# Modeling a Frequency Selective Multipath Fading channel using TDL filters

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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 frequency-selective fading channel.

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

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].

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## References:

• What do you mean by “The first propagation delay τ0τ0 has no effect on the model behavior and hence it can be removed”?

• The first delay block in the filter has no effect on the output. With or without this delay block, the output will remain same. It can be removed.

• In the TDL representation of the three ray model, shouldn’t x(t) enter tau_0, tau_1, and tau_2 blocks?

The way it was represented, shouldn’t the block labels be tau_0, (tau_1 – tau_0), and (tau_2 – tau_1 – tau_0)?

Thanks!

• The filter delays should be marked ∆τ. ∆ is missing in the diagram. Yes, it the difference between the current delay and the previous delay.

• Awais Iqbal

What is the physical significance of “channel tap” in a Raleigh fading channel ? Is it related to number of paths and how to determine the number of taps?
As the single tap flat fading channel is implemented as

Data_length=length(Data); %% Data is any signal
ray = sqrt(0.5*((randn(1,Data_length)).^2+(randn(1,PAR.Data_length)).^2));
Data_sent = Data.*ray;

Now, If I want to model a 6 tap frequency selective channel, how can I modify the above MATLAB code ?

• For a flat fading channel tap is taken as 1. For frequency selective fading, the channel taps (N>1) can be obtained from the power delay profile. Power delay profile (PDP) plot contains the intensity of the received signal in a multipath environment plotted against the delta delay (difference in travel time between multipath arrivals).

The multi-tap channel need to implemented using an FIR filter, where the number of FIR taps depend on the maximum excess delay computed from the PDP. (see here: https://www.gaussianwaves.com/2014/07/power-delay-profile/ ). Once the number of taps are determined, there are methods to determine the tap coefficients and the time delays in the FIR filter. This is topic more complex to describe in the comment section. You could refer the following paper for more details.

M. Paetzold, A. Szczepanski, N. Youssef, Methods for Modeling of Specified and Measured Multipath Power-Delay Profiles,
IEEE Trans. on Vehicular Techn., vol.51, no.5, pp.978-988, Sep.2002

• Awais Iqbal

Thank you..
Can we consider number of paths=number of taps ?