# cyclic prefix

Cyclic prefix takes the last μ values of the signal $$x$$ and prefixes them in front.

The new sequence with the prefix, $$\tilde{x}[n]$$ is defined from $$-\mu \leq n \leq N-1$$ (making it length $$N+\mu$$) as

\begin{equation} \tilde{x}[n] = x[N-\mu], x[N-\mu - 1], \dots, x[N-1], x, x, \dots, x[N-1] \end{equation}

Note the property that $$\tilde{x}[n - k] = x[n-k (mod N)]$$. This property is used to turn the linear convolution to compute the output of a linear time invariant system into a circular convolution in orthogonal frequency division multiplexing.

Cyclic prefixes are used because they eliminate inter-symbol-interference if long enough, since they are a form of guard band. However, they cost power and reduce the data rate by $$N/(\mu + N)$$ if blocks of $$N$$ symbols are prefixed.

Zero-padding on the end of each block to achieve circular conv. is a more power efficient (and apparently valid) technique, but has the disadvantage that when the reciever removes the suffix, it makes the noise colored, in the sense that the noise's spectrum isn't flat.