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Oversampling is commonly encountered in orthogonal frequency division multiplexing (OFDM) systems to ease various performance characteristics. In this paper, we investigate the performance and complexity of one tap zero-forcing (ZF) and minimum mean-square error (MMSE) equalizers in oversampled OFDM systems. Theoretical analysis and simulation results show that oversampling not only reduces the noise at equalizer output but also helps mitigate ill effects of spectral nulls. One tap equalizers therefore yield improved symbol-error-rate (SER) performance with the increase in oversampling rate, but at the expense of increased system bandwidth and modest complexity requirements.

Orthogonal frequency division multiplexing (OFDM) is an attractive multicarrier modulation technique for high data rate transmission systems [

Oversampling of the OFDM signal (or oversampled inverse fast Fourier transform (IFFT)) is adopted in many practical OFDM systems to enhance various performance characteristics. Oversampling is used in [

In wireless systems, OFDM has the ability to convert a frequency selective fading channel into parallel frequency flat fading sub-channels which are equalized using one tap zero-forcing (ZF) and MMSE equalizers. Unfortunately, no attention is given to the performance analysis and complexity of one tap equalizers in oversampled OFDM systems. The objective of this paper is therefore to fill this gap. To facilitate our analysis, we put symbol rate sampled and oversampled OFDM system models, independent of any PAR reduction technique under a common denominator as they form the basis of developing the efficient low complexity equalizers. In this study, our major point of concern is the degradation in the one tap equalizer output signal-to-noise ratio (SNR) (or equivalently decision point SNR) affecting the probability of symbol error. Based on our theoretical analysis and simulation results, we conclude that oversampling is more profitable than thought as it not only improves the per-symbol equalization performance of ZF and MMSE OFDM receivers but also makes them more resilient to channel nulls. The cost to achieve this improved performance is the increased system bandwidth and modest increase in the receiver complexity.

This paper is organized as follows: Section 2 presents the matrix data models of the oversampled and symbol based OFDM. In Section 3, corresponding one tap equalizers, probability-of-error expressions, corresponding performance analysis and complexity issues are considered. In Section 4, illustrating simulations are carried out while conclusions are drawn in Section 5.

Notation: We use boldface capital (lower case) letters to denote the matrices (column vectors), sometimes with subscripts to stress on their sizes. An N × N identity matrix is denoted as I N and an all-zero matrix of size K × N is denoted as O K × N . E [ ⋅ ] is used to denote expectation operation, ‖ ⋅ ‖ Euclidean norm, tr ( A ) for trace of a matrix A , ( ⋅ ) H for Hermitian transpose, and ( ⋅ ) * for complex conjugate. Argument i is used to denote ith block of symbols. The meaning of other variables will be clear from the context.

We consider an oversampled OFDM data transmission system, shown in

x a ( t ) = 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 , 〈 k + N 〉 denotes modulo N, s N = [ s 0 , s 1 , ⋯ , s N − 1 ] T represents the size N complex input data symbol/bock, s k represents the complex modulated symbol of the kth subcarrier, N is the number of subcarriers, and T s is the symbol period of the OFDM signal.

An oversampled version of the OFDM waveform (1) can be written as

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

where 0 ≤ n ≤ M − 1 , M = J N and J > 1 is the oversampling factor.

Let the samples associated with the ith OFDM signal be denoted as x n ( i ) . With this notation, we can now express the ith oversampled OFDM signal (2) arranged in a vector of length M as x M ( i ) = [ x 0 ( i ) , x 1 ( i ) , ⋯ , x M − 1 ( i ) ] T . The vector x M ( i ) can be obtained by using a length-M 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 x N ( i ) . Several different oversampling strategies can be defined. The approach above is our choice here as it results in trigonometric interpolation of the OFDM time domain signal [

To describe how the M − N zeros are inserted in x M ( i ) , we define the M × N precoding matrix

A ≜ ( 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 ) (3)

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

x M ( i ) = γ F M H u M ( i ) = γ F M H A s N ( i ) , (4)

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.

The OFDM signals based on the system model (4) are cyclically extended, digital-t-analogue (D/A) conversion and transmit filtering are performed, the signal is modulated to some carrier frequency, power amplified and then transmitted through the channel. At the receiving end, the reverse action takes place. After frequency down-conversion, receive filtering and analogue-to-digital (A/D) conversion, the cyclic prefix (CP) parts of the subsequent OFDM symbols are removed. We represent the overall combined effect of the spectral shaping pulse, the continuous time channel, the receive filter and the sampling through a discrete time causal, length L c finite impulse response (FIR) filter, with impulse response h c = [ h c ( 0 ) , h c ( 1 ) , ⋯ , h c ( L c − 1 ) ] T . To avoid inter-block-interference (IBI) between successive OFDM data blocks, the CP must be larger than or equal to the channel order. Assuming perfect synchronization and carrier recovery, the time domain received baseband data vector can then be expressed as:

r M ( i ) = γ C M ( h c ) F M H A s N ( i ) + n M ( i ) , (5)

where the FIR channel vector h c is represented 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 M × 1 additive white Gaussian noise (AWGN) vector with covariance E { n M n M H } = σ n 2 .

After demodulation with the M-point FFT matrix, the data model (5) becomes

F M r M ( i ) = γ F M C M ( h c ) F M H u M ( i ) + F M n M ( i ) r ˙ M ( i ) = γ diag ( ρ 0 , ρ 1 , ⋯ , ρ M − 1 ) u M ( i ) + n ˙ M ( i ) = γ D M ( h ˜ c ) u M ( i ) + n ˙ M ( i ) , (6)

where h ˜ c = [ ρ 0 , ρ 1 , ⋯ , ρ M − 1 ] = M F M h c and D M ( h ˜ c ) is a M × M diagonal matrix with entries ρ k ≡ ρ ( 2 π k / M ) = ∑ l = 0 L c − 1 h c ( l ) exp ( − j 2 π k l / M ) denoting the kth subcarrier channel frequency response.

Notice that the middle M − N elements of u M ( i ) are zeros, hence, the corresponding elements of r ˙ M ( i ) are nothing but corresponding samples of the noise data vector. These redundant noise samples do not play any role in symbol recovery due to independence of subcarriers. We therefore discard middle M − N zeros from r ˙ M ( i ) . To do this, we define the N × M receive matrix R : = A T . Pre-multiplying the received vector r ˙ M ( i ) by R , the data model (6) thus simply reduces to

R r ˙ M ( i ) = γ R D M ( h ˜ c ) u M ( i ) + R n ˙ M ( i ) r ¯ N ( i ) = γ D N ( h ⌢ c ) s N ( i ) + n ¯ N ( i ) , (7)

where D N ( h ⌢ c ) equals the matrix D M ( h ˜ c ) with all rows removed for which the corresponding entries of u M are zero, h ⌢ c = [ ρ 0 , ⋯ , ρ N / 2 − 1 , ρ M − N / 2 , ⋯ , ρ M − 1 ] = [ τ 0 , τ 1 , ⋯ , τ N − 1 ] denoting the vector on the diagonal of D N ( h ⌢ c ) and n ¯ N ( i ) = [ n 0 , ⋯ , n N / 2 − 1 , n M − N / 2 , ⋯ , n M − 1 ] = [ n ¯ 0 , n ¯ 1 , ⋯ , n ¯ N ] equals the AWGN vector n ˙ M with all entries removed from which corresponding entries of u M are zero.

The OFDM system with J = 1 is known as the Nyquist rate (or critical) sampling based OFDM, therefore: 1) no zeros are appended in the transmission; 2) an N size IFFT F N H is used and 3) we set β = 1 . As a result of this, at the transmitter side, data model (4) becomes

x N ( i ) = F N H s N ( i ) . (8)

Notice that the transmission power in oversampled and Nyquist rate based OFDM is maintained at

tr ( E { γ F M H A s N ( i ) ( γ F M H A s N ( i ) ) H } ) = tr ( E { F N H s N ( i ) ( F N H s N ( i ) ) H } = N σ s 2 ) .

i.e., the transmission power in both cases is equal and is given by the power used to transmit data. It is important to mention here that the total transmit power is the sum of power used to transmit data and the power used to transmit the CP. However, the CP power being typically small is normally ignored [

At the receiver side, the N-point FFT demodulated ith observed block is then

r ˙ N ( i ) = F N C N ( h c ) F N H s N ( i ) + F N n N ( i ) = diag ( δ 0 , δ 1 , ⋯ , δ N − 1 ) s N ( i ) + n ˙ N ( i ) = D N ( h ˜ c ) s N ( i ) + n ˙ N ( i ) , (9)

where C N ( h c ) is an N × N circulant matrix with first row [ h c ( 0 ) , 0 , ⋯ , 0 , h c ( L c − 1 ) , ⋯ , h c ( 1 ) ] , h ˜ c = [ δ 0 , δ 1 , ⋯ , δ N − 1 ] = N F N h c , with δ k ≡ ρ ( 2 π k / N ) = ∑ l = 0 L c − 1 h c ( l ) exp ( − j 2 π k l / N ) denoting the kth subcarrier channel frequency response.

For the data model (7) above, and N × N matrix G OS can be used to describe a linear equalizer that yields x ^ N ( i ) = G OS r ¯ N ( i ) . ZF and MMSE equalizers can be written respectively, as

G z f OS = 1 γ ( D N ( h ⌢ c ) ) − 1 G m m s e OS = γ σ s 2 D N H ( h ⌢ c ) ( σ n 2 I N + γ 2 σ s 2 D N ( h ⌢ c ) D N H ( h ⌢ c ) ) − 1 . (10)

Based on the data model (9), a N × N matrix G NS can be used as a linear equalizer to yield s ^ N ( i ) = G NS r ˙ N ( i ) . Its ZF and MMSE versions can be written as

G z f NS = ( D N ( h ˜ c ) ) − 1 G m m s e NS = σ s 2 D N H ( h ˜ c ) ( σ n 2 I N + σ s 2 D N ( h ˜ c ) D N H ( h ˜ c ) ) − 1 . (11)

From a unifying input-output perspective, in both the received data models (7) and (9) the kth transmitted symbol at the kth subcarrier equals the transmitted symbol scaled by the complex kth sub-channel gain that is corrupted by AWGN. The corresponding equalizers in Equations (10) and (11) are diagonal matrices, thus implying per-subcarrier/one tap equalization simplicity.

The oversampled OFDM, one tap ZF and MMSE equalization outputs can be written respectively as

s ^ k zf = s k ( i ) + 1 γ τ k n ¯ k ( i ) (12)

s ^ k mmse = γ σ s 2 | τ k | 2 ( σ n 2 + γ 2 σ s 2 | τ k | 2 ) s k ( i ) + γ σ s 2 τ k * ( σ n 2 + γ 2 σ s 2 | τ k | 2 ) n ¯ k ( i ) (13)

An important measure of performance of one tap equalizers is the SNR at the output of the equalizer. In both the cases, SNR at the kth symbol output can be written as

SNR k OS = σ s 2 σ n 2 / ( γ 2 | τ k | 2 ) . (14)

Now if: 1) M-ary QAM symbols are used, 2) ( 3 M − 1 ) SNR k ≫ 1 and 3)

SNR k = SNR k OS , then the average probability of symbol error can be approximated as [

P e ≈ 1 N ∑ k = 0 N − 1 P e k ≈ 1 N ∑ k = 0 N − 1 4 ( 1 − 1 M ) Q ( ( 3 M − 1 ) SNR k ) , (15)

where P e k denotes the probability of the kth symbol received incorrectly and the Q-function Q ( x ) ≜ 1 2 π ∫ x ∞ e − y 2 / 2 d y .

If the ZF and MMSE equalizers are used on the Nyquist rate data model, then the overall probability expression (15) is valid with the SNR at the kth output given by

SNR k = SNR k NS = σ s 2 σ n 2 / | δ k | 2 . (16)

Notice that P b ∝ SNR k . As a consequence the symbol errors on the kth subcarrier decrease as SNR k increases. Consequently, average probability of error will become low due to high average SNR at the equalizer output. In the following, we therefore focus on the SNR k Equations (14) and (16) to analyze the average probability of error performance of one tap equalizers. The SNR k equations differ in the following aspects.

1) In Equations (14) and (16), ignoring the channel fading effect, we observe that noise powers (the numerator terms) are not equal. Due to the scaling factor 1 / γ , the SNR k OS has lower noise power as compared with the noise power in SNR k NS . The noise power can be minimized and thus SNR k OS can be improved further as the scaling factor is decreased by increasing J.

2) Frequency response of a channel with tap coefficients [−0.0156 − 0.0102i, 0.0051 + 0.0033i, −0.0257 − 0.0164i]with increasing value of J is shown in

moderate null which finally disappears with increasing J. By comparing the frequency responses h ⌢ c and h ˜ c , in both the cases it can be observed that the elements of h ⌢ c i.e., τ k are relatively large in magnitude than the elements of h ˜ c (i.e. | δ k | < | τ k | ). This improvement in channel gains due to oversampling obviously causes high signal power in SNR k OS as compared with SNR k NS .

3) The disadvantage associated with the oversampled OFDM one tap equalizers is the loss of symbol detectability. If the transfer function h ⌢ c happens to have null on (or close to) the corresponding subcarrier (i.e., if | τ k | ≪ 1 is zero or close to zero), the corresponding symbol carried by that frequency will be impossible to detect due to significant drop in SNR k . Similar discussion holds for symbol based one tap equalizers. However, notice that the frequency response vector h ⌢ c is different from h ˜ c and as evident from

The above properties show that oversampling not only improves SNR but helps in avoidance of losing data due to channel nulls at the price of increased FFT complexity and bandwidth loss. Also, SNR can be improved, and equalizers can be made more resilient to channel nulls by increasing the oversampling rate. The elements of data vector s N ( i ) can thus be estimated more reliably (this lowers the probability of error) and will give oversampled OFDM one tap equalizers an edge in performance relative to symbol based OFDM one tap equalizers. We underscore here that the oversampled OFDM can be considered as a special case of precoded OFDM systems which involves data rate expansion due to insertion of zeros at the transmitter and have been devised to counteract channel effects [

O ( N ) computations are needed to calculate equalizers in G z f OS and G m m s e OS . Also, O ( N ) computations are required to detect the symbol estimates using x ^ N ( i ) = G z f / m m s e OS r ¯ N ( i ) . If we account the O ( M log M ) computations needed for FFT, only O ( log M ) computations are needed per symbol. On the other hand, similar discussion holds for G z f NS and G m m s e NS , except that O ( N log N ) computations needed for FFT. So, the complexity per symbol of conventional equalizer is O ( log N ) . Due to the increased FFT size, the per-symbol equalization complexity of the oversampled equalizers is obviously slightly higher than the Nyquist sampled equalizers.

In order to verify the performance edge of the oversampled OFDM one tap equalizers, we consider a baseband OFDM system with the number of subcarriers N = 80, and J = 1, 2, 3, 4. The randomly generated input data were modulated by 4-ary Quadrature Amplitude Modulation (4-QAM) and the quasi-static Rayleigh fading channel was assumed to have taps (3-ray channel).The carrier frequency was set at 900 MHz and the Doppler frequency was fixed at f d = 66.67 MHz. This means that the vehicle was moving at a speed of 80 Km/h. In

improve as the oversampling factor is increased. Also, observe that performance plots in case of ZF and MMSE equalization do not differ. This is obviously since SNR in both the cases does not change SNR k .

The widespread application of OFDM is mainly motivated by simple one tap equalization that it affords. In this paper, oversampling was studied from the one tap equalization performance and complexity perspective. With the theoretical analysis and simulations, we justified that oversampling not only improves equalizer output SNR but also helps in avoidance of losing data due to channel nulls. Increased oversampling thus lowers the probability of error but of course at the price of increased bandwidth and FFT size. In our simulations, we limited J to at most 4 times as in most of the practical OFDM systems it varies between 2 and 4 to achieve the desired system performance [

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

Ali, H. and Khan, F. (2018) On the Performance Analysis of One Tap Equalizers in Oversampled OFDM Systems. Int. J. Communications, Network and System Sciences, 11, 187-198. https://doi.org/10.4236/ijcns.2018.119011