EXTENDING OFDM SYMBOLS TO REDUCE POWER

CONSUMPTION

Andr´e B. J. Kokkeler and Gerard J. M. Smit

Department of Computer Science, Electrical Engineering and Mathematics, University of Twente,

PO Box 217, 7500 AE Enschede, The Netherlands

Keywords:

Correlation, Differential Phase Shift Keying, Fourier Transforms, Frequency Division Multiplexing, Modula-

tion.

Abstract:

Existing communication standards have limited capabilities to adapt to low SNR environments or to exploit

low data rate requirements in a power efﬁcient way. Existing techniques like e.g. control coding do not

reduce the computational load when reducing data rates. In this paper, we introduce differential Extended

Symbol OFDM (differential ES-OFDM) which is based on the transmission of symbols that are extended

in time. This way it can operate at low SNR. Using differential BPSK modulation, approximately 2.1 dB

SNR improvement per doubling of the symbol length (halving the bitrate) is obtained. The sensitivity to

frequency offsets of differential ES-OFDM is basically independent of symbol extension. Extending symbols

reduces the computational load on the radio modem within the transmitter which is essential to reduce overall

power consumption. The differential ES-OFDM receiver architecture also offers opportunities to reduce power

consumption.

1 INTRODUCTION

A general trend in modern society is an increasing

emphasis on considerate use of resources, a trend

which can also be observed within Wireless Com-

munications. Besides the reduction of direct energy

consumption, one of the aims is to reduce the polu-

tion of the EM spectrum as much as possible to not

disturbe other users and to minimize human exposure

to EM radiation. Considerate use of resources is ap-

plicable to both infrastructure of wireless communi-

cations systems (e.g. basestations) as well as hand-

held devices. For handhelds, an additional incentive

to reduce power consumption is increased operational

time. For both mobile phones as well as laptops,

backlight and the wireless interface use most of the

energy, see (Carroll and Heiser, 2010) and (Mahesri

and Vardhan, 2005) respectively. Backlight power

consumption is generally counteracted with agressive

backlight dimming while techniques to reduce power

consumption within the wireless interface are less ob-

vious. According to (Carroll and Heiser, 2010) and

(Perrucci et al., 2011), to save power within handheld

devices and to reduce EM radiation, one should con-

centrate on the wireless radio interface of transmit-

ters. This interface can be separated into the Power

Ampliﬁer (PA) and the radio modem which runs the

transceiver algorithms (physical layer) and Medium

Access Control protocols (MAC layer). To reduce

power consumption in a wireless radio transmitter in-

terface, the power consumption in the PA and the

power consumed by the radio modem should be re-

duced evenly (Rantala et al., 2009).

In OFDM systems, given a speciﬁc modulation

scheme (e.g. 16-QAM), minimum Signal-to-Noise-

Ratio (SNR) constraints have to be satisﬁed at the re-

ceiver to achieve acceptable Bit Error Rates (BERs).

In case data rate requirements are reduced, lower-

ing the modulation scheme leads to lower power con-

sumption and lower radiated power levels. Once ar-

rived at the lowest modulation level (BPSK), other

techniques have to be used to reduce power levels.

All options mentioned in section 2 result in more

complexity at the transmitter and/or receiver thus

leading to lower data rates against higher power con-

sumption. This conﬂicts with our conclusion that,

for effective power reduction, both the power trans-

mitted (consumed in the PA) and the power con-

sumed by the radio modems should be reduced. In

this paper we propose an OFDM technique which

enables the reduction of radiated power levels lead-

ing to lower data rates and lower power consump-

257

B. J. Kokkeler A. and J. M. Smit G..

EXTENDING OFDM SYMBOLS TO REDUCE POWER CONSUMPTION.

DOI: 10.5220/0003950002570262

In Proceedings of the 1st International Conference on Smart Grids and Green IT Systems (SMARTGREENS-2012), pages 257-262

ISBN: 978-989-8565-09-9

 2012 SCITEPRESS (Science and Technology Publications, Lda.)

tion. It is based on differential encoding of data and

the extension of symbols in time and we will refer

to it as differential Extended Symbol OFDM (differ-

ential ES-OFDM). After describing this technique in

section 3, a BER analysis is presented in section 4.

Using BPSK modulation, theoretical BER curves for

AWGN channels are presented. Estimated and simu-

lation results for both AWGN channels and frequency

selective Rayleigh fading channels are presented as

well. In section 5, the effects of frequency offsets are

studied.

2 RELATED WORK

In (Maeda et al., 2002) and (Gaffneyet al., 2005), rep-

etition of symbols in time, allowing less power to be

transmitted, is analyzed. By means of Maximum Ra-

tio Combining, multiple replicas of symbols are used

to minimize the BER. In (Medina and Kobayashi,

2000), repetition of data in the frequency domain is

elaborated. Another option to facilitate lower trans-

mit power is to add error-control coding (Haykin,

2001). Besides that, the efﬁciency of PA’s can be in-

creased by reducing the Peak-to-Average-Power Ra-

tio (PAPR) of the signal to be transmitted as is done

in e.g. (Harada et al., 2007).

3 DIFFERENTIAL ES-OFDM

A coherent OFDM receiver has to be synchronized

to the transmitter in time, frequency and phase. Es-

pecially phase synchronization is costly to obtain.

Differential modulation disregards the phase infor-

mation at the expense of SNR performance. Both

frequency- and time differential demodulation detec-

tion (FDDD and TDDD respectively) can be used in

an OFDM system (Lott, 1999). FDDD and TDDD

are compared in (Tan and Beaulieu, 2007a) and it

is concluded that FDDD is prefered if the normal-

ized doppler spread dominates the normalized delay

spread. When symbols are extended in time which we

will introduce later on, the normalized doppler spread

increases and the normalized delay spread decreases

(Tan and Beaulieu, 2007a). This makes FDDD more

suited when using time extended symbols.

In Fig. 1, we present a base-band equivalent

model of a differential ES-OFDM system.

At the transmitter, a source produces data vector

which consists of N −1 complex values (indicated

as S

, f = 1, 2, ..., N −1) where each value is a con-

stellation point from a chosen modulation scheme.

Enc

DiffS

ES,L

Diff Dec

IDFT

Channel

I −1

DFT

Figure 1: Base-band equivalent model of differential ES-

OFDM.

We require that |S

| is constant for all f. S

is used to

modulate carriers S by means of differential encoding

= S

f−1

·S

, f = 1, 2, ..., N −1 (1)

= 1 (2)

S is transformed into the time domain through the

IDFT giving s.

√

N−1

∑

f=0

2πft

, t = 0, 1, ..., N −1 (3)

I copies of s are concatenated giving the extended

symbol s

= s

MOD(t,N)

, t = 0, 1, ..., IN −1 (4)

where MOD(, N) indicates the modulo N operator.

The last L samples of s (L ≤ N) act as a cyclic preﬁx

completing s

ES,L

. The values of this extended sym-

bol are shifted out serially and transmitted through the

channel. We assume an additive white Gaussian noise

(AWGN) channel adding n to s

ES,L

resulting in r

ES,L

= s

ES,L

+ n

, t = 0, 1, ..., IN −1 (5)

In the receiver, the ﬁrst step is to remove the cyclic

preﬁx. The resulting extended symbol is r

. At the

receiver, we identify I blocks, where each block con-

tains N samples

t,i

= r

t+iN)

t = 0, 1, ..., N −1 and i = 0, 1, ..., I −1

(6)

Each individual block is converted into the frequency

domain. Since the DFT is a linear operation, the sig-

nal and noise parts can be separated.

f,i

= DFT(r

t,i

) = S

+ N

f,i

f = 0, 1, ..., N −1 and i = 0, 1, ..., I −1

(7)

Each individual block is differentially decoded.

This operation is deﬁned as

f,i

= R

f,i

·R

∗

f−1,i

, f = 1, 2, ..., N −1 (8)

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258

where

∗

indicates the complex conjugate. From the I

blocks of data, corresponding values are averaged:

I−1

∑

i=0

f,i

(9)

Neglecting the noise contributions, assuming channel

gain equals 1 and using (1) to (9), we see that the

output of the receiver equals

= |S

f−1

= S

(10)

which is the transmitter input.

To asses the computational complexity of ES-

OFDM, we note that, at the transmitter, the rate at

which IDFTs are executed is reduced with a factor

I. The same holds for the differential encoding. We

conclude that the average power consumption at the

transmitter is lowered. At the receiver however, the

number of DFTs and differential decoding operations

is independent of I. Compared to ’normal’ OFDM

(I=1), the rate of operations at the ES-OFDM receiver

is slightly increased because of the summation of I

decoded symbols leading to a slightly higher power

consumption at the receiver.

4 BIT ERROR RATES FOR

DIFFERENTIAL ES-OFDM

SNR performances of differentially modulated car-

riers in AWGN and fading environments have been

studied in e.g. (Miller and Lee, 1998). SNR perfor-

mance of differential OFDM systems has been stud-

ied in (Tan and Beaulieu, 2007a). However, in these

analyses, the symbol length is ﬁxed and the results

are not applicable in case of extended symbols. Our

approach is based on the observation that, although it

is common to describe a differential OFDM receiver

as an FFT followed by a crosscorrelation, the order

of these functions can be exchanged. Restricting our-

selves to BPSK modulation, this allows us to use the

results presented in (Simon and Divsalar, 1992). In

that paper, a continuous time domain analysis of a

DPSK demodulator in case of an AWGN channel is

presented where a bandpass ﬁltered signal is corre-

lated with a delayed version of the same signal. In

Appendix A, we rewrite (9) in such way that, for each

carrier f, R

is the result of the (cross-)correlation of

two modulated carriers. Because of the orthogonality

of the carriers, a differential ES-OFDM receiver can

be considered as N −1 parallel DPSK demodulators.

In the expression presented in (Simon and Di-

vsalar, 1992) relating SNR to BER, two parameters

are important. First, the energy per bit over noise

SNR (dB)

BER

I = 16

I = 4

I = 1

Theory

Simulations

Estimates

-10

-8 -6 -4

-2

4 6

−3

−2

−1

Figure 2: BERs for differential ES-OFDM for an AWGN

channel.

spectral density ratio (

) and second the product

of the bandwidth of the bandpass ﬁlter and the in-

tegration time (FT). For differential ES-OFDM us-

ing BPSK, the energy per bit over noise spectral den-

sity ratio equals SNR·I and the bandwidth-integration

time product equals ∆f ·

N∆ f

= I, where ∆f is the

inter-carrier spacing. The ﬁnal result of (Simon and

Divsalar, 1992), in case of differential ES-OFDM us-

ing BPSK and an AWGN channel, becomes

BER =

−(I·SNR)

I−1

∑

i=0

(I ·SNR)

I−1

∑

j=i

j+I−1

j−i

(11)

k!(n−k)!

(12)

This expression can be used to determine the theo-

retical BER for differential ES-OFDM in case of an

AWGN channel. A simulation model has been imple-

mented as well for a system with 64 carriers (N = 64),

a cyclic preﬁx length of 8 (L = 8), I = 1, 4 and 16.

The results are presented in Fig. 2 and are labeled

with ’Theory’ and ’Simulations’. We see that theory

and simulation correspond and BER performance in-

creases for increasing I.

To explain the SNR improvement as a function of

I, (11) does not give much insight except for I = 1 in

which case it reduces to the standard relation between

SNR and BER for AWGN channels (Haykin, 2001).

We therefore derived an estimate of the relation be-

tween SNR and BER as a function of I in Appendix

BER ≈ 0.5∗e

−SNR

(13)

where

SNR

I ·SNR

+ SNR

(I ·SNR)

+ 2I ·SNR+ I

2·SNR+ 1

(14)

EXTENDINGOFDMSYMBOLSTOREDUCEPOWERCONSUMPTION

259

SNR (dB)

Improvement (dB)

I = 16

I = 4

I = 1

-10

-8

-6

-4

-2

0.2

-2

Figure 3: SNR improvement for extension factors I = 1, 4

and 16.

We see that increasing I does not lead to a con-

stant equivalent improvement of SNR

but leads to

an SNR dependent improvement. The improvement

of the equivalent SNR

, for I = 1, 4 and 16 is pre-

sented in Fig. 3.

For the trivial case where I = 1, no improvement

is obtained. For high SNR, I = 4 and I = 16, the SNR

improvement is close to the maximum improvement

of 3 dB per doubling of I. For lower SNR, the im-

provement degrades because uncorrelated noise (N

and N

, see Appendix B) become more and more

dominant which will, for very low SNR lead to an im-

provementof 1.5 dB per doubling of I (or equivalently

3 dB per quadrupling of I). Based on (13), BER esti-

mates for I = 1, 4 and 16 have been added to Fig. 2.

The estimates are a bit too optimistic. This is caused

by the fact that, especially the signal and noise cross

products (S

∗

and S

∗

, see Appendix B), do not

resemble an AWGN signal as assumed. Expression

(14) has also been used to determine a BER estimate

for Rayleigh fading channels based on

BER ≈

2∗(1+ SNR

)

(15)

BER estimates, based on (15) are shown in Fig. 4

for I = 1, 4 and 16. Simulations have been done for

these cases with a 7-tap frequency selective Rayleigh

fading channel. These results are also shown in Fig.

Both the simulations and the estimated BER val-

ues show that the BER performance improves with

increasing I. We also see that the estimate is becom-

ing more and more optimistic for increasing I, similar

to the AWGN case.

SNR(dB)

BER

I = 1

I = 4

I = 16

Estimates

Simulations

-10 -8 -6

-4

-2 0 2 4

−2

−1

Figure 4: BERs for differential ES-OFDM for an AWGN

(top) and a 7-tap frequency selective Rayleigh fading chan-

nel (bottom).

5 FREQUENCY OFFSETS

One of the known disadvantages of OFDM, compared

to single carrier systems, is its high sensitivity to fre-

quency offsets. The effects of frequency offsets on

frequency-differential modulation have been studied

in (Miller and Lee, 1998) and (Tan and Beaulieu,

2007b). In these publications, approximations of the

relation between the BER and frequency offsets are

given for ﬁxed symbol lengths. In our contribution

we concentrate on an approximation of the relative

effects of symbol extension.

The frequency offset is modeled by multiplying

the extended symbol before transmission s

with a

factor e

j2πδ

, where δ

models the frequency offset.

We deﬁne the signal r

t,δ

at the receiver as

t,δ

= s

·e

2πδ

+ n

, t = 0, 1, ..., IN −1 (16)

The starting point of the analysis is the correlator out-

put z as deﬁned in (22) of Appendix A. The signal

related parts of z are rewritten using (16)

τ,δ

IN−1

∑

t=0

2πδ

·e

2πt

)

∗

2πδ

MOD(t+τ, IN)

MOD(t+τ,IN)

(17)

Ignoring the edge effect where t + τ ≥ IN,

2πδ

MOD(t+τ, IN)

is approximated by e

2πδ

(t+τ)

. Ex-

pression 17 then becomes

τ,δ

≈

2πδ

IN−1

∑

t=0

·e

2πt

)

∗

MOD(t+τ,IN)

(18)

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The distorting factor e

2πδ

does not depend on the

symbol extension factor I and differential ES-OFDM

is therefore, by approximation, insensitive to ex-

tended symbol lengths with respect to frequency off-

sets.

6 CONCLUSIONS

By extending symbols, differential ES-OFDM can

achieve acceptable BERs at low SNR. For AWGN

and frequency selective Rayleigh fading channels, the

SNR requirement at the receiver to achieve acceptable

BERs is lowered when extending symbols (and inher-

ently, lowering the data rate). In case of ﬁxed channel

conditions, doubling the symbol length reduces the

required radiated power level at the transmitter with

approximately 2.1 dB. This way power consumption

within the PA is reduced. When extending symbols,

the computational load per unit time reduces almost

proportionally for the transmitter. We thus satisfy the

requirement that the power consumed in both the PA

and the radio modem of the transmitter should be re-

duced.

The computational load at the receiver slightly in-

creases when extendingsymbols if we consider the re-

ceiver model presented in Fig. 1. However,as already

indicated in our analysis in Section 4, the receiver ar-

chitecture can also be changed into a crosscorrelation

followed by an FFT. In (Kokkeler and Smit, 2011),

it is shown that such a receiver architecture can be

simpliﬁed by reducing the resolution of the Analog-

to-Digital Converters giving ample opportunities to

lower the power consumption of the receiver. This

is still a subject of further research.

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APPENDIX A

The expressions given in section 3 describe a differ-

ential ES-OFDM receiver using a so-called FX corre-

lator. In this appendix, we will rearrange the expres-

sions in such way that the receiver describes an XF

correlator. Using the shift theorem and (8), (9) can be

EXTENDINGOFDMSYMBOLSTOREDUCEPOWERCONSUMPTION

261

rewritten as

I−1

∑

i=0

f,i

·R

∗

f−1,i

I−1

∑

i=0

∗

f−1,i

·R

f,i

I−1

∑

i=0



√

N−1

∑



2πt



∗

2πft





√

N−1

∑

−j

2πft



(19)

The summation over t

can start at any index within a

symbol. We choose to add t

to the index.

I−1

∑

i=0

N−1

∑



2πt



∗

2πft



N−1

∑

),i

−j

2πf (t

)



(20)

If we deﬁne t

+ iN = t, t

= τ and use (6), (20) be-

comes

IN−1

∑

t=0



2πt



∗

2πft



N−1

∑

τ=0

MOD(t+τ,IN)

−j

2πf (t+τ)



(21)

The exponential parts dependent on t can be removed,

and by exchanging the summation order, (21) be-

comes

N−1

∑

τ=0



IN−1

∑

t=0



2πt



∗

·r

MOD(t+τ,IN)



·e

−j

2πf τ

√

N · DFT(z

)

(22)

We see that R

is proportional to the DFT of a cross-

correlation result z

. To produce z

, the received

extended symbol r

is correlated with a frequency

shifted version of the same symbol. The frequency

shift corresponds to one carrier spacing.

APPENDIX B

The starting point is the general blockdiagram of a

DPSK demodulator presented in Fig. 5. In this ﬁg-

ure, S

and S

represent the differentially encoded sig-

nals. N

and N

are produced by uncorrelated noise

sources. We assume that the variances of the signals

and noise sources equal σ

and σ

respectively. The

SNR at each input of the multiplier thus equals

The two sums S

+ N

and S

+ N

are multiplied and

averaged over IN samples. This average is then fed

detector

+ N

BERρ

∑

IN−1

i=0

+ N

Figure 5: General blockdiagram of a DPSK receiver.

into the detector. Expression 11 describes the relation

between the SNR of the input signal and the BER at

the output of the detector as a function of I. When

keeping the SNR at both inputs of the multiplier con-

stant, increasing I will lead to a higher signal-to-noise

ratio ρ at the input of the detector and a lower BER

at the output of the detector. To estimate the effect of

increasing I, we assume that the SNR improvement at

the input of the detector is not caused by increasing I

(we assume I = 1) but by an increased SNR at both

input of the multiplier. The standard relation between

SNR and BER as given in (Haykin, 2001) is used to

estimate the effect of I.

The signal at the output of the multiplier (see

Fig. 5) consists of 4 parts: the signal related part

∗

) and three noise related parts (S

∗

, S

∗

, and

∗

). After averaging, we assume that the noise re-

lated parts show Gaussian behavior. In case I = 1, the

signal-to-noise ratio ρ at the input of the detector is

approximated by

I=1

≈

2·σ

+ σ

SNR

2·SNR+ 1

(23)

Extending the symbol increases ρ with a factor I. To

determine the signal-to-noise ratio of a non-extended

input signal (SNR

) that results in the same ρ as

when extending a symbol with I, we formulate the

following equation

(SNR

)

2·SNR

+ 1

I ·SNR

2·SNR+ 1

= ρ

(24)

Rewriting this equation leads to the following solu-

tion:

SNR

I ·SNR

+ SNR

(I ·SNR)

+ 2I ·SNR+ I

2·SNR+ 1

(25)

The estimate of the equivalent SNR can be used

within the well knownBER expressions (see (Haykin,

2001))for both AWGN and Rayleigh fading channels.

For AWGN channels, this leads to the following ex-

pression

BER ≈ 0.5∗e

−SNR

(26)

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