The Cyclic Prefix

Let’s discuss the DFT, before we attempt to equalize the impact of the wireless channel in this lesson.

Linear vs. circular convolution

In a continuous-time Fourier transform, the output of a system $y(t)$ for a given input $x(t)$ and impulse response $h(t)$ is given by:

$$y(t) = x(t) \ast h(t)$$

In the frequency domain, this results in a product between the two respective spectra.

$$Y(f) = X(f)\cdot H(f)$$

On the other hand, for a discrete-time system, such linear convolution can be implemented, but there will be no product in the discrete-frequency domain taken through a DFT.

$$Y[k]\neq X[k]\cdot H[k]$$

Instead, a circular convolution in the time domain generates a product in the frequency domain.

$$y[n] = x[n]\circledast h[n]$$

Only in this case, we get:

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

Wireless channel

Now, the wireless channel doesn’t know about digital processing and discrete domains. The output of a wireless channel is a linear convolution between the transmitted signal and the channel impulse response. So the problem is how do we perform a linear convolution between two signals in which we can only control the input so that the output produces a circular convolution result?

To answer this question, we first consider the result of convolution where one signal is periodic. We call this periodic convolution.