**2.2 Cyclic prefix**

Also known as the guard interval, the Cyclic Prefix (CP) is the overhead in the time domain of an OFDM system that utilizes the delay spread due to multipath . When available spectrum is spread into several narrow-band subcarriers the symbol time increases and the opportunity to improve spectral efficiency and robustness by introducing overhead in Time Domain is possible.

By a careful estimation of the delay spread and hence a reasonable cyclic prefix *Tg* the ISi can be eliminated or reduced to negligible levels. The net effect of the guard interval is that all multipath effects only affects the guard interval and not the actual data symbol while the *Tg* remains small enough to be ignored in useful symbol time *Ts* [7]. The symbol duration, *T* is thus composed of the useful symbol time *Ts* and the guard interval *Tg.* as illustrated in Fig 4.

In attempting to create an ISi free channel the channel must appear to provide a cyclic convolution, a major property of 1/FFT as we shall see in the next section. The idea of a cyclic prefix is thus integral to interference free multi-carrier wideband technology .

Consider a maximum channel delay spread, r = u + 1, if you add T*9* = u then we can then consider the entire bit-stream as a single OFDM symbol with L vector-lengths .

Fig. 4. OFDM Symbol

$$X = [\boldsymbol{\pi}\_1 \boldsymbol{\pi}\_2 \boldsymbol{\pi}\_3 \dots \boldsymbol{\pi}\_L] \tag{1.2}$$

$$X\_{cp} = [\mathbf{x}\_{L-u}\mathbf{x}\_{L-u+1}\mathbf{x}\_{L-u+2}\dots\mathbf{x}\_{L-1}\mathbf{x}\_0\mathbf{x}\_1\dots\mathbf{x}\_{L-1}] = \mathbf{x}\_{cp}\mathbf{x}\_d\tag{1.3}$$

*where Xep is the cyclic pref ix and xd is the original data* 

The channel output is given by, *Yep* = *h* 0 *Xep,* where h is a length *u* + l vector describing the impulse response of the channel during the OFDM symbol and the length of *Yep* is L + 2u samples with one *u* from the previous symbol discarded and the other *u* discarded at the next symbol leaving only the L symbols as, originally intended, output. The idea thus to represent the signal as a circular convolution system with CP that is at least as long as the channel delay spread results in a desired channel output *Y* to be decomposed into a simple product of the channel frequency response H = FFT{h} and the channel frequency domain input, X = FFT{x} [9].

#### **2.31/FFT**

OFDM employs an efficient computational technique known as the Fast Fourier Transform (FFT) and its inverse, the Inverse First Fourier Transform (IFFT). An FFT transforms or decomposes into its frequency components while the IFFT will reverse the signal to the special domain. The FFT is a faster algorithm of the Discrete Fourier Transform (DFT) with time savings of up to a factor of [N */logN].* 

If we consider a data sequence *X* = *(X0 ,X <sup>1</sup> , ••• Xn, ... XN- 2 ,XN\_ <sup>1</sup> )and Xk* = *Ak* + *jBk* then a DFT / IDFT representation of an OFDM signal can be expressed thus, 1.4 below

$$\propto\_{n} \frac{1}{N} \sum\_{k=0}^{N-1} X\_k e^{j2\pi \frac{kn}{N}} = \frac{1}{N} \sum\_{k=0}^{N-1} X\_k e^{j2\pi f\_k t\_n}, \ n = 0, 1, 2...N-1 \tag{1.4}$$

Where *fk* = *k* / *N lit, tn* = *nlit and lit is an arbitrary symbol duration of the* = *sequence Xn*  If we take the real part as *Sn* = *Re(xn)* 

$$\hat{\mathbf{x}} = \frac{1}{N} \sum\_{k=0}^{N-1} (A\_k \cos 2\pi f\_k t\_n - B\_k \cos 2\pi f\_k t\_{n'} \qquad n = 0, 1, 2...N-1\tag{1.5}$$

Applied to a low-pass filter we *tn* = *t intervals,* 0 ~ *t* ~ *N lit in equation* 1.5 *above.* 

The time and frequency domain representations can be given as in Eq 1.6.

$$\int\_{a}^{a} \mathbf{x}\_{i}(t) \mathbf{x}\_{j}^{\*}(t) dt = \begin{cases} 1, \; i = j \\ 0, \; i \neq j \end{cases} \\ \text{and } \int\_{a}^{a} \mathbf{X}\_{i}(f) \mathbf{X}\_{j}^{\*}(f) df = \begin{cases} 1, \; i = j \\ 0, \; i \neq j \end{cases} \tag{1.6}$$

The time-domain spreading is achieved by repeating the same information in an OFOM symbol on two different sub-bands giving frequency diversity while the frequency-domain spreading is achieved by choosing conjugate symmetric inputs to the IFFT. This also exploits frequency diversity and minimizes the transmitter complexity and improves power control.

In section 2.2 we introduced cyclic prefix and the crucial role of circular convolution applied to a linear-time invariant FIR. We shall illustrate this further to help understand the I/FFT processing in an OFOM system.

Suppose we were to compute the output y[n] of a system as a circular convolution of its impulse response h[n] and the channel input x[n] [9].

$$\mathbf{x}\left[n\right] = h\left[n\right] \oplus \mathbf{x}\left[n\right] = \mathbf{x}\left[n\right] \oplus h\left[n\right] \tag{1.7}$$

where h[n] 0 x[n] = x[n] 0 h[n] ~ Lt:J h[k]x[n - k]i with the circular function x[nh = x[nmodL] is periodic with period L.

We can thus define the output as a OFT{ y[n]} in time and frequency as given in Eq. 1.8

$$DFT\{y[n]\} = DFT\left\{h[n] \circledast x[n]\right\} \text{ and } \ Y[m] = H\{m\}X[m] \tag{1.8}$$

The L point OFT is then defined by

$$DFT\left\{\mathbf{x}\left[n\right]\right\} = X\mathbf{I}[m] \triangleq \frac{1}{\sqrt{L}} \sum\_{n=0}^{L-1} \mathbf{x}[n]e^{-j\frac{2\pi mm}{L}}\tag{1.9}$$

with the inverse, IDFT defined by

$$IDFT\left\{X\left[m\right]\right\} = \mathbf{x}\left[n\right] \triangleq \frac{1}{\sqrt{L}} \sum\_{n=0}^{L-1} \mathbf{X}[m] e^{\frac{j\left(2\pi nm\right)}{L}}\tag{2.0}$$

In summary an OFOM system may be viewed as a functional block diagram shown in Fig. 5.

Fig. 5. OFOM Functional Block Diagram

~[ ] Y[m] From Fig. 5 the estimated data symbols, X m = H[m] while X and Y represent the L transmitted and received symbols.

