Learn how the effect of a wireless channel is removed through the equalization process.

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After the insertion of the CP, the signal is ready to be sent over the wireless channel. This is done through a process known as upconversion in which the modulated signal is mixed with a sinusoid of a particular frequency intended for the signal.

Role of channel estimates

We have learned that a linear convolution occurs between the transmitted signal x[n]x[n] (that has a cyclic prefix attached) and the wireless channel h[n]h[n].

y[n]=x[n]h[n]=m=0N1x[m]h[nm]y[n]=x[n]\ast h[n]=\sum_{m=0}^{N-1}x[m]h[n-m]

This is true for n=0,1,,N1n=0,1,\cdots,N-1. We have also investigated in an earlier lesson why this results in their spectral product. At the FFT output, we have

Y[k]=X[k]H[k]Y[k]= X[k]\cdot H[k]

for k=0,1,,N1k = 0,1, \cdots,N-1. Clearly, each symbol X[k]X[k] is distorted by a complex number H[k]H[k]. To recover all NN symbols, we either need to know the wireless channel coefficient h[n]h[n] or its DFT H[k]H[k].

For this purpose, known sequences (pilots or training) are sent along with the information symbols. At locations where the symbols X[k]X[k] are known, the channel can be estimated as:

Y[k]=X[k]H[k]H^[k]=Y[k]X[k],  for all kY[k]=X[k]\cdot H[k] \quad \longrightarrow \quad \hat H[k]=\frac{Y[k]}{X[k]}, ~~\text{for all~}k


Since all the remaining locations at which the data symbols are known, these channel estimates can be used as:

Y[k]=X[k]H[k]X^[k]=Y[k]H[k],  for all kY[k]=X[k]\cdot H[k] \quad \longrightarrow \quad\hat X[k] = \frac{Y[k]}{H[k]},~~\text{for all~}k

Let’s see this process in code. To avoid complexity, we assume that channel estimates H^[k]\hat H[k] are already available at the receiver.

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