The Cyclic Prefix

Explore how linear convolution can be converted into circular convolution with the help of 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)y(t) for a given input x(t)x(t) and impulse response h(t)h(t) is given by:

y(t)=x(t)h(t)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)H(f)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]X[k]H[k]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]h[n]y[n] = x[n]\circledast h[n]

Only in this case, we get:

Y[k]=X[k]H[k]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.

Get hands-on with 1200+ tech skills courses.