh distributed – if the mean of the distribution \(E [h(t; \tau)] = 0\)
- Rician distributed – if the mean of the distribution \(E [h(t; \tau)] \neq 0\)

Respectively, these two scenarios model the * presence or absence of a Line of Sight (LOS) path* between the transmitter and the receiver. The first propagation delay \(\tau_0\) has no effect on the model behavior and hence it can be removed.

Similarly, the propagation delays can also be randomized, resulting in a more realistic but extremely complex model to implement. Furthermore, the power-delay-profile specifications with arbitrary time delays, warrant non-uniformly spaced tapped-delay-line filters, that are not suitable for practical simulation. For ease of implementation, the given PDP model with arbitrary time delays can be converted to tractable uniformly spaced statistical model by a combination of interpolation/approximation/uniform-sampling of the given power-delay-profile.

## Real-life modelling:

Usually continuous domain equations for modeling multipath are specified in standards like COST-207 model in GSM specification. Such continuous time power-delay-profile models can be simulated using discrete-time Tapped Delay Line (TDL) filter with \(N\) number of taps with variable tap gains. Given the order \(N\), the problem boils down to determining the discrete tap spacing \(\Delta t\) and the path gains \(a_n\), in such a way that the simulated channel closely follows the specified multipath PDP characteristics. A survey of method to find a solution for this problem can be found in [1].