# modelling wireless multi-path channel - is channel response sum of delta(t-path_delay) or "delta" should be smoothed somehow?

Consider the wireless channel. It is mult-path - means that signal is sum of sent signals with some delay and fadings. The standard models for "impulse response" for multi-path channel in wireless is:

$h(t) = c_1 \delta(t - t_1) + c_2 \delta(t - t_2) + ... + c_N \delta(t - t_N)$,

where $t_i$ are path-delays and $c_i$ are fadings.

Question I wonder is this model realistic ? Is it really reasonable to assume "delta"-functions for each path or may be it we need to assume something more smooth ? What is good reference, where the results of real measurements can be found ?

[EDIT] in response to comment.

I am interested in cellur telephone links. I know that the standard e.g. channel 3GPP models assume 6-9 paths with delta-functions. I am suspicious about this assumption. So I want to see results of real measurements which support this model.

[EDIT 2] More details. I am interested in channel estimation for OFDM system cellur system. Assume carrier frequency is something like 1GHz - 2GHz. One user typically may have ~100 OFDM elements. Each element is about 15kHz. So the user bandwidth 1.5MHz.

What am suspicious about is that if we have 9 paths and we have about 100 pilot symbols, then our channel estimation would be quite good , and BER will be close to ideal ch.est. Because we need to estimate ~9*2 unknowns from 100 of data - so we get good quality of estimation. This is in theory, but I am afraid if will test such system in practice we mail fail, cause in practice it might be not 9 paths but 90 paths ...

So let me stress the following part of the question:

Where to find results of real measurements in urban requirement for channel response ?

I guess that should be old stuff may be from 60-ies ? Cause many modern literature concern MIMO channels and pay attention to angular spread and so on - while I need more simple information.

Depends on the physics of the situation. The model is reasonably good if you have secular reflections and it is not very good if you have diffuse reflections. If the size and geometry details of the reflectors are large as compared to the wavelength, than its mostly specular. If the surfaces are uneven are very non flat, then its more diffuse. This is all measured in units of wavelengths. For a 5GHz link that's about 6cm. For an FM radio link that's about 3m. It also depends on the relative bandwidth of the signal. For most wireless links that's pretty narrow, i.e the behavior at the carrier frequency is a good model for the behavior of the overall system. Most natural physical transfer functions such as a reflection of a wall do not change very rapidly with frequency so that's why the singe impulse is a good starting point. Of course there are always exceptions to the rules so it depends on your specific problem.

• can you expand on what you mean by 'diffuse' reflections VS 'specular' ones? What does it mean for a channel to be 'diffuse'? Apr 25 '12 at 14:37
• Consider light reflection: "specular" is a mirror, angle of reflection = angle of incidence and the reflection is an undistorted copy of the original image. Diffuse is a regular white wall, angle of reflections is random and independent of angle of incidence. The original image is lost in the reflection. In terms of the impulse response it means that it's not a delta but spread out in time, fuzzy, and changes quickly with very small changes in the system and only partially correlated to the original signal Apr 25 '12 at 15:30
• @Hilmar Thank you very much for yours answer and comments ! I have added some backround in question, if you can add more comments it would be very helpful. Apr 26 '12 at 7:23

You may try this paper:

Asplund H., et al - How typical is the "Typical Urban" channel model? - Mobile-based Delay Spread and Orthogonality Measurements

to see an actual measurments result, though it's quite old and discusses "Typical Urban" model that aren't used now. I don't know any other results.

But in fact all these models should be used as a reference for lab testing and initial decision making. When designing a real system it should be taken to the fields and tested in actual environments, which are quite various.

Considering your estimation question. You are concerned with measuring channel power delay profile (PDP), i.e. it's fadings and delays. It is one way of estimating a multipath channel, one of many. Other approaches include frequency-domain estimation, like polynomial fitting or various time-domain estimations. If you are interested in concrete methods, there is a great survey on this topic.

Ozdemir M. K., et al - Channel Estimation for Wireless OFDM Systems