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In this paper, a linear moving average recursive filtering technique is proposed to reduce the peak-to-average power ratio (PAR) of orthogonal frequency division multiplexing (OFDM) signals. The proposed low complexity technique is analyzed in an oversampled OFDM system and a simple distribution approximation of the oversampled and linearly filtered OFDM signals is also proposed. Corresponding time domain linear equalizers are developed to recover originally transmitted data symbols. Through extensive computer simulations, effects of the new filtering technique on the oversampled OFDM peak-to-average power ratio (PAR), power spectral density (PSD) and corresponding linear equalizers on the frequency selective Rayleigh fading channel transmission symbol-error-rate (SER) performance are investigated. The newly proposed recursive filtering scheme results in attractive PAR reduction, requires no extra fast Fourier transform/inverse fast Fourier transform (FFT/IFFT) operations, refrains from transmitting any side information, and reduces out-of-band radiation. Also, corresponding linear receivers are shown to perform very close to their frequency domain counterparts.

Due to high spectral efficiency, immunity to impulse noise, robustness in multipath fading environments and ease of implementation, OFDM has emerged as an attractive multicarrier modulation technique for high speed mobile and wireless communication systems. This transmission technique has been adopted in digital video broadcasting [

One major problem associated with OFDM transmitted signal is the high PAR [

To alleviate the PAR problem in OFDM systems, several approaches have been devised. For example, selective mapping (SM) [

Deliberate amplitude clipping of the OFDM signal [

In this paper, we propose a low complexity moving average PAR reduction filtering technique. We call the technique post-IFFT filtering (PoF) because filtering is applied after IFFT at the transmitter. In the analysis of the proposed technique oversampling of original OFDM signals is considered and a simple mathematical expression to evaluate the complementary cumulative density function (CCDF) of oversampled and linearly filtered OFDM signals is presented.

The proposed technique is simple to implement, requires no extra IFFTs, and refrains from transmitting any side information. The low pass feature of the technique is effective in reducing the out-of-band radiation and thus lends itself to minimize the possible increase in PAR, SER and bandwidth due to pulse shaping filters [

This paper is organized as follows: Section 2 characterizes the OFDM transmission and proposes an approximate expression for the distribution of oversampled and linearly filtered OFDM signals. Section 3 covers the preliminaries of the moving average filter. To lay out the groundwork for developing the new technique, the matrix data model of the oversampled OFDM signal is described in Section 4. PoF implementation and corresponding time domain linear symbol recovery solutions are described in Section 5. Several practical issues and pertinent complexity and receiver performance trade-offs are considered in Section 6. In Section 7, illustrating simulations are carried out while conclusions are drawn in Section 8.

In OFDM transmission, a high speed input bit stream, after complex modulation mapping, is converted into complex data symbols of length N, which is transformed into an OFDM signal via N-point IFFT. The resulting continuous time complex baseband OFDM signal x a ( t ) can be written as

x a ( t ) = x a ( t ) I + j x a ( t ) Q = 1 N ∑ k = − N / 2 N / 2 − 1 s ( 〈 k + N 〉 ) e j 2 π k t T s (1)

where 0 ≤ t ≤ T s , x a ( t ) I and x a ( t ) Q denote the real and imaginary parts of x a ( t ) , j 2 = − 1 , 〈 k + N 〉 denotes ( k + N ) modulo N, s N = [ s ( 0 ) , s ( 1 ) , ⋯ , s ( N − 1 ) ] T represents the size N complex input data symbol, s ( k ) represents the complex modulated symbol of the kth subcarrier, N is the number of subcarriers, and T s denotes the symbol period of the OFDM signal. A cyclic prefix (CP) (i.e., guard interval) is added to the resulting signal in order to avoid the inter-block-interference (IBI) in time dispersive channels. The CP does not affect the PAR characteristics to be analyzed in this paper. Therefore, in order not to complicate the notation, the CP has been omitted here.

The PAR of x a ( t ) can be defined as

ς a ≜ max 0 ≤ t ≤ T s | x a ( t ) | 2 P a v (2)

where P a v is the average power defined as P a v ≜ E { | x a ( t ) | 2 } = 1 T s ∫ 0 T s | x a ( t ) | 2 d t and P a v = { | s ( k ) | 2 } based on Parseval’s theorem.

The Nyquist rate samples of the OFDM waveforms (1) (i.e., x a ( t ) | t = n T s / N ) can be represented as:

x ⌣ ( n ) = x ⌣ ( n ) I + j x ⌣ ( n ) Q = 1 N ∑ k = − N / 2 N / 2 − 1 s ( 〈 k + N 〉 ) e j 2 π n k N (3)

where 0 ≤ n ≤ N − 1 , and the real and imaginary parts of x ⌣ ( n ) are denoted by x ⌣ ( n ) I and x ⌣ ( t ) Q , respectively. The PAR of x ⌣ ( n ) can thus be defined as

ς ≜ max 0 ≤ n ≤ N − 1 | x ⌣ ( n ) | 2 P a v (4)

Based on (3), a simple approximate expression for the distribution of the PAR has been derived in [

Pr ( ς > ς 0 ) ≈ 1 − ( 1 − e − ς 0 ) N (5)

To approximate more accurately the PAR in (2), an oversampled version of (1) can be used [

x ( n ) = 1 N ∑ k = − N / 2 N / 2 − 1 s ( 〈 k + N 〉 ) e j 2 π n k M (6)

where 0 ≤ n ≤ M − 1 , M = J N and J > 1 is the oversampling factor. Usually, J ≥ 4 is used to capture the peaks of x a ( t ) .

The assumption in (5), that the samples are mutually independent and uncorrelated, is not strictly correct when oversampling is applied. Hence, an approximation was proposed in [

Pr ( ς > ς 0 ) ≈ 1 − ( 1 − e − ς 0 ) α N (7)

In order to reduce the PAR of OFDM signals, we introduce a linear moving average filter before adding the CP. If the complex Gaussian baseband signal (6) is passed through the proposed linear filter, the output is also Gaussian. So, the envelope of the complex OFDM signal after simple filtering has the Rayleigh distribution. However, correlation of the output signal samples will increase due to filter memory; hence the approximation (7) is not valid. This difficulty can be neatly resolved by a similar assumption as for the oversampling case above. We thus propose an empirical approximation by assuming that the distribution of N carriers with oversampling and filtering can be given by α β N subcarriers, without oversampling and filtering. Consequently, the CCDF of PAR reduced OFDM signal can be given by

Pr ( ς > ς 0 ) ≈ 1 − ( 1 − e − ς 0 ) α β N (8)

where β is greater than one and can be determined by exhaustive computer simulations. We remark here that straightforward application of (8) is complicated to work with as the parameter β depends not only on the filter length L f but also on the OFDM signal size M.

A moving average filter [

h f ( n ) = { 1 / L f 0 ≤ n ≤ L f − 1 0 otherwise (9)

where L f is the filter length. Using (6) as input to the filter, the output of the filter can be expressed as

y ( n ) = h f ( n ) ∗ x ( n ) = ∑ l = 0 L f − 1 h f ( l ) x ( n − l ) = 1 L f [ x ( n ) + x ( n − 1 ) + ⋯ + x ( n − L f + 1 ) ︸ ICI ] (10)

Notice the smoothing window operation performed by the filter. Each smoothed value is computed as the average of a number of preceding data values and the degree of smoothing increases with L f . The filtering output (10), contains interfering terms from other subcarriers, the so-called inter-carrier-inter-ference (ICI) effect.

This loss of orthogonality due to ICI is an undesirable effect and is well known to result in SER degradation at the receiver side.

The filter can be described in the frequency domain by the Fourier transform of the rectangular pulse

H f ( ω ) = 1 L f ∑ l = 0 L f − 1 e − j ω l = D L f ( ω ) e − j ω ( L f − 1 ) / 2 (11)

where D L f ( ω ) is the Drichlet function [

D L f ( ω ) ≜ sin ( ω L f / 2 ) L f sin ( ω / 2 ) (12)

Let the samples associated with the ith data vector of size M be denoted as x M , i ( n ) . With this notation, we can now express the ith oversampled OFDM signal (6) as

x M ( i ) = [ x M , i ( 0 ) , ⋯ , x M , i ( M − 1 ) ] T (13)

The vector x M ( i ) can be obtained by using a M-point IFFT on the extended data vector u M ( i ) = [ s 0 ( i ) , ⋯ , s N / 2 − 1 ( i ) , 0 , ⋯ , 0 , s N / 2 ( i ) , ⋯ , s N − 1 ( i ) ] T obtained by inserting M − N zeros in the middle of the information symbol vector s N ( i ) = [ s N , i ( 0 ) , ⋯ , s N , i ( N − 1 ) ] T [

P ≜ [ I N / 2 × N / 2 O ( M − N ) × N / 2 O N / 2 × N / 2 O N / 2 × N / 2 O ( M − N ) × N / 2 I N / 2 × N / 2 ] (14)

Pre-multiplying the data vector s N ( i ) by the precoding matrix P yields the extended data vector u M ( i ) = P s N ( i ) . The relationship between x M ( i ) and s N ( i ) , and thus the associated data matrix model for the signal x M , i ( n ) can be written as:

x M ( i ) = γ F M H u M ( i ) = γ F M H P s N ( i ) (15)

where F M is the M × M FFT matrix with ( m , n ) entry ( 1 / M exp ( − j 2 π m n / M ) ) , and the power loss factor γ = N / M is used here to retain the same power before and after the oversampling.

If the oversampled OFDM signals are received by an FIR moving average filter h f = [ h f ( 0 ) , h f ( 1 ) , ⋯ , h f ( L f − 1 ) ] T , with 1 < n ≤ M , the expression for the ith filter output symbol block can be written as

y ⌣ M ( i ) = H M ( h f ) x M ( i ) + H ˜ M ( h f ) x M ( i − 1 ) (16)

where H M ( h f ) is the M × M lower triangular Toeplitz filtering matrix with first column [ h f ( 0 ) , ⋯ , h f ( L f − 1 ) , 0 , ⋯ , 0 ] T and H ˜ M ( h f ) is the M × M upper triangular Toeplitz matrix with first row [ 0 , ⋯ , 0 , h f ( L f − 1 ) , ⋯ , h f ( 1 ) ] .

Due to the filter memory, IBI arises between successive blocks and renders y ⌣ M ( i ) in (16) dependent on both x M ( i ) and x M ( i − 1 ) . To avoid IBI and thus to process data blocks independently at the transmitter (and also at the receiver), we assume that all input data prior to x M , i ( 0 ) is zero, and make no assumptions about the input data after x M , i ( M − 1 ) . That way, we can write the expression for the IBI free ith output symbol block as

y M ( i ) = [ y M , i ( 0 ) , ⋯ , y M , i ( M − 1 ) ] T = H M ( h f ) x M ( i ) (17)

Notice that

y M , i ( 0 ) = x M , i ( 0 ) / L f , y M , i ( 1 ) = ( x M , i ( 0 ) + x M , i ( 1 ) ) / L f , ⋮ y M , i ( M − 1 ) = ( x M , i ( M − L f ) + ⋯ + x M , i ( M − 1 ) ) / L f

Therefore, a tremendous advantage of this filtering implementation is its efficient recursion

y M , i ( n ) = y M , i ( n − 1 ) + 1 L f [ x M , i ( n ) − x M , i ( n − L f ) ] (18)

As compared with the matrix vector product (17), the recursion (18) is much faster, requiring fewer additions/subtractions regardless of the filter length L f . It is not necessary at the transmitter to wait for all the samples of an OFDM symbol before the first filtered outputs are produced. Of course, this is ideal for delay limited systems.

The OFDM signals based on the system model (17) are cyclically extended, digital-to-analogue (D/A) conversion and transmit filtering are performed, the signal is modulated, power amplified and then transmitted through the channel. At the receiver, following the frequency down-conversion, receive filtering and analogue-to-digital (A/D) conversion, the CP of the subsequent OFDM symbols are discarded. We use a discrete time length L c , FIR filter with impulse response h c = [ h c ( 0 ) , h c ( 1 ) , ⋯ , h c ( L c − 1 ) ] T to represent the overall combined effect of the spectral shaping pulse, the continuous time channel, the receive filter and the sampling. At the output of the demodulator, the time domain received baseband data vector can be written as:

r M ( i ) = C M ( h c ) y M ( i ) + n M ( i ) = γ C M ( h c ) H M ( h f ) F M H P s N ( i ) + n ( i ) (19)

where the FIR channel vector h c is denoted by the M × M circular channel matrix C M ( h c ) with first row [ h c ( 0 ) , 0 , ⋯ , 0 , h c ( L c − 1 ) , ⋯ , h c ( 1 ) ] and n M ( i ) = [ n 0 ( i ) , n 1 ( i ) , ⋯ , n M − 1 ( i ) ] T is the complex AWGN vector.

Since the filtering removes the subcarrier orthogonality at the transmitter, therefore, the data model based on FFT of (19) does not allow us to use standard frequency domain equalizers. We therefore suffice on the time domain data model (19) for symbol recovery. According to the ZF criterion, the equalizer is chosen to assure perfect symbol recovery in the absence of noise. In Equation (19), the matrices P , H M ( h f ) and F M H are full rank by design. Therefore, by assuming that the matrix H M ( h c ) is full rank, the ZF solution is unique and is given by s ^ N ( i ) = G zf PoF r M ( i ) , where G zf PoF is the ZF equalizing matrix, which can be found in two steps. First, we obtain the estimate of y M ( i ) = γ H M ( h f ) F M H P s N ( i ) as y ^ M ( i ) = ( C M ( h f ) ) − 1 r M ( i ) ; and then we find s ^ N ( i ) = ( γ H M ( h f ) F M H P ) † y ^ M ( i ) , which leads to

G zf PoF = W ( C M ( h c ) ) − 1 (20)

where W = ( γ H M ( h f ) F M H P ) † is an N × M matrix representing pseudo inverse of the overall combined effect of filtering, IFFT modulation and oversampling operations.

At low signal-to-noise ratio (SNR), a vector MMSE equalizer can lead to improved receiver performance. According to MMSE criterion, the equalizer is chosen to minimize the mean square error (MSE) E { ‖ e ( i ) ‖ 2 } = E { ‖ s ^ N ( i ) − s N ( i ) ‖ 2 } . The MSE can be written as a function of the equalizing matrix G as:

J ( G ) : = E { tr [ ( γ G C M ( h c ) H M ( h f ) F M H P − I N ) s N ( i ) + G n M ( i ) ] × [ ( γ G C M ( h c ) H M ( h f ) F M H P − I N ) s N ( i ) + G n M ( i ) ] H } (21)

By setting gradient ∇ G J ( G ) = 0 and solving for G , MMSE equalizing matrix G = G mmse PoF (yielding the MMSE estimate s ^ N ( i ) = G mmse PoF r M ( i ) ) is given by

G mmse PoF = γ σ s 2 V C M H ( h c ) ( σ n 2 I M + γ 2 σ s 2 C M ( h c ) Q C M H ( h c ) ) − 1 (22)

where V = ( H M ( h f ) F M H P ) H and Q = H M ( h f ) H M H ( h f ) . Furthermore, in deriving (22) it is assumed that the correlation matrices R s s : = E { s N ( i ) s N H ( i ) } = σ s 2 I N and R n n : = E { n M ( i ) n M H ( i ) } = σ n 2 I M .

Since the matrix W in Equation (20), and matrices V and Q in Equation (22), are not channel dependent and remain fixed, therefore, they can be pre-computed and straightforwardly embedded in the receiver.

A block diagram of an OFDM system involving the proposed PoF transform and corresponding time domain linear equalizers is illustrated in

The average power of the PoF OFDM signals is

E { ‖ y M ( i ) ‖ 2 } M = tr ( E { y M ( i ) y M H ( i ) } ) M = tr ( E { H M ( h f ) x M ( i ) x M H ( i ) H M H ( h f ) } ) M = σ s 2 [ tr ( E { H M ( h f ) H M H ( h f ) } ) M ] = σ s 2 [ ( 1 + 2 + ⋯ + L f L f 2 + M − L f L f ) / M ] (23)

The sum of an arithmetic series consisting of n terms a 1 , a 2 , ⋯ , a n with common difference d is given by a 1 + a 2 + ⋯ + a n = n ( a 1 + a n ) / 2 . This implies that ( 1 + 2 + ⋯ + L f ) / L f 2 = ( 1 + L f ) / 2 L f . Therefore (23) can be written as

E { ‖ y M ( i ) ‖ 2 } M = σ s 2 ( 2 M − L f + 1 ) 2 M L f (24)

It is thus clear from (23) that the PoF PAR reduction process also involves reduction in the average power of OFDM signals^{1}. Apart from PAR, the reduction in average transmit power affects the system performance in two ways, one positive and one negative. Firstly, it will result in a more desirable spectrum. Secondly, it will decrease SNR at the receiver, which means an increase in SER. We will present these simulations in the next section.

The oversampling can be seen to increase the size of IFFT/FFT matrices in the proposed linear transceivers, resulting in increased computational complexity and bandwidth requirements. Though proposed filtering and symbol recovery schemes can straightforwardly work on Nyquist sampled OFDM signals, however, oversampling must be provided in the process to approximate more accurately the PAR and it is for this reason several PAR reduction approaches and corresponding PAR distribution studies based on oversampled OFDM signals have appeared in literature [

The simple recursive PoF approach requires only 2M additions per OFDM data vector (2 additions per sample). Generally, computing time for one addition is much less than that for one multiplication (which requires 4 real multiplications and 2 real additions). This shows that the PoF technique is a computationally efficient approach.

The matrix-vector product G PoF r M ( i ) , for obtaining s ^ N ( i ) requires O ( N 2 ) computations. So the computational complexity per symbol of our linear equalizers in Equations (20) and (22) is O ( N ) . Of course, this is higher than the standard frequency domain equalizers which have per symbol complexity of order O ( log M ) , but computationally much heavier Viterbi like approaches and iterative^{2} techniques involving multiple FFTs to recover symbols are not required.

In practical OFDM systems, error control codes are usually used to combat channel nulls (or deep fades). Our proposed filtering and symbol recovery approaches are applicable if the coded symbols are transmitted. The methods do not capitalize on any particular type of constellation, hence they are directly applicable to the cases where transmitted information is drawn from any signal constellation.

The time domain ZF equalizer, assures symbol recovery if the circulant matrix C M ( h c ) is full rank. The matrix C M ( h c ) is full rank if and only if the transfer function h c has no zeros on the FFT grid. Although we may adopt the MMSE equalizer when the channel has nulls on (or close to) the FFT grid, but lack of equalizability will result in an error floor in the resulting SER performance.

In order to verify the performance of the proposed schemes, we consider (unless otherwise specified) a baseband OFDM system with the number of subcarriers N = 80 , the oversampling factor J = 4 , and randomly generated input data are modulated by quaternary phase-shift keying (QPSK). Furthermore, we call proposed time domain equalizers: PoF-ZF and PoF MMSE equalizers for convenience, in the rest of this paper.

^{5} random realizations of corresponding signals. As can be seen, filtering results in a more desirable statistical characteristic.

At excess probability of 10^{−4}, the PAR reduction is 3.5 dB for L f = 2 , whereas, diminishing effect in PAR returns can be observed for L f > 2 . From (23) [or equivalently (24)], L f = 2 also yields the minimum reduction in the average transmit power. Furthermore, notice that as L f decreases, so does the ICI and corresponding improvement in SNR. This shows that filter with L f = 2 not only yields better PAR but also will result in better SER performance at the receiver side as compared with higher values of L f . We therefore limit suitability of the PoF scheme with L f = 2 . Since L f = 2 is now fixed, the receiver need not to be notified of the filter length whenever filtering is applied, thereby eliminating the need of side information overhead.

To demonstrate the effectiveness of the low complexity (here called high-speed) filtering option in Equation (18) against the matrix vector product option in Equation (17) (here called direct),

In

The unwanted sideband power is normally reduced through the use of pulse shaping filters [

later approach offers lower MSE) and all equalizers are able to correct all transmitted symbols above SNR of 6 dB. A performance gap emerges between the PoF and frequency domain equalizers; however, no significant performance loss can be seen due to the filtering operation.

In this paper, a computationally efficient PAR reduction technique based on linear moving average filtering (that we called PoF) was proposed. We also proposed the distribution function of oversampled and linearly filtered OFDM signals. The proposed filtering technique relies on time domain recursive approach for efficient implementation and requires the filter length L f = 2 . The scheme results in attractive PAR reduction, requires no extra FFT/IFFT operations, and refrains from transmitting any side information. A key feature of the technique is the reduction in out-of-band radiation and problems related to the pulse shaping. The effect of multicarrier modulation, oversampling, filtering and channel dispersion is modelled as a linear transformation. Therefore, to recover the originally transmitted symbols, we proposed corresponding time domain linear ZF and MMSE receivers which were seen to perform very close to their frequency domain counterparts. The new filtering and equalization schemes do not capitalize on a particular coding or constellation technique and can be used for any number of subcarriers.

The authors declare no conflicts of interest regarding the publication of this paper.

Ali, H. and Yaqub, R. (2018) A Low Complexity Linear Moving Average Filtering Technique for PAR Reduction in OFDM System. Int. J. Communications, Network and System Sciences, 11, 199-215. https://doi.org/10.4236/ijcns.2018.1110012