OFDM Schemes with Non-Overlapping Time Waveforms - CiteSeerX

OFDM Schemes with Non-Overlapping Time Waveforms Slimane Ben Slimane

Radio Communication Systems Department of Signals, Sensors and Systems Royal Institute of Technology, 100 44 Stockholm, Sweden Tel. +46 8 790 9353, Fax +46 8 790 9370, Email: [email protected] Abstract | 1 For Orthogonal Frequency Division Multiplexing (OFDM) schemes, the time waveform plays a major role not only in shaping the spectrum of the transmitted signal but also in separating the signal subcarriers at the receiver. The selection of such waveform is then very important and can make a big dierence in spectrum utilization and system performance. The diculty in using pulse shaping with OFDM appears in the detector design since smooth time waveforms destroy the orthogonality between subcarriers and introduce Inter-Channel Interference (ICI). This paper investigates the use of non-overlapping signal waveforms for OFDM schemes. It is shown that pulse shaping for such schemes can be equivalently achieved using a discrete shaping matrix. Such shaping structure is very suitable for digital implementation and can be combined with OFDM schemes without aecting their IDFT/DFT operators. It therefore makes the use of guard time interval (cyclic pre x) for OFDM schemes possible with any kind of nonoverlapping shaping waveform. I. Introduction

For a conventional OFDM scheme, the used waveform is a rectangular pulse shape. Such a waveform is optimum in separating the signal subcarriers at the receiver when perfect timing and frequency synchronizations are achievable. However, its side lobes fall o only as 1=f and therefore does not shape the signal spectrum. OFDM schemes with rectangular pulse shapes are also very sensitive to synchronization imperfections and the system performance degrades rapidly even for very small frequency errors [1, 2]. Ideal synchronization situations are quite dicult to achieve especially when the signal is to be transmitted over mobile radio channels, where the channel is time varying making the received signal a mixture of several multipath components. Frequency variations and timing errors are therefore very likely to occur and in a random manner. Using time waveforms having low side-lobes reduce adjacent subcarrier interference [3, 4], reduce the sensitivity of OFDM schemes to frequency variations [4, 5, 6], and make the transmitted spectrum more compact. In this paper we consider OFDM schemes that use nonoverlapping time waveforms (i.e., time waveforms having the same duration as the OFDM symbol interval). We show that the system can be equivalently represented by a shaping processor followed by a regular OFDM scheme. The shaping 1

VTC '98, Ottawa, Canada, May 1998

processor consists of multiplying the input sequence by a matrix, referred to as shaping matrix, and then modulated by the subcarriers. This structure is similar to that of MultiCarrier-CDMA (MC-CDMA) schemes [7, 8], which gives the possibility to use the same MC-CDMA receivers. It also introduces a dierent approach in designing pulse shaping waveforms and allows the system to get all the bene ts of pulse shaping without aecting the OFDM structure. II. System Model

We consider an OFDM scheme with a total of N orthogonal subcarriers. Each subcarrier is modulated with a low rate sequence of pulses (symbols) of duration T . The system block diagram is shown in Figure 1. For a given OFDM symbol interval, the equivalent lowpass of the transmitted signal can be written as follows:

x(t) =

NX ?1 m=0

sm;k pm (t ? kT )ej2 mT t ; kT  t  (k + 1)T (1)

with sk = [s0;k ; s1;k ;


; sN ?1;k ]T is a sequence of complex baseband modulated signals related to the used modulation scheme and pm(t) is a pulse shape of duration T used at subcarrier m. Since the waveform, pm (t), is de ned over the interval 0  t  T , using extension by periodicity we can de ne a periodic waveform as follows:

wm (t) =

+1 X

i=?1

pm(t ? iT ) =

+1 X

i=?1

cm;i ej2i Tt ;

where Z

T





pm (t)e?j2i Tt dt = T1 Pm Ti : 0 is the exponential Fourier series coecient and Pm (f ) is the Fourier transform of pm (t). In this case the waveform pm (t) can be rewritten as cm;i = T1

pm (t) =

+1 X

i=?1

cm;i ej2i Tt uT (t ? T=2)

(2)

where

e j2π0t/T

uT (t) =



1; ?T=2  t  T=2 0; elsewhere

(3)

Most practical shaping waveforms can be represented by a nite number of harmonics. Without loss of generality we assume that the total number of harmonics for the pulse shape pm (t) is L + U +1. In this case, pm (t) can be rewritten as follows:

pm (t) =

+U X

i=?L

p0 (t)

cm;i ej2i Tt uT (t ? T=2)

(4)

e j2πt/T input Baseband data Modulator

serial to parallel converter

p1 (t) Multiplexing pN-2 (t) e j2π(N-2)t/T pN-1 (t) e j2π(N-1)t/T

Fig. 1. block of the OFDM scheme using time shaping waveforms. OFDM modulator

e-j2 π Lt/T

Consider the signal space with dimension N + L + U and having as orthogonal basis

e-j2 π (L-1)t/T





e?2L Tt ; e?2(L?1) Tt ;


; e2(N +U ?1) Tt ; e2(N +U ) Tt ;

then the pulse shape pm (t) can be represented by the following signal vector in this space

input Baseband data Modulator

serial to parallel converter

Shaping

+1

m=?L

where

s~m;k =

s~m;k ej2 mT t ; 0  t ? kT  T; min(NX ?1;m+L) i=max(0;m?U )

ci;m?i si;k :

(5)

(6)

By de ning the following sequence

~sk = [~s?L;k ; s~?L+1;k ;


; s~N +U ?1;k ]T ;

we can write

~sk = PT sk where sk is as previously de ned, 2

P=

6 6 6 6 6 4

p Dp 0

.. .

1

DN ?2 pN ?2 DN ?1 pN ?1

(7) 3 7 7 7 7 7 5

;

xl (t)

e j2π (N+U-1)t/T

1

Replacing pm (t) by its expression in (1), the equivalent lowpass of the OFDM signal becomes

x(t) =

Multiplexing

e j2π (N+U-2)t/T

pm = [cm;?L; cm;?L ;


; cm;U ? ; cm;U ; 0;


; 0] N +X U ?1

xl (t)

(8)

is an N  (N + L + U ) matrix, and Di pm is the ith right cyclic shift of the vector pm . We refer to P as the shaping matrix. This means that the OFDM transmitted signal with shaping waveforms can be generated using the block diagram shown in Figure 2. The shaping processor consists simply by multiplying the sequence sk by the shaping matrix P. The output of the shaping process is then modulated by a

Fig. 2. Equivalent block of the OFDM scheme using time shaping waveforms.

regular OFDM scheme with N + L + U subcarriers giving the signal x(t) as de ned in (5). Notice that each subcarrier in the block diagram of Figure 2 is now modulated by a rectangular pulse (symbol). This indicates that it is possible to insert a guard time interval (cyclic pre x) between consecutive symbols and then remove it at the receiver as it is done in a conventional OFDM scheme to deal with any ISI caused by the channel. Assuming a guard time interval of duration Tg , the modi ed shaping waveform including the guard interval can be written as follows

pm (t) = =

 P+U

0;

+U X

i=?L

i=?L cm;i e

j 2i Tt ;

?Tg  t  T

elsewhere

cm;i ej2i Tt uT +Tg (t ? (T ? Tg )=2): (9)

Since any continuous waveform over the interval 0  t  T can be de ned by the expression given in (4), the above representation is quite general and be applied to any kind of shaping waveform of duration T . Example 1 { Regular OFDM For a conventional OFDM scheme the time waveform is a rectangular pulse shape and all subcarriers use the same waveform

pm (t) = uT (t ? T=2): In this case L = U = 0, and the corresponding vector has length N . The shaping matrix is then an N  N

identity matrix, P = I.

Example 2 { Multi-Carrier CDMA Considering the expression of the transmitted sequence ~sk (eq. (7), we notice that if the shaping matrix is an orthogonal Hadamard matrix, P = [a ] ; a =  p1 ; (10) ij

ij

N

the sequence then becomes that of a MC-CDMA signal with a total of Nu = N users [7, 8] and the block diagram in Figure 2 becomes a MC-CDMA transmitter. Therefore, the same signal can be obtained using the block diagram of Figure 1, with each subcarrier using the following pulse shape

pm (t) =

N ?X m?1 i=?m

a(i+m)m ej2i Tt uT (t ? T=2):

(11)

In other words the MC-CDMA scheme can be seen as a regular OFDM scheme with non-overlapping time waveforms. The shaping in this case is introduced by the Hadamard codewords. Notice that all these waveforms are complex and that a dierent waveform is assigned to each subcarrier. For subcarrier m we have, L = ?m; U = N ? m ? 1. Example 3 { OFDM with Hanning Window The Hanning window is de ned as follows:   pm (t) = p1 ?e?j2 Tt + 2 ? ej2 Tt uT (t ? T=2); 6 indicating that L = U = 1 and the corresponding vector has length N + 2. The shaping matrix is then an N  (N + 2) matrix and is given by 2 6

6 P = p16 666 4

?1 2 ?1 ?1 2 ?1

0

0

... ... ... ?1 2 ?1 ?1 2 ?1

3 7 7 7 7 7 5

It is observed that when identical pulse shapes are used for all subcarriers then the rows of the matrix P are cyclic shifts of each other. Since any time waveform over the interval 0  t  T can be de ned by the expression given in (4), this matrix representation is quite general and can be applied to any kind of shaping waveform of duration T . It also allows simple digital implementation and more freedom in designing optimum set of pulse shaping waveforms without altering the OFDM IDFT/DFT operators. III. Receiver Design

We have seen that an OFDM transmitter with nonoverlapping time waveforms consists of multiplying the input sequence by a shaping matrix followed by a regular OFDM modulator. Therefore, an optimum receiver can be seen as

r(t)

rk OFDM demodulator (FFT)

Signal detection

Equalizer

estimated data

Channel estimation

Fig. 3. OFDM receiver

an OFDM demodulator (correlator receiver) followed by signal separation and decision as shown in Figure 3. Our task is then to design the second part of this receiver which is obviously dependent on the transmitted set of time waveforms. For an ideal channel with Additive White Gausian Noise (AWGN) only, the received baseband OFDM signal can be written as

r(t) = x(t) + n(t);

(12)

where x(t) is as previously de ned and n(t) is a zero-mean complex Gaussian process. For subcarrier m, the kth received sample at the output of the OFDM demodulator takes the following form

rm;k = s~m;k + nm;k (13) where s~m;k is as given in (6) and nm;k is a zero-mean complex Gaussian random variable. Considering the output samples of all subcarriers, we obtain a matrix format of the received signal as follows

rk = PT sk

+ nk

(14)

where rk = [r?L;k ; r?L+1;k ;


; rN +U ?1;k ]T is the received vector at time instant kT . The receiver then needs to deal with this shaping matrix in order to separate the subcarrier signals and make a decision. A. Orthogonal Waveforms To ensure zero ICI in an OFDM system, the set of waveforms fpm(t)ej2mt=T g should form an orthogonal set over the interval 0  t  T , i.e., T

Z 0

pm (t)pn (t)ej2(m?n)t=T dt =



T; n = m 0; n = 6 m

(15)

Using the expression of pm (t) given by (6) in the above condition and carrying out the integration we obtain the following

P
PT  = I:

(16)

where (
)T  denotes the transpose conjugate. It is easy to

verify that examples 1 and 2 fall in this category. Therefore, any set of time waveforms having a shaping matrix that satis es (16) ensures zero ICI and forms an orthogonal set for the OFDM scheme. The subcarrier signals at each OFDM block can then be separated by just multiplying the received vector in (14) by the transpose conjugate of P

r0k = P rk = sk

+ P  nk :

(17)

Notice that this same result is obtained if instead the receiver uses a lter at each subcarrier matched to its transmitted pulse shape. That is, a lter with impulse response

hm(t) = pm (T ? t); m = 0; 1; 2;


; N ? 1

(18)

B. Non-Orthogonal Waveforms When the condition in (15) is not satisfyed there will be ICI between subcarriers. In this case the matrix product of (16) produces a symmetric matrix. This means that each subcarrier will receive interference from its neighbouring subcarriers of both sides. Such a situation makes equalization or sequence estimation too complicated. For instance for the pulse shape given in Example 2, the product in (16) produces the following symmetric matrix 2

P
PT  = 16

6 6 6 6 6 6 6 6 6 4

6 ?4 1 0 ?4 6 ?4 1 1 ?4 6 ?4 1 ... ... ... ... ... 1 ?4 6 ?4 1 1 ?4 6 ?4 0 1 ?4 6

3 7 7 7 7 7 7 7 7 7 5

We notice that a subcarrier around the middle of the spectrum receives interference from four neighbouring subcarriers, two from each side. Therefore, to obtain an ICI free OFDM scheme with the Hanning window and the same receiver structure of (17), one can only use every 4th subcarrier. In this case the above matrix product reduces to an identy matrix but at the expense of a considerable reduction in system eciency. It is then necessary to have a receiver that can take into account the eect of this interference and allows the use of all subcarriers. Let us consider the shaping matrix P given in (8). Using the Gram-Schmidt orthogonalization procedure we can derive an orthonormal set of vectors, call them v0 ; v1;


; vN ?2; vN ?1, from the matrix P, thus

vi = (vi; ; vi; ;


; vi;N 0

vm = where

m = and and

1

L U ?1 ); with

+ +

( p0

(

0 ; P ?1 Dm pm ? m i=0 m;i vi ; m

pE ; q

0

Pm?1

Em ? i=0

2

m=0 m = 1; 2;


; N ? 1

m=0 jm;i j2 ; m = 1; 2;


; N ? 1

m;i = Dm pm
vi = Em =

jjvi jj = 1;

Z 0

T

N +LX +U ?1 j =0

: cm;j?L vi;j

jpm (t)j dt = T; 2

Let us de ne a matrix V having as rows the vectors vi 's. That is, 3 2

V=

6 6 6 4

v v

0 1

.. .

vN ?

7 7 7 5

1

The shaping matrix can then be rewritten as the product of two matrices as follows P = A
V; (19) where 2 3 0 0


0 6 .. 77 ... . 7 A = 666 1..;0 . .1 (20) 7 . . 4 . . 0 5 .

N ?1;0


N ?1;N ?2 N ?1

is a lower triangular matrix with all its elements obtained from the used set of time shaping waveforms. For the particular case of orthogonal set of waveforms, the matrix A is simply an N  N identity matrix. Using the above transformation, the received vector at time kT becomes rk = VT AT sk + nk Multipyling rk by the transpose conjugate of V we get (21) r0k = AT sk + n0k where n0k is a vector of N uncorrelated zero-mean complex Gaussian random variables. From the expression of the matrix A we can identify two alternative receiver solutions:  It is observed that at any given subcarrier of the OFDM scheme the interference now comes only from one side. In this case a MLSE receiver based on the Viterbi algorithm can be implemented. The algorithm runs along the subcarriers starting with subcarrier N ? 1 as the rst input up to subcarrier 0.  The expression of the matrix A also indicates that the received signal at subcarrier N ? 1 is now free of interference. Therefore, an alternative and simpler solution would be to use a Feedback Equalizer (FE) along the subcarriers starting from subcarrier N ? 1 up to subcarrier 0. The second block in the OFDM receiver can now be implemented using any of the above solutions. The OFDM system can then use all its subcarriers with any chosen set of time waveforms. However, the system performance will still be depend on the correlation coecient  between the chosen time waveforms. Example { A Modi ed Hanning Window As an illustrative example, we consider a modi ed Hanning window as follows:  p p pm(t) = 1 ? 2a ? 2 a cos(2t=T ) uT (t ? T=2); (22)

Modified Hanning Window 1.8

OFDM with BPSK Modulation, N=64

1.6 −1

10 1.4

a=0

1

−2

10 a=0.05

0.8

Bit Error Probability

p_m(t)

1.2

0.6 a=0.1

0.4 a=0.167

0.2 0 0

0.1

0.2

0.3

0.4

0.5 t/T

0.6

0.7

0.8

0.9

−3

10

1 −4

Fig. 4. The modi ed Hanning window.

where 0  a  0:5 is parameter controlling the correlation between subcarriers. As shown in Figure 4, a = 1=6 gives the Hanning window and a = 0 gives the rectangular window. Following the decomposition procedure introduced earlier, the entries of the matrix A are obtained with 8 > m ? i > 2; < 0; a ; m ?i=2 m;i = >  m??2  : 1;0 m?1;m?2 m;m?2 ; m ? i = 1 m?1 A MLSE receiver with a total of M 2 states can then be used for the detection of an OFDM scheme with M -level modulation PSK (QAM) scheme. If instead a feedback equalizer is used then it requires two taps. Figure 5 shows the performance of the MLSE receiver for an OFDM scheme with BPSK modulation. Such a receiver has a total of 4 states. As shown in this gure, the system performance is dependent on the correlation factor between neighbouring subcarriers. For instance the Hanning window introduces about 2 dB degradation as compared to the ideal (orthogonal) case. By properly choosing the correlation factor it is possible to design shaping waveforms for OFDM schemes without aecting their performance over the AWGN channel. IV. Conclusions

It has been shown in this paper that time shaping waveforms used with OFDM schemes is equivalent to spreading the information over dierent subcarriers. The system was equivalently represented by a discrete shaping matrix followed by a regular OFDM scheme. This representation is very suitable for digital implementation and is easily used with OFDM without aecting its IDFT/DFT operators. The system can then get the extra bene t from using pulse shaping without aecting any of its original structure. This also makes the design of time waveforms for OFDM easier and more exible. It was shown that the optimum receiver is one that takes into account the memory introduced by the time waveform. For that a MLSE receiver was derived. Such a receiver is blockwise and operates along the subcarriers. Concerning the system performance, it was shown that the average bit

10

a=0.0 a=0.05 a=0.10 a=0.167 −5

10

1

2

3

4

5

6 7 Eb/No, dB

8

9

10

11

Fig. 5. Performance of OFDM schemes with BPSK modulation and using the modi ed Hanning window over AWGN channels.

error probability over AWGN channels is dependent on the correlation between subcarrier signals. Strong correlation degrades the system performance but with moderate correlation it is possible to achieve the same performance as that of the ideal (orthogonal) case. References

[1] T. Pollet, M. V. Bladel, and M. Moeneclaey,\BER sensitivity of OFDM systems to carrier frequency oset and wiener phase noise," IEEE Trans. Comm., vol. 43, pp. 191-193, 1995. [2] G. Malmgren, \Impact of carrier frequency oset, Doppler spread and time synchronisation errors in OFDM based SFN," Globecom '96, London, U.K., November 1996. [3] M. Gudmundson and P.-O. Anderson, \Adjacent channel interference in an OFDM system," VTC '96, May 1996. [4] G. Malmgren, Single Frequency Broadcasting Networks, PhD. Thesis, TRITA-S3-RST-9701, Royal Intitute of Technology, Sweden, May 1997. [5] P. K. Remvik and N. Holte, \Carrier frequency oset robustness for OFDM systems with dierent pulse shaping lters," Globecom '97, Pheonix, Arizona, 1997. [6] K. Matheus and K.-D. Kammeyer, \Optimum design of multicarrier systems with soft impulse shaping including equalization in time or frequency," Globecom '97, Pheonix, Arizona, November 1997. [7] K. Kazel, \Performance of CDMA/OFDM for mobile communication systems," ICUPC '93, pp. 975-979, Ottawa, Canada, 1993. [8] N. Yee, J. Linnartz, and G. Fettweis, \Multi-carrier CDMA in indoor wireless radio networks," PIMRC '93, pp. 109-113, Yokohama, Japan, 1993.