**2. OFDM**

Orthogonal Frequency-Division Multiplexing is multicarrier modulation technique with dedicated carrier spacing in the frequency domain . The orthogonality aims to address the Inter-Symbol Interference (ISi) while bringing better data rates with increased numbers of sub-carriers. If the modulated carrier is represented by a *sin x* function, then the subchannels peak at the zero crossings where modulation/ demodulation occurs.

OFDM systems provide amongst other advantages robustness to multi-path and frequency selective fading, simple equalization and a better spectral efficiency due to enhanced modulation techniques, optimized capacity due to flexibility in time and frequency domains, an expanded coverage provided by sub-channelization and a scalable design to support extra demand giving preserved radio performance .

#### **2.1 Sub-carriers**

To achieve an OFDM system two conditions must apply:


This condition is satisfied by the equation 1.1 with the property expressed as a complex conjugate function.

$$\int\_{t\_s}^{t\_s + T\_s} e^{-j2\pi \frac{k}{T\_s}(t - t\_s)} \cdot \sum\_{n=0}^{N-1} d\_n e^{j2\pi \frac{n}{T\_s}(t - t\_s)} dt = \sum\_{n=0}^{N-1} d\_n \int\_{t\_s}^{t\_s + T\_s} e^{j2\pi \frac{n-k}{T\_s}(t - t\_s)} dt = \mathcal{S}\_{nk} \mid -T\_g \le t \le T\_s \tag{1.1}$$

*where N is the number of subcarriers, T*5 *is symbol time, IS is the Kronecker delta function* 

The adjacent sub-carrier separation shown in Fig. 3 is given by 1 fr and the peak values are <sup>s</sup> found at n [ 1 /rJ where *n* = 1,2,3 ... L is an integer with a maximum L value equivalent to the maximum possible number of sub-carriers.

Fig. 3. OFDM Spectra of Sub-Carriers

Ideally, in a high data rate system the channel delay spread , is much smaller than the symbol time *Ts* to avoid ISi. In wideband communications we require a much smaller symbol time. It's therefore, crucial to modulate the available spectrum into L sub-carriers, several discrete narrowband channels, to reduce ISi to ensure r « *Ts* or *Ts/ L* » r. The *L*  concurrent sub-channels are then used to send the total desired data rate ISi free, see Fig 3 [9]. Usually Inter-Carrier Interference (ICI) is easily eliminated by transmitter-receiver synchronization
