# Spectrally Efficient OFDMA Lattice Structure via Toroidal Waveforms on the Time-Frequency Plane

- Sultan Aldirmaz
^{1}Email author, - Ahmet Serbes
^{1}and - Lutfiye Durak-Ata
^{1}

**2010**:684097

https://doi.org/10.1155/2010/684097

© Sultan Aldirmaz et al. 2010

**Received: **2 January 2010

**Accepted: **29 June 2010

**Published: **20 July 2010

## Abstract

We investigate the performance of frequency division multiplexed (FDM) signals, where multiple orthogonal Hermite-Gaussian carriers are used to increase the bandwidth efficiency. Multiple Hermite-Gaussian functions are modulated by a data set as a multicarrier modulation scheme in a single time-frequency region constituting toroidal waveform in a rectangular OFDMA system. The proposed work outperforms in the sense of bandwidth efficiency compared to the transmission scheme where only single Gaussian pulses are used as the transmission base. We investigate theoretical and simulation results of the proposed methods.

## Keywords

## 1. Introduction

As the demand for mobility and high performance in multimedia services increase, efficient spectrum usage becomes a critical issue in wireless communications. Orthogonal frequency division multiplexing (OFDM) is a multicarrier modulation that has been employed by a number of current and future wireless standards including a/g, a-d, long term evolution (LTE) downlink, and next generation networks. OFDM has been a popular method because of its robustness in frequency selective fading characteristics of broadband wireless systems. OFDM technology transmits data by dividing it into parallel streams to be modulated by subchannels each having a different carrier frequency. Basically, data is carried on narrow-band subcarriers in frequency domain and the carrier spacing is carefully selected so that each subcarrier is orthogonal to the others.

There have been numerous studies in the literature to provide efficient spectrum usage in high-performance wireless applications, including video streaming [1, 2]. Different coding methods and modulation types are developed for this task, such as hexagonal QAM structure [3, 4]. In [5], linear combinations of Hermite-Gaussian functions are used in the generation of two orthogonal pulse shapes to increase the system throughput. However, such a linear combination increases the bandwidth of a single subchannel significantly.

Pulse shape design is an important research issue because of the corruption eects of channels, such as Doppler shifts, fading, and noise. Conventional OFDM uses rectangular-pulse shapes for each data. Since this pulse shape has *sinc* shape in the frequency domain, its energy is dispersed to other subcarriers when the channel is dispersive. For this purpose, there is a ceaseless pursuit for dierent pulse shapes. Nyquist pulses with raised cosine spectra, Hermite-Gaussian functions based pulses [5, 6] and an optimized combination of Slepian sequences [7–9] have been investigated in the literature. In [7], prolate spheroidal wave functions (PSWFs) which have maximum energy concentration within a given time interval are used to design a time-frequency division multiplexing (TFDM) system for multiple users. If the transmitted pulses have maximum concentration on the joint time-frequency (T-F) domain, then the transmitted signal shall be preserved against the channel affects better. Intersymbol interference (ISI) and inter-carrier interference (ICI) are the main problems of OFDM systems. When the transmission channel is time and frequency dispersive, the transmitted signal spreads in both domains. Thus symbols corrupt each other. To avoid ISI and ICI, pulse shape design and hexagonal lattice structures are introduced. In a hexagonal lattice structure, time and frequency distances are increased, so the spread signal may not be affected by the other symbols. Ambiguity function (AF) is also a useful time-frequency tool to analyze ISI and ICI which provides us to observe pulse spread both in time and frequency due to the channel effects [10, 11]. In [5, 6], a linear combination of Hermite-Gaussian functions is optimized to construct a desired pulse shape against the Doppler affect. In [12], Hermite functions are used in UWB communication systems with different modulation types such as PPM, BPSK, and pulse shape modulation. Linear combinations of Hermite pulses of orders
to
are obtained to construct a single pulse shape which obeys FCC constraints in [13].

*N*of them to form a symbol. Figure 1 presents the general framework of the proposed system. For example, assuming a BPSK modulated data set , the proposed algorithm generates a pulse , where is the th order Hermite-Gaussian function. As Hermite-Gaussian functions are orthogonal to each other, the demodulation process is carried out easily.

The remainder of this paper is organized as follows. In Section 2, a preliminary is given about Hermite-Gaussian pulses and their time-frequency localization. System model and the receiver part are described in Section 3. The simulation results are discussed by presenting both SNR versus BER values for different overlapping percentages of the signal waveforms on the lattice form in Section 4. Finally, conclusions are drawn in Section 5.

## 2. Preliminaries

### 2.1. Hermite-Gaussian Functions

### 2.2. Time-Frequency Localization of Hermite-Gaussian Functions

The zeroth-order Hermite-Gaussian function, or equivalently the conventional Gaussian function, is the best localized function in both time and frequency domain having the lowest TBP equal to .

## 3. The System Model

### 3.1. Toroidal-Rectangular Lattice Structure

*N*consecutive-order Hermite-Gaussian functions is

Proposition 1.

TBP of linear combination of Hermite-Gaussian signals of order to is times greater than the zeroth-order Hermite-Gaussian signal.

Proof.

*N*Hermite-Gaussian functions. These pulses are added together to construct a toroidal waveform. Hermite pulses for , and are expressed as

### 3.2. Channel Model and the Receiver Structure

where , , and represent path number, Doppler spread and delay, respectively. In simulations, both AWGN and Rayleigh channel models are considered as the channel effect.

*M*multiplicative windows to select the desired time interval and

*N*convolutional band-pass filters to separate frequency bands. Hence, we filter only a single rectangular time-frequency region in advance. Afterwards, we shift the signal to the baseband and obtain the toroidal lattice structure which contains multiple data. We take inner products of the toroidal-data with Hermite-Gaussian pulses to estimate the transmitted data by taking advantage of orthogonality between different orders of Hermite-Gaussian functions. Let us assume be the received toroidal signal with AWGN

*n(t)*after filtering in time-frequency. The estimated data can be obtained as,

Hermite-Gaussian functions and their linear combinations are orthogonal to each other. Thus, their time and frequency shifted versions are used in the proposed OFDM system simplifying the detection process.

## 4. Simulation Results

## 5. Conclusions

We have proposed a new toroidal waveform in a rectangular-lattice OFDMA structure. The system includes *N* different-order Hermite-Gaussian pulses. Each of these pulses is modulated by different data, and they are combined together to construct a toroidal structure which increases the data rate up to *N* times. However, in case of a four-layer Hermite-Gaussian toroidal structure, as the third-order function covers almost twice the TF region compared to zeroth-order one, the overall data rate increased more than twice. As part of future works, system robustness against doubly dispersive channel effects and ICI will be investigated.

## Declarations

### Acknowledgment

The authors are supported by the Scientific and Technological Research Council of Turkey, TUBITAK under the grant of Project no. 105E078.

## Authors’ Affiliations

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