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IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 52, NO. 6, JUNE 2004 1711

Interleaved Orthogonal Frequency DivisionMultiplexing (IOFDM) System

V. G. S. Prasad and K. V. S. Hari , Senior Member, IEEE

Abstract—In orthogonal frequency division multiplexing(OFDM) systems, for every block of data samples, an overheadof samples of cyclic prefix (CP) or zero padding (ZP) is addedto combat frequency selective channels. The code rate, which isdefined as the ratio

( + )

, is a measure of the efficiencyof transmitting user information. In this paper, a new systemis proposed to increase the code rate without increasing thenumber of subcarriers and without increasing the bandwidth. Theproposed system considers appending the zeros (ZP) once forevery blocks of data samples, which would increase the coderate to ( + ) . It is assumed that the channel is notvarying over the transmission of consecutive data blocks. Inorder to recover the data blocks in a computationally efficient

manner, an interleaving scheme is proposed, and the proposedsystem is called the interleaved OFDM (IOFDM) system. Variousissues such as computational complexity, peak-to-average powerratio (PAPR), and the effect of synchronization errors on theperformance of the IOFDM system are also presented. Based ona numerical simulation study, the average bit-error-rate (BER)performance of the IOFDM system is shown to beveryclose to thatof the OFDM system for a moderate increase in computationalcomplexity and delay.

Index Terms—Code rate, cyclic prefix, interblock interference,interleaving, OFDM, zero padding.

I. INTRODUCTION

I N recent years, orthogonal frequency division multiplexing(OFDM) has been adopted as a standard for various applica-

tions like digital audio/video broadcasting (DAB/DVB), wire-

less LANs, etc. Conventional OFDM systems transform infor-

mation symbol blocks and then insert redundancy in the form

of either cyclic prefix (CP) or zero padding (ZP) [ 1]–[3]. The

length of CP/ZP should be longer than the channel delay spread

to avoid interblock interference (IBI) arising due to the fre-

quency-selective nature of the channel.

Itwas proposedin [4]and[5] that ZP assures symbolrecovery

even when channel nulls occur on some subcarriers, which is

not possible with the use of CP. However, there is an increase in

receiver complexity.

The redundancy due to the CP/ZP causes reduction in thecode rate of the communication system. The loss in the code

rate may be around 10–25%. The loss is more when the finite

Manuscript received June 20, 2002; revised June 21, 2003. The associate ed-itor coordinating the review of this manuscript and approving it for publicationwas Dr. Sergios Theodoridis.

V. G. S. Prasad is with Emuzed India Pvt. Ltd., Bangalore, India.K. V. S. Hari is with the Department of Electrical Communication En-

gineering, Indian Institute of Science, Bangalore 560 012, India (e-mail:[email protected]).

Digital Object Identifier 10.1109/TSP.2004.827179

impulse response (FIR) channel order is comparable with the

number of subcarriers used in the OFDM system.

In this paper, we propose a scheme called interleaved or-

thogonal frequency division multiplexing (IOFDM), which has

a higher code rate than the conventional OFDM. IOFDM en-

hances the code rate without bandwidth expansion and without

increasing the number of subcarriers but with a moderate in-

crease in computational complexity and delay.

In the proposed IOFDM system, it is assumed that the

channel is not varying over the transmission of consecutive

data blocks. It is proposed that ZP is used once for every

transmission of consecutive data blocks, which wouldincrease the code rate. However, there is an introduction of

IBI. In order to recover the data symbols in a computationally

efficient manner, an interleaving scheme is proposed. In [6],

a scheme called vector OFDM (VOFDM) was presented to

increase the code rate. Even though the VOFDM transmitter

and the IOFDM transmitter look similar, the receiver sections

are entirely different. The IOFDM system is computationally

more efficient than the VOFDM system, as shown later.

Peak-to-average power ratio (PAPR) of the IOFDM system and

the effect of synchronization errors on the performance of the

IOFDM system are also presented.

The paper is organized as follows. In Section II, the conven-

tional OFDM system is briefly described. Section III presentsthe proposed IOFDM system. Section IV presents the perfor-

mance of theIOFDM systemusingsimulationresults. Thepaper

is concluded in Section V.

A. Notation

Underlined lowercase letters are used to represent vectors,

and boldfaced upper case letters are reserved for matrices. The

superscripts , , and stand for transpose, complex conju-

gate transpose, and inverse operation, respectively. repre-

sents the identity matrix, and represents the

discrete Fourier transform (DFT) matrix. denotes

the convolution of and . denotes the expecta-tion operator on , and represents Kronecker delta func-

tion. denotes the estimate of .

II. OFDM SYSTEM

A. OFDM Transmitter

The block diagram in Fig. 1 describes the discrete-time base-

band model of an OFDM system [7]. The bit stream is mapped

to an information symbol sequence using a modulation

scheme like binary phase shift keying (BPSK), quaternary phase

1053-587X/04$20.00 © 2004 IEEE



1712 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 52, NO. 6, JUNE 2004

Fig. 1. Block diagram of an OFDM system.

shift keying (QPSK), 16-QAM, etc. The sequence is parsed

into blocks of length

. The symbol block is then mapped to a precoded block of length through

a precoder such that

(1)

The role of is to effectively convert a frequency-selective

channel into a number of flat channels. It is assumed that the

frequency-selective channel is FIR in nature and that the upper

bound on its order is available, i.e., ,

where denotes the channel between the transmitter and the

receiver. A transmitted block

of length is formed as follows:

(2)

Setting the last elements of to zeros is known as zero

padding. The ZP will avoid IBI between the transmitted blocks.

After parallel-to-serial conversion of , the sequence

is then serially transmitted through a transmitting antenna.

B. OFDM Receiver

At the receiver, the received sequence in the presence of

noise is given by

(3)

where denotes complex additive white Gaussian noise(AWGN). It is assumed that the impulse response is con-

stant over the transmission of . Since the channel induced

IBI between the transmitted blocks is avoided due to the ZP, one

can focus at each received block

of length separately. After serial-to-parallel con-

version of , the received block is given by

(4)

where is the channel convolution matrix with the

th element being and

.

Define the -transforms:1 ,

, and . Then,

the -transform of is given by

(5)

Choosing distinct points on the complex -plane,

the precoder is designed such that

(6)

where is chosen to impose power constraint, i.e.,

. Equation (6) enables

proper symbol recovery at the receiver. The precoder can be

constructed by considering to be an FIR filter of order

with zeros being and [7]. The

polynomial can be uniquely determined by Lagrange

interpolation through the points and is obtained as

follows:

(7)

Evaluating (5) at , we get

(8)

Using (6), (8) is simplified as

(9)

Equation (9) reveals that OFDM converts the frequency-selec-

tive channel into a number of flat channels.

1c ( m ) denotes the element present in the ( m + 1 ) st row and ( k + 1 ) st

column of C , and C ( z ) denotes the Z -transform of the ( k + 1 ) st column of C .



PRASAD AND HARI: IOFDM SYSTEM 1713

In order to decode , we define

and write the set of equations given by (9) in a matrix/vector

form as follows:

(10)

where diag , and

......

. . ....

It is assumed that perfect channel state information (CSI) is

available at the receiver. Using the available CSI, we obtain the

decision vector of

length as follows:

(11)

where . After detecting from

, the estimated symbol sequence can be retrieved.

C. Improving the Code Rate

Even though OFDM simplifies equalization at the receiver,

the redundancy due to the CP/ZP causes reduction in the code

rate. The code rate of the OFDM system, which is denoted by

, is given by

(12)

One way to increase the code rate is by increasing the number of

subcarriers in the OFDM system. Let us consider the number of

subcarriers in the OFDM system to be , where . The

code rate would then be equal to , which

is greater than . Fig. 2 shows the improvement in the code

rate, which is denoted by , as a function of , for

different values of and . It is to be noted that the code rate

improvement flattens out after a moderate value of . The code

rate improvement for moderate values of is between 16 and

32%, which is significant. Since the OFDM systems are verysensitive to frequency offsets, the larger the number of subcar-

riers implies a decrease in the frequency spacing, and therefore,

frequency accuracy becomes more and more critical. In addi-

tion, the PAPR increases with the number of subcarriers.

In VOFDM,2 vectors of length are used as

the cyclic prefix once for every transmission of consecu-

tive symbol vectors of length . The code rate in the VOFDM

system, which is denoted by , is given by

(13)

2d a e stands for the smallest integer b such that b a .

Fig. 2. 0 versus P .

D. Problem Statement

Given the constraint on the number of subcarriers and thebandwidth, obtain a scheme that improves the code rate of an

OFDM system.

III. IOFDM SYSTEM

We now present the idea of processing consecutive data

blocks of length using the ZP of length . This would yield

the same code rate as given by . The proposed IOFDM

system is presented in this section.

For the sake of illustrating the principle behind the

proposed method, we will first consider the case when

. Considering two consecutive precoded blocks ,

, and zeros, we construct the transmitted block of length

as follows:

(14)

It is assumed that the channel is constant over the trans-

mission of .

At the receiver, the corresponding received block

of length in the absence

of noise is given by

(15)

where is the channel convolution matrix, where theth element is . The -transform of is given

by

(16)

Evaluating (16) at , we get

(17)




Therefore, we have equations with unknowns. We need

additional equations to retrieve the information symbols

. Considering additional

distinct points on the complex -plane, we get

(18)


and write the set of equations given by (17) and (18) in a ma-

trix/vector form as follows:

(19)

where diag ,

and

......

. . ....

. . .

......

. . ....

. . .

Solving the set of equations given by (19) is an expensive

process because we need to invert the matrix . In

addition, the choice of and, therefore, the inversionof is not straightforward to establish.

To address this issue, we propose to construct the transmitted

block of length by interleaving the elements

of and as follows:

(20)

The corresponding -transform of the received block of

length in the absence of noise is given by

(21)

Evaluating (21) at and and then

using (6), we obtain the equations needed to retrieve

as follows:

(22)

Solving the set of equations given by (22) is not an expensive

process because number of 2 2 Vandermonde matrix in-

versions are needed to retrieve

[8]. We can further reduce the computational complexity of

the receiver by preprocessing the symbol block at

the transmitter. The preprocessing results in the precoded block

being

(23)

where diag is the preprocessing matrix.

Because of preprocessing, (22) becomes

(24)

Solving the set of equations given by (24) needs two-point

inverse discrete Fourier transform (IDFT) operations. Next, we

generalize the above idea by considering consecutive symbolblocks.

A. IOFDM Transceiver

We consider consecutive symbol blocks

and then preprocess the symbol block

as follows:

(25)

where diag is the preprocessing matrix.

Now, the preprocessed block is the effective symbol block.

It is linearly mapped to a precoded block through the pre-coder such that

(26)

A transmitted block of length is formed

by interleaving the elements of and

padding zeros as follows:

(27)

After parallel-to-serial conversion of , the sequence

is then serially transmitted through a transmitting antenna. Be-cause of interleaving, we call the proposed scheme interleaved

orthogonal frequency division multiplexing (IOFDM). The

block diagram in Fig. 3 describes the discrete-time baseband

model of an IOFDM transmitter. The interleaving process at

the IOFDM transmitter for is shown in Fig. 4.

At the receiver, the received sequence is given by

(28)

where is complex AWGN. Here, it is assumed that the

channel is constant over the transmission of . Since

IBI between the transmitted blocks is avoided due to the ZP,




Fig. 3. Block diagram of an IOFDM transmitter.

Fig. 4. Interleaving process at the IOFDM transmitter for P = 3 .

one can focus at each received block of length , which

is given by

(29)

where is the channel convolutionmatrix, and

. The -transform of the re-

ceived block is given by

(30)

Let be the distinct points on the complex

-plane, where they are related to as follows:

and

(31)

Evaluating (30) at , we get and

(32)

The -transforms evaluated at are

obtained as follows:

......

.... . .

...

(33)Using (6) and (31), (32) is simplified as follows:

(34)


and write the set of equations given by (34), for a fixed , in a

matrix/vector form as follows:

(35)

where diag ,

, and

. After frequency

domain equalization and -point IDFT, we obtain the decision

vector

of length as follows:

(36)

where . After detecting

from , the estimated symbol sequence can be re-

trieved. The block diagram in Fig. 5 describes the discrete-time

baseband model of an IOFDM receiver. Fig. 6 shows the dein-

terleaving process at the IOFDM receiver for .

Remarks:

• For , the IOFDM system is equivalent to the con-

ventional OFDM system.

• In OFDM, if subcarriers are used, the precoder

would be designed using distinct points

on the complex -plane. It would also be assumed that

the channel nulls do not occur at




Fig. 5. Block diagram of an IOFDM receiver.

Fig. 6. Deinterleaving process at the IOFDM receiver for P = 3 .

, for proper data recovery. In IOFDM, the condition on

the channel nulls remains similar, but the precoder is

designed using only distinct points .

• Equation (36) reveals that IDFT operations are

required to retrieve . This

is less expensive than the matrix inversion.

• There is an increase in the processing delay as data

blocks have to be received.

B. Example of the IOFDM System

Let be the distinct points on

the complex -plane. For this choice of , the points, the matrices , , , , and the constant are

as shown in the equation at the bottom of the page, where is

the overlap-add operation matrix, which is defined as

follows:

where is the th column of . For this choice of ,

we can exploit the computational advantage of fast Fourier

transform (FFT) to implement the IOFDM system.

Remarks:

• The form of receiver considered is not robust to channel

nulls at subcarrier locations . Other solutions

(like the pseudoinverse solution [4] and [5]) can be used

for ZP-IOFDM, which are robust to channel nulls at sub-

carrier locations, but these methods may not give the com-

putational advantage. However, the benefit of code rate

improvement and PAPR improvement (discussed in a later

section) still exists with the use of IOFDM.

C. Computational Complexity

Even though IOFDM improves the code rate, there is a mod-

erate increase in computational complexity. In the conventional

OFDM system, for a carrier system, the transmitter has to

operate a general precoder matrix of size , which

could result in a computation cost of complex multi-

plications (CMs). In case of the IOFDM system, it would re-

quire CMs for precoder operation and an additional

CMs for diagonal transformation, as in (25), resulting in a total

number of CMs to be , which would be less than the

previous case. For the case when is the IDFT matrix (as in the

example above), the reduction in CMs is analogous to to savings

obtained due to splitting the -point IDFT into smaller but-

terflies.

The number of CMs required to retrieve consecutivesymbols at the receiver is

taken as the complexity measure. Let and beinteger powers

of 2, i.e., and . The number of CMs required

to find the -point DFT is .

For the choice of , we compare

the complexity of IOFDM with the complexities of OFDM and

VOFDM. In the conventional OFDM system, the number of

CMs required to evaluate is .

The number of CMs required to evaluate and to get

from as in (11) is . Therefore, the com-

plexity of the conventional OFDM system, which is denoted by

, is given by

(37)

diag

diag




In the proposed IOFDM system, the number of CMs required

to evaluate both and is

. In order to get

from as in (36), we need CMs. Thus,

the complexity of the IOFDM system, which is denoted by ,

is given by

(38)

To obtain the computational complexity of the VOFDM

system, let and be the transmitted block and the

received block, respectively. The -transform of the received

block in the absence of noise is given by

(39)

Rewriting (39) in terms of polyphase components [9], we get

(40)

In VOFDM, the symbol vector is given by

(41)

where

......

. . ....

and .

The number of CMs required to evaluate both

and is .

In order to get from as in (41), we need CMs.

Therefore, the complexity of the VOFDM system, which is

denoted by , is given by

(42)

Table I shows the comparison of the IOFDM system’s com-plexity and the VOFDM system’s complexity for .

Remarks:

• It is clear that the IOFDM system requires more com-

plexity than the OFDM system to decode data sym-

bols.

• The IOFDM system is computationally more efficient

than the VOFDM system.

D. Peak-to-Average Power Ratio

For the choice of , the dis-

crete-time baseband representation of the transmitted signal in

TABLE ICOMPUTATIONAL COMPLEXITY FOR K = 2 5 6

the conventional OFDM system, consisting of subcarriers,

is given by

(43)

Because of the superposition of many statistically independent

subcarrier signals, the OFDM signal exhibits high instantaneous

peak signal level with respect to average signal level. This high

peak-to-average nature of the OFDM signal results in harmonic

distortion, unless the transmitter’s power amplifier exhibits lin-

earity throughout the dynamic range of the signal. PAPR is used

as a measure to compare the instantaneous peak signal level

with the average signal level. PAPR of the conventional OFDM

system, which is denoted by , is defined as

(44)

where is the average symbol energy. The worst-case PAPR

that can happen in the OFDM system is

(45)

where is the symbol having largest power in the consid-

ered modulation alphabet.

In an IOFDM system that is implemented by interleavingconsecutive data blocks of length , the transmitted signal

is given by

and (46)

The PAPR of the IOFDM system, which is denoted by , is

defined as

(47)

The worst-case PAPR that can happen in the IOFDM system is

(48)

Comparing (45) and (48), we can say that both the IOFDM

system and the OFDM system, which consist of subcarriers,

have the same worst-case PAPR. Therefore, the IOFDM system

with interleaving data blocks of length has less worst-case




PAPR compared with the conventional OFDM system with

subcarriers.

E. Effect of Synchronization Errors

In this section, we present the effect of synchronization errors

on the performance of both the OFDM system and the IOFDM

system. Let be the distinct points on the complex

-plane, which are uniformly separated on the unit circle.In the conventional OFDM system, the transmitted contin-

uous-time baseband signal over an OFDM symbol period

is described by

(49)

where is the symbol duration on each subcarrier, is the

ZP duration, is the sampling period which is defined as

, and

else.

In the presence of a carrier frequency offset , the received

signal in the absence of noise is given by

(50)

where denotes the FIR channel

between the transmitter and the receiver. The received signal

is sampled at the time instances

, where denotes the normalizedtimingoffset.

The discrete-time baseband representation of the received signal

is

(51)

where denotes the normalized carrier frequency

offset, and

Denoting as the -transform of , we get

(52)

Evaluating (52) at , we get,

(53)

where

In an OFDM system consisting of subcarriers, (53) be-

comes

(54)

Assuming and then

performing zero-forcing equalization, we get

(55)where

The sequence denotes the inter carrier interference (ICI)

and has an average power of

(56)

In the proposed IOFDM system, the continuous-time base-

band representation of the transmitted signal over an

IOFDM symbol period is given by

(57)

In the presence of a carrier frequency offset and a normal-

izedtiming offset , the received signal in the absence noise

is given by

(58)

where

and

Denoting as the -transform of and then evaluating

at , we get, and

, (59), shown at the bottom of the page. After

(59)




performing zero-forcing equalization and -point IDFT, we get,

and

(60)

where we have (61), shown at the bottom of the page. The se-

quence denotes ICI and has an average power of

(62)

Since , both the IOFDM system im-

plemented by interleaving data blocks of length and the

OFDM system consisting of subcarriers are equally af-

fected by synchronization errors.

IV. SIMULATION RESULTS

In this section, we present the average bit-error-rate

(BER) performance of the IOFDM system in various

channel models. QPSK modulation technique is used tomap the bit stream into an information symbol sequence. Let

be the distinct points on the

complex -plane. It is assumed that perfect CSI is available at

the receiver.

A. BER Performance in GSM Channels

In our simulations, the channel bandwidth is assumed

to be 5 MHz. The entire bandwidth is divided into

. To make subcarriers orthogonal

to each other, the symbol duration on each subcarrier, which

is denoted by , is s. Zero padding of duration

is used to avoid IBI arising due to the frequency-selective

nature of the channel. The value of is chosen according to

the channel used in simulation. The 12-tap GSM typical urban

(TU) channel and the hilly terrain (HT) channel are used.

We choose s in the TU channel and s

in the HT channel. The channel taps are characterized by the

Jakes’ Doppler spectrum [10]. In order to gain insight into the

average BER performance, we have taken 4000 information

symbol blocks of length per trial and averaged over

25 trials. In effect, for each (signal energy per informa-

tion bit to noise power spectral density ratio) value considered,

the BER values are obtained based on 51.2 bits.

Example 1: We study the performance of the IOFDM system

in the TU channel with a Doppler spread of 20 Hz. In Fig. 7, the

performance of the IOFDM system with different values of is

presented. The performance of the IOFDM system is very close

to that of the OFDM system ( ). It is slightly worse at low

values.

Fig. 7. Performance comparison of IOFDM systems in TU channel.

Fig. 8. Performance comparison of IOFDM systems in HT channel.


in the HT channel with a Doppler spread of 20 Hz. Fig. 8 shows

the performance comparison of the IOFDM systems. As in the

previous example, the performance of the IOFDM system is

very close to that of the conventional OFDM system.

B. BER Performance in HIPERLAN/2 Channels

In the HIPERLAN/2 standard, the available channel band-

width is 20 MHz. The entire spectrum is divided into

subchannels. The symbol duration on each subcarrier is

s. A fixed duration of 0.8 s is used as ZP to avoid IBI,

(61)




Fig. 9. Performance comparison of IOFDM systems in channel model B.

Fig. 10. Performance comparison of IOFDM systems in channel model E.

i.e., s and . Jakes’ Doppler spectrum corre-

sponding to a terminal speed of 3 m/s is assumed for all taps in

the channel. To gain insights into the average BER performance,

we have taken 8000 information symbol blocks, of length

, per trial and averaged over 25 trials. In effect, for each

value considered, the BER values are obtained based on

25.6 bits. Channel model B and channel model E are con-sidered in our simulations.


in channel model B. Fig. 9 shows the performance comparison

of IOFDM systems in channel model B.

Example 4: We consider the case of using IOFDM for chan-

nels that are longer than the ZP used in an OFDM system. In

channel model E, the minimum length of the ZP needed to avoid

IBI is . The HIPERLAN/2 standard uses in gen-

eral. Because of the reduced ZP, the performance of the con-

ventional OFDM system is poor. To avoid this degradation, we

use the complete ZP, i.e., , and interleave more than two

precoded blocks, i.e., . Fig. 10 shows the performance

Fig. 11. Effect of normalized frequency offset of 1 = 0 : 0 5 on theperformance of various systems.

comparison of IOFDM systems in channel model E. As shown

in Fig. 10, the IOFDM system with and perform

better than the conventional OFDM system. This demonstrates

that for channels having longer lengths compared with the ZP,

we can improve the performance as well as increase the code

rate by adopting IOFDM.

The simulation results demonstrate that the IOFDM system

hasthe potentialto be an effective systemfor improvingthe code

rate of an OFDM system without loss of performance. Fig. 11

shows the effect of normalized frequency offset of on

the performance of various systems in the TU channel.

V. CONCLUSION

A new OFDM system based on interleaving the elements of

consecutive precoded blocks has been presented. The pro-

posed IOFDM system promises higher code rate compared with

the conventional OFDM system without bandwidth expansion,

without increasing number of subcarriers, and with moderate

increase in computational complexity. The IOFDM system im-

plemented by interleaving data blocks of length has less

worst-case PAPR compared with the OFDM system consisting

subcarriers. Both systems are equally affected by synchro-

nization errors. Simulation results verify that the performance

of the IOFDM system in terms of BER is very close to that of

the conventional OFDM system.

ACKNOWLEDGMENT

The authors would like to thank the reviewers for their valu-

able comments, which have helped the technical content of the

paper significantly.

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V. G. S. Prasad was born in Andhra Pradesh, India,in 1977. He received the B.Tech. degree from theNational Institute of Technology, Warangal, India,in 2000 and the M.E. degree from the Departmentof Electrical Communication Engineering, IndianInstitute of Science, Bangalore, India, in 2002.

Since March 2002, he has been a DSP engineer atEmuzed India Pvt. Ltd., Bangalore. His research in-terests include signal processing for communicationsystems and space-time and space-frequency codedtechniques for frequency selective channels.

K. V. S. Hari (SM’97) received the B.E. degreefrom Osmania University, Hyderabad, India, in1983, the M.Tech. degree from the Indian Instituteof Technology, New Delhi, India (IIT Delhi), in1985, and the Ph.D. degree from the University of California San Diego, La Jolla, in 1990.

He has been an Associate Professor with the De-partment of Electrical Communication Engineering,Indian Institute of Science (IISc), Bangalore, India,

since February 1998. His research interests arein statistical signal processing. He has worked onspace-time signal processing algorithms for direction-of-arrival estimation,acoustic signal separation using microphone arrays, and MIMO wirelesscommunication systems. He has also worked on MIMO wireless channel mea-surements and modeling and is the co-author of the IEEE 802.16 standard onwireless channel models for fixed broadband wireless communication systems.He was a visiting faculty at Helsinki University of Technology, Espoo, Finland,from May to July 2002; Stanford University, Stanford, CA, from September1999 to December 2000; and the Royal Institute of Technology, Stockholm,Sweden, from July to September 1995. He was an Assistant Professor withthe Department of Electrical and Computer Engineering, IISc, from February1992 to January 1998 and a scientist at Osmania University, from December1990 to January 1991. He was also with the Defence Electronics Research Lab,Hyderabad, from December 1985 to July 1987.

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