**3.1. System model**

The OFDM implementation of multicarrier modulation is shown in Figure 3. The input data stream is modulated by a QAM modulator, resulting in a complex symbol stream *X*[0],*X*[1],*...,X*[*N −* 1] of length *N*. This symbol stream is passed through a serial-to-parallel converter, whose output is a set of *N* parallel QAM symbols *X* [0]*,..., X* [*N−*1] corresponding to the symbols transmitted over each of the subcarriers. Thus, the *N* symbols output from the serial-to-parallel converter are the discrete frequency components of the OFDM modulator output *s*(*t*). In order to generate *s*(*t*), these frequency components are converted into time samples by performing an inverse DFT on these *N* symbols, which is efficiently implemented using the IFFT algorithm. The IFFT yields the OFDM symbol consisting of the sequence *x*[*n*] = *X*[0]*,..., x*[*N −* 1] of length *N*, where

$$\mathbf{x}(n) = \sum\_{i=0}^{N-1} \mathbf{X}(i)\mathbf{e}^{j[2\pi mi/N]}, 0 \le n \le N-1. \tag{4}$$

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

equalizers suffer less from noise enhancement than linear equalizers, but typically entail higher

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

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

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

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

The OFDM implementation of multicarrier modulation is shown in Figure 3. The input data stream is modulated by a QAM modulator, resulting in a complex symbol stream *X*[0],*X*[1],*...,X*[*N −* 1] of length *N*. This symbol stream is passed through a serial-to-parallel converter, whose output is a set of *N* parallel QAM symbols *X* [0]*,..., X* [*N−*1] corresponding to the symbols transmitted over each of the subcarriers. Thus, the *N* symbols output from the serial-to-parallel converter are the discrete frequency components of the OFDM modulator output *s*(*t*). In order to generate *s*(*t*), these frequency components are converted into time samples by performing an inverse DFT on these *N* symbols, which is efficiently implemented using the IFFT algorithm. The IFFT yields the OFDM symbol consisting of the sequence *x*[*n*]

, 0 () ( ) 1. 2 / *<sup>N</sup>*

å £ £= - (4)

*<sup>n</sup> <sup>X</sup> <sup>i</sup> nx <sup>N</sup> j ni N e* p

complexity.

42 Selected Topics in WiMAX

equalizers.

other equalization techniques.

channel is just a division.

**3.1. System model**

system will be investigated over AWGN.

= *X*[0]*,..., x*[*N −* 1] of length *N*, where

1


0

*i*

=

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

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

To simplify our derivation, we will choose *N* =8 subcarriers, prefix-length=2. Assume channel impulse response is *: h*0*,h*1*,h*2,0,0….

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

This is equivalent with:

$$
\begin{bmatrix} r\_m^0 \\ r\_m^1 \\ r\_m^2 \\ r\_m^3 \\ r\_m^4 \\ r\_m^5 \\ r\_m^6 \\ r\_m^7 \end{bmatrix} = \begin{bmatrix} h\_0 & 0 & 0 & 0 & 0 & 0 & h\_2 & h\_1 \\ h\_1 & h\_0 & 0 & 0 & 0 & 0 & 0 & h\_2 \\ h\_2 & h\_1 & h\_0 & 0 & 0 & 0 & 0 & 0 \\ 0 & h\_2 & h\_1 & h\_0 & 0 & 0 & 0 & 0 \\ 0 & 0 & h\_2 & h\_1 & h\_0 & 0 & 0 & 0 \\ 0 & 0 & 0 & h\_2 & h\_1 & h\_0 & 0 & 0 \\ 0 & 0 & 0 & 0 & h\_2 & h\_1 & h\_0 & 0 \\ \hline \\ 0 & 0 & 0 & 0 & 0 & h\_2 & h\_1 & h\_0 \\ \text{modified channel matrix} \end{bmatrix} \begin{bmatrix} \mathbf{x}\_m^0 \\ \mathbf{x}\_m^1 \\ \mathbf{x}\_m^1 \\ \mathbf{x}\_m^2 \\ \mathbf{x}\_m^3 \\ \mathbf{x}\_m^4 \\ \mathbf{x}\_m^6 \\ \mathbf{x}\_m^7 \\ \mathbf{x}\_m^8 \end{bmatrix} \tag{6}
$$

nonselective within the bandwidth of each individual carrier. Focusing on one particular

*k j e* q = r

Designate its value within the bandwidth of the *kth* carrier. Equalization of the channel requires that at the DFT output in the receiver, the *kth* carrier signal be multiplied by a complex

This is the result of an optimization based on the zero-forcing (ZF) criterion [14], which aims at canceling ISI regardless of the noise level. To minimize the combined effect of ISI and additive noise, the equalizer coefficients can be optimized under the minimum mean-square

H

+

\* 2 2 2 *k*

*k na* s s

Channel equalization in OFDM systems thus takes the form of a complex multiplier bank at

The ZF criterion does not have a solution if the channel transfer function has spectral nulls in the signal bandwidth. Inversion of the channel transfer function requires an infinite gain and leads to infinite noise enhancement at those frequencies corresponding to spectral nulls. In general, the MMSE solution is more efficient, as it makes a trade-off between residual ISI (in the form of gain and phase mismatchs) and noise enhancement. This is particularly attractive

Analyzing the operation principle of OFDM, Frequency domain equalization is illustrated in Figure 4a which shows the baseband equivalent model of a single-carrier system employing this equalization technique. The received signal samples are passed to an N-point DFT, each

to transform the signal back to the time domain. Now, if we take the system sketched in Figure 4a and place it between an IDFT operator and a DFT operator, we obtain an OFDM system

C

2: is the variance of additive noise, and *σ<sup>a</sup>*

for channels with spectral nulls or deep amplitude depressions.

output sample is multiplied by a complex coefficient Ci

*k*

=

symbols. Note that the MMSE solution reduces to the ZF solution for of *σ<sup>n</sup>*

H

(9)

PAPR Reduction in WiMAX System http://dx.doi.org/10.5772/55380 45

1/ *C H k k* = (10)

2 is the variance of the transmitted data

2= 0.

, and the output is passed to an IDFT

(11)

carrier, the influence of multipath fading reduces to attenuation and a phase rotation.

H . *k k*

Referring back to the channel transfer function H(w), we let

error (MMSE) criterion. This optimization yield

coefficient

Where *σ<sup>n</sup>*

the DFT output in the receiver.

**3.2. Frequency-domain equalization**

The modified channel matrix is a so-called "circulant" matrix (constant along the diagonals & wrapped around). For every circulant matrix C is diagonalized by a DFT & I-DFT matrix:

*C* =(*IDFT* ).(diagonal matrix).(*DFT* )

Diagonal matrix has DFT of first column of C on its main diagonal

By substituting this:

$$\begin{aligned} DFT\_{\begin{subarray}{c}r\_{m}^{3}\\r\_{m}^{2}\\r\_{m}^{3}\\r\_{m}^{4}\\r\_{m}^{5}\\r\_{m}^{6}\\r\_{m}^{6}\end{subarray}} &= \begin{bmatrix}H\_{0}&0&0&0&0&0&0&0\\0&H\_{1}&0&0&0&0&0&0\\0&0&H\_{2}&0&0&0&0&0\\0&0&0&H\_{3}&0&0&0&0\\0&0&0&0&H\_{4}&0&0&0\\0&0&0&0&0&H\_{5}&0&0\\0&0&0&0&0&0&H\_{6}&0\\0&0&0&0&0&0&0&H\_{7}\\\end{bmatrix} \begin{bmatrix}X\_{m}^{0}\\X\_{m}^{1}\\X\_{m}^{2}\\X\_{m}^{3}\\X\_{m}^{4}\\X\_{m}^{5}\\X\_{m}^{6}\\X\_{m}^{7}\\X\_{m}^{8}\end{bmatrix} \end{aligned} \tag{7}$$

which means that after removing the prefix-samples and performing a DFT in the receiver, the obtained samples are equal to the transmitted (`frequency-domain') symbols, up to a channel attenuation *H*<sup>i</sup> (for tone-i). Hence channel equalization may be performed in the frequency domain, by component-wise divisions (divide by *H*<sup>i</sup> for tone-i) (1-taps FDE).

The multi-path fading channel can be written as:

$$
\sigma = H\_{\mathfrak{e}} \mathfrak{x} + \mathfrak{v} \tag{8}
$$

The channel equalization issue will be investigated in OFDM. Let *h*(t) designate the channel impulse response and H(w) its Fourier transform, i.e., the channel transfer function. If the number of carriers is sufficiently large, the channel transfer function becomes virtually nonselective within the bandwidth of each individual carrier. Focusing on one particular carrier, the influence of multipath fading reduces to attenuation and a phase rotation.

Referring back to the channel transfer function H(w), we let

{

0

é ù

*m m m m m m m m*

ê ú ê ú ê ú ê ú <sup>=</sup> ê ú ê ú ê ú ê ú ê ú

3

received symbol in freq.domain

domain, by component-wise divisions (divide by *H*<sup>i</sup>

The multi-path fading channel can be written as:

<sup>14243</sup>ê ú ë û

*C* =(*IDFT* ).(diagonal matrix).(*DFT* )

By substituting this:

44 Selected Topics in WiMAX

attenuation *H*<sup>i</sup>

mo received samples for symbol m

Diagonal matrix has DFT of first column of C on its main diagonal

<sup>0</sup> <sup>1</sup> <sup>1</sup> <sup>2</sup>

*<sup>r</sup> <sup>H</sup> r H r H r H DFT <sup>r</sup> <sup>H</sup>*

ê ú <sup>=</sup> ê ú

0

é ù

*m m m m m m m m*

ê ú ê ú ê ú

3

ê ú ê ú ê ú ê ú

<sup>0</sup> 2 1 <sup>1</sup> 1 0 <sup>2</sup> <sup>2</sup> 210

ê ú é ù ê ú ê ú

*<sup>r</sup> <sup>h</sup> h h r h h h*

00000 00000 00000

0 0000 0 0 000 000 0 0 0000 0

dified channel matrix

ê ú <sup>14444444244444443</sup> ê ú ë û

é ù ê ú ê ú ê ú ê ú ê ú ê ú ê ú ê ú ê ú ê ú ê ú

*x x x x x x x x*

*m m m m m m m m*

é ù ê ú ê ú ê ú ê ú ê ú ê ú ê ú ê ú ê ú ê ú ê ú

*X X X X X X X X*

*m m m m m m m m*

.

for tone-i) (1-taps FDE).

*<sup>c</sup> r Hx v* = + (8)

(6)

(7)

.

210

00000

The modified channel matrix is a so-called "circulant" matrix (constant along the diagonals & wrapped around). For every circulant matrix C is diagonalized by a DFT & I-DFT matrix:

2

ê ú é ù ê ú ê ú

000 0000 . 0000 000

*<sup>H</sup> <sup>r</sup> <sup>H</sup> <sup>r</sup>*

ê ú ë û

which means that after removing the prefix-samples and performing a DFT in the receiver, the obtained samples are equal to the transmitted (`frequency-domain') symbols, up to a channel

The channel equalization issue will be investigated in OFDM. Let *h*(t) designate the channel impulse response and H(w) its Fourier transform, i.e., the channel transfer function. If the number of carriers is sufficiently large, the channel transfer function becomes virtually

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

*r H*

3

00000 00 000000 0 0000000

matrix in freq.domain

(for tone-i). Hence channel equalization may be performed in the frequency

ê ú <sup>1444444442444444443</sup> ê ú ë û

0000000 0 000000 00 00000

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

ê ú ë û ê ú ë û

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

*r hhh r hhh r hhh hhh r*

$$\mathbf{H}\_k = \rho\_k.e^{\mathbf{j}\theta\_k} \tag{9}$$

Designate its value within the bandwidth of the *kth* carrier. Equalization of the channel requires that at the DFT output in the receiver, the *kth* carrier signal be multiplied by a complex coefficient

$$\mathbf{C}\_{k} = \mathbf{1}/\mathbf{H}\_{k} \tag{10}$$

This is the result of an optimization based on the zero-forcing (ZF) criterion [14], which aims at canceling ISI regardless of the noise level. To minimize the combined effect of ISI and additive noise, the equalizer coefficients can be optimized under the minimum mean-square error (MMSE) criterion. This optimization yield

$$\mathbf{C}\_{k} = \frac{\mathbf{H}\_{k}^{\ast}}{\left|\mathbf{H}\_{k}\right|^{2} + \sigma\_{n}^{2}\sqrt{\sigma\_{a}^{2}}}\tag{11}$$

Where *σ<sup>n</sup>* 2: is the variance of additive noise, and *σ<sup>a</sup>* 2 is the variance of the transmitted data symbols. Note that the MMSE solution reduces to the ZF solution for of *σ<sup>n</sup>* 2= 0.

Channel equalization in OFDM systems thus takes the form of a complex multiplier bank at the DFT output in the receiver.

The ZF criterion does not have a solution if the channel transfer function has spectral nulls in the signal bandwidth. Inversion of the channel transfer function requires an infinite gain and leads to infinite noise enhancement at those frequencies corresponding to spectral nulls. In general, the MMSE solution is more efficient, as it makes a trade-off between residual ISI (in the form of gain and phase mismatchs) and noise enhancement. This is particularly attractive for channels with spectral nulls or deep amplitude depressions.
