- Open Access
Single-carrier fractional Fourier domain equalization system with zero padding for fast time-varying channels
© Chen and Chu; licensee Springer. 2014
- Received: 25 February 2014
- Accepted: 16 April 2014
- Published: 7 May 2014
Single-carrier frequency domain equalization (SC-FDE) has been shown to be an attractive transmission scheme for broadband wireless channels. However, its performance would degrade a lot if the channel is fast time-varying. In this paper, we analyzed the single-carrier fractional Fourier domain equalization (SC-FrFDE) system and applied it to the fast time-varying channel. It can solve the time-varying problem by selecting the optimal fractional Fourier transform order. To this SC-FrFDE system, its transmitter uses chirp-periodic circular prefix to eliminate ISI and this has an evident disadvantage that the receiver need to feedback the optimal fractional Fourier transform order to the transmitter through a feedback channel. To simplify the system, we propose to use zero padding (ZP) at the transmitter. There is already the overlap-add method as ZP in SC-FDE system. But the overlap-add method cannot be used in fast time-varying channels. Thus, we propose a new method as ZP. Simulation results show that our proposed method can significantly improve the system performance.
- Single-carrier fractional Fourier domain equalization
- Chirp-periodic circular prefix
- Fast time-varying channels
Single-carrier frequency-domain equalization (SC-FDE) has been shown to be an attractive equalization scheme for broadband wireless channels which have very long impulse response memory . SC radio modems with frequency domain equalization have very similar performance, efficiency, and low signal processing complexity advantages as OFDM, and in addition are less sensitive than OFDM to RF impairments . So the SC-FDE system has the favorable advantages than the OFDM system. And this arises the use of SC modulation [2, 3]. However, when met with fast-moving channels in broadband wireless communications, the performance of SC-FDE will degrade a lot because orthogonality among different subchannels at the receiver is destroyed and ICI arises.
The concept of fractional Fourier transform (FrFT) was first introduced by N. Wiener in 1929. Then it appears as a mathematical tool for solving quantum mechanics problems [4, 5], but this method did not gain much attention. Until fractional Fourier transform was introduced in the field of image analysis in optics and signal processing by Mendlovic , Ozaktas , and W. Lohmann [8, 9] and had it attracted great attention. Much research has been done in the multicarrier system. FrFT was first applied in the multicarrier system by M. Martone in 2001 . Afterwards, Tomaso Erseghe proposes to use affine Fourier transform to remove ICI in the multicarrier system under time-varying channel models in 2005 . Sufficient research [12, 13] has been done in the multicarrier system compared to the researches done in the single-carrier (SC) system. In 2012, the FrFT was applied in the SC system , but it mainly discusses the FrFT to cope with the deep fading problem. The detailed procedure of the system is not discussed. Based on this, we will discuss the SC-FrFDE system in detail and expand this system to the fast time-varying channels.
In SC-FrFDE, discrete fractional Fourier transform (DFrFT) and inverse discrete fractional Fourier transform (IDFrFT) replace fast Fourier transform (FFT) and inverse fast Fourier transform (IFFT) correspondingly. Because of the chirp periodicity of the sampled signals, it is needed to add a chirp-periodic circular prefix (chirp CP) instead of the traditional cyclic prefix at the transmitter. By selecting the optimal fractional Fourier transform order, the chirp CP can remove the ISI as well as transform the linear convolution of the signal and the time-varying channel into fractional circular convolution. Thus, the SC-FrFDE system can cope with fast time-varying channels well.
The SC-FrFDE with chirp CP has an evident disadvantage that the calculation formula of chirp CP includes the transform order. So there needs a feedback channel between the transmitter and receiver to feedback the optimal fractional Fourier transform order. This will increase the complexity of the whole system. To simplify the system, we propose to use zero padding (ZP) at the transmitter. There is already the overlap-add method as ZP in SC-FDE system. But the overlap-add method cannot be used in fast time-varying channels. Thus, we propose a new method as ZP.
The rest of the paper is organized as follows: In section 2, we introduce the basic definitions of DFrFT and fractional circular convolution. In section 3, the SC-FrFDE system model with chirp CP is detailedly described and represent that this system can deal with fast time-varying channels. In section 4, the expression of ZP is given. And the principle of selecting the optimal fractional Fourier transform order is introduced in section 5. To show the validity of the proposed scheme, Monte Carlo simulation results are presented in section 6. Lastly, in section 7, we make a conclusion.
2.1 Discrete fractional Fourier transform
2.1.1 Fractional Fourier transform
In that expression, . Since the FrFT is periodic with the period of 4, the transform order p can be limited to the interval of [−2, 2].
2.1.2 Discrete time fractional Fourier transform (DTFrFT)
The above is the FrFT of the continuous signal. However, in the SC system, the signal that we are going to deal with is discrete signal. So, it is required to know the DFrFT of the discrete signal.
Δt is the sampling interval of time domain. Δt = 1/ml/sr. Among which ml represents the modulation method. For example, ml = 1 corresponds to the BPSK modulation; ml = 2 corresponds to QPSK modulation. sr is the symbol transmission rate. In this paper, the symbol transmission rate is set to be sr = 250,000 symbol/s.
It is readily seen that the sampled signal is discrete in the time domain. But its FrFT is still analog in the FrFD. To get the DFrFT of the time domain discrete signal, it is required to sample in the FrFD.
2.1.3 Discrete fractional Fourier transform (DFrFT)
Assuming the sampling points in the time domain and the fractional Fourier domain are N and M. Generally M = N. We can get X p (m) := X p (mΔu), m = 0, 1, …, N − 1. The FrFD sampling interval is .
This chirp periodicity is very important because it will be used in the guard interval (GI).
2.2 Discrete fractional Fourier transform
In which x2((n))p,N is the p th-order chirp periodic extension sequence with period of N and R N (n) is used to get the principal value interval.
This is the fractional circular convolution theorem.
Equation 26 indicates that by adding and removing chirp CP, the linear convolution is turned into circular convolution (It is not fractional circular convolution). It is the circular convolution about h(n, l) and one chirp periodic extension sequence x((n))p,N.
In which . Y p (k) and X p (k) is the fractional Fourier domain data of y(n) and x(n). The fractional circular convolution is very useful in the FrFD equalization.
If the transmitter uses chirp CP as guard interval, the receiver need to feedback the optimal fractional Fourier transform order to the transmitter through a feedback channel. It will increase the complicity of the system. To solve this problem, the transmitter could adopt ZP which has no information about the optimal fractional Fourier transform order. If the channel is slow time-varying, we could use the well-known overlap-add technique as ZP to make the circulant Toeplitz matrix . But if the channel is fast time-varying, this method cannot be applied. First, let us show the invalid of overlap-add method under fast time-varying channels.
4.1 The invalid of the traditional overlap-add method
For simplicity, we will consider the overlap-add method under fast time-varying channels in the SC-FDE system. Just like channel matrix for zero padding in , we will observe the variations of the channel matrix H. For simple case N = 4, N c = 2. Channel matrix for zero padding in  can be rewritten as
If the channel is invariant or varies slowly during one data block, h(4, 1) ≈ h(0, 1) and can be approximately seen as a circular matrix and thus can be diagonalized through the FFT/IFFT. But if the channel is fast time-varying, is obvious not a circular matrix and would not be diagonalized. This situation is exactly the same with the sC-FrFDE system. So the overlap-add method is not suitable for fast time-varying channels. We need to think of other ways to add ZP.
4.2 Proposed method of zero padding
Then the signal is transmitted to the fast time-varying channel H. H is shown in Equation 19.
γzp is the additive white Gaussian noise.
In Equation 43 . If h p (n, l) ≈ h p (l), then
By taking IDFrFT, this signal will be transformed to time domain. When the signals get into the QPSK demodulation module, it is only essential for us to demodulate the first N − N c data (the last N c signals of the transmitting signal are zeros). And its BER performance is measured by its useful signals which are the first N − N c elements of the transmitted signals.
In conventional SC-FDE system, the signal is modulated by the exponential bases. When met with fast-fading channels, the Fourier-domain matrix cannot be diagonalized. To solve this problem, we can think of the fast-fading channel as a channel with ‘time-varying’ frequency response. The optimal transmission/reception methodology should be able to ‘diagonalize’ nonstationary signals . So, we propose to use the chirp basis instead of traditional exponential basis. By selecting the optimal fractional Fourier transform order, it can effectively cope with fast time-varying channels. In this section, we will give the principle that selects the optimal order.
and represent desired signal and the inter-carrier interference signal.
In this section, we present the simulation results of the ZP-SC-FrFDE, compared with chirp CP-SC-FrFDE and the conventional cyclic prefixed SC-FDE. The performance of the system is measured by bit error rate (BER) with 1,000 realizations. We consider an uncoded single-carrier transmission with N = 64 and QPSK constellation. The time domain sampling interval is Δt = 2 × 10−6 s. The length of the guard interval is 16 which is larger than the channel's maximum propagation delay.
Simulation parameters of channels
In this paper, we introduce the SC-FrFDE system to fast time-varying channels to solve the fast time-varying problem. Because this system needs to know the fractional Fourier transform order at the transmitter and there needs to be a feedback channel between the transceiver. The complexity forced us to find the zero padding method. The traditional overlap-add method cannot be used in fast time-varying channels. And we propose a new method to add ZP. The ZP-SC-FrFDE is discussed afterwards. Both of the two systems need to know the optimal fractional Fourier transform order, so we discussed the method of choosing the optimal fractional Fourier transform order. By using the optimal order, the fast time-varying channel matrix can be diagonalized in the FrFT domain. Simulation results show that the ZP-SC-FrFDE system outperforms the other two systems despite of the channel model and the max Doppler frequency.
This work was supported partly by the National Natural Science Foundation of China under Grants No. 61201251, No.61172086, No.61210005 and No.61331021.
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