Symbol by Symbol Doppler Rate Estimation for HighlyMobile Underwater OFDM

Nathan Parrish, Sumit RoyDepartment of Electrical Engineering

University of WashingtonSeattle, Washington 98195

nparrish,sroy@u.washington.edu

Payman ArabshahiApplied Physics Lab

University of WashingtonSeattle, Washington 98105

payman@apl.washington.edu

ABSTRACTWe propose an OFDM receiver capable of estimating andcorrecting, on a symbol-by-symbol basis, the subcarrier de-pendent Doppler shifting due to the movement of source andreceiver in an underwater acoustic network. We proposetwo methods of estimation: one of which is based upon themarginal maximum likelihood principle, and one of which isad-hoc. We compare the performance of both estimators tothe Cramer-Rao lower bound. We show through simulationthat the proposed receiver design performs well for a sourcethat is accelerating at 0.29 m/s2.

Categories and Subject DescriptorsC.3 [Special Purpose and Application-Based Systems]:Signal Processing Systems

General TermsPerformance, Reliability

KeywordsUnderwater Acoustic Networks, OFDM, Doppler Rate

1. INTRODUCTIONThe wideband modulation scheme Orthogonal Frequency

Division Multiplexing (OFDM) has recently generated muchinterest in the context of underwater acoustic communica-tions for its potential to increase achievable data rates [11,2]. OFDM - through the use of a cyclic prefix (CP) orguard interval - transforms the frequency selective underwa-ter channel into multiple frequency non-selective orthogonalchannels. Additionally, OFDM is easily implemented viathe Fast Fourier Transform (FFT). However, because of therelatively low propagation speed of acoustic sound in water,the motion of the source and receiver causes non-uniformDoppler shifting among the subcarriers at different frequen-cies. If not compensated, this Doppler shift destroys the

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orthogonality of the subcarriers causing severe inter-carrierinterference (ICI) at the receiver. For networks of highly mo-bile underwater vehicles, such as the Remus line of AUVs[12] which can travel at speeds of up to 2.6 m/s, receiveralgorithms must be able to correct for this Doppler shift.

For underwater channels in which the Doppler shift on allcomponent multipaths is approximately equal, it has beenshown that the resulting (non-uniform) Doppler among sub-carriers described above can be estimated via a single chan-nel parameter - the Doppler rate (vehicle speed divided bythe speed of sound in the medium). Li et al. [6] recentlyshowed that by re-sampling the received signal at a periodproportional to the Doppler rate, the frequency dependentDoppler shift can be eliminated. Therefore, conventionalpilot assisted estimation methods for joint channel and car-rier frequency offset estimation can be used to recover thetransmitted data symbols.

Doppler rate estimation has been accomplished in [6] and[7] via a preamble/postamble structure that assumes thatthis rate is constant over a block of OFDM symbols. Sharifshowed in [9] that the Doppler rate can be estimated bya pair of LFM chirps, and Li extended this method foruse in OFDM. In [7], the Doppler rate is estimated overa block of OFDM symbols via a preamble that contains arepeated OFDM symbol with a cyclic prefix. Both of theabove designs estimate the Doppler rate independent of theOFDM symbols and assume that the rate remains constant.This poses limitations for a network of highly mobile un-derwater nodes. First, if the underwater nodes change di-rection and/or speed often, then the Doppler rate must bere-estimated often which requires re-transmission of the pre-amble and a consequent loss in data rate. Furthermore, ifthe Doppler rate changes significantly over the course of anOFDM block, then reception will fail due to increased ICI.

In this paper, we propose a receiver structure that allevi-ates these problems by estimating (and correcting for) theDoppler rate on a symbol-by-symbol basis. A block dia-gram of our proposed receiver structure is shown in Figure(1). The transmitted block of OFDM symbols is precededby a preamble that allows for initial timing and Dopplerrate estimation. Joint carrier frequency offset and channelestimation are then performed on the first symbol assumingthat the the Doppler rate estimated during the preamble isequal to the actual Doppler rate during the first OFDMsymbol. The recovered data is used to update the esti-mate of the Doppler rate which is then used for decodingof the next OFDM symbol; this process is continued on asymbol-by-symbol basis. We develop two algorithms for the

Figure 1: Receiver Block Diagram

Doppler rate update: one ad-hoc, and the other based on theMarginal Maximum Likelihood principle. We show throughsimulation that such a receiver design can support increaseddata rates with reduced probability of symbol error whenthe relative velocity between the transmitter and receiverchanges over a velocity range commensurate with the Re-mus 6000 AUV.

Notation: Superscript (.)(i) represents the i-th OFDMsymbol vector. Lowercase bold face letters indicate columnvectors, and uppercase bold face letters indicate matrices.The superscript (.)T indicates transpose and (.)H indicatesconjugate transpose. E[.] indicates the expected value. Fi-nally . indicates a value acquired via estimation.

2. OFDM SIGNAL AND CHANNEL MODELA baseband representation of CP-OFDM for a transmit

pulse shape g(t) is given by

x(t) =

∞X

i=0

1√K

K2−1

X

k=−K2

d(i)k ej2π k

T K(t−Tg−iTs)g(t − iTs) (1)

where d(i)k represents the symbol on the k-th sub-carrier

within the i-th OFDM symbol. K is the total number ofsubcarriers and Ts is the OFDM symbol period, given byTs = T (Ng + K), where Ng is the number of symbols inthe cyclic prefix and T = 1/B is the sampling period for achannel bandwidth of B. The separation between OFDMsubcarriers is ∆f = 1

TK.

The baseband symbol is modulated at carrier frequencyfc to yield the corresponding transmitted passband signalf(t) = Rex(t)ej2πfct which is transmitted over an acousticchannel with time varying channel impulse response

h(τ, t) =

L−1X

l=0

αl(t)δ(t − τl(t)) (2)

where τl(t) and αl(t) are the delay and complex gain ofpath l at time t. In (2), there exist L discrete multipathcomponents.

The received signal r(t) plus additive white Gaussian noise,w(t), is

r(t) = Re

(

L−1X

l=0

αl(t)x(t− τl(t))ej2πfc(t−τl(t)) + w(t)

)

(3)

which is downconverted to give

y(t) =

L−1X

l=0

αl(t)x(t − τl(t))e−j2π(fcτl(t)+∆fct) + w(t) (4)

In (4), ∆fc is the carrier frequency offset between thesource and receiver. We assume that the time variation inthe delay is due primarily to the motion of the source/receiver,and is approximately equal on all paths. Therefore, for rel-ative velocity v(t) between the source-receiver pair, τl(t) =

τl +R t

0s(t)dt where s(t) = v(t)

c, where the first term repre-

sents the delay at a reference time t = 0.Substituting for τl(t) and x(t) into equation (4), the re-

ceived signal (neglecting the noise term) can be expressedas

y(t) =

∞X

i=0

e−j2π(fc

R

t0 s(t)dt+∆fct) 1√

K· (5)

K/2−1X

k=−K/2

d(i)k ej2π k

TK(t−Tg−iTs−

R

t0 s(t)dt)H(fk, t)

where H(fk, t) is the channel transfer function at time t onthe kth subcarrier.

H(fk, t) =

L−1X

l=0

bl(t)e−j2π

kτlT K (6)

bl(t) = αl(t)g

„

t − iTs − τl −Z t

0

s(t)dt

«

e−j2πfcτl (7)

We denote the time when the leading edge of the ith sym-bol arrives at the receiver as t(i). We also assume that s(t)and αl(t) are slowly time varying with respect to the OFDM

symbol interval, therefore over the interval t = [t(i), t(i+1)]

s(t) = s(i)

al(t) = a(i)l

The propagation of symbols from source to receiver isshown in figure (2). In this figure we also indicate the delayspread of the channel, denoting the time that the trailingedge of the ith symbol arrives at the receiver as t′(i + 1).From equation (7), we can see that symbol 0, transmit-ted over interval [0, Ts] arrives at the receiver over interval

Figure 2: Propagation of OFDM Symbols from Transmitter to Receiver under the assumption that theDoppler Rate is slowly time varying. Black lines (red lines) from source to receiver represent the propagationon the path with the minimum (maximum) delay. Notice that all ISI from symbol i − 1 arrives during thecyclic prefix of symbol i arrive within the Cyclic Prefix.

[t(0), t′(1)] = [ τmin

1−s(0) , Ts+τmax

1−s(0) ]. It can also be shown that:

t(i) = t(i−1) +Ts

1 − s(i−1)(8)

t′(i) = t′(i−1) +Ts

1 − s(i−1)

Additionally, we assume that NgT > τmax − τmin, ie. thatall the ISI is contained within the cyclic prefix (as indicatedin figure (2)). Since the cyclic prefix will be discarded bythe OFDM receiver, we disregard the ISI and express thereceived signal over the interval [t(i), t(i+1)] as:

y(i)(t′) = y(t(i) + t′) (9)

= e−j2π(fcs(i)+∆fc)t′ 1√K

·

K/2−1X

k=−K/2

d(i)k ej2π k

T K((t′)(1−s(i))−Tg)H(i)(fk)

where H(i)(fk) is:

H(i)(fk) =L−1X

l=0

b(i)l e−j2π

k(τl−τmin)

T K (10)

b(i)l = α

(i)l g

`

t′ − (τl − τmin)´

e−j2π(fc(τl+t(i))+∆fct(i)) (11)

We note the three effects of the Doppler Rate in (9) whichwere also cited in [6]. The first is that the i-th receivedOFDM symbol which was transmitted over a time period ofTs seconds is received over a time period of Ts

1−s(i) + τmax −τmin seconds. This indicates a spreading of the received sig-nal due to the multipath of the channel as well as a ’dilation’due to the Doppler rate. The second effect is the subcarrierdependent Doppler shift ∆fk = −s(i)k/KT . Finally, there

is an effective carrier frequency offset of fcs(i) + ∆fc across

all subcarriers.

3. DISCRETE TIME MODELWe develop an equivalent discrete time model by sampling

Eq. (9). In order to completely eliminate the subcarrier de-pendent Doppler shift from the received signal samples, the

ith OFDM symbol must be sampled at rate T (i) = T

1−s(i) .

However, since we do not a-priori know s(i), we sample atrate T (i) = T

1−s(i−1) (we show later how the estimate of s is

generated from OFDM symbol i − 1). By sampling symboli at this rate, the received discrete time samples are givenas:

y(i)n = e−j2πǫ(i)nT 1√

K

K/2−1X

k=−K/2

d(i)k ej2π k

K(nβ(i)−Ng)H(i)(fk)+w(i)

n

(12)

where ǫ(i) = fcs(i)

1−s(i−1) + ∆fc

1−s(i−1) and β(i) = 1−s(i)

1−s(i−1) . The

final term in (12) is a zero-mean, circularly symmetric com-plex Gaussian noise with variance σ2

w that is i.i.d in the indexn.

After the first Ng samples (which form the cyclic prefix)are removed, the remaining K samples in Equation (12) areexpressed in vector form as

y(i) = Mǫ(i)WHβ(i)D

(i)h(i) + w(i) (13)

where y(i) is the vector of the K discrete time samples,

Mǫ(i) = diag[1, e−j2πǫ(i)T , ..., e−j2πǫ(i)(K−1)T ] is the effec-tive carrier frequency offset,

h = [H(i)(f−K/2), ..., H(i)(fK/2−1)]

T

is the channel transfer function, Wβ(i) is a matrix with entry

[Wβ(i) ]k,n = 1√K

e−j2πk((n+Ng)β(i)

−Ng)

K , D(i) is a diagonal

matrix of the data symbols, and w(i) is a zero-mean iidGaussian noise vector with covariance matrix σ2

wI.Assuming that we know the effective carrier frequency off-

set, we form

y(i) = MHǫ(i)y

(i)

= WHβ(i)D

(i)h(i) + MHǫ(i)w

(i) (14)

Taking the DFT of the received data, we get

W1y(i) = W1W

Hβ(i)D

(i)h(i) + q(i) (15)

q(i) = W1(MHǫ(i))w

(i) (16)

−600 −400 −200 0 200 400 60010

15

20

25

30

35

40

45

50

Subcarrier Index

Sig

nal t

o IC

I pow

er (

dB)

Figure 3: Eh

|Sigm|2|ICIm|2

i

, ratio of signal power to ICI

power per subcarrier K = 1024 subcarriers, ∆f = 2.8Hz, and ∆v(i) = 0.156 m/s (solid line) ∆v(i) = 0.302m/s (dashed line).

It is noted that the vector q(i) is a unitary transform ofthe vector w(i) and therefore, has the same statistics. Wedenote the covariance matrix of q(i) as σ2

qI. From (15) themth received subcarrier symbol on the ith received OFDMsymbol, for m = [−K/2, ..., K/2 − 1] is:

z(i)m =

1

K

K/2−1X

k=−K/2

H(i)(fk)d(i)k e(j2π k

KNg(β(i)−1)) ·

K−1X

n=0

ej2π nK

(kβ(i)−m) + q(i)m (17)

We define the matrix G(β) with elements

[G(β(i))]m,k =1

K

K−1X

n=0

ej2π nK

(k(β(i))−m) (18)

[G(β(i))]m,k =1

Ke

jπ(K−1)(k(β(i) )−m)K

sin(π(k(β(i)) − m))

sin(π(k(β(i))−m)K

)(19)

and the matrix NG(β(i)) as a diagonol matrix with the (k, k)

element given as ej2π kK

(Ng(β(i)−1)). Finally, we can write thereceived signal vector as:

z = G(β(i))NG(β(i))D(i)h(i) + q(i) (20)

The matrix G(β(i)), therefore, creates an ICI pattern, wherethe ICI on the mth subcarrier is:

ICIm =X

k 6=m

H(i)(fk)d(i)k [G(β(i))NG(β(i))](m,k) (21)

and the useful signal is:

Sigm = H(i)(fm)d(i)m [G(β(i))NG(β(i))](m,m) (22)

Figure (3) plots the ratio of signal power to ICI power,

Eh

|Sigm|2|ICIm|2

i

, for two different values of β(i), assuming that

E[|H(i)(fk)|2] = E[|d(i)k |2] = 1. However, rather than con-

sidering β(i), it is more instructional to consider the relativeacceleration of the transmitter/receiver directly. Therefore,

Figure 4: Receiver Design

the figure indicates the ∆v(i), which is the change in velocityfrom one OFDM symbol to the next.

4. RECEIVER DESIGNThe receiver schematic is shown in Figure (4). The incom-

ing symbol is sampled at rate T (i) = T

1−s(i−1) , where s(i−1)

is the estimate of the Doppler Rate for the previous symbolor from initial synchronization. Our receiver operates in twostages. The first stage involves transmitted symbol recov-ery via joint carrier frequency offset and channel estimationunder the assumption that β(i) = 1. The second stage usesthe carrier frequency offset estimate, transmitted symbol es-timates, and the channel estimates found in the first stageto estimate the Doppler rate. The next symbol is sampledthen at rate T (i+1) = T

1−s(i) given by the updated Dopplerrate.

In order to perform joint channel and carrier frequency off-set estimation, we first assume that the channel is a tappeddelay line with Ng taps spaced at T (i) intervals. By furtherassuming β = 1, the model for the received samples (13)becomes (we drop the superscript (i) for simplicity)

y = MǫWH1 DVb + w (23)

where V is a K by Ng matrix equal to the first Ng columnsof W1 and b is the vector of tap delay line channel coeffi-cients. A subset of the transmitted symbols are known pilotsymbols, whose location is given by the matrix S. If thereare p pilot symbols, then S is a p by K matrix with rowp having a 1 in the column corresponding to the pth pilotlocation and zeros elsewhere.

We perform joint channel and carrier frequency offset es-timation by the method proposed in [5]. We first assume arange of values for ǫ. For each candidate, which we denoteas ǫ, we can form a least squares estimate of the channelimpulse response as:

b = (SDV)†SW1MHǫ y (24)

where (.)† indicates the pseudoinverse. From this estimateof the channel impulse response, we can form the estimateof the channel transfer function as

h = Vb (25)

Finally, we recover the transmitted data symbols via

dk =zk

hk

(26)

For each candidate ǫ, the mean square error (MSE) be-

tween the known pilot symbols and the dk is computed. Theǫ which gives the lowest MSE is chosen as ǫ.

The data symbols in an underwater communications sys-tem are typically channel coded, so that the soft estimatesgiven by (26) are forwarded down the receiver chain for de-interleaving and decoding before a final symbol decision isreached. However, we also immediately form a hard decisionof the transmitted data symbols by simply decoding the softestimates to the nearest neighbor in the transmitted symbolconstellation, and we denote these estimates as dk.

5. DOPPLER RATE ESTIMATION

5.1 Initial EstimationIn [1] it was shown that the joint maximum likelihood

estimate of the Doppler rate and delay s and τ0 for a non-dispersive channel (ie, a channel for which L = 1) can becomputed by transmitting a known training signal and usinga bank of sliding correlators at the receiver. Each correla-tor is tuned to a different Doppler rate, and the maximumlikelihood estimate is a 2-D grid search over time (for τ0)and the correlators (for s). In [7], the estimation is moreeffectively carried out for dispersive, underwater channel bytransmitting a repeated OFDM symbol with a Cyclic Pre-fix, and using a sliding bank of self-correlators, each tunedto a different Doppler rate. In [7] the Doppler rate was as-sumed constant over packet of several OFDM symbols. Weadopt this method of initial Doppler rate and delay estima-tion; however, we add a method for tracking the change inDoppler rate from OFDM symbol to symbol.

5.2 Marginal Maximum Likelihood EstimatorWe wish to develop a maximum likelihood estimator for

β from (20). In order to form (20), we use ǫ given by thejoint channel and carrier frequency offset estimation. We canrepresent F(β) = G(β)NG(β)DV, where we form D froma combination of known pilot symbols and our estimatesgiven in section (4). Since the noise is assumed iid, zeromean Gaussian, the maximum likelihood estimate of (β,b)is achieved by maximizing the probability:

p(z|β, b) =1

(πσ2q)K

exp

− 1

σ2q

(z− F(β)b)H(z− F(β)b)

ff

(27)

over all possible pairs β, b. This is achieved by maximizing:

Λ(z|β, b) = exp

− 1

σ2q

(|z|2 − zHF(β)b − bHF(β)Hz

+ bHF(β)HF(β)b)

ff

(28)

The first term in (28) has no impact on the maximization.Furthermore, in order to estimate β independently of b, wedevelop a marginal likelihood estimator, as shown in [8], by

averaging over the probability distribution of b. We do thisby performing the operation

R ∞−∞ Λ(z|β, b)p(b) db. We sub-

stitute (28) into this equation. Assuming that the elementsof the channel impulse response are independent Rayleighfading arrivals, the marginal maximum likelihood estimate

of β is given by:

Λ(z|β) =1

det[I + 1σ2 R

bF(β)HF(β)]

·

exp

„

1

σ2

«2

zHF(β) ·»

Rb

−1 +1

σ2F(β)HF(β)

–−1

F(β)Hz]

ff

(29)

At high SNRs, it can be shown that the maximum of (29)can be found by maximizing:

Λ(z|β) = zHF(β)[F(β)HF(β)]−1F(β)Hz (30)

We denote the value of β which maximizes (30) as βMML.

5.3 An Ad-hoc EstimatorThe marginal maximum likelihood estimator presented in

the preceding section requires a grid search. In this sec-tion, we develop an ad-hoc estimator of β, which does notrequire a grid search, but which performs sub-optimally incomparison to the MML estimator.

From section 3, it is shown that the received sample onthe kth subcarrier can be expressed as:

zk = ckejπk(β−1)

K(K−1+2Ng)dkhk + qk + ICIk (31)

The first term in (31) is comprised of ck, a real attenuationfactor given by the sin terms in element (k, k) of (19), thephase offset due to [G(β)Ng(β)](k,k), the transmitted databit, and the channel transfer function. The other terms arethe AWGN and ICIk as defined in equation (21). If weassume that the the ICI is additional zero-mean AWGN,then we can estimate the angle in the above equation as:

πk(β − 1)

K(K − 1 + 2Ng) = ∠

„

d∗k(

zk

hk)

«

(32)

and

βk =∠(d∗

k( zk

hk))K

(K − 1 + 2Ng)πk+ 1 (33)

Therefore, β can be found by averaging the βk.

β =1

K

X

βk (34)

Note that this estimate of β is similar to that used by Sto-janovic in [11].

If we assume that the ICI term is additional Gaussiannoise, then β is an unbiased estimator of β as follows.

E

»

∠(d∗k(

zk

hk))K

–

= ∠(d∗k(

E[zk]

hk))K

=πk(β − 1)

K(K − 1 + 2Ng)

therefore

E[βk] =∠(d∗

k(E[zk]hk

))K

(K − 1 + 2Ng)πk+ 1

= β

and clearly the expected value of (34) is β.

5.4 Cramer-Rao Lower BoundWe use the methodology given in [3] to derive the Cramer-

Rao lower bound (CRLB) for the joint estimation of β andb using equation (14). We define the vector of unknowns as

θ = [b0, ..., bL−1, b0, ..., bL−1, β]T , where b and b are the realand imaginary parts of the channel impulse. From (14), wecan see that y is a vector of random variables with distribu-tion N(WH

β DVb, σ2qI). Therefore, the joint probability of y

and θ is:

p(y|θ) =1

(πσ2q)K

exp

− 1

σ2q

(y − WHβ DVb)H ·

(y − WHβ DVb)

ff

(35)

Since the covariance of the random samples is not depen-dent on the θ, we can derive the (r, c) element of the Fisherinformation matrix as:

[I(θ)]r,c =2

σ2n

Re[∂µ(θ)

∂θr]H [

∂µ(θ)

∂θc] (36)

µ(θ) = WHβ DVb (37)

Substituting (37) into (36), the Fisher Information matrixis:

FI =2

σ2q

2

6

4

ReFHF −ImFHF −ImFHFbImFHF ReFHF ReFHFb

Im(Fb)HF Re(Fb)

HF Re(Fb)

HFb

3

7

5

(38)

where we define F = WHβ DV and F = j2π

KNDWH

β KDDVbwith KD = diag[−K/2, ..., 0, ..., K/2−1] and ND = diag[Ng ,..., Ng + n− 1]. Note that this is similar to the Fisher Infor-mation matrix given in Appendix II of [4] for OFDM witha carrier frequency offset.

The CRLB for the ith element of θ is given by the (i, i)element of FI−1. Note that (38) shows that the CRLB is de-pendent upon the specific channel realization b. The modi-fied CRLB (MCRLB) is formed by averaging the CRLB overthe channel realizations.

In figure (5) we compare through simulation the perfor-mance of the MML and Ad-hoc Doppler rate estimators withthe MCRLB. In our derivation for both estimators of β, wehave assumed that we are able to perfectly estimate andcompensate for ǫ. However, it is expected that in practicethere will be some residual carrier frequency offset due toimperfect estimation of ǫ. Speth [10] gives an approxima-tion to the additional noise (in the form of ICI) caused by

a residual CFO of ∆ǫ as σ2Ω ≈ π2

3(∆ǫTK)2. We show the

performance of the two estimators in the presence of resid-ual CFO, with the performance given as a function of SINR.The curves in figure (5) show that the non-Gaussian natureof the CFO error imposes a limit on the performance of theMML estimator at high SINR. Furthermore, we can see thatat low SINR the MML estimator does not meet the MCRLB,which we expect from the approximation in (30). Finally,we can see that the MML estimator outperforms the Ad-hocestimator.

6. SIMULATION SETUPWe simulate the performance of three receiver algorithms

for OFDM with a changing Doppler rate. In the first receiveralgorithm, we perform initial estimation of the Doppler rate

0 5 10 15 20 25 3010

−10

10−9

10−8

10−7

10−6

10−5

10−4

Estimator Variance

SINR

MCRBMML EstimatorAd−Hoc Estimator

0 5 10 15 20 25 3010

−10

10−9

10−8

10−7

10−6

10−5

10−4

Estimator Variance

SINR

MCRBMML EstimatorAd−Hoc Estimator

0 5 10 15 20 25 3010

−10

10−9

10−8

10−7

10−6

10−5

10−4

Estimator Variance

SINR

MCRBMML EstimatorAd−Hoc Estimator

Figure 5: Performance of the Ad-Hoc and MMLDoppler Rate estimators compared to the ModifiedCRLB. In the top figure, the residual CFO is 0 andin the middle and bottom figures it is .5% and 1%of the subcarrier spacing, respectively. The varianceis plotted vs SINR, which includes the noise due toboth AWGN and residual CFO

0 50 100 150 2000

0.5

1

1.5

2

2.5

3

Time (s)

Vel

ocity

(m

/s)

Figure 6: Receiver Velocity

0 1 2 3 4 5 6 7 8 9−13

−12

−11

−10

−9

−8

Tap Index

Pow

er (

dB)

Figure 7: Power Delay Profile

via a pre-amble and then assume that the Doppler rate isconstant over some duration, at which point we re-transmitthe pre-amble to estimate a new Doppler rate. The sec-ond and third receiver designs track the Doppler rate, usingthe marginal maximum likelihood and Ad-hoc algorithmsdescribed in the preceding sections. We will denote thesethree approaches as init, MML, and Ad-hoc.

Our simulation set-up is as follows. Received packets aregenerated according to (13). The Doppler rate during packeti is determined by the velocity of the source/receiver duringpacket i by the equation si = vi/c where c = 1500 m/sis the speed of sound in seawater. Figure (6) shows thevelocity profile that we use in the simulation. The velocityranges from 0.5 m/s to 2.6 m/s. We assume that the craftaccelerates and decelerates at approximately 0.29 m/s2.

In our simulation, the channel coefficients and the Dopplerrate are constant over an OFDM symbol. The Doppler ratechanges from symbol to symbol as described in the previ-ous paragraph. We simulate a channel with 10 taps withan exponentially decaying power delay profile, the powerdelay profile is shown in figure (7). Without loss of gen-erality, the tap weights are chosen so that E[|H(fk)|2] =PL−1

l=0 E[|bl|2] = 1. The channel taps are modeled as iid andindividually Rayleigh fading on each tap.

We transmit OFDM symbols with K = 256 subcarriersand Ng = 10 guard symbols. The data symbols are sub-modulated using 4-PSK modulation, and every fourth sub-carrier contains a probe symbol. The frequency band thatwe use is 10 - 14 KHz, and we use a raised cosine pulseshaping filter with a rolloff factor of 0.4. Therefore, the Ts

0 5 10 15 20 25 3010

−12

10−10

10−8

10−6

10−4

10−2

SNR (dB)

Doppler Rate Estimator Variance

MCRBMML EstimatorAd−Hoc Estimator

Figure 8: Doppler Rate Estimator Variance

is approximately 93 ms, and the velocity changes by +/- .027m/s over each OFDM symbol with a corresponding changein the Doppler rate of 1.8 ∗ 10−5.

For each receiver, we assume that the initial Doppler rateis estimated perfectly during the pre-amble. The perfor-mance of each receiver is given as a function of the SNR,which we define as:

SNR =E[|dkH(fk)|2]

σ2q

(39)

We show the results in figures (8), (9), and (10). Figure(8) shows that the variance of both the MML estimator andAd-Hoc estimator exceeds the MCRLB; however, since weare not accounting for the additional noise imposed by theresidual CFO, this is to be expected. It is also noted thatat 0 dB, the performance of the MML estimator is closestto the MCRLB, which is not expected. This is most likelydue to the resolution of the MML Doppler Rate estimator(ie, the range of β over which the estimator is searching).This figure also shows that we do not see the lower boundon the MML estimate due to CFO error which was observedin figure (5). The explanation for this is given in figure (9)where it can be seen that as the SNR improves, so does theCFO estimate. The net effect of this is that MML DopplerRate estimate continuously improves with SNR.

Finally, we show the uncoded symbol error rate in fig-ure (10). At low SNR, the Ad-hoc estimator performs verypoorly because it makes estimates of the Doppler rate thatdeviate further from the true Doppler rate than does theinitial estimate. This causes an increase in ICI, and thuspoor performance. However, at high SNR both the of thereceivers with Doppler rate tracking perform better thanthat using only an initial estimate of the Doppler rate. Ofcourse, the performance of the ’initial only’ estimator couldbe improved by re-estimating the Doppler rate more often,but this would also incur increased overhead and a lowerdata rate.

7. CONCLUSIONWe have described OFDM symbol reception in an under-

water channel where the Doppler rate varies between sym-bols. The receiver design that we have proposed buildsupon previous works which estimated the Doppler rate via

0 5 10 15 20 25 3010

−2

10−1

100

101

102

103

SNR (dB)

CFO Estimator Mean Square Error

MML EstimatorAd−Hoc EstimatorInitial Est. Only

Figure 9: Carrier Frequency Offset estimator MSE

0 5 10 15 20 25 3010

−3

10−2

10−1

100

Uncoded Symbol Error Rate

SNR (dB)

MML EstimatorAd−Hoc EstimatorInitial Only Estimator

Figure 10: Uncoded Symbol Error Rate

a pre/post-amble and assumed that it was constant over ablock duration. Of the two Doppler rate estimators thatwe have proposed, the MML estimator shows better per-formance, but is more complex. Future work will invovletesting the proposed receiver design via experimentation inunderwater channels.

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