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4404 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 7, NO. 11, NOVEMBER 2008

Generalized BICM for Block Fading ChannelsSantosh Nagaraj, Member, IEEE

AbstractIn this paper, we propose a novel form of bitinterleaved coded modulation (BICM) that we call generalizedBICM (GBICM) for block fading channels. The achievablediversity order on block fading channels is limited by the channelitself and not by the code used. Conventional BICM attempts tomaximize the Hamming free distance of the underlying convolu-tional code. Although this maximizes diversity on independentlyfading channels with ideal interleaving, its effect on block fadingchannels is not clear. In this paper, we analyze code performanceon block fading channels and seek to optimize it by suitablymodifying the BICM paradigm. The resulting GBICM system isjust as simple and as flexible as BICM is when coding for differentmodulations and different kinds of channels. In fact, the samedecoder can be used to decode both BICM and GBICM. We showcode design principles and analysis techniques with examples.Simulation results supporting the arguments are shown.

Index TermsBlock fading channels, bit interleaved codedmodulation, multivariate transfer function.

I. INTRODUCTION

BANDWIDTH efficient error control coding techniquesfor communication systems have been investigated byseveral authors (see [1]). Trellis coded modulation (TCM) wasfirst proposed by Ungerboeck [2], [3] for additive white Gaus-sian noise (AWGN) channels. TCM involves joint coding-modulation wherein modulation is an inherent part of the codeitself. Bit interleaved coded modulation (BICM) [4][10] isa well known technique for bandwidth efficient coding onfading channels. BICM is today used in several systems (IEEE802.11g, for example) to provide coding and diversity gains athigh spectral efficiencies. BICM is also extremely flexible inthe sense that encoding and decoding operations are indepen-dent of the modulation used. BICM is ideal for independentlyfading Rayleigh channels with perfect interleaving so that eachcodebit in an error event experiences an independently fadingchannel.

BICM separates the aspects of coding and modulation,thereby allowing concatenation of any convolutional code toany modulation. BICM is preferred on fading channels for thereason that it achieves a higher order of diversity than othertechniques of equal decoding complexity [5]. For techniqueswith joint coding-modulation, diversity order is equal to theminimum number of differing symbols between two distinctcode sequences and is always smaller than that achieved byBICM. With BICM, diversity order is equal to the minimumnumber of differing bits (Hamming free distance) between twodistinct code sequences.

Manuscript received August 20, 2007; revised October 24, 2007 and De-cember 29, 2007; accepted January 6, 2008. The associate editor coordinatingthe review of this paper and approving it for publication was H. Nguyen.

Dr. Nagaraj is with the Department of Electrical and Computer Engineering,San Diego State University (e-mail: [email protected]).

Digital Object Identifier 10.1109/T-WC.2008.070927

There are several applications today where the channelfalls under neither the independently fading channel modelor the additive white Gaussian noise (AWGN) channel model.Such channels exhibit fading and provide diversity, but thenumber of independent fading realizations within a code blockis small. Several codebits of an error event experience thesame fading gain. For example, the IEEE 802.11g standardinvolves encoding of data blocks that are confined to oneOFDM symbol. Due to correlated fading across the OFDMsubcarriers, the maximum diversity order achievable is limitedby the coherence bandwidth of the channel [11]. Performanceof codes over block fading channels have been considered in[12], [13] among others. Similar channels are also encounteredin systems with slow time-frequency hopping (like GSM,EDGE, etc. [12]) and are called quasi-static or block fadingchannels.

Block fading channels have received considerable attentionin the recent past. Information theoretic aspects and codingtheorems have been provided in [11], [12], [14], [15]. Per-formance of convolutional codes over block fading channelshave been analyzed in papers such as [16], [17]. Block fadingchannels have also been considered in a different scenario(not applicable to this paper) by other authors recently. Theyfocus on the multiantenna or the multiuser scenarios [18], [19].Over block fading channels, the diversity order achieved bythe BICM system is limited by the channel itself and not bythe convolutional code used. Increasing the code complexity(thereby, the free distance) has no direct effect on performance.This is because, the code optimization criterion for blockfading channels is not equivalent to increasing the free distance[20]. The question arises if it would be possible to modify theBICM model to optimize it for the block fading channel whilestill maintaining the flexibility that BICM offers.

In this paper, we propose a novel form of BICM that we callgeneralized BICM (GBICM) for block fading channels. Thedecoding complexity of GBICM is identical to that of BICM.In fact, the same decoder can be used to decode both BICMand GBICM. GBICM is just as flexible as BICM is whenencoding for different modulations. We show code designprinciples with examples that achieve performance gains of12 dB over BICM. We also show performance analysistechniques for GBICM. Finally, we introduce a novel measureof code performance on block fading channels to evaluateBICM and GBICM.

In Section II, we present the block fading channel modeland some preliminaries. In Section III, we present the GBICMsystem model and code design principles with examples andcomparisons with BICM. In Section IV, we present perfor-mance bounding techniques for GBICM.

1536-1276/08$25.00 c 2008 IEEE

NAGARAJ: GENERALIZED BICM FOR BLOCK FADING CHANNELS 4405

xj,k

C

X

Fig. 1. System model of a BICM encoder with a binary convolutional code C, an interleaver , and, M -ary modulator X

II. PRELIMINARIES

A. BICM System Model

We will first describe the model of a BICM system (Figure1) before proceeding to GBICM. Consider a block fadingchannel with L independent Rayleigh fading realizationshj , 1 j L over each code block. A rate rc = kc/ncconvolutional encoder C encodes a block of information bitsand the resulting codebits of the codeword c are interleavedby . Groups of m interleaved codebits are then modulatedonto M -ary signal constellations x X (either M -PSK orM -QAM), where M = 2m. The average transmitted energyper symbol is S, so that E[|x|2] = S. We will use a doubleindex notation (j, k) to emphasize the block fading nature ofthe channel. The k-th M -ary symbol transmitted on the j-thchannel is represented by xj,k and is received at the receiveras a faded and AWGN corrupted symbol yj,k = hjxj,k +j,k,where j,k are Gaussian noise samples with variance N0/2.The transmitted and received signals can be considered to bematrices represented by x = [xj,k] and y = [yj,k], respectivelyfor simplicity of notation.

B. Diversity Limit (from Knopp and Humblet [14])

Let L be the maximum diversity order afforded by thechannel. The maximum diversity order achievable by anyrate rc code is

= L(1 rc)+ 1. (1)This result follows from the Singleton Bound and is valid forany code of any complexity.

III. GENERALIZED BICM

With the description of a BICM system above, it is nowvery easy to describe a GBICM system. GBICM differs fromconventional BICM in that the interleaver is no longer a bitinterleaver, but is a b-interleaver (b-) for some integer1 < b < m. The interleaver first groups b successive bits ofthe codeword c and then interleaves the b-bit groups. That is,the b-bit groups are not broken up by the interleaver. Groups ofm/b interleaved bit groups are modulated onto M -ary signalsX . Equivalently, GBICM is simply BICM with a modifiedinterleaver structure. The b bits of any one bit group aretransmitted on the same M -ary symbol.

NOTE: The difference between GBICM and either BICMor TCM is in the fact that 1 < b < m. With b = 1, GBICMis equivalent to BICM and with b = m, GBICM is equivalentto TCM.

GBICM can be pictured in either of the following twoways: (a) In a conventional BICM system, the bit inter-leaver ensures that the correlation introduced by the M -arymodulator is completely broken. With GBICM, since b-bitgroups of successive codebits are transmitted on the same

symbol, the correlation introduced by the modulator is onlypartially broken. However, the b-interleaver ensures that thereis no modulation-introduced correlation between successiveb-bit groups. This fact differentiates GBICM from TCM,where the entire correlation introduced by the modulation ispreserved. GBICM is, therefore, a hybrid of BICM and TCM;(b) Alternatively, one can simply consider a GBICM system asyet another BICM system, but, with a different bit interleaverthan the conventional one. Note that the b-interleaver is alsoa specific case of a bit interleaver.

From a practical perspective, the 2nd picture is attractive.It shows us that GBICM can be implemented in a system thatcurrently runs BICM with only one small change the inter-leaver. Everything else remains the same. From an analysisperspective, the 1st approach is more relevant. Incorporatingthe partial correlation that exists between successive codebitsallows us to obtain tighter performance bounds than thatpossible by completely ignoring the correlation.

The receiver we use is identical to the BICM receiver in [5],but with the modified deinterleaver. We want to mention herethat it is possible to design a better receiver for GBICM byconsidering the partial correlation and obtaining metrics for b-bit groups instead of bit metrics. However, we choose to workwith the conventional BICM receiver itself in order to maintainan identical complexity with BICM. The performance loss dueto this is small. We will also assume that the receiver hasperfect channel state information.

A. Why GBICM Works Product Euclidean Distance (PED)

Let c and c be any two distinct codewords that differ inNj symbol positions along the j-th channel and the squaredEuclidean distance at the k-th position be d2j,k for 1 k Nj .The squared Euclidean distance between c and c as seen bythe receiver is S

j=1 h

2j

Njk=1 d

2j,k. We have assumed here

that dj,k correspond to distances on a unit symbol energysignal set, i.e., for S = 1. Let

Njk=1 d

2j,k = fj . Applying

the Chernoff bound [21] to the conditional pairwise errorprobability P2(c c|h), we get

P2(c c|h) exp( SN0

j=1

h2jfj), and, after averaging,

P2(c c) = Eh [P2(c c|h)]

(

1j=1 fj

)(N0S

)(2)

for Rayleigh gains hj . Among codes that achieve the samediversity order , the one with the higher product Euclideandistance (PED)

i=1(

Njk=1 d

2j,k) for all pairs of distinct

codewords achieves the lower average bit error rate Pb onthe block fading Rayleigh channel [20], [22].


nearest error inunderlined position

underlined positionnearest error in

45

1

0

25

Fig. 2. Distance for single bit error c c 1 with 16-QAM signal setsdepends on the other three bits in the label; If 0010 is transmitted, an error inthe underlined position is at least 4

5away, whereas, if 1010 is transmitted,

an error in the underlined position is only 25

away.

Minimum PED: The Euclidean distance d associated withthe nearest single bit error (c c 1) is fixed with BPSKor QPSK modulations. However, with 16-QAM modulation,it is not. The distance depends on the position i of the bit con the 16-QAM label and on the values of the other 3 bitsin the symbol. By enumerating the m2m1 possible distancesassociated with a single bit error, we can obtain the probabilitymass function (PMF) of the distance d. For the unit energyGray-mapped 16-QAM constellation used in IEEE 802.11g(see Figure 2), it is

pd(c c 1) = 34(d25) +

14(d 6

5), (3)

where, () is the Kronecker delta function.For GBICM, single bit group errors c c q are more

relevant than single bit errors. There are 2b1 kinds of singlebit group errors. For example, with b = 2, the single bit grouperrors can be (c c 01), or, (c c 11), or, (c c 10)for any bit group c. Euclidean distance for any bit group errorq depends on the position of the bit group as well as thevalues of the m/b 1 other bit groups in the symbol. By anenumeration of the bit group errors, it is possible to obtain thePMFs of the distances.The distance PMFs for Gray-mapped16-QAM are:

pd(c c 01) = 34(d25) +

14(d 6

5)

pd(c c 11) = (d 45)

pd(c c 10) = 34(d25) +

14(d 6

5). (4)

We define the minimum PED (MPED) of an error event as

MPED(c c) = mindj,k

j=1

(Njk=1

d2j,k). (5)

Fig. 3. Shortest error event (in bold) with the 2-state convolutional code(generator polynomials (1, 3) in octal) spans two trellis stages.

Let w be the Hamming weight of an error event. Sup-pose that wj of these bits occur over the j-th channel for1 j so that w = 1 wj . The MPED for BICM issimply d2min

1 wj , where, dmin is the minimum distance

of the signal constellation X . With GBICM, let wj,q be thenumber of bit-groups of the q-th kind on the j-th channel for1 q 2b 1 so that wj =

q wj,qf(q) where f(q) is the

Hamming weight of the b-bit error label q. Now, if dmin,q isthe minimum Euclidean distance between any two constella-tion points differing by a Hamming weight of f(q), MPEDfor GBICM is

j=1

q wj,qd

2min,q . If dmin,q >

f(q)dmin

for the considered signal constellation and labeling map, theMPED of GBICM exceeds that of BICM. Consequently,GBICM has a lower bit error probability than BICM.

Intuitively, the constellation labeling must be such thatneighboring points differ in only one bit-group for goodGBICM performance. Further, the requirement that dmin,q

f(q)dmin necessitates that f(q) = 1 for all immediateneighbors of any constellation point. Only Gray mappingsatisfies these requirements. It appears that Gray-maps areideal for use in GBICM due to their ability to provideincreased dmin,q with increasing f(q) for commonly usedsignal constellations like QPSK, 8-QAM, 16-QAM, and 64-QAM.

Example 1: Consider a channel with L = 2 independentfading realizations h1 and h2 inside every code block. Alsoconsider BICM and GBICM generated by the 2-state con-volutional code with generator polynomials (1, 3) in octal.Although the free distance of this code is 3, the diversityorder BICM achieves is only = 2 since L itself is 2. Themodulation used is 16-QAM with b = 2 for the GBICMsystem. Assume that the all-zero codeword is transmitted. Theshortest error event that can occur with the 2-state code isshown in Figure 3. This error event corresponds to the decodedcodeword 1110. We will next evaluate the PED for this errorevent. As will be shown later, the corresponding GBICM codealso achieves = 2.

BICM: The Hamming weight of the error event is 3 andthe two codewords will differ in three symbols due to bitinterleaving. Since the code achieves diversity = 2, at leastone symbol must be transmitted on each channel hj for j =1, 2. So, N1 = 2 and N2 = 1, or, N1 = 1 and N2 = 2. In


11 12 13 14 15 16 17 18 19 20104

103

102

101

Eb/N0 (dB)

P b2state BICM, L=22state GBICM, L=2

Fig. 4. Performance of 2-state BICM and GBICM on L = 2 channels.

either case, the MPED between c and c would be

MPEDB(000 111) = min[(d2(c c 1)+

d2(c c 1)) d2(c c 1)]=

3225

, for 16-QAM. (6)

GBICM: The b-interleaver (with b = 2) ensures that the two2-bit groups of the error event are transmitted on only twodifferent symbols. Again, since the code achieves diversity = 2, at least one of the symbols must be transmitted oneach channel hj for j = 1, 2. So, N1 = 1 and N2 = 1. TheMPED between c and c in that case would be

MPEDG(00 00 11 10) = min [d2(c c 11)d2(c c 10)]

=6425

, (7)

for the Gray-mapped 16-QAM signal set. Gray mapping alsomaximizes the MPED for the GBICM system. The aboveanalysis shows a 10 log10(MPEDG/MPEDB)

1 = 1.5 dB

SNR gain over BICM with 2-state codes and 16-QAM setson L = 2 channels. Although we considered only the shortesterror event in our analysis above, we expect similar gains forother error events as well. We have simulated the performancesof BICM and GBICM on L = 2 channels and the results areshown in Figure 4. GBICM provides about 1 dB gain overBICM. This has been supported by our simulations.

Example 2: We have also simulated BICM and GBICM onL = 4 and L = 5 channels with the industry standard, rc =0.5, convolutional code of complexity 64 states. The resultsare shown in Figure 5 for 16-QAM signal sets and b = 2.GBICM provides about 1.52.0 dB improvement over BICM.Note that the gains are higher at higher code complexities.

IV. PERFORMANCE BOUNDS FOR GBICM

GBICM is inherently a non-linear code since there is partialmodulation-induced correlation between successive codebits.For any two distinct codewords c and c, the pairwise errorprobability P2(c c) depends on (c, c) and not on c c.

10 12 14 16 18 20105

104

103

102

101

Eb/N0 (dB)

P b

64state BICM, L=464state GBICM, L=464state BICM, L=564state GBICM, L=5

Fig. 5. Performance of 64-state BICM and GBICM on L = 4 and L = 5channels.

TABLE IDIVERSITY ORDER () FOR TCM, GBICM, AND BICM WITH 16-QAM

SETS.

Diversity Order Trellis Complexity

TCM GBICM (b = 2) BICM

2 states 1 2 3

4 states 2 3 5

8 states 2 4 6

16 states 3 5 7

The analysis can be simplified by linearizing this code with theintroduction of an additional parameter u such that u = 0 oru = 1 for each symbol randomly. If u = 1, the M -ary labelingfor the symbol is complemented. The receiver is expected toknow the sequence u so that decoding is not affected.

A. Transfer Function

The generalized transfer function (TF) [16], [21] forGBICM is different from that for BICM in that, the distanceproperties are now required separately for each of the 2b 1kinds of bit group errors (see Section III-A).

Consider the rate kcnc code as a ratekc/bnc/b

code in termsof the number of bit groups. To simplify the analysis, wewill assume that nc/b = L. That is, each bit group generatedby the encoder during one trellis transition is transmitted ona different channel. Note that in many cases, this can beachieved by considering an equivalent code instead of theoriginal code. Each of the nc/b bit groups corresponding to atrellis transition is represented by the term Dj,q , where index jrepresents the channel over which the bit group is transmittedand index q can take values from 1, . . . , 2b 1 to denote theparticular kind of non-zero bit group error. For example, arate 1/2 code on an L = 2 channel can be expressed as anequivalent rate 2/4 code.

Example 3: Consider the GBICM code generated in Ex-ample 1 for 16-QAM with L = 2, b = 2 and rc = 1/2. Sincewe need nc/b = L, we get nc = 4, i.e., the code shouldbe considered as an equivalent rc = 2/4 code with generator


00/00 00 + 10/11 01

01/01 11 + 11/10 10

00/01 00 + 10/10 01

01/00 11 + 11/11 10

0 1 0D1,1 + ND1,2D2,1

ND1,3D2,1

N2D1,2D2,2 + ND1,1D2,3

ND2,3 + N2D1,3D2,2

Fig. 6. Trellis and state transition diagram for GBICM with b = 2 and rate2/4, 2-state convolutional code with generator polynomials (1, 3, 4, 14) inoctal; parameter N denotes the presence of an information bit 1 (see [21]).

polynomials (1, 3, 4, 14) in octal. This means that the basicGBICM trellis stage corresponds four output bits or two 2-bitgroups (see Figure 6). The first bit group is transmitted overthe j = 1 channel and the second over the j = 2 channel, asis the case with interleaving over block fading channels [15].The GBICM transfer function can be obtained by solving thestate diagram in Figure 6.

Diversity Order: The maximum diversity order that can beachieved by GBICM of a given b is the minimum number ofbit groups an error event spans and is always less than that ofBICM. So, GBICM is not suitable for use on independentlyfading channels with ideal interleaving. However, GBICMmay be an excellent choice for indoor wireless LANs wherethe achievable diversity orders are limited by the channelcoherence bandwidth. Table I lists the maximum diversityorders of BICM, GBICM and TCM for rc = 1/2 and 16-QAM.

Let w = (w1,1, . . . , wL,2b1) be the weight profile ofan error event with wj,q bit groups of the q-th kind onthe j-th fading channel. An expansion of the generalizedtransfer function gives the weight enumeration factors a(w)that correspond to the total number of information bit errorsassociated with error events of profile w as

T (D1,1, . . . , DL,2b1, N)N

N=1

=w

a(w)L

j=1

2b1q=1

Dwj,qj,q

(8)The union bound on the bit error probability of the convolu-tional code conditioned on the channel realization h is givenby

Pb(h) bkc

w

a(w)P2(w|h), (9)

where, P2(w|h) is the conditional pairwise error probabilityof two code sequences c and c that are separated by a weightprofile of w. Consequently, the average bit error probability

can be upper bounded as

Pb

min (0.5, Pb(h)) p(h)dh, (10)

where, p(h) is the probability density function of the channelgains. The min (0.5, Pb(h)) term is necessary to make thebound tight as the union bound may not converge if h valuesare small [16]. Owing to the presence of the min operatorinside the integral, Pb can only be obtained numerically. Aclosed form solution does not exist. We will next solve forP2(w|h).

B. Expurgated Bound

Let c and c be any two distinct codewords that differ inwj,q bit groups of the q-th kind along the j-th channel sothat cj,k = cj,k q for 1 k wj,q . Since GBICM with aGray mapped signal set can be considered to be BICM with aspecific kind of interleaver, it satisfies all the conditions listedin [5] for expurgation of the union bound

Pb,EX(w|h) = Eu,i[2

L1 2b1

q=1 wj,q(bf(q))

x

P (x z|h, i, u)], (11)

where, the summation is over all signal sequences x with bitgroup cj,k in the i-th bit group position of the (j, k)th symboland z is the sequence nearest to x with cj,k in the i-th bitgroup position of the (j, k)th symbol. Here, f(q) denotes thenumber of error bits (ones) in the binary label of q. Equation(11) can be greatly simplified for GBICM on block fadingchannels.

Consider the term (12)

P (x z|h, i, u)

= Q

SL

j=1

2b1q=1

wj,qk=1 h

2jd

2j,q,k

2N0

(12)

under ML decoding with perfect channel state information atthe receiver. Here dj,q,k is the Euclidean distance between xand z at the k-th symbol of the q-th kind of bit group error onthe j-th channel. Note that dj,q,k > 0 for all j, q, k consideredin the sum and can take only a few different values dependingon i and u. With some algebra, (11) can be simplified as [7]

Pb,EX(w|h) = Edj,q,k

Q

S

j,q,k h2jd

2j,q,k

2N0

.(13)

Now, dj,q,k is not fixed for a given q (see Section III-A).


Using the fact that dj,q,k dmin,q , (12) can be written as

P (x z|h) Q

SL

j=1 h2j

2b1q=1 wj,qd

2min,q

2N0

exp

SLj=1 h2j 2b1q=1 wj,qd2min,q

4N0

exp

S

Lj=1

2b1q=1

wj,qk=1 h

2jd

2j,q,k

4N0

,(14)

by using the relation Q(x + y) Q(x)ey/2. By using(14) in (13), we get

Pb,EX(w|h) Q

SL

j=1 h2j

2b1q=1 wj,qd

2min,q

2N0

exp


4N0

Edj,q,k

[exp

(S

j,q,k h

2jd

2j,q,k

4N0

)]

Q

SL

j=1 h2j

2b1q=1 wj,qd

2min,q

2N0

exp


4N0

L

j=1

2b1q=1

wj,qq (Sh2j4N0

), (15)

where, q() is the characteristic function of pd(c c q)from equation (4).

Note: The above analysis has to be modified if the non-nearest neighbors cannot be expurgated from the union boundfor the considered signal constellation. In that case, the BICMUnion Bound (BICM-UB) from [5] should be the startingpoint for the analysis. We, however, ignore such a scenario inthis paper as it was argued in Section III that GBICM worksbest with Gray mapping which lends itself to expurgation ofnon-nearest neighbors.

C. Transfer Function Bound

Unlike ideal interleaving channels considered in [5], blockfading channels allow us to obtain transfer function basedbounds and approximations to the conditional error probabil-ity. Using the upper bound Q(y) 12ey/2 [21] in (12),

P (x z|h) 12

exp

S

Lj=1

2b1q=1 wj,qh

2jd

2min,q

4N0

,(16)

11 12 13 14 15 16 17 18 19 20104

103

102

101

Eb/N0 (dB)

P b

Pb,BB upper boundPb,TF upper boundPb,EX upper boundSimulation

Fig. 7. Simulation and analysis results for 2-state GBICM and L = 2channels with 16-QAM signal sets and b = 2.

which is independent of i, u, etc. Equation (16) can be usedto bound the conditional error probability

Pb,TF (h) b2kc T (D1,1 . . . , N)

N

N=1,Dj,q=e

Sh2

jd2min,q

4N0

(17)and Pb can be obtained by limiting and averaging numericallyas shown in (10).

Figure 7 shows the performance of 2-state GBICM onL = 2 channels of Example 1, along with the numerically av-eraged (using (10)) transfer function bound Pb,TF and expur-gated bound Pb,EX . Also graphed is the Bhattacharya boundPb,BB from [12] which is obtained by ignoring the partialmodulation-induced correlation between successive codebits.

V. CONCLUSIONS

We have shown in this paper that when coding for blockfading channels, BICM in its conventional form may notprovide the best performance. We proposed a novel form ofcoding called GBICM, which provided 12 dB improvementin performance over BICM. A key aspect here is that im-plementing GBICM in a system that is already implementingBICM is very simple. Only the interleaver has to be modified.The same decoder can be used. We feel that GBICM is feasibleto be implemented in a practical system if the transmitter wereto know the kind of channel the receiver faces. Reasonableperformance improvements can be expected with GBICMthen. We have shown that the right measure of performanceon block fading channels is not the Hamming free distance,but, the product Euclidean distance.

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Santosh Nagaraj received his B. Tech. degree inElectrical Engineering from the Indian Institute ofTechnology, Madras, India, in 2000, and his PhDdegree from Purdue University, West Lafayette, INin 2005.

Since 2005, he has been on the faculty of SanDiego State University, San Diego, CA, where iscurrently an Assistant Professor of Electrical andComputer Engineering. His research interests areprimarily in the areas of communication systemdesign, broadband modulation and demodulation

techniques, and signal processing. He is currently working on adaptivemodulation and coding (AMC) for orthogonal frequency division multiplexing(OFDM) systems with feedback and on design of efficient coding techniquesfor block fading channels. His other research interests include multiple inputmultiple output (MIMO) systems and power control.

Dr. Nagaraj has served as a reviewer for several prestigious journals such asIEEE TRANSACTIONS ON COMMUNICATIONS, IEEE TRANSACTIONS ONWIRELESS COMMUNICATIONS, etc. He has served as a reviewer for severalprofessional conferences as well. He has also served on the technical programcommittee for IEEE WIRELESS TELECOMMUNICATIONS SYMPOSIUM.

In addition to his active research and professional activities, Dr. Nagarajhas been a dedicated teacher committed to excellence in undergraduate andgraduate teaching. In 2007, he received the Outstanding Faculty Award fromSan Diego State Universitys College of Engineering.

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