Wireless Pers Commun DOI 10.1007/s11277-017-4933-5

BER Efficient Interleaved OFDM System Ekant Sharma1 • Sahil Rane1 • Kasturi Vasudevan1

Ó Springer Science+Business Media, LLC 2017

Abstract Precoded orthogonal frequency division multiplexed (OFDM) systems have low bit error rate (BER) performance compared to conventional OFDM systems. In this work, we have proposed a new buffer-interleaver based system architecture. The proposed system is found to have a better BER performance than the conventional precoded OFDM system when minimum mean square error (MMSE) equalization is performed. Using block-diagonal matrices, we have mathematically modelled buffer-interleaver architecture to derive a closed-form expression for the BER performance under frequency-selective fading channels, for zero-forcing (ZF) and MMSE equalizers. Our simulation results match the theoretical expression. It is shown that, as the buffer size increases, BER performance of MMSE equalizer improves, whereas in the case of ZF equalizer the BER performance nearly remains the same as the buffer size increases. A theoretical argument to show the BER performance enhancement is also provided. Keywords OFDM  Precoding  Interleaver  ZF  MMSE

1 Introduction Current wireless communication systems require high data rate transmission which leads to inter symbol interference (ISI). Orthogonal frequency division multiplexing (OFDM) has been found to be the most promising candidate to mitigate this effect by converting the & Ekant Sharma [email protected] Sahil Rane [email protected] Kasturi Vasudevan [email protected] 1

Department of Electrical Engineering, Indian Institute of Technology, Kanpur, Kanpur, UP 208016, India

123

E. Sharma et al.

frequency selective channel to frequency flat subchannels [1]. One of the major drawbacks with OFDM systems, however, is the high peak-to-average power ratio (PAPR) of the transmitted signals. The high PAPR introduces intermodulation distortion and undesired out-of-band radiation due to the non-linearity of the high power amplifier (HPA) which results in higher BER and adjacent channel interference [2]. Hence, many PAPR reduction techniques have been proposed in the literature [3]. In the past few years, various new transforms have been applied in OFDM to replace both precoder as well as IFFT/FFT operations. Precoders when combined with the IFFT/ FFT operation, give a new transform which has lesser complexity than the conventionally precoded systems. In addition, all these transforms are channel independent. Due to the sparse nature of these transforms, along with complexity, some of these also reduce PAPR, for example combination of Walsh–Hadamard matrix and IDFT to form T-Transform [4] and C-Transform formed by combining discrete cosine transform (DCT) and Hadamard transform [5]. In the literature, it has been shown that, interleavers used at the transmitter increases the extrinsic diversity and hence yields a better BER performance [6, 7]. In [7], authors have discussed the precoded OFDM system, in which they have performed interleaving in frequency domain. Also the derivation of post-equalization SNR is carried out on a single block of precoded data after de-interleaving and hence does not take into account interleaving operation. To reduce the effects of channel correlation, data interleaving along with turbo coding is used in [8, 9]. Use of interleaver also reduces the variability of the noise power [10]. The idea is to divide the N symbols to be transmitted in smaller blocks, individually precode them and then interleave them among each other before mapping them on the subcarriers. In [11], to achieve maximum diversity gain, optimal subcarrier grouping scheme has also been derived. In our work, we have extended the conventionally precoded OFDM system to exploit the transmission diversity with the help of interleaving. Instead of dividing the N symbols in smaller block, we transform the entire block of N symbols in the time domain and then store them in a buffer. Once a fixed number of OFDM symbols are stored in the buffer, they are interleaved among each other before transmitting in block of size N. This ensures that the transform size is same and its complexity is not increased. With the increase in the buffer size, more number of symbols would be interleaved, hence more diversity and better BER performance. In this paper, we have modelled the interleaver using block-diagonal matrices, to derive the SNR expressions for zero-forcing (ZF) and minimum mean square error (MMSE) equalizers. The simulation results are found to be almost in agreement with the analytical results. This modelling has resulted in permuted form of precoding matrix which mathematically shows how diversity is achieved through means of interleaving. We have also provided an argument to show that the upper bound on SNR for proposed architecture is higher than the upper bound for non-interleaved precoded OFDM system. The paper is organised as follows. Notations used in this paper are given in Sect. 2. In Sect. 3, the proposed system model is presented. The BER performance analysis for the proposed system is provided in Sects. 4 and 5 discusses the simulation results. Finally the conclusion is given in Sect. 6.

123

BER Efficient Interleaved OFDM System

2 Notation The vectors and matrices are denoted by bold letters, Crs denotes a complex matrix of dimension r  s. The superscript ð:ÞT and ð:ÞH denotes transpose and Hermitian transpose, respectively. The diagðxÞ denotes a (square) diagonal matrix with x on its main diagonal, I denotes an nN  nN identity matrix, and 0 denotes a zero vector. A block diagonal matrix with n number of matrices of size N  N in its diagonal is denoted by A€ 2 CnNnN .

3 System Model Consider a system model as shown in Fig. 1 with N number of subcarriers. The signal sðiÞ 2 CN1 at the input of the buffer is sðiÞ ¼ GSðiÞ

ð1Þ

h iT ðiÞ ðiÞ ðiÞ ðiÞ where SðiÞ ¼ S0 S1 S2 . . . SN1 and the transform matrix G 2 CNN is the product of ðiÞ

the precoder matrix H and the IFFT matrix QH , i.e., G ¼ QH H. The input symbols Sj , 0  j  N  1; 0  i  n  1 are drawn from an M-ary constellation, and are assumed to be independent with zero mean and variance r2s . The buffered symbols, s 2 CnN1 at the transmitter are represented as 3 2 ð0Þ 3 2 ð0Þ 3 2 G 0 ... 0 s S 6 ð1Þ 7 6 0 G . . . 0 7 6 ð1Þ 7 s 76 S 6 7 6 7 € 7 6 6 7 s ¼ 6 ð2Þ .. 7 .. . . 7 6 .. 7 ¼ GS 6 .. 7 ¼ 6 .. 4 . 5 4 . . . 54 . 5 . sðn1Þ

0

0

... G

Sðn1Þ

Fig. 1 Proposed system block diagram

123

E. Sharma et al.

where each sðiÞ 2 CN1 ; 0  i  n  1, is the transformed symbol vector obtained using (1), € 2 CnNnN denotes a block diagonal matrix with n number of identical matrices G in its G h iT diagonal, and S ¼ Sð0Þ Sð1Þ Sð2Þ    Sðn1Þ 2 CnN1 . The interleaving operation as shown in Fig. 2 is modelled using interleaver matrix P 2 CnNnN . Hence, the nN number of interleaved symbols are represented as € S ð3Þ sint ¼ PG h i ð0Þ ð1Þ ðn1Þ where sint ¼ sint sint    sint 2 CnN1 consists of n number of the interleaved transformed symbol frames. ðiÞ

Further, each sint 2 CN1 , 0  i  n  1 is transmitted over frequency selective multipath fading channel as a frame after the addition of cyclic prefix of length q samples. The ðiÞ

ðiÞ

ðiÞ

T

ðiÞ

channel is modelled as q-tap FIR filter cðiÞ ¼ ½c0 ; c1 ; . . .; cq1  where cl , 0  l  q  1, 0  i  n  1 are assumed to be independent and identically distributed complex Gaussian random variables with zero mean and unit variance. Assuming perfect synchronization at the receiver, the received ith signal frame rðiÞ 2 CN1 , after discarding cyclic prefix is expressed as [12] ðiÞ

ðiÞ

ðiÞ

rðiÞ ¼ cðiÞ  sint þ wðiÞ ¼ Ccirc sint þ wðiÞ

ð4Þ

where  denotes the circular convolution, 0  i  n  1, and wðiÞ 2 CN1 is a circularly symmetric Gaussian random vector with each element having variance N 0 =2 per  T ðiÞ dimension. The circulant matrix Ccirc 2 CNN consists of a vector cðiÞ 01Nq of size N  1 as the first column with other rows being circularly shifted versions of the first ðiÞ

column. A circulant matrix can be diagonalized using FFT matrix as Ccirc ¼ QH !ðiÞ Q,

Fig. 2 Row–column interleaving

123

BER Efficient Interleaved OFDM System

n o ðiÞ ðiÞ ðiÞ where !ðiÞ ¼ diag C0 C1    CN1 is an N  N diagonal matrix with each element being the FFT coefficient of the time domain channel cðiÞ . The received signal frame rðiÞ is equalized in the frequency domain using either zeroforcing (ZF) or minimum mean square error (MMSE) equalizer CðiÞ 2 CNN followed by an IFFT operation as shown in Fig. 1, which is expressed as ðiÞ

yint ¼ QH CðiÞ QrðiÞ ðiÞ

¼ QH CðiÞ !ðiÞ Qsint þ QH CðiÞ QwðiÞ :

ð5Þ

ðiÞ

The equalized signal frames yint 2 CN1 , 0  i  n  1 are stored in buffer before they are de-interleaved, as follows 2 ð0Þ 3 yint 6 ð1Þ 7 6 y 7 6 int 7 € H C€! € H C€Q €Q € sint þ Q €w  yint ¼ 6 ð6Þ 7¼Q 6 .. 7 4 . 5 ðn1Þ

yint

€ !, € C€ represents the block diagonal matrices of the size where yint 2 CnN1 , and Q, nN  nN, constructed from their counterparts Q, !ðiÞ ; CðiÞ , respectively as shown below 2 3 2 ð0Þ 3 Q 0 ... 0 ! 0 ... 0 6 0 Q ... 0 7 6 7 !ð1Þ . . . 0 7 6 7 6 0 € € ¼6 . 7 6 ; ! ¼ Q .. 7 .. .. 6 . 6 .. 7 ... . . . ... 7 4 . 5 4 . . 5 . . 0 0 ... Q 0 0 . . . !ðn1Þ 2 ð0Þ 3 C 0 ... 0 6 7 Cð1Þ . . . 0 7 6 0 C€ ¼6 .. 7 .. .. 6 .. 7: 4 . . 5 . . 0

0

...

Cðn1Þ

 2 CnN1 is written as The noise vector w 2

wð0Þ

6 ð1Þ 6 w ¼6 w 6 .. 4 .

3 7 7 7: 7 5

ð7Þ

wðn1Þ Now, on the buffered frames de-interleaving operation is performed. Under the assumptions that PPT ¼ I, we perform de-interleaving operation as

123

E. Sharma et al.

2

yð0Þ

6 yð1Þ 6 y ¼ 6 6 .. 4 .

3 7 7 7 ¼ PT yint : 7 5

ð8Þ

yðn1Þ After de-interleaving, yðiÞ , 0  i  n  1 is multiplied by inverse transform GH to recover the modulated symbols. S^i ¼ GH yðiÞ :

ð9Þ

h iT ^ ¼ S^ð0Þ S^ð1Þ S^ð2Þ . . . S^ðn1Þ 2 CnN1 is expressed as The detected input symbol S  ¼G € H y S^ €r € H C€Q € H PT Q ¼G H

H

ð10Þ H

H

€QP € G € w: € C€! € C€Q € S þ G € P Q € P Q  ¼G T

T

4 Periodic Interleaver In this paper, the permutation matrix periodically interleave s which contains nN elements, as shown in Fig. 2. Let, l denotes an index of an vector of dimension nN  1 to be interleaved, such that l 2 f0; 1; . . .; N  1; N; . . .; 2N  1; . . . . . .; ðn  1ÞN; . . .; nN  1g:

ð11Þ

Now assume that, m denotes an index of the corresponding periodically interleaved vector m 2 f0; N; . . .; ðn  1ÞN; 1; N þ 1; . . .; ðn  1ÞN þ 1; . . .; N  1; . . .; nN  1g:

ð12Þ

The relation between l and m is given as m ¼ ðl mod nÞN þ

  l : n

ð13Þ

The interleaving function in matrix form is represented using a permutation matrix P of size nN  nN which contains only zeros and ones, and satisfies PPT ¼ I. The position of ones and zeros in P is calculated using (13) as follows:   8 < 1; for n ¼ ðm mod nÞN þ m Pm;n ¼ ð14Þ n : 0; otherwise where 0  m  nN  1, 0  n  nN  1 are the row and column variable respectively.

123

BER Efficient Interleaved OFDM System

5 BER Performance Analysis The architecture proposed in this paper, results in a better BER performance when MMSE equalizer is used. Initially, we derive the SNR for the proposed architecture for both ZF and MMSE equalizers.

5.1 Permuted Precoding Matrix For the sake of mathematical analysis, we introduce a permuted version of precoder matrix, € G € p ¼ QP € 2 CnNnN . The permuted precoder matrix is also uniwhich is expressed as H € p does not have equal magnitude. Note that tary, however, unlike H all the elements of H € p reduces to block when no interleaving is introduced in the system i.e. P ¼ I, then H diagonal form of H i.e. 2 3 H 0 ... 0 6 0 H ... 0 7 6 7 €¼6 . ð15Þ H .. 7 .. . . 6 . 7: 4 . . . 5 . 0

0

... H

During the mathematical analysis in the following sections, it will be observed that Hp is the primary reason why the detection of symbol reduces to block of nN elements and not N elements.

5.2 ZF Equalizer ^ using the ZF equalizer ðC€ Þ can be Using (10), the demodulated data symbol vector S z expressed as € H C€z ! €QP € G € H C€z Q €w  ¼G € H PT Q € S þ G € H PT Q  S^ € H C€z Q €w € H PT Q  ¼ S þ G ¼ S þ

ð16Þ

€ H C€z W  H p

 T €w  ¼Q  ¼ Wð0Þ ; Wð1Þ ; . . .; Wðn1Þ and WðiÞ is the FFT of the corresponding where W ðiÞ time domain n noise vector wo . The ZF equalizer for the ith frame is given as ðiÞ ðiÞ ðiÞ ðiÞ ðiÞ € In the case , where c ¼ 1=C , which implies C€z ¼ 1=!. CðiÞ ¼ diag c ; c ; . . .; c z

0

1

N1

k

k

 in the uth of ZF, the error is mainly due to the noise in channel. The error e ¼ S^  S, symbol is expressed as [12] eu ¼

nN1 X

v Hp;u;v C€z;v;v W

ð17Þ

v¼0

where 0  u  nN  1, Hp;u;v denotes an element corresponding to uth row and vth column  v is the vth of matrix Hp , C€z;v;v is the diagonal element of equalizer matrix C€z and W  Further, due to the unitary property of FFT matrix, and uncorelement of the vector W.  v , the error variance of the uth symbol is written as related W

123

E. Sharma et al.



r2eu

¼ N0

nN1 X v¼0

2 Hp;u;v    €v;v 2

ð18Þ

ði Þ The elements €v;v are related to the channel DFT coefficients Ck as ðiÞ €v;v ¼ Ck where; i ¼ bv=N c; k ¼ v mod N:

ð19Þ

The SNR of the uth symbol is expressed as SNRzf ;u ¼

f PnN1 jHp;u;v j2 2 v¼0 j€v;v j

ð20Þ

where f ¼ r2s =N0 and r2s ¼ E½Su Su  is the power in each transmitted symbol Su .

5.3 MMSE Equalizer To derive the SNR expression for MMSE equalizer, consider (10) with equalizer C€m . For ith frame the MMSE equalizer is expressed as n o ðiÞ ðiÞ ðiÞ CðiÞ ð21Þ m ¼ diag k0 ; k1 ; . . .; kN1 where [12] ðiÞ 

C ðiÞ : kk ¼  2k  ðiÞ  Ck  þ 1=f

ð22Þ

^ using the MMSE equalizer, is expressed as The demodulated data symbol vector S € H C€m ! €QP € G € H C€m Q €w € S þ G € H PT Q ¼G € H PT Q  S^ € H C€m Q €H € p S þ H €w € H C€m !  ¼H p p  ¼ A S þ Bw

ð23Þ

where A; B 2 CnNnN are expressed as € H C€m ! €H €p A ¼H p € H C€m Q: € B ¼H p  is written as Hence, the uth element of S^ u ¼ Au;u Su þ Wu S^

ð24Þ

ð25Þ

where Au;u is (u, u)th element of A and Wu consists of ISI from other symbols and channel noise, given by [12] Wu ¼

nN1 X v¼0;v6¼u

123

u : Au;v Sv þ ½Bw

ð26Þ

BER Efficient Interleaved OFDM System

Now, estimation error e in the uth symbol is expressed as   eu ¼ Au;u  1 Su þ Wu :

ð27Þ

Since, Su and Wu are uncorrelated, variance of eu is expressed as  2 r2eu ¼ Au;u  1 r2s þ r2Wu :

ð28Þ

Further, using the orthogonality property [12], we have h 

i     S r2eu ¼ E eu eu ¼ E eu S^ u u     ¼ E eu Su      ¼ E  Au;u  1 Su þ Wu Su   ¼ 1  Au;u r2s : Henceforth, from (28) and (29) the variance of Wu is   r2Wu ¼ Au;u 1  Au;u r2s :

ð29Þ

ð30Þ

The SNR corresponding to the uth symbol is expressed as SNRmmse;u ¼

A2u;u r2s r2Wu

Au;u ¼ 1  Au;u

ð31Þ

where Au;u ¼

nN1 X

 H € p;u;v 2

v¼0

 2 f€v;v   2 : 1 þ f€v;v 

ð32Þ

The probability of bit error for M-PSK and M-QAM modulation, is given as [13] pffiffiffiffiffiffiffiffiffiffiffiffiffiffi p

2 Pe;MPSK ¼ Q 2SNRu sin log2 M M

rffiffiffiffiffiffiffiffiffiffiffiffiffiffi! ð33Þ 4 1 3SNRu Pe;MQAM ¼ 1  pffiffiffiffiffi Q log2 M M1 M R 1 t2 e 2 dt denotes the Q function and SNRu is the SNR expressions where QðxÞ ¼ p1ffiffiffiffi 2p x derived in (20) and (31) for ZF and MMSE receivers, respectively.

5.4 BER Performance Without Interleaving To eliminate the interleaving action, we set interleaver matrix as P ¼ I. Hence (10) reduces to € H C€! € H C€Q €Q €G € w: € S þ G €H Q ¼G €H Q  S^

ð34Þ

123

E. Sharma et al.

 all are block diagonal matrices and hence same  and S, Note that, in (34) except for w equation also holds true for their diagonal counterparts each of size N  N and is expressed as S^i ¼ GH QH Ci !i QGSi þ GH QH Ci Qwi ¼ HH Ci !i HSi þ HH Ci Qwi

ð35Þ

where 0  i  n  1 and S^i 2 CN1 . Note that, in this case the precoding matrix is H and all pffiffiffiffi the elements of this precoder have same magnitude 1= N . Using this fact and following the similar procedure explained earlier, for ZF equalizer SNR corresponding to uth symbol within frame S^i is ~ zf ;u ¼ P f SNR N1 1 v¼0

N

1

:

ðiÞ 2 Cv

ð36Þ

j j

Similarly, for MMSE equalizer SNR is expressed as ~ mmse;u ¼ SNR

~ u;u A ~ u;u 1A

ð37Þ

~ u;u is expressed as where A ~ u;u A

N 1  X  Hu;v 2 ¼ v¼0

N 1 1X ¼ N v¼0

 2   fCðiÞ v   2   1 þ fCðiÞ v 

 2   fCðiÞ v   2 :   1 þ fCðiÞ v 

ð38Þ

ð39Þ

Using (37) and (39), the final expression for SNR is

~ mmse;u SNR

2 PN1 fjCðiÞ v j v¼0 ðiÞ 2 1þfjCv j ¼ PN1 : 1

v¼0

ð40Þ

ðiÞ 2 Cv

1þfj

j

The SNR expression obtained above is exactly same as that of precoded OFDM systems [12, 14]. Since P is not a block diagonal matrix in (10), dividing the equation in their counterparts of size N  N is not possible. Hence, for the proposed architecture decoding a symbol requires a received frame of size nN  1. This implies that because of interleaving, information about each transmitted symbol is scattered in nN number of received symbols. As opposed to this, in (35) decoding requires received frame of size N  1 and hence looses the diversity as compared to the proposed architecture and results in reduction in BER performance, as we will see in simulation results.

123

BER Efficient Interleaved OFDM System

5.5 BER Performance Improvement This section proposes an argument to show that in case of MMSE receiver, upper bound on SNR for the proposed architecture SNRmmse;u is higher than the upper bound on SNR for   ~ mmse;u . non-interleaved precoded OFDM SNR ~ u;u for the From (31) and (37) it is evident that SNR is directly proportional to Au;u and A proposed and non-interleaved case, respectively. Now, Au;u can also be expressed as the dot product of two vectors i.e., Au;u ¼ E T F

ð41Þ

where h      iT € p;u;0 2 ; H € p;u;1 2 ; . . .; H € p;u;nN1 2 E ¼ H

ð42Þ

nN1

and "

 2  2  2 # T f€0;0  f€1;1  f€nN1;nN1  F ¼  2 ;  2 ; . . .;  2 1 þ f€0;0  1 þ f€1;1  1 þ f€nN1;nN1 

ð43Þ

:

nN1

Using Cauchy-Schwartz inequality [13], we have   Au;u   kE kkF k:

ð44Þ

On the similar lines, for non-interleaving case ~ u;u ¼ E~T F ~ A

ð45Þ

where

−2

Fig. 3 BER performance analysis of the proposed system considering MMSE equalizer with QPSK and N = 256

10

ξ=1 ξ=2 ξ=4

−3

Bit Error Rate

10

ξ=8 ξ = 16

−4

10

−5

10

−6

10

10

11

12

13

14

15

16

17

18

19

20

21

22

23

SNR per bit (Eb/No)

123

E. Sharma et al. 100

Fig. 4 BER performance analysis of the proposed system considering ZF equalizer with QPSK and N = 256

ξ=1 ξ=2 ξ=4 ξ=8 ξ = 16

Bit Error Rate

10−1

10−2

10−3

10−4 0

3

6

9

12

15

18

21

24

27

30

33

36

SNR per bit (Eb/No)

h 2  2  2 iT E~ ¼ Hu;0  ; Hu;1  ; . . .; Hu;N1 

ð46Þ

N1

and 2

 2  2   3T  ðiÞ   ðiÞ   ðiÞ 2 f C 0  f C 1  fCN1  7 ~ ¼6 F 4  2 ;  2 ; . . .;   5  ðiÞ   ðiÞ   ðiÞ 2 1 þ f C 0  1 þ f C 1  1 þ fCN1 

:

ð47Þ

N1

~ both are vectors of size N  1 while E and F are of size nN  1. Note that E~ and F   € Now, since Hp and H both are unitary matrices we have kE k ¼ E~ ¼ 1. Therefore,   Au;u   kF k    ð48Þ ~   ~ : Au;u   F €v;v and CðiÞ (19), it is observed that first N elements of F Using the relation between ! k ~ and all the remaining ðn  1ÞN elements of F are positive, are same as that of F   ~ : ð49Þ kF k F ~ u;u , we conclude that the upper bound Hence, using (49) and expression for Au;u and A ~ u;u . Moreover from (31), the same is valid for on the value attained by Au;u is higher than A ~ mmse;u . Henceforth, SNR, i.e. upper bound on SNRmmse;u is higher than upper bound on SNR the proposed architecture yields better BER performance than non-interleaving case and our simulation results also confirms the same, as discussed in next Section.

123

BER Efficient Interleaved OFDM System 0

Fig. 5 BER performance of the proposed system with MMSE equalizer comparing theoretical and simulation results for n ¼ 2 with QPSK and N = 256

10

Theoretical ξ =2 Simulation ξ =2

−1

10

Bit Error Rate

−2

10

−3

10

−4

10

−5

10

−6

10

5

10

15

20

25

SNR per bit (Eb/No)

6 Simulation Results In this section, the BER performance of the proposed system is discussed. The BER performance is evaluated for QPSK under the assumption of perfect time and frequency synchronization, and channel information at the receiver. For simulations, we have chosen N ¼ 256 subcarriers and Rayleigh fading channel having L ¼ 10 taps with uniform channel power delay profile. In [14], a transform matrix is proposed which we have used for our simulations in this paper. Each column of this transform matrix G is a cyclically shifted version of Gaussian integer perfect sequence. The primary aim of this transform is to decrease the computational complexity. Moreover due to the sparse nature of the transform matrix it results in the reduction of PAPR. Figure 3 shows the BER performance of the proposed system considering MMSE equalizers with different buffer sizes (n ¼ 1; 2; 4; 8; 16). Note that n ¼ 1 is the non-buffered case. As the buffer size n increases, the BER performance is improved. Moreover, the BER is greatly reduced when buffer size is increased to four, while marginal improvement in performance is observed for higher values of buffer sizes. This suggests that the proposed architecture exhibits a better performance at the cost of negligible amount of latency. It is clear from Fig. 3 that for the buffer length of n ¼ 2, around 1:0 dB and 1:5 dB improvements are observed at BER ¼ 105 and BER ¼ 106 , respectively as compared to n ¼ 1. The improvement in BER is more in high SNR region while there is no improvement in the low SNR region. The BER performance of the proposed system considering ZF equalizer is shown in Fig. 4. It is observed that for different buffer lengths the BER performance more or less remains same. Although interleaver provides more diversity, in the case of ZF since the equalizer directly inverts the channel, it results in noise enhancement, hence there is no performance improvement in terms of BER for different buffer size. Figures 5 and 6 compares the theoretical and the simulated BER of the proposed system considering ZF and MMSE equalizers, respectively for n ¼ 2. The theoretical BER is pffiffiffiffiffiffiffiffiffiffiffiffi plotted using (33) which get reduced to Qð SNRu Þ for QPSK, where SNRu for ZF and

123

E. Sharma et al. 0

Fig. 6 BER performance of the proposed system with ZF equalizer comparing theoretical and simulation results for n ¼ 2 with QPSK and N = 256

10

Simulation, ξ = 2 Theoretical, ξ = 2

−1

Bit Error Rate

10

−2

10

−3

10

0

2

4

6

8

10

12

14

16

18

20

22

2425

SNR per bit (in dB)

MMSE equalizers are derived in (20) and (31), respectively. It is observed that simulations closely follow the theoretical results obtained.

7 Conclusion In this paper, an interleaver based buffered OFDM system using low complexity transform is proposed. The proposed system model exhibits BER performance improvement for MMSE equalizer without affecting its PAPR. A closed-form expression for the BER performance for both ZF and MMSE equalizers is derived. It is observed that, simulations results approximately follows the analytical BER.

References 1. Vasudevan, K. (2010). Digital communications and signal processing. India: Universities Press. 2. Jiang, T., & Wu, Y. (2008). An overview: Peak-to-average power ratio reduction techniques for OFDM signals. IEEE Transactions on Broadcasting, 54(2), 257. 3. Rahmatallah, Y., & Mohan, S. (2013). Peak-to-average power ratio reduction in OFDM systems: A survey and taxonomy. IEEE Communications Surveys & Tutorials, 15(4), 1567–1592. 4. Ahmed, M. S., Boussakta, S., Sharif, B. S., & Tsimenidis, C. C. (2011). OFDM based on low complexity transform to increase multipath resilience and reduce PAPR. IEEE Transactions on Signal Processing, 59(12), 5994–6007. 5. Leftah, H., Boussakta, S., et al. (2014). Novel OFDM based on C-transform for improving multipath transmission. IEEE Transactions on Signal Processing, 62(23), 6158–6170. 6. Huang, X. (2007). Diversity performance of precoded OFDM with MMSE equalization. In International Symposium on Communications and Information Technologies, 2007. ISCIT’07. IEEE, 2007, (pp. 802–807). 7. Hasegawa, F., Plataniotis, K. N., & Pasupathy, S. (2010). A novel cyclic shift interleaver design for linearly precoded OFDM systems. IEEE Transactions on Wireless Communications, 9(11), 3330–3334.

123

BER Efficient Interleaved OFDM System 8. Vasudevan, K. (2013). Coherent detection of turbo coded OFDM signals transmitted through frequency selective rayleigh fading channels. In IEEE International Conference on Signal Processing, Computing and Control (ISPCC), 2013. IEEE, (pp. 1–6). 9. Vasudevan, K. (2015). Coherent detection of turbo-coded OFDM signals transmitted through frequency selective Rayleigh fading channels with receiver diversity and increased throughput. Wireless Personal Communications, 82(3), 1623–1642. 10. Ortı´n, J., Garcı´a, P., Gutie´rrez, F., & Valdovinos, A. (2009). Channel independent precoder for OFDMbased systems over fading channels. IEEE Transactions on Broadcasting, 55(4), 818–825. 11. Tran, N. H., Nguyen, H. H., & Le-Ngoc, T. (2007). Subcarrier grouping for OFDM with linear constellation precoding over multipath fading channels. IEEE Transactions on Vehicular Technology, 56(6), 3607–3613. 12. Lin, Y.-P., Phoong, S.-M., & Vaidyanathan, P. (2010). Filter bank transceivers for OFDM and DMT systems. Cambridge: Cambridge University Press. 13. Proakis, J . G. (2011). Digital communications. New York City, NY: McGraw-Hill. 14. Wang, S.-H., Li, C.-P., Lee, K.-C., & Su, H.-J. (2015). A novel low-complexity precoded OFDM system with reduced PAPR. IEEE Transaction on Signal Processing, 63(6), 1366–1376. Ekant Sharma received the M.Tech degree in Electrical Engineering from Indian Institute of Technology, Kanpur, India, in 2011, and the B.E. degree in Electronics and Communication Engineering from CSIT, Durg, India, in 2009. From 2011 to 2012, he was with IBM-ISL and worked as an software engineer. He has worked as an assistant professor at Graphic Era University, Dehradun and SSTC, Bhilai, during 2012 to 2013 and 2013 to 2014, respectively. Since July 2014, he is a Ph.D. student of the Signal processing, Communication and Networks group in the Department of Electrical Engineering at Indian Institute of Technology, Kanpur, India. His current research interests include wireless communication and massive MIMO.

Sahil Rane received his B.E. (Bachelor of Engineering) degree in Electronics from University of Mumbai, India in 2010 and M.Tech (Master of Technology) in Electrical Engineering from Indian Institute of Technology, Kanpur, India, in 2015. Since November 2015, he has worked on developing MIMO-OFDM channel estimation algorithms, MIMO channel modelling, SNR estimation as a part of LTE physical layer implementation with Sasken Communication Technologies Limited, Bengaluru, India. His current research area includes MIMO wireless communication, channel estimation, signal to noise ratio estimation.

123

E. Sharma et al. K. Vasudevan completed his Bachelor of Technology (Honours) from the department of Electronics and Electrical Communication Engineering, IIT Kharagpur, India, in the year 1991, and his M.S. and Ph.D. from the department of Electrical Engineering, IIT Madras, in the years 1996 and 2000 respectively. During 1991–1992, he was employed with Indian Telephone Industries Ltd, Bangalore, India. He was a Post Doctoral Fellow at the Mobile Communications Lab, EPFL, Switzerland, between Dec. 1999 and Dec. 2000, and an engineer at Texas Instruments, Bangalore, between Jan 2001 and June 2001. Since July 2001, he has been a faculty at the Electrical department at IIT Kanpur, where he is now an Associate Professor. His interests lie in the area of communications and signal processing.

123