**3. Equalization in wireless channel**

Equalization defines any signal processing technique used at the receiver to alleviate the ISI problem caused by delay spread. Signal processing can also be used at the transmitter to make the signal less susceptible to delay spread: spread spectrum and multicarrier modulation fall in this category of transmitter signal processing techniques. ISI mitigation is required when the modulation symbol time *Ts* is on the order of the channel's rms delay spread *σTm*. Higher data rate applications are more sensitive to delay spread, and generally require high-perform‐ ance equalizers or other ISI mitigation techniques. In fact, mitigating the applications are more sensitive to delay spread, and generally require high-performance equalizers or other ISI mitigation techniques. In fact, mitigating the applications are more sensitive to delay spread, and generally require high-performance equalizers or other ISI mitigation techniques. In fact, mitigating the impact of delay spread is one of the most challenging hurdles for high-speed wireless data systems.

Equalizer design must typically balance ISI mitigation with noise enhancement, since both the signal and the noise pass through the equalizer, which can increase the noise power. Nonlinear equalizers suffer less from noise enhancement than linear equalizers, but typically entail higher complexity.

Equalization techniques fall into two broad categories: linear and nonlinear. The linear techniques are generally the simplest to implement and to understand conceptually. However, linear equalization techniques typically suffer from more noise enhancement than nonlinear equalizers.

Among nonlinear equalization techniques, decision-feedback equalization (DFE) is the most common, since it is fairly simple to implement and generally performs well. However, on chan‐ nels with low SNR, the DFE suffers from error propagation when bits are decoded in error, leading to poor performance. The optimal equalization technique is maximum likelihood se‐ quence estimation (MLSE). Unfortunately, the complexity of this technique grows exponential‐ ly with the length of the delay spread, and is therefore impractical on most channels of interest.

However, the performance of the MLSE is often used as an upper bound on performance for other equalization techniques.

This sequence corresponds to samples of the multicarrier signal: i.e. the multicarrier signal consists of linearly modulated subchannels, and the right hand side of (4) corresponds to samples of a sum of QAM symbols *X*[*i*] each modulated by carrier frequency. The cyclic prefix is then added to the OFDM symbol, and the resulting time samples *x*˜ [*n*] = *x*˜[*−μ*]*,..., x*˜ [*N −* 1]

Where *h(n)* is impulse response of channel with length µ + 1 = Tm/Ts, where Tm is the channel delay spread and Ts the sampling time associated with the discrete time sequence, *v*[n] is

To simplify our derivation, we will choose *N* =8 subcarriers, prefix-length=2. Assume channel

0000000

ê ú <sup>ù</sup> ê ú <sup>ú</sup> ê ú <sup>ú</sup> ê ú <sup>ú</sup> ê ú <sup>ú</sup> ê ú <sup>ú</sup> ê ú <sup>ú</sup> ê ú <sup>ú</sup> ê ú <sup>ú</sup> ê ú <sup>ú</sup> ê ú <sup>ú</sup> ê ú ê úû ê ú

channel matrix <sup>7</sup>

{

ê ú ê ú ë û (5)

transmitted sequence for symbol m

PAPR Reduction in WiMAX System http://dx.doi.org/10.5772/55380 43

é ù ê ú

*x x x x x x x x x x*

*m m m m m m m m m m*

= *x*[*N − μ*],*..., x*[0]*,..., x*[*N −* 1] are ordered by the parallel-to-serial converter[13].

210

0000000

*hhh <sup>r</sup> hhh <sup>r</sup>*

<sup>4</sup> <sup>210</sup> <sup>5</sup> <sup>210</sup> 6 210 7 210

210

0 000000 0 0 00000 000 0000 0000 000 00000 0 0 000000 0

144444444424444444443

The received signal is *r*[*n*] = *x*˜[*n*] *\* h*[*n*] + *ν*[*n*],*−μ ≤ n ≤ N −* 1.

Received samples for symbol *m*, after removing prefix:

<sup>210</sup> <sup>1</sup> <sup>210</sup> <sup>2</sup>

*<sup>r</sup> hhh r hhh r hhh r hhh r hhh hhh r*

AWGN.

impulse response is *: h*0*,h*1*,h*2,0,0….

**Figure 3.** Model for studied OFDM-system.

ê ú é ê ú ê ê ú ê ê ú ê ê ú ê ê ú <sup>ê</sup> <sup>=</sup> ê ú <sup>ê</sup> ê ú ê ê ú ê ê ú ê ê ú ê ê ú <sup>ë</sup> ê ú ë û

{

received samples for symbol m

This is equivalent with:

0

é ù

*m m m m m m m m*

3

It is clear that equalization in OFDM can be very simple. This is one of the major advantages of using OFDM over single carrier systems. Channel equalization in OFDM actually can be done by just a simple division in the frequency domain. This is because the channel as a filter is convolved with the input signal in the time domain on transmission. This operation is equivalent to multiplication in the frequency domain and thus undoing the effects of the channel is just a division.

This section studies the performance of OFDM system over multipath SUI channels which are not clarified until now. Moreover, the performance of this system will be compared with the performance of the system with the frequency domain equalizer (FDE) using MMSE. Also this system will be investigated over AWGN.
