Wavelet Based Multicarrier Modulation (MCM) Systems: PAPR Analysis

*Jamaluddin Zakaria and Mohd Fadzli Mohd Salleh*

#### **Abstract**

Orthogonal frequency division multiplexing (OFDM) is a prominent system in transmitting multicarrier modulation (MCM) signals over selective fading channel. The system offers to attain a higher degree of bandwidth efficiency, higher data transmission, and robust to narrowband frequency interference. However, it incurs a high peak-to-average power ratio (PAPR) where the signals work in the nonlinear region of the high-power amplifier (HPA) results in poor performance. Besides, an attractive dynamic wavelet analysis and its derivatives such as wavelet packet transform (WPT) demonstrates almost the same criteria as the OFDM in MCM system. Wavelet surpasses Fourier based analysis by inherent flexibility in terms of windows function for non-stationary signal. In wavelet-based MCM systems (wavelet OFDM (WOFDM) and Wavelet packet OFDM (WP-OFDM)), the constructed orthogonal modulation signals behaves similar to the fast Fourier transform (FFT) does in the conventional OFDM (C-OFDM) system. With no cyclic prefix (CP) need to be applied, these orthogonal signals hold higher bandwidth efficiency. Hence, this chapter presents a comprehensive study on the manipulation of specified parameters using WP-OFDM, WOFDM and C-OFDM signals together with various wavelets under the additive white Gaussian noise (AWGN) channel.

**Keywords:** multicarrier modulation (MCM), orthogonal frequency division multiplexing (OFDM), peak-to-average power ratio (PAPR), wavelet transform, wavelet packet transform (WPT)

#### **1. Introduction**

Orthogonal Frequency Division Multiplexing (OFDM) technique provides a number of advantages: In OFDM since the subcarriers are overlapped, accomplishes a higher degree of spectral efficiency that results in higher transmission data rates. Considering the use of the efficient FFT technique, the process is considered computationally lower. Besides, in the Single-Carrier Modulation (SCM) the ISI problem which commonly occurs the use of the cyclic prefix (CP) greatly eliminates the problem [1]. The division of a channel into several narrowband flat fading (subchannels) results in the subchannels being more resilient towards frequency selective fading. The loss of any subcarrier(s) due to channel frequency selectivity, proper channel coding schemes can recover the lost data [1]. Thus, this technique

offers robust protection against channel impairments without the need to implement an equalizer as in the SCM, and this greatly reduces the overall system complexity. However, the high Peak-to-Average Power Ratio (PAPR) has been the major drawback in the OFDM system. This situation happens when the peak OFDM signals surpass the specified threshold and as a result the high-power amplifier (HPA) operates in a nonlinear region. This produces spectral regrowth of the OFDM signals and broken the orthogonality among the subcarriers. Thus, the effect on bit error rate (BER) performance at the receiver is poor.

therefore the product of ð Þ Δ*ta*,*<sup>b</sup>* Δ *f <sup>a</sup>*,*<sup>b</sup>*

*DOI: http://dx.doi.org/10.5772/intechopen.94579*

is defined as

expressed as [17, 18].

space of *L*<sup>2</sup>

**65**

where the terms *<sup>v</sup>*þ<sup>∞</sup> <sup>¼</sup> *<sup>L</sup>*<sup>2</sup>

� � is independent of the dilation parameter *<sup>a</sup>*.

*φβ* <sup>¼</sup> *<sup>φ</sup>*ð Þ *<sup>t</sup>* � *<sup>β</sup>* , *βϵZ*, *φϵL*<sup>2</sup> (2)

ð Þ is the vector space of square

*<sup>β</sup>* , *<sup>β</sup>* <sup>∈</sup>*<sup>Z</sup>* (3)

*αβφβ*ð Þ*t* (4)

**<sup>2</sup>***<sup>φ</sup>* **<sup>2</sup>***<sup>α</sup>* ð Þ *<sup>t</sup>* � *<sup>β</sup>* (5)

*αβφ* **<sup>2</sup>***<sup>α</sup>* ð Þ *<sup>t</sup>* <sup>þ</sup> *<sup>β</sup>* (7)

… ∁*v*�**<sup>2</sup>** ∁*v*�**<sup>1</sup>** ∁*v***<sup>0</sup>** ∁*v***<sup>1</sup>** ∁*v***<sup>2</sup>** ∁ … *:* (8)

, and *v*�<sup>∞</sup> ¼ f g0 indicate that within the same vector

*<sup>β</sup>* (6)

If any time resolution gain is obtained, this inversely effect the cost of frequency resolution and vice versa. Therefore, this detains the Heisenberg uncertainty principle for the dilated and translated wavelet *ψCWTa*,*<sup>b</sup>* ð Þ*t* and the mother wavelet *ψCWT*ð Þ*t* . The most natural choice for dilation step is 2 that results in octave bands or dyadic scales. The wavelet is compressed in frequency domain by a factor of 2 for each successive value of scale index. This produces the stretched in time domain by the same factor. The translation factor is set to the value of "1" to get the dyadic

integrated function. The parameter *v*<sup>0</sup> is a space spanned by scaling function, which

*<sup>v</sup>*<sup>0</sup> <sup>¼</sup> *Span φβ*ð Þ*<sup>t</sup>* � �

*x t*ð Þ¼ <sup>X</sup> þ∞

*φα*,*<sup>β</sup>* ¼ **2**

Then, the new function for the expanded subspace *vα* is given as

*<sup>v</sup><sup>α</sup>* <sup>¼</sup> *span φβ* <sup>2</sup>*<sup>α</sup>* ð Þ*<sup>t</sup>* � �

*x t*ð Þ¼ <sup>X</sup> þ∞

*β*¼�∞

One can increase the size of the subspace by changing the time scale of the scaling functions. The two-dimensional parameterization (time and scale) of scaling

> *α=*

In the extended subspace, whenever *x t*ð Þ ∈*va*, then it can be expressed as

*β*¼�∞

From (Eq. (7)), the span *v<sup>α</sup>* is larger than *v*0, for *α* >0 and *φα*,*<sup>β</sup>*ð Þ*t* able to represent the finer detail (due to its finer scale). For *α* <0 this condition is true that

, there exist both high resolution and low-resolution coefficients.

represents for the coarse scale. Wavelet obeys to multi-resolution concept's requirement, where every signal is decomposed into finer detail gradually as

*<sup>β</sup>* <sup>¼</sup> *span φα*,*<sup>β</sup>*ð Þ*<sup>t</sup>* � �

sampling fashion. The time-shift and scaling function are set as [16];

*Wavelet Based Multicarrier Modulation (MCM) Systems: PAPR Analysis*

where *Z* is the set of all integer numbers, and *L*<sup>2</sup>

In this subspace, if *x t*ð Þ∈*v*0, it can be expressed as

function *φ*ð Þ*t* from *v*<sup>0</sup> to *v<sup>α</sup>* can be expressed as

To deliver massive high-speed data over a wireless channel, Multi-carrier-modulation (MCM) scheme has been widely used transmission technique. Despite its advantages, the MCM scheme is prone to high PAPR signal transmission, which has been single out as the main difficulty. In the MCM scheme, the conventional way to obtain orthogonal subcarrier signals is by using a Fourier transform. The emergence of wavelet transforms has paved the way for new promising techniques to obtain orthogonal subcarrier signals in future MCM systems. Wavelet transforms have been testified practical for the MCM system due to the orthogonal overlapping symbols property that they possess in time and frequency domains, respectively.

In order to mitigate PAPR, there have been many techniques proposed in literature either to reduce the peak power with fixed average power or alter the distribution so that the average power produced has smaller peak power [2–6]. Due to this, there are two categories of PAPR reduction techniques which are called as signal distortion technique and signal scrambling technique. A prominent technique known as Partial Transmit Sequence (PTS) has been first introduced in [7]. This technique categorized as signal scrambling offers big potential for further exploration as explored in the works [8–11].

This chapter presents the analysis of various wavelet families in their applicability towards MCM systems and their PAPR profiles. Details analysis is presented for obtaining the BER results for various Wavelets.

#### **2. Background**

#### **2.1 Wavelet transform**

In this section the basic concept of wavelet and wavelet packet transform (WPT) are presented. The WPT is constructed based on the continuous wavelet transform (CWT) and wavelet transform (WT) theory. For ease of reading, all the following equations in these subsections are mostly taken from [12–15].

#### *2.1.1 Discrete wavelet transform (DWT)*

The computation cost for wavelet coefficients in the CWT is high since they are highly redundant data, which is not desirable for real application. Therefore, discrete wavelets offer as the alternative for practical applications. In discrete wavelets, the scalable and translatable wavelets are discrete. The process of discrete scaling and translation of the mother wavelet can be expressed as

$$
\psi\_{a,\emptyset} = \sqrt{a\_0^a} \nu \left( a\_0^a t - \beta\_{b\_0} \right) \tag{1}
$$

where *a*<sup>0</sup> represents the fixed step of dilation and *b*<sup>0</sup> indicates the translation factor. The integer *α* and *β* signify the indices scale and translation, respectively. The scaling in time domain correlates with an inverse scaling in frequency domain, *Wavelet Based Multicarrier Modulation (MCM) Systems: PAPR Analysis DOI: http://dx.doi.org/10.5772/intechopen.94579*

therefore the product of ð Þ Δ*ta*,*<sup>b</sup>* Δ *f <sup>a</sup>*,*<sup>b</sup>* � � is independent of the dilation parameter *<sup>a</sup>*. If any time resolution gain is obtained, this inversely effect the cost of frequency resolution and vice versa. Therefore, this detains the Heisenberg uncertainty principle for the dilated and translated wavelet *ψCWTa*,*<sup>b</sup>* ð Þ*t* and the mother wavelet *ψCWT*ð Þ*t* . The most natural choice for dilation step is 2 that results in octave bands or dyadic scales. The wavelet is compressed in frequency domain by a factor of 2 for each successive value of scale index. This produces the stretched in time domain by the same factor. The translation factor is set to the value of "1" to get the dyadic sampling fashion. The time-shift and scaling function are set as [16];

$$
\rho\_{\beta} = \varrho(t - \beta), \beta \epsilon Z, \varrho \epsilon L^2 \tag{2}
$$

where *Z* is the set of all integer numbers, and *L*<sup>2</sup> ð Þ is the vector space of square integrated function. The parameter *v*<sup>0</sup> is a space spanned by scaling function, which is defined as

$$w\_0 = \frac{\overline{\text{Span}\{\rho\_\beta(t)\}}}{\beta}, \beta \in Z \tag{3}$$

In this subspace, if *x t*ð Þ∈*v*0, it can be expressed as

$$\mathfrak{x}(\mathfrak{t}) = \sum\_{\mathfrak{\beta} = -\infty}^{+\infty} \mathfrak{a}\_{\mathfrak{\beta}} \mathfrak{q}\_{\mathfrak{\beta}}(\mathfrak{t}) \tag{4}$$

One can increase the size of the subspace by changing the time scale of the scaling functions. The two-dimensional parameterization (time and scale) of scaling function *φ*ð Þ*t* from *v*<sup>0</sup> to *v<sup>α</sup>* can be expressed as

$$
\varphi\_{a,\beta} = \mathfrak{Z}^{a\_2} \mathfrak{q} (\mathfrak{Z}^a \mathfrak{t} - \beta) \tag{5}
$$

Then, the new function for the expanded subspace *vα* is given as

$$w\_a = \frac{\overline{\operatorname{span}\left\{\rho\_\beta(\mathfrak{Z}^a t)\right\}}}{\beta} = \frac{\overline{\operatorname{span}\left\{\rho\_{a,\beta}(t)\right\}}}{\beta} \tag{6}$$

In the extended subspace, whenever *x t*ð Þ ∈*va*, then it can be expressed as

$$\mathfrak{x}(\mathfrak{t}) = \sum\_{\mathfrak{\mathfrak{f}} = -\infty}^{+\infty} a\_{\mathfrak{\mathfrak{f}}} \mathfrak{q} (\mathfrak{2}^{\mathfrak{a}} \mathfrak{t} + \mathfrak{f}) \tag{7}$$

From (Eq. (7)), the span *v<sup>α</sup>* is larger than *v*0, for *α* >0 and *φα*,*<sup>β</sup>*ð Þ*t* able to represent the finer detail (due to its finer scale). For *α* <0 this condition is true that represents for the coarse scale. Wavelet obeys to multi-resolution concept's requirement, where every signal is decomposed into finer detail gradually as expressed as [17, 18].

$$\dots \mathbb{C} \boldsymbol{\nu}\_{-2} \; \mathbb{C} \boldsymbol{\nu}\_{-1} \; \mathbb{C} \boldsymbol{\nu}\_{0} \; \mathbb{C} \boldsymbol{\nu}\_{1} \; \mathbb{C} \boldsymbol{\nu}\_{2} \; \mathbb{C} \dots \tag{8}$$

where the terms *<sup>v</sup>*þ<sup>∞</sup> <sup>¼</sup> *<sup>L</sup>*<sup>2</sup> , and *v*�<sup>∞</sup> ¼ f g0 indicate that within the same vector space of *L*<sup>2</sup> , there exist both high resolution and low-resolution coefficients.

offers robust protection against channel impairments without the need to implement an equalizer as in the SCM, and this greatly reduces the overall system complexity. However, the high Peak-to-Average Power Ratio (PAPR) has been the major drawback in the OFDM system. This situation happens when the peak OFDM signals surpass the specified threshold and as a result the high-power amplifier (HPA) operates in a nonlinear region. This produces spectral regrowth of the OFDM signals and broken the orthogonality among the subcarriers. Thus, the effect

To deliver massive high-speed data over a wireless channel, Multi-carrier-modulation (MCM) scheme has been widely used transmission technique. Despite its advantages, the MCM scheme is prone to high PAPR signal transmission, which has been single out as the main difficulty. In the MCM scheme, the conventional way to obtain orthogonal subcarrier signals is by using a Fourier transform. The emergence of wavelet transforms has paved the way for new promising techniques to obtain orthogonal subcarrier signals in future MCM systems. Wavelet transforms have been testified practical for the MCM system due to the orthogonal overlapping symbols property that they possess in time and frequency domains, respectively. In order to mitigate PAPR, there have been many techniques proposed in literature either to reduce the peak power with fixed average power or alter the distribution so that the average power produced has smaller peak power [2–6]. Due to this, there are two categories of PAPR reduction techniques which are called as signal distortion technique and signal scrambling technique. A prominent technique known as Partial Transmit Sequence (PTS) has been first introduced in [7]. This technique categorized as signal scrambling offers big potential for further explora-

This chapter presents the analysis of various wavelet families in their applicability towards MCM systems and their PAPR profiles. Details analysis is presented for

In this section the basic concept of wavelet and wavelet packet transform (WPT) are presented. The WPT is constructed based on the continuous wavelet transform (CWT) and wavelet transform (WT) theory. For ease of reading, all the

The computation cost for wavelet coefficients in the CWT is high since they are highly redundant data, which is not desirable for real application. Therefore, discrete wavelets offer as the alternative for practical applications. In discrete wavelets, the scalable and translatable wavelets are discrete. The process of discrete

where *a*<sup>0</sup> represents the fixed step of dilation and *b*<sup>0</sup> indicates the translation factor. The integer *α* and *β* signify the indices scale and translation, respectively. The scaling in time domain correlates with an inverse scaling in frequency domain,

<sup>0</sup>*t* � *β<sup>b</sup>*<sup>0</sup>

� � (1)

following equations in these subsections are mostly taken from [12–15].

scaling and translation of the mother wavelet can be expressed as

*ψα*,*<sup>β</sup>* <sup>¼</sup> ffiffiffiffiffi *aα* 0 p *ψ a<sup>α</sup>*

on bit error rate (BER) performance at the receiver is poor.

tion as explored in the works [8–11].

*2.1.1 Discrete wavelet transform (DWT)*

**2. Background**

*Wavelet Theory*

**64**

**2.1 Wavelet transform**

obtaining the BER results for various Wavelets.

Consequently, if *x t*ð Þ∈*va*, then *x*ð Þ 2*t* ∈*va*þ1. Additionally, the *ϕ*ð Þ*t* term is expressed as the weighted sum of the time-shifted scaling function

$$\mathfrak{gl}(\mathfrak{t}) = \sum\_{n=-\infty}^{+\infty} h(n) \sqrt{2} \mathfrak{gl}(2\mathfrak{t} - n), n \in \mathbb{Z} \tag{9}$$

where *g*(*n*) is called the wavelet function coefficient. The relationship between

Both coefficients are restricted by the orthogonality condition. If *h*(*n*) has a

The wavelet function coefficients *g n*ð Þ is normally required by the orthonormal perfect reconstruction (PR) process. In the communication system point of view, this PR process offers advantage to the receiver whereby the received signals can be reconstructed perfectly. For example, Haar wavelet below is analyzed with the

> 1 0≤*t* <0*:*5 �1 0*:*5≤ *t*<1 0 *otherwise*

1 0 ≤*t*≤ 1 <sup>0</sup> *otherwise* �

Furthermore, the basic version of Haar wavelet for wavelet and scaling function

ffiffi 2 p , �1 ffiffi 2 p

ffiffi 2 <sup>p</sup> , <sup>1</sup> ffiffi 2 p

The Haar filter coefficients are obtained by applying (Eq. (9)) and (Eq. (12)).

*g n*ð Þ¼ <sup>1</sup>

*h n*ð Þ¼ <sup>1</sup>

*h*ð Þ 1 � *n* (13)

*h N*ð Þ � 1 � *n* (14)

� � (17)

� � (18)

ð Þ has its discrete wavelet expansion

(15)

(16)

wavelet filter *g*(*n*) and scaling filter *h*(*n*) can be expressed as [19];

*Wavelet Based Multicarrier Modulation (MCM) Systems: PAPR Analysis*

finite even length *N*, then the (Eq. (13)) can be rewritten as

wavelet function *ψ*ð Þ*t* can be expressed as

*DOI: http://dx.doi.org/10.5772/intechopen.94579*

and its scaling function is

Furthermore, the signal *x t*ð Þ∈*L*<sup>2</sup>

*Haar Wavelet transform (a) mother wavelet function, (b) scaling function.*

is shown in **Figure 2**.

given as [14].

**Figure 2.**

**67**

*g n*ð Þ¼ �ð Þ<sup>1</sup> *<sup>n</sup>*

*g n*ð Þ¼ �ð Þ<sup>1</sup> *<sup>n</sup>*

*ψ*ðÞ¼ *t*

*φ*ðÞ¼ *t*

8 ><

>:

where the term *h*(*n*) represents the scaling function coefficients (sequence of real or imaginary numbers). The *v<sup>α</sup>*þ**<sup>1</sup>** is the expanded space of *v<sup>α</sup>* and *w<sup>α</sup>* represents the corresponding orthogonal complement. Therefore, a new set of spaces is produced. Suppose that *w<sup>α</sup>*þ**<sup>1</sup>** be the subspace spanned by the wavelet, the enlargement of *v*<sup>1</sup> and *v*<sup>2</sup> space are written as (Eq. (10)) below and as illustrated as in **Figure 1** [19].

$$\begin{aligned} v\_1 &= v\_0 \bigoplus w\_0 \\ v\_2 &= v\_1 \bigoplus w\_1 = (v\_0 \bigoplus w\_0) \bigoplus w\_1 \\ &\vdots \\ v\_{a+1} &= v\_a \bigoplus w\_a = v\_0 \bigoplus\_{l=0}^a w\_l, \forall a \in \mathbb{Z} \end{aligned} \tag{10}$$

The definition of the wavelet function *ψ*ð Þ*t* is the same as the scaling space *v*0. Let the space spanned by the wavelet function *ψβ*ð Þ*t* be *w*0, and the expanded space spanned by *ψα*,*<sup>β</sup>*ð Þ*t* be *w<sup>α</sup>* that is obtained after using (Eq. (3)) to (Eq. (6)). The *w<sup>α</sup>* term is orthogonal to *v<sup>α</sup>* and thus the orthogonality between *φ*ð Þ*t* and *ψ*ð Þ*t* is given as [19];

$$\left<\varphi\_{a,\emptyset}(\mathbf{t}), \varphi\_{a,\emptyset}(\mathbf{t})\right> = \int \varphi\_{a,\emptyset}(\mathbf{t})\varphi\_{a,\emptyset}(\mathbf{t})d\mathbf{t} = \mathbf{0} \tag{11}$$

Due to these wavelets are in space spanned by the next finer scaling function, the wavelet function *ψ*(*t*) can be expressed by the sum of the weighted time-shifted wavelet function given as

$$\psi(t) = \sum\_{n=-\infty}^{+\infty} \mathbf{g}(n)\sqrt{2}\rho(2t - n), n \in \mathbb{Z} \tag{12}$$

**Figure 1.** *Wavelet vector spaces and scaling function.*

*Wavelet Based Multicarrier Modulation (MCM) Systems: PAPR Analysis DOI: http://dx.doi.org/10.5772/intechopen.94579*

where *g*(*n*) is called the wavelet function coefficient. The relationship between wavelet filter *g*(*n*) and scaling filter *h*(*n*) can be expressed as [19];

$$\mathbf{g}(n) = (-1)^{n} h(\mathbf{1} - n) \tag{13}$$

Both coefficients are restricted by the orthogonality condition. If *h*(*n*) has a finite even length *N*, then the (Eq. (13)) can be rewritten as

$$\mathbf{g}(n) = (-\mathbf{1})^n h(N - \mathbf{1} - n) \tag{14}$$

The wavelet function coefficients *g n*ð Þ is normally required by the orthonormal perfect reconstruction (PR) process. In the communication system point of view, this PR process offers advantage to the receiver whereby the received signals can be reconstructed perfectly. For example, Haar wavelet below is analyzed with the wavelet function *ψ*ð Þ*t* can be expressed as

$$\varphi(t) = \begin{cases} 1 & 0 \le t < 0.5 \\ -1 & 0.5 \le t < 1 \\ 0 & otherwise \end{cases} \tag{15}$$

and its scaling function is

Consequently, if *x t*ð Þ∈*va*, then *x*ð Þ 2*t* ∈*va*þ1. Additionally, the *ϕ*ð Þ*t* term is expressed

*h n*ð Þ ffiffiffi **2**

where the term *h*(*n*) represents the scaling function coefficients (sequence of real or imaginary numbers). The *v<sup>α</sup>*þ**<sup>1</sup>** is the expanded space of *v<sup>α</sup>* and *w<sup>α</sup>* represents the corresponding orthogonal complement. Therefore, a new set of spaces is produced. Suppose that *w<sup>α</sup>*þ**<sup>1</sup>** be the subspace spanned by the wavelet, the

enlargement of *v*<sup>1</sup> and *v*<sup>2</sup> space are written as (Eq. (10)) below and as illustrated as

*v*<sup>2</sup> ¼ *v*<sup>1</sup> ⨁*w*<sup>1</sup> ¼ ð Þ *v*<sup>0</sup> ⨁*w*<sup>0</sup> ⨁*w*<sup>1</sup>

The definition of the wavelet function *ψ*ð Þ*t* is the same as the scaling space *v*0. Let the space spanned by the wavelet function *ψβ*ð Þ*t* be *w*0, and the expanded space spanned by *ψα*,*<sup>β</sup>*ð Þ*t* be *w<sup>α</sup>* that is obtained after using (Eq. (3)) to (Eq. (6)). The *w<sup>α</sup>* term is orthogonal to *v<sup>α</sup>* and thus the orthogonality between *φ*ð Þ*t* and *ψ*ð Þ*t*

ð

Due to these wavelets are in space spanned by the next finer scaling function, the wavelet function *ψ*(*t*) can be expressed by the sum of the weighted time-shifted

> *g n*ð Þ ffiffi 2

*α*

*wl*, ∀*α*∈ *Z*

*φα*,*<sup>β</sup>*ð Þ*t ψα*,*<sup>β</sup>*ð Þ*t dt* ¼ 0 (11)

<sup>p</sup> *<sup>φ</sup>*ð Þ <sup>2</sup>*<sup>t</sup>* � *<sup>n</sup>* , *<sup>n</sup>*<sup>∈</sup> *<sup>Z</sup>* (12)

*l*¼0

<sup>p</sup> *<sup>φ</sup>*ð Þ **<sup>2</sup>***<sup>t</sup>* � *<sup>n</sup>* , *<sup>n</sup>*<sup>∈</sup> *<sup>Z</sup>* (9)

(10)

as the weighted sum of the time-shifted scaling function

*<sup>φ</sup>*ð Þ¼ *<sup>t</sup>* <sup>X</sup> þ∞

in **Figure 1** [19].

*Wavelet Theory*

is given as [19];

**Figure 1.**

**66**

*Wavelet vector spaces and scaling function.*

wavelet function given as

*n*¼�∞

*v*<sup>1</sup> ¼ *v*<sup>0</sup> ⨁*w*<sup>0</sup>

*vα*þ<sup>1</sup> ¼ *v<sup>α</sup>* ⨁*w<sup>α</sup>* ¼ *v*<sup>0</sup> ⨁

. . .

*φα*,*<sup>β</sup>*ð Þ*<sup>t</sup>* , *ψα*,*<sup>β</sup>*ð Þ*<sup>t</sup>* � � <sup>¼</sup>

*<sup>ψ</sup>*ðÞ¼ *<sup>t</sup>* <sup>X</sup> þ∞

*n*¼�∞

$$\varphi(t) = \begin{cases} 1 & 0 \le t \le 1 \\ 0 & \text{otherwise} \end{cases} \tag{16}$$

Furthermore, the basic version of Haar wavelet for wavelet and scaling function is shown in **Figure 2**.

The Haar filter coefficients are obtained by applying (Eq. (9)) and (Eq. (12)).

$$\mathbf{g}(n) = \left(\frac{1}{\sqrt{2}}, \frac{-1}{\sqrt{2}}\right) \tag{17}$$

$$h(n) = \left(\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right) \tag{18}$$

Furthermore, the signal *x t*ð Þ∈*L*<sup>2</sup> ð Þ has its discrete wavelet expansion given as [14].

**Figure 2.** *Haar Wavelet transform (a) mother wavelet function, (b) scaling function.*

$$\mathbf{x}(t) = \sum\_{\beta=-\infty}^{+\infty} c\_{a\_0}(\beta)\boldsymbol{\rho}\_{a\_0,\beta}(t) + \sum\_{\beta=-\infty}^{+\infty} \sum\_{a=a\_0}^{+\infty} d\_a(\beta)\boldsymbol{\nu}\_{a,\beta}(t) \tag{19}$$

(wavelet function) are substituted into (Eq. (19)) (reconstruction function), thus

X þ∞

*dα*ð Þ *β ψα*,*β*ð Þ*t*

*m*

*<sup>φ</sup>* <sup>2</sup>*<sup>α</sup>*þ<sup>1</sup> � <sup>2</sup>*<sup>β</sup>* � *<sup>n</sup>* � �

*<sup>φ</sup>* <sup>2</sup>*<sup>α</sup>*þ<sup>1</sup> � <sup>2</sup>*<sup>β</sup>* � *<sup>n</sup>* � � (22)

*dα*ð Þ *m g*ð Þ *β* � 2*m* (23)

*α*¼*α*<sup>0</sup>

*g n*ð Þ ffiffi 2 � � p *<sup>α</sup>*þ<sup>1</sup>

By multiplying both sides of (Eq. (22)) with *<sup>φ</sup>* <sup>2</sup>*<sup>α</sup>*þ<sup>1</sup> � *<sup>β</sup>* � � and taking the integral produces the lower scale of DWT coefficients [18], the scaling DWT coefficients of

*<sup>c</sup>α*ð Þ *<sup>m</sup> <sup>h</sup>*ð Þþ *<sup>β</sup>* � <sup>2</sup>*<sup>m</sup>* <sup>X</sup>

This implies that the scaling DWT coefficients at a certain value ð Þ *α* þ 1 can be computed by taking the weighted sum of wavelet DWT coefficients that are multiplied with the scaling DWT coefficients at scale *α*. **Figure 5** illustrates this process which is known as a 2-channel synthesis filter bank. The scaling DWT coefficients ð Þ *cα*ð Þ *β* and wavelet DWT coefficients ð Þ *dα*ð Þ *β* at scale *α* are first up-sampled by factor 2. Then, the scaling DWT coefficients ð Þ *<sup>c</sup>α*ð Þ *<sup>β</sup>* are filtered with a LPF *<sup>H</sup>*^ , and the wavelet DWT coefficients ð Þ *<sup>d</sup>α*ð Þ *<sup>β</sup>* are filtered with a HPF *<sup>G</sup>*^ respectively. Finally, the two filtered signals are added together to form the scaling DWT

þ∞

*β*¼�∞

*h n*ð Þ ffiffi 2 � � p *<sup>α</sup>*þ<sup>1</sup>

*n*¼�∞

*<sup>d</sup>α*ð Þ *<sup>β</sup>* <sup>X</sup> þ∞

*<sup>c</sup>α*<sup>0</sup> ð Þ *<sup>β</sup> φα*0,*β*ðÞþ*<sup>t</sup>* <sup>X</sup>

*Wavelet Based Multicarrier Modulation (MCM) Systems: PAPR Analysis*

*DOI: http://dx.doi.org/10.5772/intechopen.94579*

*n*¼�∞

*<sup>c</sup>α*<sup>0</sup> ð Þ *<sup>β</sup>* <sup>X</sup> þ∞

> X þ∞

*α*¼�∞

*<sup>c</sup>α*þ<sup>1</sup>ð Þ¼ *<sup>β</sup>* <sup>X</sup>

*m*

produces

**Figure 4.**

*x t*ðÞ¼ <sup>X</sup> þ∞

*DWT decomposition (three level).*

higher scale is given as

**Figure 5.**

**69**

*DWT reconstruction (single level).*

*β*¼�∞

*β*¼�∞

<sup>þ</sup> <sup>X</sup> þ∞

*β*¼�∞

coefficients at scale *α* þ 1 i.e. ð Þ *c<sup>α</sup>*þ<sup>1</sup>ð Þ *β* .

<sup>¼</sup> <sup>X</sup> þ∞

where *α*, *β*, ∈*Z* which *Z* is real integer. The *α*<sup>0</sup> is an arbitrary integer, and *L*<sup>2</sup> ð Þ is the vector space of the square integrated function. The frequency (or scale) and time localizations are provided by the parameters *α* and *β* respectively. The approximation coefficient and the detail coefficient have been deduced as *cα*ð Þ *β* and *dα*ð Þ *β* respectively.

In the wavelet expansion, by manipulating (Eq. (9)) and (Eq. (19)), the higher scale (i.e. *α* þ 1) can also be obtained that results the approximation coefficient as

$$\mathcal{L}\_a(\boldsymbol{\beta}) = \left< \mathbf{x}(t), \rho\_{a,\boldsymbol{\beta}}(t) \right> = \int \mathbf{x}(t) \mathcal{Z}^{a/2} \boldsymbol{\rho} (\mathcal{Z}^a \mathbf{t} - \boldsymbol{\beta}) d\mathbf{t} = \sum\_m h(m - 2\boldsymbol{\beta}) c\_{a+1}(m) \tag{20}$$

while the detail coefficient is expressed as

$$d\_a(\beta) = \left< \mathbf{x}(t), \boldsymbol{\mu}\_{a,\beta}(t) \right> = \left< \mathbf{x}(t) 2^{a/2} \boldsymbol{\mu}(2^a t - \beta) dt = \sum\_m \mathbf{g}(m - 2\beta) c\_{a+1}(m) \tag{21}$$

Both the terms of *cα*ð Þ *β* and *dα*ð Þ *β* in (Eq. (20) and (21)) are computed by taking the weighted sum of DWT coefficients of higher scale ð Þ *α* þ 1 . In order to obtain the scaling of the DWT coefficients ð Þ *cα*ð Þ *β* at scale *α*, the scaling function coefficient *h n*ð Þ is convolved with the scaling DWT coefficients ð Þ *cα*þ<sup>1</sup>ð Þ *β* at scale *α* þ 1, followed by subsampling with a factor of 2. Similarly, to obtain the wavelet DWT coefficients ð Þ *dα*ð Þ *β* at scale *α*, the wavelet function coefficient *g n*ð Þ is convolved with the scaling DWT coefficients ð Þ *cα*þ<sup>1</sup>ð Þ *β* at scale *α* þ 1, followed by subsampling with a factor of 2. Hence, as shown in **Figure 3**, that both of these expressions can be illustrated as 2-channel filter banks analysis [20].

The input signal to the 2-channel filter bank is split into two parts. The first portion of the signal goes to filter *H* and second goes to filter *G*. Subsequently, both the filtered signals are subsampled by 2. Each filtered signal contains half of the number of original samples and spans half of the frequency band. However, the number of samples at the output of the filter bank is the same as the original signal since there are two filters used. The decomposition process starts at the largest scale of *c*ð Þ *β* . If there are three level of decompositions involved, this implies the term *c*3ð Þ *β* exist and produces the terms *c*0ð Þ *β* , *d*0ð Þ *β* , *d*1ð Þ *β* and *d*2ð Þ *β* at the decomposition branches, as illustrated in **Figure 4**.

On the other hand, the reconstruction of the DWT coefficients process is expressed by (Eq. (19)). If (Eq. (9)) (for scaling refinement) and (Eq. (12))

**Figure 3.** *DWT decomposition (single level).*

*Wavelet Based Multicarrier Modulation (MCM) Systems: PAPR Analysis DOI: http://dx.doi.org/10.5772/intechopen.94579*

**Figure 4.** *DWT decomposition (three level).*

*x t*ðÞ¼ <sup>X</sup> þ∞

respectively.

*Wavelet Theory*

*<sup>c</sup>α*ð Þ¼ *<sup>β</sup> x t*ð Þ, *φα*,*<sup>β</sup>*ð Þ*<sup>t</sup>* � � <sup>¼</sup>

*<sup>d</sup>α*ð Þ¼ *<sup>β</sup> x t*ð Þ, *ψα*,*<sup>β</sup>*ð Þ*<sup>t</sup>* � � <sup>¼</sup>

branches, as illustrated in **Figure 4**.

**Figure 3.**

**68**

*DWT decomposition (single level).*

*β*¼�∞

ð

ð

while the detail coefficient is expressed as

be illustrated as 2-channel filter banks analysis [20].

*x t*ð Þ2*<sup>α</sup>=*<sup>2</sup>

*x t*ð Þ2*<sup>α</sup>=*<sup>2</sup>

*<sup>c</sup><sup>α</sup>*<sup>0</sup> ð Þ *<sup>β</sup> φα*0,*<sup>β</sup>*ð Þþ*<sup>t</sup>* <sup>X</sup>

where *α*, *β*, ∈*Z* which *Z* is real integer. The *α*<sup>0</sup> is an arbitrary integer, and *L*<sup>2</sup>

is the vector space of the square integrated function. The frequency (or scale) and time localizations are provided by the parameters *α* and *β* respectively. The approximation coefficient and the detail coefficient have been deduced as *cα*ð Þ *β* and *dα*ð Þ *β*

In the wavelet expansion, by manipulating (Eq. (9)) and (Eq. (19)), the higher scale (i.e. *α* þ 1) can also be obtained that results the approximation coefficient as

*<sup>φ</sup>* <sup>2</sup>*<sup>α</sup>* ð Þ *<sup>t</sup>* � *<sup>β</sup> dt* <sup>¼</sup> <sup>X</sup>

*<sup>ψ</sup>* <sup>2</sup>*<sup>α</sup>* ð Þ *<sup>t</sup>* � *<sup>β</sup> dt* <sup>¼</sup> <sup>X</sup>

Both the terms of *cα*ð Þ *β* and *dα*ð Þ *β* in (Eq. (20) and (21)) are computed by taking the weighted sum of DWT coefficients of higher scale ð Þ *α* þ 1 . In order to obtain the scaling of the DWT coefficients ð Þ *cα*ð Þ *β* at scale *α*, the scaling function coefficient *h n*ð Þ is convolved with the scaling DWT coefficients ð Þ *cα*þ<sup>1</sup>ð Þ *β* at scale *α* þ 1, followed by subsampling with a factor of 2. Similarly, to obtain the wavelet DWT coefficients ð Þ *dα*ð Þ *β* at scale *α*, the wavelet function coefficient *g n*ð Þ is convolved with the scaling DWT coefficients ð Þ *cα*þ<sup>1</sup>ð Þ *β* at scale *α* þ 1, followed by subsampling with a factor of 2. Hence, as shown in **Figure 3**, that both of these expressions can

The input signal to the 2-channel filter bank is split into two parts. The first portion of the signal goes to filter *H* and second goes to filter *G*. Subsequently, both the filtered signals are subsampled by 2. Each filtered signal contains half of the number of original samples and spans half of the frequency band. However, the number of samples at the output of the filter bank is the same as the original signal since there are two filters used. The decomposition process starts at the largest scale of *c*ð Þ *β* . If there are three level of decompositions involved, this implies the term *c*3ð Þ *β* exist and produces the terms *c*0ð Þ *β* , *d*0ð Þ *β* , *d*1ð Þ *β* and *d*2ð Þ *β* at the decomposition

On the other hand, the reconstruction of the DWT coefficients process is expressed by (Eq. (19)). If (Eq. (9)) (for scaling refinement) and (Eq. (12))

þ∞

X þ∞

*dα*ð Þ *β ψα*,*<sup>β</sup>*ð Þ*t* (19)

*h m*ð Þ � 2*β c<sup>α</sup>*þ1ð Þ *m* (20)

*g m*ð Þ � 2*β cα*þ<sup>1</sup>ð Þ *m* (21)

ð Þ

*α*¼*α*<sup>0</sup>

*m*

*m*

*β*¼�∞

(wavelet function) are substituted into (Eq. (19)) (reconstruction function), thus produces

$$\begin{split} x(t) &= \sum\_{\beta=-\infty}^{+\infty} c\_{a\_0}(\beta)\varphi\_{a\_0}\beta(t) + \sum\_{\beta=-\infty}^{+\infty} \sum\_{a=a\_0}^{+\infty} d\_a(\beta)\varphi\_{a,\beta}\beta(t) \\ &= \sum\_{\beta=-\infty}^{+\infty} c\_{a\_0}(\beta) \sum\_{n=-\infty}^{+\infty} h(n) \left(\sqrt{2}\right)^{a+1} \varphi(2^{a+1} - 2\beta - n) \\ &+ \sum\_{\beta=-\infty}^{+\infty} \sum\_{a=-\infty}^{+\infty} d\_a(\beta) \sum\_{n=-\infty}^{+\infty} g(n) \left(\sqrt{2}\right)^{a+1} \varphi(2^{a+1} - 2\beta - n) \end{split} \tag{22}$$

By multiplying both sides of (Eq. (22)) with *<sup>φ</sup>* <sup>2</sup>*<sup>α</sup>*þ<sup>1</sup> � *<sup>β</sup>* � � and taking the integral produces the lower scale of DWT coefficients [18], the scaling DWT coefficients of higher scale is given as

$$c\_{a+1}(\beta) = \sum\_{m} c\_a(m)h(\beta - 2m) + \sum\_{m} d\_a(m)\mathbf{g}(\beta - 2m) \tag{23}$$

This implies that the scaling DWT coefficients at a certain value ð Þ *α* þ 1 can be computed by taking the weighted sum of wavelet DWT coefficients that are multiplied with the scaling DWT coefficients at scale *α*. **Figure 5** illustrates this process which is known as a 2-channel synthesis filter bank. The scaling DWT coefficients ð Þ *cα*ð Þ *β* and wavelet DWT coefficients ð Þ *dα*ð Þ *β* at scale *α* are first up-sampled by factor 2. Then, the scaling DWT coefficients ð Þ *<sup>c</sup>α*ð Þ *<sup>β</sup>* are filtered with a LPF *<sup>H</sup>*^ , and the wavelet DWT coefficients ð Þ *<sup>d</sup>α*ð Þ *<sup>β</sup>* are filtered with a HPF *<sup>G</sup>*^ respectively. Finally, the two filtered signals are added together to form the scaling DWT coefficients at scale *α* þ 1 i.e. ð Þ *c<sup>α</sup>*þ<sup>1</sup>ð Þ *β* .

**Figure 5.** *DWT reconstruction (single level).*

In short, the DWT decomposes signals into coefficients. The IDWT reconstructs the original signals from coefficients which can be implemented efficiently by iterating the 2-channel synthesis filter bank.

*ζ* 2*p*þ1

*DOI: http://dx.doi.org/10.5772/intechopen.94579*

coefficients *ζ*

*p*

orthogonality is preserved.

**Figure 7.**

**71**

*Decomposition and bandwidth division for (a) DWT and (b) WPT.*

*ζ p*

*<sup>l</sup>* ð Þ¼ *<sup>β</sup>* <sup>X</sup>*<sup>ζ</sup>*

**2.2 Multicarrier modulation (MCM) system**

*<sup>l</sup>*þ<sup>1</sup> ð Þ¼ *<sup>β</sup>* <sup>X</sup>

*Wavelet Based Multicarrier Modulation (MCM) Systems: PAPR Analysis*

*<sup>l</sup>* ð Þ *β* at any level *l* can be expressed as

2*p*

*m*

*<sup>l</sup>*þ1ð Þ *<sup>m</sup> <sup>h</sup>*ð Þþ *<sup>β</sup>* � <sup>2</sup>*<sup>m</sup>* <sup>X</sup>*<sup>ζ</sup>*

speed serial signals into multiple low-speed parallel signals with *N* overlapping subcarriers. This special multicarrier modulation scheme was introduced by Chang [21], and is known as the orthogonal frequency division multiplexing (OFDM). The technique is widely used in various applications such as in European Digital Audio Broadcasting (DAB), IEEE 802*.*11 (WiFi) and IEEE 802*.*16 (WiMAX). OFDM has high spectral efficiency and consecutive subcarriers experience no crosstalk if the

Multicarrier modulation (MCM) scheme is a technique that transforms the high-

In this study two wavelet-based MCM systems are used i.e. the wavelet-based OFDM (WOFDM) and wavelet packet-based OFDM (WP-OFDM) systems. As seen above. The primary difference between these two MCM systems is the way the wavelet tree being expanded. Therefore, in wavelet-based OFDM (WOFDM), the decomposition process expands the branches in dyadic way. In wavelet packetbased OFDM (WP-OFDM), the decomposition process expands the nodes as a full binary tree. Hence, wavelet packet process possesses richer signal analysis than wavelets process and for the detail analysis, wavelet packet process is capable to focus on any of the tree nodes. This main difference of the two MCM systems is illustrated in **Figure 7**. Notice that the wavelet decomposition produces different range of bandwidth divisions. The wavelet bandwidth is in form of dyadic division, while wavelet packet bandwidth is uniform. Therefore, the use of wavelet packet

In WPT, the number of iterations by the 2-channel filter bank increases exponentially as the number of levels increased. Therefore, WPT has higher computational complexity than the regular DWT. The WPT requires *O*(*Nlog*(*N*)) operation (by using fast filter bank algorithm), while Fast Fourier Transform (FFT) requires only *O*(*N*) operations to complete DWT [16]. The reconstruction (inverse WPT) is executed by taking the reverse direction of the tree in **Figure 6**. The wavelet packet

*g m*ð Þ � 2*β ζ*

*p*

2*p*þ1

*<sup>l</sup>* ð Þ *m* (26)

*<sup>l</sup>*þ<sup>1</sup> ð Þ *<sup>m</sup> <sup>g</sup>*ð Þ *<sup>β</sup>* � <sup>2</sup>*<sup>m</sup>* (27)

#### *2.1.2 Wavelet packet transform (WPT)*

In DWT decomposition, the direction of decomposition is heading towards the low pass branches, i.e. the sequence of iteration for the 2-channel filter bank is always taking the low pass filters. At the end of decomposition, the low frequencies portion contains fewer numbers of coefficients, hence occupying a narrow bandwidth. The high frequencies portion contains larger number of coefficients, hence occupying a wide bandwidth.

On the other hand, wavelet packet transform (WPT) executes the iteration of 2 channel filter bank on both sides, i.e. on the low pass and high pass filter branches for decomposition. Therefore, the WPT has evenly space frequency resolution and similar bandwidth size since both the high frequency and low frequencies components are decomposed. In WPT, the filter bank structure is expanded into a full binary tree. A set of WPT coefficients is labeled by *ζ* and the level that corresponds to the depth a node in the tree structure is indicated by *l* and parameter *p* indicates the position at current node. Every parent node is split by the WPT in two orthogonal subspaces *W<sup>P</sup> <sup>l</sup>* which is located at the next recursive level, and is given as [19];

$$\mathcal{W}\_l^p = \mathcal{W}\_{l+1}^{2p} \oplus \mathcal{W}\_{1+1}^{2p+1}, \mathcal{W}\_l^p = \overline{\text{Span}\left\{ 2^{l/2} \zeta\_l^p \left( 2^l t - \beta \right) \right\}}\tag{24}$$

In WPT, the scaling WPT coefficients are denoted as *ζ* 2*p <sup>l</sup>*þ<sup>1</sup> and wavelet WPT coefficients are labeled as *ζ*<sup>2</sup>*p*þ<sup>1</sup> , given as in the following expressions, and are depicted as in **Figure 6**.

$$
\zeta\_{l+1}^{2p}(\beta) = \sum\_{m} h(m - 2\beta) \zeta\_l^p(m) \tag{25}
$$

**Figure 6.** *WPT decomposition at single level.*

*Wavelet Based Multicarrier Modulation (MCM) Systems: PAPR Analysis DOI: http://dx.doi.org/10.5772/intechopen.94579*

$$\zeta\_{l+1}^{2p+1}(\beta) = \sum\_{m} \mathbf{g}(m - 2\beta) \zeta\_l^p(m) \tag{26}$$

In WPT, the number of iterations by the 2-channel filter bank increases exponentially as the number of levels increased. Therefore, WPT has higher computational complexity than the regular DWT. The WPT requires *O*(*Nlog*(*N*)) operation (by using fast filter bank algorithm), while Fast Fourier Transform (FFT) requires only *O*(*N*) operations to complete DWT [16]. The reconstruction (inverse WPT) is executed by taking the reverse direction of the tree in **Figure 6**. The wavelet packet coefficients *ζ p <sup>l</sup>* ð Þ *β* at any level *l* can be expressed as

$$\zeta\_l^p(\beta) = \sum \zeta\_{l+1}^{2p}(m)h(\beta - 2m) + \sum \zeta\_{l+1}^{2p+1}(m)\mathbf{g}(\beta - 2m) \tag{27}$$

#### **2.2 Multicarrier modulation (MCM) system**

In short, the DWT decomposes signals into coefficients. The IDWT reconstructs

In DWT decomposition, the direction of decomposition is heading towards the low pass branches, i.e. the sequence of iteration for the 2-channel filter bank is always taking the low pass filters. At the end of decomposition, the low frequencies portion contains fewer numbers of coefficients, hence occupying a narrow bandwidth. The high frequencies portion contains larger number of coefficients, hence

On the other hand, wavelet packet transform (WPT) executes the iteration of 2 channel filter bank on both sides, i.e. on the low pass and high pass filter branches for decomposition. Therefore, the WPT has evenly space frequency resolution and similar bandwidth size since both the high frequency and low frequencies components are decomposed. In WPT, the filter bank structure is expanded into a full binary tree. A set of WPT coefficients is labeled by *ζ* and the level that corresponds to the depth a node in the tree structure is indicated by *l* and parameter *p* indicates the position at current node. Every parent node is split by the WPT in two orthog-

*<sup>l</sup>* which is located at the next recursive level, and is given as [19];

*ζ p <sup>l</sup>* 2*<sup>l</sup>*

, given as in the following expressions, and are

*p*

2*p*

*<sup>t</sup>* � *<sup>β</sup>* � � � � (24)

*<sup>l</sup>* ð Þ *m* (25)

*<sup>l</sup>*þ<sup>1</sup> and wavelet WPT

*<sup>l</sup>* <sup>¼</sup> *Span* <sup>2</sup>*<sup>l</sup>=*<sup>2</sup>

*h m*ð Þ � 2*β ζ*

the original signals from coefficients which can be implemented efficiently by

iterating the 2-channel synthesis filter bank.

*2.1.2 Wavelet packet transform (WPT)*

occupying a wide bandwidth.

*W<sup>p</sup>*

coefficients are labeled as *ζ*<sup>2</sup>*p*þ<sup>1</sup>

depicted as in **Figure 6**.

**Figure 6.**

**70**

*WPT decomposition at single level.*

*<sup>l</sup>* <sup>¼</sup> *<sup>W</sup>*<sup>2</sup>*<sup>p</sup>*

*<sup>l</sup>*þ<sup>1</sup> <sup>⨁</sup>*W*<sup>2</sup>*p*þ<sup>1</sup>

In WPT, the scaling WPT coefficients are denoted as *ζ*

*<sup>l</sup>*þ<sup>1</sup>ð Þ¼ *<sup>β</sup>* <sup>X</sup>

*ζ* 2*p* <sup>1</sup>þ<sup>1</sup> ,*W<sup>p</sup>*

*m*

onal subspaces *W<sup>P</sup>*

*Wavelet Theory*

Multicarrier modulation (MCM) scheme is a technique that transforms the highspeed serial signals into multiple low-speed parallel signals with *N* overlapping subcarriers. This special multicarrier modulation scheme was introduced by Chang [21], and is known as the orthogonal frequency division multiplexing (OFDM). The technique is widely used in various applications such as in European Digital Audio Broadcasting (DAB), IEEE 802*.*11 (WiFi) and IEEE 802*.*16 (WiMAX). OFDM has high spectral efficiency and consecutive subcarriers experience no crosstalk if the orthogonality is preserved.

In this study two wavelet-based MCM systems are used i.e. the wavelet-based OFDM (WOFDM) and wavelet packet-based OFDM (WP-OFDM) systems. As seen above. The primary difference between these two MCM systems is the way the wavelet tree being expanded. Therefore, in wavelet-based OFDM (WOFDM), the decomposition process expands the branches in dyadic way. In wavelet packetbased OFDM (WP-OFDM), the decomposition process expands the nodes as a full binary tree. Hence, wavelet packet process possesses richer signal analysis than wavelets process and for the detail analysis, wavelet packet process is capable to focus on any of the tree nodes. This main difference of the two MCM systems is illustrated in **Figure 7**. Notice that the wavelet decomposition produces different range of bandwidth divisions. The wavelet bandwidth is in form of dyadic division, while wavelet packet bandwidth is uniform. Therefore, the use of wavelet packet

**Figure 7.** *Decomposition and bandwidth division for (a) DWT and (b) WPT.*

transform in MCM system is preferable since its major characteristic resembles the conventional OFDM [22].

(QMF) that contain half-band of the low and high-pass filters, i.e. *h n*½ � and *g n*½ � respectively of length *L*. The relationship of the filters is described as the following;

*g L*½ �¼ � � <sup>1</sup> � *<sup>n</sup>* ð Þ<sup>1</sup> *<sup>n</sup>*

wavelet packet carriers for modulation at the end of the transmitter, while pair of *h n*½ � and *g n*½ � is the analysis filter-pair for demodulation at the end of the receiver.

> 2 p X *m*

> > 2 p X *m*

**3. PAPR profile of wavelet-based multicarrier modulation signals**

This section presents a comprehensive study on the PAPR profile of multicarrier modulation (MCM) signals. The performance of the transmitted signal is measured by the ratio of peak power signal to its corresponding average power signal within similar MCM frame, known as the peak-to-average power ratio (PAPR). It is desired to have a minimum PAPR as possible in order to reduce the complexity of high power amplifier (HPA) and at the same time, the average transmitting power can be boosted up efficiently as maximum as possible in a linear region of a HPA. Besides, it is disadvantageous of having high PAPR as the signals may be distorted in the nonlinear region of the HPA and results in poor reception and bit error rate (BER) performance. In order to cope with high PAPR, this chapter provides a study that investigates the wavelet-based OFDM (WOFDM), wavelet packet-based OFDM (WP-OFDM) and conventional OFDM systems performances. This investigation is carried out by replacing different orthogonal base modulations, which is normally used in Fourier based MCM (as the conventional OFDM system).

This section presents the general multicarrier modulation system model structures for implementation. The condition for determining the initial data value and maximum potential number of symbols to be carried by system subcarriers are also

The three evaluated multicarrier modulation (MCM) system models are represented by a single general MCM model as illustrated in **Figure 10**. The information bits are generated based on the uniform random distribution binary

*h n*½ �Υ*<sup>p</sup>*

*g n*½ �Υ*<sup>p</sup>*

The complex conjugate time reversed variant is given by [24];

*Wavelet Based Multicarrier Modulation (MCM) Systems: PAPR Analysis*

½ �¼ *<sup>n</sup> <sup>h</sup>*<sup>∗</sup> ½ � �*<sup>n</sup> and g*<sup>0</sup>

*h*0

Υ<sup>2</sup>*<sup>p</sup>*

Υ<sup>2</sup>*p*þ<sup>1</sup>

where *p* is subcarrier index at any tree depth *l*.

**3.1 Multicarrier modulation system models**

discussed.

**73**

*3.1.1 System models descriptions*

*<sup>l</sup>*þ<sup>1</sup>ðÞ¼ *<sup>t</sup>* ffiffi

*<sup>l</sup>*þ<sup>1</sup> ðÞ¼ *<sup>t</sup>* ffiffi

½ � *n* and *g*<sup>0</sup>

*DOI: http://dx.doi.org/10.5772/intechopen.94579*

The wavelet packet coefficients Υ*<sup>p</sup>*

The pair of *h*<sup>0</sup>

via MRA as [24];

*h n*½ � (28)

½ �¼ *<sup>n</sup> <sup>g</sup>* <sup>∗</sup> ½ � �*<sup>n</sup>* (29)

*<sup>l</sup>* ð Þ 2*t* � *n* (30)

*<sup>l</sup>* ð Þ 2*t* � *n* (31)

½ � *n* is the synthesis filter-pair which is used to produce

*<sup>l</sup>* are obtained from QMF filters which are derived

In wavelet packet-based OFDM (WP-OFDM) scheme that wavelet packet transform is utilized to change a series of parallel signals into a single composite signal. Both OFDM and WP-OFDM possess high spectral efficiency since their subcarriers are orthogonal that overlap between each other. The only difference between the two schemes is in term of the shape of the subcarriers produced. In ordinary OFDM the Fourier bases are used i.e. the sine or cosine terms. However, in WP-OFDM scheme the wavelet packet provides flexibility for modification of the filter banks property to suit the characteristic of system transmission [14]. The general multicarrier modulation system is shown in **Figure 8**.

WP-OFDM is implemented by the using the inverse orthogonal transform at the transmitter which is known as the inverse discrete wavelet packet transform (ID-WPT) as illustrated in **Figure 9** (left-hand side). The forward orthogonal transform is implemented at the receiver called as discrete wavelet packet transform (DWPT) as depicted in **Figure 9** (right-hand side). The implementation of WP-OFDM that utilizes the wavelet packet transform has been derived from MRA concept [23]. It is commenced by introducing a pair of filters called as quadrature mirror filters

#### **Figure 8.**

*General schemes for multicarrier modulation.*

**Figure 9.** *IDWPT and DWPT in MCM scheme.*

*Wavelet Based Multicarrier Modulation (MCM) Systems: PAPR Analysis DOI: http://dx.doi.org/10.5772/intechopen.94579*

(QMF) that contain half-band of the low and high-pass filters, i.e. *h n*½ � and *g n*½ � respectively of length *L*. The relationship of the filters is described as the following;

$$\lg[L - 1 - n] = (-1)^n h[n] \tag{28}$$

The complex conjugate time reversed variant is given by [24];

$$h'[n] = h^\*\left[-n\right] \text{ and } \mathbf{g'}[n] = \mathbf{g}^\*\left[-n\right] \tag{29}$$

The pair of *h*<sup>0</sup> ½ � *n* and *g*<sup>0</sup> ½ � *n* is the synthesis filter-pair which is used to produce wavelet packet carriers for modulation at the end of the transmitter, while pair of *h n*½ � and *g n*½ � is the analysis filter-pair for demodulation at the end of the receiver. The wavelet packet coefficients Υ*<sup>p</sup> <sup>l</sup>* are obtained from QMF filters which are derived via MRA as [24];

$$\Upsilon\_{l+1}^{2^p}(t) = \sqrt{2} \sum\_{m} h[n] \Upsilon\_l^p(2t - n) \tag{30}$$

$$\Upsilon\_{l+1}^{2p+1}(t) = \sqrt{2} \sum\_{m} \mathbf{g}[n] \Upsilon\_{l}^{p}(2t - n) \tag{31}$$

where *p* is subcarrier index at any tree depth *l*.

#### **3. PAPR profile of wavelet-based multicarrier modulation signals**

This section presents a comprehensive study on the PAPR profile of multicarrier modulation (MCM) signals. The performance of the transmitted signal is measured by the ratio of peak power signal to its corresponding average power signal within similar MCM frame, known as the peak-to-average power ratio (PAPR). It is desired to have a minimum PAPR as possible in order to reduce the complexity of high power amplifier (HPA) and at the same time, the average transmitting power can be boosted up efficiently as maximum as possible in a linear region of a HPA. Besides, it is disadvantageous of having high PAPR as the signals may be distorted in the nonlinear region of the HPA and results in poor reception and bit error rate (BER) performance. In order to cope with high PAPR, this chapter provides a study that investigates the wavelet-based OFDM (WOFDM), wavelet packet-based OFDM (WP-OFDM) and conventional OFDM systems performances. This investigation is carried out by replacing different orthogonal base modulations, which is normally used in Fourier based MCM (as the conventional OFDM system).

#### **3.1 Multicarrier modulation system models**

This section presents the general multicarrier modulation system model structures for implementation. The condition for determining the initial data value and maximum potential number of symbols to be carried by system subcarriers are also discussed.

#### *3.1.1 System models descriptions*

The three evaluated multicarrier modulation (MCM) system models are represented by a single general MCM model as illustrated in **Figure 10**. The information bits are generated based on the uniform random distribution binary

transform in MCM system is preferable since its major characteristic resembles the

In wavelet packet-based OFDM (WP-OFDM) scheme that wavelet packet transform is utilized to change a series of parallel signals into a single composite signal. Both OFDM and WP-OFDM possess high spectral efficiency since their subcarriers are orthogonal that overlap between each other. The only difference between the two schemes is in term of the shape of the subcarriers produced. In ordinary OFDM the Fourier bases are used i.e. the sine or cosine terms. However, in WP-OFDM scheme the wavelet packet provides flexibility for modification of the filter banks property to suit the characteristic of system transmission [14]. The

WP-OFDM is implemented by the using the inverse orthogonal transform at the transmitter which is known as the inverse discrete wavelet packet transform (ID-WPT) as illustrated in **Figure 9** (left-hand side). The forward orthogonal transform is implemented at the receiver called as discrete wavelet packet transform (DWPT) as depicted in **Figure 9** (right-hand side). The implementation of WP-OFDM that utilizes the wavelet packet transform has been derived from MRA concept [23]. It is commenced by introducing a pair of filters called as quadrature mirror filters

general multicarrier modulation system is shown in **Figure 8**.

conventional OFDM [22].

*Wavelet Theory*

**Figure 8.**

**Figure 9.**

**72**

*IDWPT and DWPT in MCM scheme.*

*General schemes for multicarrier modulation.*

**Figure 10.**

number. The data are arranged (in every frame) in a horizontal matrix 1 *ninit*, and are converted into *base*16 number format. Subsequently they are encoded by Reed-Solomon (RS) codes, and converted into *baseM* number format, where *M* corresponds to the total mapping points in a particular QAM constellation. In this work, the Reed-Solomon of RS(*n*, *k*) is used, where *n* is the encoded data, and *k* is original data. In particular, the RS(15,11) is used throughout the work for protecting the original data. This channel coding scheme is compatible with hexadecimal number for encoding and decoding processes. In addition, RS encoded symbols are converted to the *baseM* symbols to achieve the same configuration that adapts with modulation constellation mapping.

**Table 1** shows four possible of *baseM* number format types associated with the number of bits per symbol, *Nbps* as well as the corresponding constellation mapping modulation types. Then, the data frames are transformed into time-domain MCM signals by using a particular inverse transform prior transmission and they are retrieved back by the corresponding forward transform at the receiver. The particular inverse and forward transforms applied are labeled in block diagrams as shown in **Figures 11** and **12**.

Each frame must contain *P* symbols and is always less than or equal to the total number of subcarriers *N,* i.e. *P*≤ *N*. The specified number of base for every modulation type is fixed as in **Table 1**. The number of initial binary information *ninit* increases as the number of bits per symbol *Nbps* increased with constant value of *N*. In this work, the RS(15, 11) is used, and the encoded data ð Þ *n* ¼ 15 , and the original data ð Þ *k* ¼ 11 respectively. This implies that each time a sequence of symbols to be encoded, the number of original symbols taken is eleven and this produces total fifteen encoded symbols afterwards. Therefore, during the encoding process, the raw binary data (*base*2) need to be converted to *base*16 symbols to suites the requirement of RS(15, 11) coding scheme where each encoded symbol should have a

This section describes how the transmission parameters values of *P* is obtained by manipulating the base number of the symbols. **Figure 10** above shows the block diagram of the of S/S encodes where raw input data bits are converted into *baseM*

**Figure 13** denotes three conversion processes for the initial input bit *ninit* which are indicated as the *α*, *β* and *γ* processes. The *α* process converts every four bits (*v* ¼ 4Þ of binary source data to a *base*16 symbol. For example, if

*ninit*,*base*<sup>2</sup> ¼ f1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1g then, the output from process *α* is *n<sup>α</sup>* ¼ f g 9, 3, 3, 2, 8, 0, 7, 1, 7, 2, 9 . This means the

Next, *β* process converts every eleven *base*16 symbols ð Þ *k* ¼ 11 into fifteen *base*16 encoded symbols ð Þ *n* ¼ 15 . For instance, when *n<sup>α</sup>* ¼ f g 9, 3, 3, 2, 8, 0, 7, 1, 7, 2, 9 then, the output of *β* process is *n<sup>β</sup>* ¼ f g 9, 3, 3, 2, 8, 0, 7, 1, 7, 2, 9, 15, 2, 7, 11 which increases

value between 0 to 15.

*Model for conventional OFDM scheme.*

**Figure 11.**

**Figure 12.**

**75**

*3.1.2 Determination transmission parameter*

*Model for wavelet- and wavelet packet-based OFDM schemes.*

*DOI: http://dx.doi.org/10.5772/intechopen.94579*

*Wavelet Based Multicarrier Modulation (MCM) Systems: PAPR Analysis*

symbols. **Figure 13** shows further details of this process.

number of overall symbols is reduced to one-fourth.

In **Figure 11**, there are two types of wavelet-based MCM models to be considered i.e. the wavelet-based OFDM (WOFDM) and wavelet packet-based OFDM (WP-OFDM) systems. At the transmitter, either the inverse discrete wavelet transform (IDWT) or inverse discrete wavelet packet transform (IDWPT) is used. At the receiver, either the discrete wavelet transform (DWT) or discrete wavelet packet transform (DWPT) is used. These modulation techniques offer higher spectral efficiency since there is no for the system to use the cyclic pre-fix (CP) codes as in the conventional OFDM.

**Figure 12** shows the conventional OFDM (C-OFDM) which is included for comparison system model. This model utilizes the inverse fast Fourier transform (IFFT) and fast Fourier transform (FFT). Additional blocks are required for appending and re-moving the CP codes where 25 percent of the OFDM frames tail are copied and appended to OFDM frames head [25, 26].


**Table 1.** BaseM *and its appropriate constellation mapping.* *Wavelet Based Multicarrier Modulation (MCM) Systems: PAPR Analysis DOI: http://dx.doi.org/10.5772/intechopen.94579*

**Figure 11.**

number. The data are arranged (in every frame) in a horizontal matrix 1 *ninit*, and are converted into *base*16 number format. Subsequently they are encoded by Reed-Solomon (RS) codes, and converted into *baseM* number format, where *M* corresponds to the total mapping points in a particular QAM constellation. In this work, the Reed-Solomon of RS(*n*, *k*) is used, where *n* is the encoded data, and *k* is original data. In particular, the RS(15,11) is used throughout the work for protecting the original data. This channel coding scheme is compatible with hexadecimal number

converted to the *baseM* symbols to achieve the same configuration that adapts with

**Table 1** shows four possible of *baseM* number format types associated with the number of bits per symbol, *Nbps* as well as the corresponding constellation mapping modulation types. Then, the data frames are transformed into time-domain MCM signals by using a particular inverse transform prior transmission and they are retrieved back by the corresponding forward transform at the receiver. The particular inverse and forward transforms applied are labeled in block diagrams as shown

In **Figure 11**, there are two types of wavelet-based MCM models to be considered i.e. the wavelet-based OFDM (WOFDM) and wavelet packet-based OFDM (WP-OFDM) systems. At the transmitter, either the inverse discrete wavelet transform (IDWT) or inverse discrete wavelet packet transform (IDWPT) is used. At the receiver, either the discrete wavelet transform (DWT) or discrete wavelet packet transform (DWPT) is used. These modulation techniques offer higher spectral efficiency since there is no for the system to use the cyclic pre-fix (CP) codes as in

**Figure 12** shows the conventional OFDM (C-OFDM) which is included for comparison system model. This model utilizes the inverse fast Fourier transform (IFFT) and fast Fourier transform (FFT). Additional blocks are required for appending and re-moving the CP codes where 25 percent of the OFDM frames tail

*BaseM* **No. of bits per symbol,** *Nbps* **Suitable mapping type**

*Base*2 1 BPSK *Base*4 2 QAM *Base*16 4 16QAM *Base*64 6 64QAM

are copied and appended to OFDM frames head [25, 26].

BaseM *and its appropriate constellation mapping.*

for encoding and decoding processes. In addition, RS encoded symbols are

modulation constellation mapping.

*Model for general MCM scheme with data sequence details.*

in **Figures 11** and **12**.

**Figure 10.**

*Wavelet Theory*

the conventional OFDM.

**Table 1.**

**74**

*Model for wavelet- and wavelet packet-based OFDM schemes.*

#### **Figure 12.** *Model for conventional OFDM scheme.*

Each frame must contain *P* symbols and is always less than or equal to the total number of subcarriers *N,* i.e. *P*≤ *N*. The specified number of base for every modulation type is fixed as in **Table 1**. The number of initial binary information *ninit* increases as the number of bits per symbol *Nbps* increased with constant value of *N*. In this work, the RS(15, 11) is used, and the encoded data ð Þ *n* ¼ 15 , and the original data ð Þ *k* ¼ 11 respectively. This implies that each time a sequence of symbols to be encoded, the number of original symbols taken is eleven and this produces total fifteen encoded symbols afterwards. Therefore, during the encoding process, the raw binary data (*base*2) need to be converted to *base*16 symbols to suites the requirement of RS(15, 11) coding scheme where each encoded symbol should have a value between 0 to 15.

#### *3.1.2 Determination transmission parameter*

This section describes how the transmission parameters values of *P* is obtained by manipulating the base number of the symbols. **Figure 10** above shows the block diagram of the of S/S encodes where raw input data bits are converted into *baseM* symbols. **Figure 13** shows further details of this process.

**Figure 13** denotes three conversion processes for the initial input bit *ninit* which are indicated as the *α*, *β* and *γ* processes. The *α* process converts every four bits (*v* ¼ 4Þ of binary source data to a *base*16 symbol. For example, if *ninit*,*base*<sup>2</sup> ¼ f1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1g then, the output from process *α* is *n<sup>α</sup>* ¼ f g 9, 3, 3, 2, 8, 0, 7, 1, 7, 2, 9 . This means the number of overall symbols is reduced to one-fourth.

Next, *β* process converts every eleven *base*16 symbols ð Þ *k* ¼ 11 into fifteen *base*16 encoded symbols ð Þ *n* ¼ 15 . For instance, when *n<sup>α</sup>* ¼ f g 9, 3, 3, 2, 8, 0, 7, 1, 7, 2, 9 then, the output of *β* process is *n<sup>β</sup>* ¼ f g 9, 3, 3, 2, 8, 0, 7, 1, 7, 2, 9, 15, 2, 7, 11 which increases

#### **Figure 13.**

*S/S encoding into* baseM *symbols block diagram.*


#### **Table 2.**

*Output of* γ *process based on mapping type selection.*

the number of symbols along the encoding processes with the additional redundancy required for channel error protection. This implies the number of overall symbols now has been increased by 15/11. After that, *γ* process converts every single *base*16 symbol into four *base*2 symbols (*w* ¼ 4Þ. Then this follows by converting every *Nbps* <sup>¼</sup> log <sup>2</sup>*<sup>M</sup>* number of *base*2 symbols back to a single *baseM* symbol. Suppose that *M* ¼ 16, continuing the above example, when *n<sup>β</sup>* ¼ f g 9, 3, 3, 2, 8, 0, 7, 1, 7, 2, 9, 15, 2, 7, 11 then, the output of *γ* process is *nγ*,*base*<sup>16</sup> ¼ f g 9, 3, 3, 2, 8, 0, 7, 1, 7, 2, 9, 15, 2, 7, 11 *:* The rest of other mappings are as listed in **Table 2**. The number of transmission symbols *P*, thus can be expressed as

$$P = n\_{init} \times \left(\frac{\mathbf{1}}{\nu}\right) \times \left(\frac{n}{k}\right) \times w \times \left(\frac{\mathbf{1}}{N\_{bps}}\right) \tag{32}$$

total subcarriers f g *N* ¼ 64; 128; 256 . It can be observed that the subcarriers are not fully occupied for the whole slot positions with frames, hence the remaining positions, *R* is filled up with zeros. There are four zero frames ð Þ *N* ¼ 64 , eight zero

*MCM partition with respect to* N. *Partition of the occupied slot: the encoded data (P) and the remaining slot*

This section presents the results and discussions on the PAPR profile performances based on several important parameters i.e. modulation types, number of subcarriers, the orthogonal bases (Fourier/Wavelets) and filter length. The BER performance is also included to investigate the efficiency of the system models. The

The effect of modulation constellation mapping on PAPR is analyzed in the following paragraphs. The list of parameters involved are shown as in **Table 5**. **Figure 15** shows that both the conventional C-OFDM and WP-OFDM systems are having almost the same PAPR profiles, regardless of the modulation mapping types used. The reason for the PAPR profiles of the wavelet based OFDM (WOFDM) outperform the PAPR profiles of the WP-OFDM, is that the WOFDM system contains a smaller number of signal analysis than the WP-OFDM system. The PAPR profile for WOFDM system is superior since the decomposition and reconstruction signals are only involved the low pass branches. Thus, there is lower probability for the peak to be above the average signals leading to slightly superior PAPR profile.

common parameters used in the experiments are as list in **Table 4** below.

**Parameter Descriptions**

Encoder type RS(15,11) Channel model AWGN CP for conventional MCM 25% of total subcarriers

System Model WOFDM, WP-OFDM, C-OFDM

frames ð Þ *N* ¼ 128 and six zero frames ð Þ *N* ¼ 256 respectively.

*Wavelet Based Multicarrier Modulation (MCM) Systems: PAPR Analysis*

**4. PAPR profile: results and analysis**

*positions (R) for total subcarriers {N = 64; 128; 256}.*

*DOI: http://dx.doi.org/10.5772/intechopen.94579*

**Figure 14.**

**Table 4.**

**77**

*Common parameters for experiments.*

where *P*≤ *N*.

Using (Eq. (32)), the number of transmission symbols *P* and initial input bit *ninit* can be obtained after defining number of subcarriers ð Þ *N* . Thus, number of bits per symbol, *Nbps* can be obtained and the quantitative relationships between these parameters are shown in **Table 3**.

**Figure 14** shows the partitions between the occupied slot positions of the encoded data ð Þ *P* and the remaining slot positions ð Þ *R* for three different values of


**Table 3.**

*The relationship between* ninit*,* P*,* N *and* Nbps.

*Wavelet Based Multicarrier Modulation (MCM) Systems: PAPR Analysis DOI: http://dx.doi.org/10.5772/intechopen.94579*

#### **Figure 14.**

the number of symbols along the encoding processes with the additional redundancy required for channel error protection. This implies the number of overall symbols now has been increased by 15/11. After that, *γ* process converts every single *base*16 symbol into four *base*2 symbols (*w* ¼ 4Þ. Then this follows by converting

0,0,0,0,0,1,1,1,0,0,0,1,0,1,1,1,0,0,1,0, 1,0,0,1,1,1,1,1,0,0,1,0,0,1,1,1,1,0,1,1}

0,1,1,3,0,2,1,3,3,0,2,1,3,2,3}

16QAM *Base*16 *nγ,base*<sup>16</sup> = {9*,* 3*,* 3*,* 2*,* 8*,* 0*,* 7*,* 1*,* 7*,* 2*,* 9*,* 15*,* 2*,* 7*,* 11} 15 64QAM *Base*64 *nγ,base*<sup>64</sup> = {36*,* 51*,* 10*,* 0*,* 28*,* 23*,* 10*,* 31*,* 9*,* 59} 10

every *Nbps* <sup>¼</sup> log <sup>2</sup>*<sup>M</sup>* number of *base*2 symbols back to a single *baseM* symbol. Suppose that *M* ¼ 16, continuing the above example, when *n<sup>β</sup>* ¼ f g 9, 3, 3, 2, 8, 0, 7, 1, 7, 2, 9, 15, 2, 7, 11 then, the output of *γ* process is *nγ*,*base*<sup>16</sup> ¼ f g 9, 3, 3, 2, 8, 0, 7, 1, 7, 2, 9, 15, 2, 7, 11 *:* The rest of other mappings are as listed in **Table 2**. The number of transmission symbols *P*, thus can be expressed as

> 1 *v* � *n k*

� *w* �

Using (Eq. (32)), the number of transmission symbols *P* and initial input bit *ninit* can be obtained after defining number of subcarriers ð Þ *N* . Thus, number of bits per symbol, *Nbps* can be obtained and the quantitative relationships between these

**Figure 14** shows the partitions between the occupied slot positions of the encoded data ð Þ *P* and the remaining slot positions ð Þ *R* for three different values of

No. of subcarrier, *N* 64 128 256 No. of bits per symbol *Nbps* 124 6 1 2 4 6 1 2 4 6 No. of initial binary information, *ninit* 44 88 176 264 88 176 352 528 176 352 748 1100 No of symbols per frame *P* 60 60 60 60 120 120 120 120 250 250 250 250

1 *Nbps* 

**Output of** *γ* **process Number of symbols** ð Þ *P* **at**

**output** *γ* **process**

60

30

(32)

*P* ¼ *ninit* �

**Parameters Value**

BPSK *Base*2 *nγ,base*<sup>2</sup> = {1,0,0,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0,0,0,

QAM *Base*4 *nγ,base*<sup>4</sup> = {2,1,0,3,0,3,0,2,2,0,0,0,1,3,

where *P*≤ *N*.

**Table 3.**

**76**

**Figure 13.**

**Table 2.**

**Mapping type**

*Wavelet Theory*

*S/S encoding into* baseM *symbols block diagram.*

*Output of* γ *process based on mapping type selection.*

**Base number**

parameters are shown in **Table 3**.

*The relationship between* ninit*,* P*,* N *and* Nbps.

*MCM partition with respect to* N. *Partition of the occupied slot: the encoded data (P) and the remaining slot positions (R) for total subcarriers {N = 64; 128; 256}.*

total subcarriers f g *N* ¼ 64; 128; 256 . It can be observed that the subcarriers are not fully occupied for the whole slot positions with frames, hence the remaining positions, *R* is filled up with zeros. There are four zero frames ð Þ *N* ¼ 64 , eight zero frames ð Þ *N* ¼ 128 and six zero frames ð Þ *N* ¼ 256 respectively.

#### **4. PAPR profile: results and analysis**

This section presents the results and discussions on the PAPR profile performances based on several important parameters i.e. modulation types, number of subcarriers, the orthogonal bases (Fourier/Wavelets) and filter length. The BER performance is also included to investigate the efficiency of the system models. The common parameters used in the experiments are as list in **Table 4** below.

The effect of modulation constellation mapping on PAPR is analyzed in the following paragraphs. The list of parameters involved are shown as in **Table 5**. **Figure 15** shows that both the conventional C-OFDM and WP-OFDM systems are having almost the same PAPR profiles, regardless of the modulation mapping types used. The reason for the PAPR profiles of the wavelet based OFDM (WOFDM) outperform the PAPR profiles of the WP-OFDM, is that the WOFDM system contains a smaller number of signal analysis than the WP-OFDM system. The PAPR profile for WOFDM system is superior since the decomposition and reconstruction signals are only involved the low pass branches. Thus, there is lower probability for the peak to be above the average signals leading to slightly superior PAPR profile.


**Table 4.** *Common parameters for experiments.*


**Table 5.**

*Parameters used for studying the effect of different type of mapping modulation.*

**Figure 15.** *CCDF of the PAPR with variation of mapping type.*

However, it can clearly be seen in **Figure 16**, the BER performances are indeed worse for all three MCM systems as the type of mapping changes from QAM towards 16QAM and 64QAM. The BER performance is highly related with the type of the signal mapping used. Theoretically, the error probability at the receiver increases as the number of constellation points increased. In order to reduce the error probability, in general higher *Eb/N*<sup>0</sup> is required. **Table 6** shows for probability of bit error at 10<sup>6</sup> the corresponding of *Eb/N*<sup>0</sup> (dB) values for all modulation mapping types. The channel impairment used for evaluating the performance is using the AWGN channel. From **Figure 16**, the 64QAM modulation mapping type that can hold higher bit information, where each symbol represents 6 bits, even though it requires much higher transmitting power.

model outperforms the PAPR profile of C-OFDM and WP-OFDM models by 1.5 dB at the CCDF level of 10<sup>5</sup> for fixed *N* = 64. The PAPR profiles for C-OFDM and WP-OFDM systems are similar. The PAPR profile for WOFDM system is superior since the decomposition and reconstruction signals are only involved the low pass branches. Thus, there is lower probability for the peak to be above the average

**Mapping type** *Eb/N***<sup>0</sup> (dB) at BER level of 10<sup>6</sup>**

QAM 9.0 16QAM 15.5 64QAM 21.5

**Parameter Descriptions** Mapping type 64QAM Number of subcarriers 64, 128, 256 Orthogonal bases Fourier, wavelet (Haar)

**Figure 18** shows that there is no significant different in term of BER performances, although different numbers of subcarriers are used for modulation. At BER

, it is found that the difference between the lowest and highest value of

signals leading to slightly superior PAPR profile.

*Parameters used for studying the effect of different No. of subcarriers.*

*Corresponding BER performances due to variation of mapping type.*

*Wavelet Based Multicarrier Modulation (MCM) Systems: PAPR Analysis*

*DOI: http://dx.doi.org/10.5772/intechopen.94579*

*Common value of* Eb/N*<sup>0</sup> for the corresponding mapping type.*

of 10<sup>6</sup>

**79**

**Figure 16.**

**Table 6.**

**Table 7.**

The following paragraphs analyze the effect on varying the number of subcarriers on the PAPR profiles. **Table 7** lists all parameters required for this experiment. It is found that, when the number of subcarriers *N* decreases i.e. from *N* = 256 until *N* = 64, the PAPR profile (CCDF) of any modulation scheme is gradually improves as shown in **Figure 17**. Explicitly, the PAPR profile of WOFDM *Wavelet Based Multicarrier Modulation (MCM) Systems: PAPR Analysis DOI: http://dx.doi.org/10.5772/intechopen.94579*

#### **Figure 16.**

*Corresponding BER performances due to variation of mapping type.*


#### **Table 6.**

*Common value of* Eb/N*<sup>0</sup> for the corresponding mapping type.*


**Table 7.**

However, it can clearly be seen in **Figure 16**, the BER performances are indeed

worse for all three MCM systems as the type of mapping changes from QAM towards 16QAM and 64QAM. The BER performance is highly related with the type of the signal mapping used. Theoretically, the error probability at the receiver increases as the number of constellation points increased. In order to reduce the error probability, in general higher *Eb/N*<sup>0</sup> is required. **Table 6** shows for probability of bit error at 10<sup>6</sup> the corresponding of *Eb/N*<sup>0</sup> (dB) values for all modulation mapping types. The channel impairment used for evaluating the performance is using the AWGN channel. From **Figure 16**, the 64QAM modulation mapping type that can hold higher bit information, where each symbol represents 6 bits, even

**Parameter Descriptions** Mapping type QAM, 16QAM, 64QAM

Orthogonal bases Fourier, wavelet (Haar)

Number of subcarriers 128

*Parameters used for studying the effect of different type of mapping modulation.*

**Table 5.**

*Wavelet Theory*

**Figure 15.**

**78**

The following paragraphs analyze the effect on varying the number of subcarriers on the PAPR profiles. **Table 7** lists all parameters required for this experiment. It is found that, when the number of subcarriers *N* decreases i.e. from *N* = 256 until *N* = 64, the PAPR profile (CCDF) of any modulation scheme is gradually improves as shown in **Figure 17**. Explicitly, the PAPR profile of WOFDM

though it requires much higher transmitting power.

*CCDF of the PAPR with variation of mapping type.*

*Parameters used for studying the effect of different No. of subcarriers.*

model outperforms the PAPR profile of C-OFDM and WP-OFDM models by 1.5 dB at the CCDF level of 10<sup>5</sup> for fixed *N* = 64. The PAPR profiles for C-OFDM and WP-OFDM systems are similar. The PAPR profile for WOFDM system is superior since the decomposition and reconstruction signals are only involved the low pass branches. Thus, there is lower probability for the peak to be above the average signals leading to slightly superior PAPR profile.

**Figure 18** shows that there is no significant different in term of BER performances, although different numbers of subcarriers are used for modulation. At BER of 10<sup>6</sup> , it is found that the difference between the lowest and highest value of

**Figure 17.** *CCDF of the PAPR with variation of number of subcarriers.*

*Eb/N*<sup>0</sup> is less than 1 dB. Thus, it can be deduced that, the number of subcarriers gives less impact to the BER performance. The *Eb/N*<sup>0</sup> is quite high (i.e. nearly 22 dB for all profiles). The increase in the number of subcarriers worsen the PAPR profile. Therefore, for practical application, the number of subcarriers *N* = 128 is selected since it is a moderate choice as compare to the other number of subcarriers.

The following paragraphs analyze the influence of different orthogonal bases, wavelet types and their filter lengths on the PAPR profile. Several wavelet families applied includes the Daubechies, Symlet, Coiflet and Meyer wavelets with various lengths of coefficients. The parameters are briefly listed as in **Table 8**. This analysis is mainly focuses on the wavelet OFDM and wavelet packet-based OFDM systems. However, the C-OFDM scheme is also included as a performance reference. Additional information regarding the characteristic of the wavelet families are included in **Table 9**.

**Figure 19** shows the PAPR profiles for the three OFDM systems, where Daubechies wavelet with different filter lengths are used (Fourier based OFDM profile is just for reference only). In analyzing the effect of wavelet filter length, various filter lengths are used in the experiment. From **Figure 19**, looking at the WOFDM profiles (red color), as the filter length increases, the PAPR profiles become worse. In other words, the Daubechies wavelet (in WOFDM) with higher filter length produces inferior PAPR profiles. This is due to the fact that with higher filter length, the wavelet has richer signal analysis. Thus, there is higher probability for the peak to be above the average signals leading to slightly inferior PAPR profile.

However, for WP-OFDM profiles (blue color), there is no significant difference in the PAPR performance. Since the signal analysis in WP-OFDM is in full binary tree analysis rather than dyadic (lower-half band) analysis in WOFDM system. There is already high amount of data involved in decomposition and reconstruction which makes the effect of wavelet filter length insignificant.

In **Figures 20** and **21**, different wavelet types (Daubechies, Symlet, Coiflet and

128

*coif3, coif5, dmey*)

Orthogonal bases Fourier, wavelet (*db1, db2, db3, db5, db10, db20, sym2, sym3, sym5, sym10, coif1,*

**Full name Abbreviated name Vanishing order Length,** *L* Haar Haar 1 2 Daubechies *dbN N* 2 *N* Symlets *symN N* 2 *N* Coiflet *coifN N* 6 *N* Discrete Meyer *dmey* — 62

Discrete Meyer wavelets) are used but the filter length is fixed *L* = 6 (short category). For long category the filter lengths are mixed, i.e. *L* ¼ f18ð Þ *coi f* 3 , 20 ð*db*10, *sym*10Þ, 62ð Þ *dmey* g respectively. From these figures, there can be observed

*Corresponding BER performance due to variation of number of subcarriers.*

*Wavelet Based Multicarrier Modulation (MCM) Systems: PAPR Analysis*

*DOI: http://dx.doi.org/10.5772/intechopen.94579*

**Parameter Descriptions** Mapping type 64QAM

*Parameters used for studying the effect of different bases and filter length.*

**Figure 18.**

Number of subcarriers

**Table 8.**

**Table 9.**

**81**

*Wavelet family characteristic [23].*

*Wavelet Based Multicarrier Modulation (MCM) Systems: PAPR Analysis DOI: http://dx.doi.org/10.5772/intechopen.94579*

#### **Figure 18.**

*Eb/N*<sup>0</sup> is less than 1 dB. Thus, it can be deduced that, the number of subcarriers gives less impact to the BER performance. The *Eb/N*<sup>0</sup> is quite high (i.e. nearly 22 dB for all profiles). The increase in the number of subcarriers worsen the PAPR profile. Therefore, for practical application, the number of subcarriers *N* = 128 is selected since it is a moderate choice as compare to the other number of subcarriers.

*CCDF of the PAPR with variation of number of subcarriers.*

The following paragraphs analyze the influence of different orthogonal bases, wavelet types and their filter lengths on the PAPR profile. Several wavelet families applied includes the Daubechies, Symlet, Coiflet and Meyer wavelets with various lengths of coefficients. The parameters are briefly listed as in **Table 8**. This analysis is mainly focuses on the wavelet OFDM and wavelet packet-based OFDM systems. However, the C-OFDM scheme is also included as a performance reference. Additional information regarding the characteristic of the wavelet families are included

**Figure 19** shows the PAPR profiles for the three OFDM systems, where Daubechies wavelet with different filter lengths are used (Fourier based OFDM profile is just for reference only). In analyzing the effect of wavelet filter length, various filter lengths are used in the experiment. From **Figure 19**, looking at the WOFDM profiles (red color), as the filter length increases, the PAPR profiles become worse. In other words, the Daubechies wavelet (in WOFDM) with higher filter length produces inferior PAPR profiles. This is due to the fact that with higher filter length, the wavelet has richer signal analysis. Thus, there is higher probability for the peak to be above the average signals leading to slightly inferior PAPR profile. However, for WP-OFDM profiles (blue color), there is no significant difference in the PAPR performance. Since the signal analysis in WP-OFDM is in full binary tree analysis rather than dyadic (lower-half band) analysis in WOFDM system. There is already high amount of data involved in decomposition and reconstruction

which makes the effect of wavelet filter length insignificant.

in **Table 9**.

**80**

**Figure 17.**

*Wavelet Theory*

*Corresponding BER performance due to variation of number of subcarriers.*


#### **Table 8.**

*Parameters used for studying the effect of different bases and filter length.*


#### **Table 9.**

*Wavelet family characteristic [23].*

In **Figures 20** and **21**, different wavelet types (Daubechies, Symlet, Coiflet and Discrete Meyer wavelets) are used but the filter length is fixed *L* = 6 (short category). For long category the filter lengths are mixed, i.e. *L* ¼ f18ð Þ *coi f* 3 , 20 ð*db*10, *sym*10Þ, 62ð Þ *dmey* g respectively. From these figures, there can be observed

**Figure 21.**

**Figure 22.**

**83**

*CCDF of the PAPR with different orthogonal bases modulation and long filter lengths.*

*Wavelet Based Multicarrier Modulation (MCM) Systems: PAPR Analysis*

*DOI: http://dx.doi.org/10.5772/intechopen.94579*

*Corresponding BER performances for Daubechies wavelet with different filter lengths.*

**Figure 19.** *CCDF of the PAPR with Daubechies wavelet with different filter lengths.*

**Figure 20.** *CCDF of the PAPR with different orthogonal bases modulation and short filter lengths.*

**Figure 21.** *CCDF of the PAPR with different orthogonal bases modulation and long filter lengths.*

**Figure 22.** *Corresponding BER performances for Daubechies wavelet with different filter lengths.*

**Figure 19.**

*Wavelet Theory*

**Figure 20.**

**82**

*CCDF of the PAPR with Daubechies wavelet with different filter lengths.*

*CCDF of the PAPR with different orthogonal bases modulation and short filter lengths.*

that no explicit difference found from PAPR profiles of WP-OFDM signals either by changing the wavelet's type or length.

**References**

[1] Van Nee R, Prasad R. OFDM for Wireless Multimedia Communications. Boston, London: Artech House; 2000.

*DOI: http://dx.doi.org/10.5772/intechopen.94579*

*Wavelet Based Multicarrier Modulation (MCM) Systems: PAPR Analysis*

for WOFDM PAPR reduction. In: Proceeding of the 2nd International Conference on Advanced Technologies for Signal and Image Processing (ATSIP '16), March 21–23, 2016; Monastir, Tunisia: P. 710–714, doi: 10.1109/

[9] Zakaria J, Salleh, M. F. M. PAPR reduction scheme: wavelet packet-based PTS with embedded side information data scheme. IET Communications.

[10] Yoon E, Hwang D, Jang C, Kim J, Yun U. Blind Selected Mapping with Side Information Estimation Based on the Received Pilot Signal, Wireless Communications and Mobile

Computing. 2018*;* (5):1–9, DOI: 10.1155/

[11] Ahmed M S, Boussakta S, Al-Dweik A, Sharif B, Tsimenidis C C. Efficient Design of Selective Mapping and Partial Transmit Sequence Using T-OFDM. IEEE Transactions on Vehicular Technology. 2020; 69(3), 2636–2648, doi: 10.1109/TVT.2019.2928361.

[12] Daubechies I. Ten Lectures on Wavelets. Society for Industrial and Applied Mathematics. 1992. doi.org/

[13] Li A, Shieh W, and Tucker R. Wavelet Packet Transform-Based OFDM for Optical Communications, Journal of Lightwave Technology. 2010*;*

[14] Torun B. Peak-to-Average Power Ratio Reduction Techniques for

Radar (MTSR), Department of Telecommunications, Faculty of

Wavelet Packet Modulation. PhD thesis, Microwave Technology and Systems for

Electrical Engineering, Mathematics and Computer Science, Delft University of

10.1137/1.9781611970104

28(24): 3519–3528.

Technology. 2010.

ATSIP.2016.7523183.

2017*;* 11 (1), 127–135

2018/8523680

[2] Zakaria J, Salleh M. F. M. Waveletbased OFDM analysis: BER performance and PAPR profile for various wavelets. In: Proceedings of the IEEE Symposium

on Industrial Electronics and Applications (ISIEA '12); 23–26 September 2012; Bandung. Indonesia:

[3] Chafii M, Palicot J, Gribonval R, Burr A. G. Power spectral density limitations of the wavelet-OFDM system. In: Proceedings of the *24th European Signal Processing Conference (EUSIPCO 16);* 28 August-2 September 2016; Budapest, Hungary: 2016. P.

[4] Hsu C. Y., Liao, H. C. Generalised precoding method for PAPR reduction with low complexity in OFDM systems. IET Communications. 2018; 12 (7): 796– 808. DOI: 10.1049/iet-com.2017.0824

incorporated power-efficient Radio over Fibre system, Optics Communications.

[5] Kaur J, Sharma V. A-STBC

[6] Sarowa S, Kumar N, Agrawal, Balwinder S S*.* Evolution of PAPR Reduction Techniques: A Wavelet Based OFDM Approach, Wireless Personal Communications. 2020; https://doi.org/

10.1007/s11277-020-07643-1

[7] Muller S, Huber J. A Comparison of Peak Power Reduction Schemes for OFDM. In: Proceedings of the IEEE Global Telecommunications Conference (GLOBECOM '97), 3–8 Nov. 1997*;* Phoenix, Arizona: 1, P. 1–5.

[8] Bouhlel A, Sakly A, Mansouri M. N. Partial Transmit Sequence technique based on Particle Swarm Optimization

2019; 441: 84–89.

**85**

IEEE; 2012. P. 29–33.

1428–1432, doi: 10.1109/ EUSIPCO.2016.7760484.

The BER performances are shown in **Figure 22**, where the experiment is carried out on the Daubechies wavelet with different filter lengths. It can be observed that no significant difference between BER performances. For example, at BER 10<sup>6</sup> all profiles having the same value of *Eb/N*0.

#### **5. Conclusion**

The phenomenon of high PAPR in MCM system cannot be avoided since the signals consist of multiple low-rate parallel signals, which can be seen as the composite subcarriers in time domain representation. It is expected by using different orthogonal base for modulation, the PAPR profile can be reduced. Hence, discrete wavelet transform (DWT) and discrete wavelet packet transform (DWPT) are used for this purpose instead of fast Fourier transform (FFT). In comparison to the C-OFDM system, WOFDM and WP-OFDM systems do not need any cyclic prefix (CP) codes for their MCM frame in order to avoid intercarrier interference (ICI) and inter symbol interference (ISI).

Although, WOFDM system provides superior PAPR performance than other systems, data are lost at higher frequencies branches since signals decomposition are in dyadic (lower half-band) fashion. On the other hand, WP-OFDM system decomposes the signals in both lower and upper-band frequencies, that enrich signals analysis. The results obtained in Section 4 proves the characteristics. In addition, applying various wavelet bases do not offer much improvement in PAPR profile.

#### **Acknowledgements**

The authors would like to acknowledge the USM RU grant (Grant No. 1001/PELECT/814100), for funding this research work.

#### **Author details**

Jamaluddin Zakaria and Mohd Fadzli Mohd Salleh\* School of Electrical and Electronic Engineering, Universiti Sains Malaysia, Seri Ampangan, 14300 Nibong Tebal, Pulau Pinang, Malaysia

\*Address all correspondence to: fadzlisalleh@usm.my

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

*Wavelet Based Multicarrier Modulation (MCM) Systems: PAPR Analysis DOI: http://dx.doi.org/10.5772/intechopen.94579*

#### **References**

that no explicit difference found from PAPR profiles of WP-OFDM signals either by

The phenomenon of high PAPR in MCM system cannot be avoided since the signals consist of multiple low-rate parallel signals, which can be seen as the composite subcarriers in time domain representation. It is expected by using different orthogonal base for modulation, the PAPR profile can be reduced. Hence, discrete wavelet transform (DWT) and discrete wavelet packet transform (DWPT) are used for this purpose instead of fast Fourier transform (FFT). In comparison to the C-OFDM system, WOFDM and WP-OFDM systems do not need any cyclic prefix (CP) codes for their MCM frame in order to avoid intercarrier interference (ICI)

Although, WOFDM system provides superior PAPR performance than other systems, data are lost at higher frequencies branches since signals decomposition are in dyadic (lower half-band) fashion. On the other hand, WP-OFDM system decomposes the signals in both lower and upper-band frequencies, that enrich signals analysis. The results obtained in Section 4 proves the characteristics. In addition, applying various wavelet bases do not offer much improvement in PAPR

The authors would like to acknowledge the USM RU grant (Grant No.

School of Electrical and Electronic Engineering, Universiti Sains Malaysia,

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

1001/PELECT/814100), for funding this research work.

Jamaluddin Zakaria and Mohd Fadzli Mohd Salleh\*

\*Address all correspondence to: fadzlisalleh@usm.my

provided the original work is properly cited.

Seri Ampangan, 14300 Nibong Tebal, Pulau Pinang, Malaysia

The BER performances are shown in **Figure 22**, where the experiment is carried out on the Daubechies wavelet with different filter lengths. It can be observed that no significant difference between BER performances. For example, at BER 10<sup>6</sup> all

changing the wavelet's type or length.

profiles having the same value of *Eb/N*0.

and inter symbol interference (ISI).

**5. Conclusion**

*Wavelet Theory*

profile.

**Acknowledgements**

**Author details**

**84**

[1] Van Nee R, Prasad R. OFDM for Wireless Multimedia Communications. Boston, London: Artech House; 2000.

[2] Zakaria J, Salleh M. F. M. Waveletbased OFDM analysis: BER performance and PAPR profile for various wavelets. In: Proceedings of the IEEE Symposium on Industrial Electronics and Applications (ISIEA '12); 23–26 September 2012; Bandung. Indonesia: IEEE; 2012. P. 29–33.

[3] Chafii M, Palicot J, Gribonval R, Burr A. G. Power spectral density limitations of the wavelet-OFDM system. In: Proceedings of the *24th European Signal Processing Conference (EUSIPCO 16);* 28 August-2 September 2016; Budapest, Hungary: 2016. P. 1428–1432, doi: 10.1109/ EUSIPCO.2016.7760484.

[4] Hsu C. Y., Liao, H. C. Generalised precoding method for PAPR reduction with low complexity in OFDM systems. IET Communications. 2018; 12 (7): 796– 808. DOI: 10.1049/iet-com.2017.0824

[5] Kaur J, Sharma V. A-STBC incorporated power-efficient Radio over Fibre system, Optics Communications. 2019; 441: 84–89.

[6] Sarowa S, Kumar N, Agrawal, Balwinder S S*.* Evolution of PAPR Reduction Techniques: A Wavelet Based OFDM Approach, Wireless Personal Communications. 2020; https://doi.org/ 10.1007/s11277-020-07643-1

[7] Muller S, Huber J. A Comparison of Peak Power Reduction Schemes for OFDM. In: Proceedings of the IEEE Global Telecommunications Conference (GLOBECOM '97), 3–8 Nov. 1997*;* Phoenix, Arizona: 1, P. 1–5.

[8] Bouhlel A, Sakly A, Mansouri M. N. Partial Transmit Sequence technique based on Particle Swarm Optimization

for WOFDM PAPR reduction. In: Proceeding of the 2nd International Conference on Advanced Technologies for Signal and Image Processing (ATSIP '16), March 21–23, 2016; Monastir, Tunisia: P. 710–714, doi: 10.1109/ ATSIP.2016.7523183.

[9] Zakaria J, Salleh, M. F. M. PAPR reduction scheme: wavelet packet-based PTS with embedded side information data scheme. IET Communications. 2017*;* 11 (1), 127–135

[10] Yoon E, Hwang D, Jang C, Kim J, Yun U. Blind Selected Mapping with Side Information Estimation Based on the Received Pilot Signal, Wireless Communications and Mobile Computing. 2018*;* (5):1–9, DOI: 10.1155/ 2018/8523680

[11] Ahmed M S, Boussakta S, Al-Dweik A, Sharif B, Tsimenidis C C. Efficient Design of Selective Mapping and Partial Transmit Sequence Using T-OFDM. IEEE Transactions on Vehicular Technology. 2020; 69(3), 2636–2648, doi: 10.1109/TVT.2019.2928361.

[12] Daubechies I. Ten Lectures on Wavelets. Society for Industrial and Applied Mathematics. 1992. doi.org/ 10.1137/1.9781611970104

[13] Li A, Shieh W, and Tucker R. Wavelet Packet Transform-Based OFDM for Optical Communications, Journal of Lightwave Technology. 2010*;* 28(24): 3519–3528.

[14] Torun B. Peak-to-Average Power Ratio Reduction Techniques for Wavelet Packet Modulation. PhD thesis, Microwave Technology and Systems for Radar (MTSR), Department of Telecommunications, Faculty of Electrical Engineering, Mathematics and Computer Science, Delft University of Technology. 2010.

[15] Gargour C, Gabrea M, Ramachandran V, Lina J M. A Short Introduction to Wavelets and Their Applications, IEEE Circuits and Systems Magazine. 2009; 9(2): 57–68.

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[18] Mallat S G. Multiresolution Approximations and Wavelet Orthonormal Bases of *L*<sup>2</sup> (*R*), Transactions of the American Mathematical Society. 1989; 315(1): 69–87.

[19] Li A, Shieh W, Tucker R. Wavelet Packet Transform-Based OFDM for Optical Communications, Journal of Lightwave Technology. 2010; 28(24): 3519–3528.

[20] Goswami J C, Chan A K. Fundamentals of Wavelets: Theory, Algorithms, and Applications, Second Edition, John Wiley & Sons, Inc; 2011.

[21] Chang R W. Synthesis of Band-Limited Orthogonal Signals for Multichannel Data Transmission, Bell Systems Technical Journal. 45: 1775– 1796. see also U.S. Patent 3,488,445, Jan. 6, 1970.

[22] Erdol N, Bao F, Chen Z. Wavelet Modulation: A Prototype for Digital Communication Systems. In: Proceeding of the IEEE Conference Record Southcon '95, 7–9 March, Fort Lauderdale, FL: 1995. P. 168–171.

[23] Jamin A, Mähönen P. Wavelet Packet Modulation for Wireless Communications: Research Articles, Wireless Communications and Mobile Computing. 2005; 5(2): 123–137.

[24] Vetterli M, Kovacevic J. Wavelets and Subband Coding, Prentice Hall Signal Processing Series, Prentice Hall PTR. 1995.

[25] Jr R M W. IEEE 802.16 Broadband Wireless Access Working Group. Technical report. SciCom, Inc. 2001.

[26] Zhang L. A Study of IEEE 802.16a OFDM-PHY Baseband. Master's thesis, Department of Electrical Engineering, Linköping Institute of Technology. 2005

Section 3

Signal Processing

and Wavelet Theory

**87**

Section 3
