Channel models suitable for studying DSSS

The typical channels that model the multipath behaviour of the wireless link are either a combination of the exponential channel model, or the Rayleigh channel model, together with AWGN [1]. These models present the most suitable conditions for the wireless environment studied in the DSSS systems.
1. AWGN channel
System simulations for this channel correspond to the simplest case: an AWGN channel which was programmed as a random signal in [2]. This signal is characterized by having no statistical correlation between two values in the time domain. As a result, its power spectral density is a constant. Therefore, it is a flat graph, which means that it contains all the frequencies and all of them have the same power. This is the reason why it is named “white”. As its name indicates, its probability density function follows a Gaussian or normal distribution. Hence, both characteristics describe this type of channel. In addition, it is additive because is added to the transmitted signal in the receiver’s input. This channel does not introduce delays or attenuation in the transmitted signal; it only adds the random sequence described above. Its power its described by the following relation:
awgn power
where
noise density
is the noise density, which depends on the energy per bit, Eb.
2. Rayleigh channel and equal power multipath channel 
In the multipath Rayleigh channel model, taps are distributed with uniform random phase and Rayleigh distribution for the amplitudes. In this model, the delay is considered smaller than the sample rate and it simulates an environment where fading occurs. An important parameter to characterize this channel is the rms delay spread, which in comparison with the symbol duration, indicates whether the channel introduces flat fading (when it is smaller) or selective fading (when it is greater). The rms can be computed by applying the following expression:
rms

As mentioned, the paths in this channel are equal power, which means that they are independent and identically distributed (i.i.d.) Gaussian variables.

3. Exponentially decaying channel 
The exponential model is characterized by independent Gaussian variables which represent taps that have an exponentially decaying average power profile. This model is consistent for narrowband systems. In order to model this channel, we need to describe the following parameters:
  • The standard deviation is used to normalize the taps power to 1 and it is defined as:
standard deviation expression
K is the number of paths introduced by the channel, and according to the IEEE802.11b, it can be truncated in order to have a finite impulse response as:
kmax
The variance is expressed as:
variance expression

Ts is the sampling period, and σk is a normalization factor, defined as:

standard deviation

The expression above has a real and an imaginary part. Therefore, the standard deviation results in:

std
We can fix τrms to typical values for indoor environments, such as 30 ns.
REFERENCES
[1] S. Halford, K. Halford, and M. Websteri (2000). Evaluating the performance of HRb proposals in the presence of multipath, Intersil Corporation, doc.: Ieee 802.11-00/xxx

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