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Fundamentals of multipath channel modeling for OFDM
OFDM technology promotes signal processing in the frequency domain as a key feature. In order to implement OFDM receiver algorithms, i.e. equalization, we need to create a channel model in the frequency domain. This article constructs a simple frequency domain multipath channel model, following {WalzmanFDM}. Consider
that we have a system with N sub-carriers with a spacing of \Delta f. The input data-vector, after modulation is a sequence of complex
numbers \{c_k\}. The OFDM transmitter transmits them by modulating the nth sub-carrier with the nth symbol from the sequence \{c_k\}.
The complex representation of this is given as
s_t = \sum_{n=1}^N c_k e^{j 2 \pi (f_0 + f_n)t}
If we sample this symbol
at the time instant kT_s, where T_s = \frac{1}{\delta f} and noting that the nth frequency is at (f_0 + n*\delta f) = f_0 + n/T_s and
ignoring the constant carrier offset f_0 i.e. moving the signal to base-band, we note that
\begin{eqnarray}
s(kT_s) &=& \sum_{n=1}^N c_k e^{ j 2 \pi ( n/T_s ).k T_s } \\
&=& \sum_{n=1}^N c_k e^{ j 2 \pi n k }
\end{eqnarray}
which is exactly equivalent to the IFFT of the input vector c_k receiver.
Now, following the model in {channelEstSVD} and others, we assume that the transmitted signal reaches the receiver via L
different paths. Each path has an associated attenuation factor \alpha_i and a path delay \tau_i. This means that the receive signal is
given by
r(t) = \sum_L \alpha_i s(t - \tau_i)
Combining this with equation above, we get the final form.
\begin{eqnarray}
r(kT_s) &=& \sum_{i=1}^L \alpha_i s(kT_s - \tau_i) + n(kT_s)\\
&=& \sum_{i=1}^L \alpha_i \sum_{n=1}^N c_k e^{ j 2 \pi ( n/T_s ).(k T_s - \tau_i) } + n(kT_s)\\
&=& \sum_{n=1}^N c_k \{ \sum_{i=1}^L \alpha_i e^{-j 2 \pi (n/T_s \tau_i)} \} e^{ j 2 \pi ( n/T_s ).(k T_s) } + n(kT_s)\\
R(f_k) &=& R(k/T_s) = DFT(r(kT_s)) = c_k \{ \sum_{i=0}^L \alpha_i e^{-j 2 \pi (k/T_s \tau_i)} + N(kT_s)\}
\end{eqnarray}
The symbol received on the kth sub-carrier is the original symbol, multiplied by
h_k = \sum_{1 \le i \le L} \alpha_i e^{-j(k/T_s \tau_i)}
As we can see, the attenuation factor h_k is frequency sensitive, since it depends on the factor e^{-j k/T_s}. The underlying
physics is as follows: for a given time delay \tau_i, for each frequency, the phase shift caused by the time delay is slightly different,
since each frequency has its own cycle time. The different signals (each corresponding to one path) will combine constructively or
destructively at the end, depending on how much of a phase shift they have undergone relative to each other.
Note that n(t) is the additive noise and $N$ is the representation of the same in the time-domain.
Further, it can be seen that h_k is not independent across sub-carriers, since they are derived from the same path equation as given in equation \eqref{eq:multipath}.
References
{Sari} H.Sari, G.Karam, and I.Jeanclaude., "An analysis of orthogonal frequency division multiplexing for mobile radio applications."
In Vehicular Technology Conference, 1994 IEEE 44th, pages 1635 --1639, June 1994.
{WalzmanFDM} T.Walzman and M.Schwarz., "Automatic equalization using discrete frequency domain",
Information Theory, IEEE Transactions on, 19(1):59 -- 68, Jan 1973.
LTE, Wireless
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