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Excessively Long Channel Estimation for CDD

OFDM Systems Using Superimposed Pilots

Weikun Hou and Xianbin Wang

Department of Electrical and Computer Engineering

The University of Western Ontario

London, Ontario N6A5B9, Canada

Abstract—To fully exploit frequency diversity in cyclic de-

lay diversity orthogonal frequency division multiplexing (CDD-

OFDM) system, accurate channel estimation is crucial. Due to the

excessive channel delay spread in CDD, traditional in-band pilot

assisted channel estimation with limited frequency resolution fails

to track the considerable variation in the frequency domain.

Alternatively, increasing pilot overhead will degrade the system

throughput significantly. In this paper, we propose to use super-

imposed pilots to estimate highly frequency selective channels

in CDD-OFDM systems. Compared to in-band pilot based

channel estimations, the proposed scheme with pilot symbols

superimposed over each subcarrier has full frequency resolution,

hence it is more robust to severe frequency selectivity from

the excessively long CDD channel. An Expectation-Maximization

(EM) algorithm is employed to estimate the channel iteratively

based on superimposed pilots and tentative soft decisions. At the

end of each iteration, to exploit the inherent channel sparsity and

refine the estimate, channel taps are sorted and selected according

to the power. Simulation results show that the performance of the

proposed scheme is promising in time varying fading channels

without an increase in pilot overhead.

I. INTRODUCTION

Cyclic delay diversity (CDD) is one transmit diversity

scheme suitable for block transmission systems. As an im-

proved variant of delay diversity, CDD does not introduce any

extra inter-symbol interference [1] [2]. Due to its simple cyclic

shift operation at each antenna before transmission, multiple

input channels can be transformed into an equivalent single in-

put channel with long channel length, and consequently spatial

diversity is transformed into additional frequency diversity.

Compared to space time block coding (STBC) and other

transmit diversity schemes, CDD is standard compatible,

which means no receiver modification is required when it is

implemented. Furthermore, CDD has no rate loss when the

number of transmit antennas Nt > 2, while STBC scheme

suffers rate loss when Nt> 2 [3] [4]. Due to its advantages,

CDD has been introduced into OFDM systems [1]- [3]. The

flexible deployment and robust data throughput with increasing

transmit antennas make CDD attractive in many OFDM based

wireless systems [3]- [8].

One major disadvantage of CDD-OFDM is the cost for

realizing the diversity gain. As the coherence bandwidth is

inversely proportional to the channel length, the excessively

long equivalent fading channel introduced by CDD will

lead to severe frequency selectivity. It will degrade the data

throughput due to the increased pilot overhead for channel

estimation. In [9] [10], two pilot symbol aided schemes for

CDD-OFDM were proposed. These schemes require additional

pilot overhead and pre-known cyclic delay parameters to

estimate each individual channel. Another pilot-aided channel

estimation algorithm by alternating the cyclic delay parameter

over adjacent symbols was proposed in [11]. However, this

scheme assumes that the fading channel remains constant over

Ntconsecutive symbols.

In this paper, we propose to use superimposed pilots to

tackle the lingering problem between channel estimation ac-

curacy and pilot overhead. The benefit of using superimposed

pilots is two folds. First, it has high immunity to severe

frequency selectivity caused by the extremely long equiva-

lent channel. As pilot symbols are superimposed over each

subcarrier [12], the channel frequency response across the

whole band can be estimated. Consequently, it can track the

drastically varying frequency response caused by the large

delay spread, which can be as long as the discrete fourier

transform (DFT) size N. Second, the superimposed pilots can

be assigned with a relatively low power to keep the system

throughput reasonable. In our simulation, it is shown that

the proposed channel estimation scheme can retain the CDD-

OFDM system performance gain without an increase in pilot

overhead.

The rest of the paper is organized as follows. In Section II,

the system model of CDD-OFDM with superimposed pilots

is provided. In Section III, the proposed iterative channel

estimation scheme based on an EM algorithm is presented.

Then the procedure for choosing the sparse channel taps during

each EM iteration is discussed to reduce the noise level.

Numerical simulation results are given in Section IV. Finally,

the conclusion is made in Section V.

Notations: Bold upper (lower) case letters denote matrices

(column vector). Diag(h) denotes diagonal matrix with column

vector h as its main diagonal. The DFT matrix of size N

is represented as F with the (k,l)-th entry given by [F]k,l=

exp(−j2π(k − 1)(l − 1)/N) where j =√−1. d denotes the

mean value of d.

II. CDD-OFDM SYSTEM WITH SUPERIMPOSED PILOT

A. Transmission model

Fig.1 shows the block diagram of the CDD-OFDM system

with Nttransmit antennas. At the transmitter, data bits after

channel coding are mapped to QAM symbols d(k). Pilot

978-1-61284-231-8/11/$26.00 ©2011 IEEE

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OFDM

Modulation

(IDFT)

QAM

Mapper

S/PP/S

Power

Normalization

Pilot

Symbols

...

...

...

...

...

...

Coded Bits

0

0

CP

CP

Cyclic shift

Cyclic shift

Cyclic shift

1

t

N

CP?

i

CP

i?

1

tN ?

?

0 ?

1( )n

?

tN S

0( )S n

( )

iS n

( )S n

x(k)

p(k)

d(k)

Fig. 1. Proposed CDD-OFDM system with superimposed pilots.

symbols p(k),0 ≤ k ≤ N − 1, are then added to these

data symbols. The transmitted symbol over each subcarrier

is represented as x(k) = d(k) + p(k),k = 0,··· ,N − 1.

After S/P conversion and the inverse discrete fourier transform

(IDFT), the time domain signal can be expressed as

s(n) =

1

√N

N−1

?

k=0

x(k)ej2π

Nkn,

0 ≤ n ≤ N − 1.

(1)

The transmitted signal si(n) for the i-th antenna is a cyclic

shift version of the original waveform in (1) with delay

parameter δi. It is normalized to keep the total transmission

power unchanged, that is

1

√Nt

where Nt is the number of transmit antennas and (·)N is

the modulo-N operation. To maximize frequency diversity,

the delay parameter difference between two adjacent antennas

should be set to larger than the maximum channel delay [7].

Finally, the CP for each branch is inserted before transmission.

si(n) =

s(n − δi)N,

0 ≤ i ≤ Nt− 1,

(2)

B. Equivalent channel in CDD-OFDM

The received signal is the combination of each transmitted

signal from separate antennas, each of which experiences

independent fading. After CP removal, it can be expressed as

the sum of cyclic convolutions between each cyclic shifted

transmitted signal and the corresponding channel impulse

response (CIR),

y(n)=

Nt−1

?

s(n) ⊗

i=0

1

√Nt

s(n − δi)N⊗ hi(n) + w(n)

=

1

√Nt

Nt−1

?

i=0

hi(n − δi) + w(n),

(3)

where hi(n) is the fading channel between the i-th transmit

and receive antenna with length Li, and w(n) is independent

and identically distributed (i.i.d.) complex Gaussian noise with

zero mean and variance N0. From (3), the combined received

signal can be treated as one signal which experiences a channel

with a large delay spread. This equivalent fading channel

results in additional frequency diversity. It is expressed as

heq(n) =

1

√Nt

Nt−1

?

i=0

hi(n − δi),

0 ≤ n ≤ N − 1.

(4)

When the difference of cyclic delay parameters Δi =

δi+1−δi> Li, the multipaths from different transmit antennas

will not be overlapped and potential diversity gain can be

obtained. Fig.2 shows an instance of an equivalent CIR for

a CDD system with two transmit antennas. In contrary to

conventional OFDM system, the delay spread of the equiv-

alent channel heq(n) is very large and consequently severe

frequency selectivity is introduced.

At the receiver, after OFDM demodulation, the frequency

domain signal of the k-th subcarrier is given by

r(k) =

1

√Nt

Nt−1

?

i=0

ci(k)e−j2π

Nkδix(k) + v(k),

0 ≤ k ≤ N − 1,

(5)

where ci(k) is the complex channel gain corresponding to the

i-th transmit antenna over the k-th subcarrier and v(k) is the

complex white noise in the frequency domain.

If the pilot and data symbol vector are denoted as p =

[p(0),p(1),··· ,p(N − 1)]Tand d = [d(0),d(1),··· ,d(N −

1)]Trespectively, the received signal in the frequency domain

can be expressed in matrix form as shown below:

r = (D + P)Fh + v

= Diag(Fh)(d + p) + v,

(6)

where r and v are the frequency domain signal and noise

vectors, D = Diag(d) and P = Diag(p) are the diago-

nal matrix of data and pilot symbols respectively. h =

[heq(0),heq(1),··· ,heq(N −1)]Tis the column vector repre-

senting the equivalent CIR with a length as long as the DFT

size N. Diag(Fh) is a diagonal matrix of complex channel

gain in the frequency domain.

III. EM BASED ITERATIVE CHANNEL ESTIMATION

Since the frequency noise vector v is white, from (6), the

likelihood function of the channel is written as

1

(πN0)Nexp(−?r − (D + P)Fh?2

The maximum likelihood (ML) estimate of the channel can

be determined as

p(r|D,h) =

N0

).

(7)

argmax

h

{lnp(r|D,h)}

{?r − (D + P)Fh?2}.

⇔

argmin

h

(8)

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Page 3

0 10

Channel tap delay normalized with sampling interval

2030405060

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Power of channel tap

Δ=δ1−δ0

Multipaths from 2nd Tx. antenna

Multipaths from 1st Tx. antenna

Fig. 2.

L0= 6, L1= 6, Δ0= 20, the equivalent channel length is Δ0+L1= 26.

CIR instance for CDD-OFDM system with two transmit antennas.

The above ML channel estimate needs the complete set (r,d)

containing the unknown data symbols.

A. EM algorithm

Instead of solving (8) directly, we resort to the Expectation-

Maximization (EM) algorithm to estimate the channel by using

tentative soft demodulation results at each iteration.

E-step: In the (q+1)-th iteration, based on (8) and the previ-

ous CIR estimate hq, maximizing expectation E{lnp(r;h)|hq}

is equivalent to minimizing the formula given below [13]:

Q(h|hq) =

|Ω|N−1

?

i=0

Pr(di|r;hq) ?r − (Di+ P)Fh?2,

(9)

where Diis the diagonal matrix with the symbol vector di(i.e.

Di= Diag(di)) and Pr(di|r;hq) is the posterior probability

of the symbol vector dibased on the received signal r and the

previous estimate hq. The symbol vector diis an N ×1 vector

and its components di(k),k = 0,··· ,N − 1, come from the

discrete modulation alphabet Ω with cardinality |Ω|. When the

same modulation scheme is deployed over all the subcarriers,

the total number of symbol vector samples is |Ω|N.

M-step: By minimizing (9), the CIR estimate in the (q+1)-

th iteration can be obtained as

1

NFH[PHP + Rd+ 2?(DHP)]−1(P + D)Hr, (10)

where D denotes diagonal matrix with symbol mean value

hq+1=

D = Diag(d(0),d(1),··· ,d(N − 1))

|Ω|N−1

?

and Rdis the diagonal matrix with the mean of symbol power

Rd= Diag(σ2

=

i=0

Pr(di|r;hq) Di,

(11)

0,σ2

1,··· ,σ2

N−1)

=

|Ω|N−1

?

i=0

Pr(di|r;hq)DH

iDi.

(12)

The derivation of (10) can be found in Appendix A.

B. Symbol posterior probability

From (11) and (12), the calculations of D and Rdrely on

the posterior probability of the symbols. As indicated in (6),

data symbol over each subcarrier is independent. Given the

specific CIR estimate hq, Pr(di|r;hq) can be factorized as

N−1

?

Therefore each diagonal element in D and Rd can be

calculated independently

⎧

⎪

Pr(di|r;hq) =

k=0

Pr(di(k) | r;hq).

(13)

⎪

⎪

⎪

⎩

The above calculations rely on the posteriori probability

based on r and hq,

⎨

d(k) =

?

d(k)∈Ω

?

d(k) Pr(d(k)|r;hq)

σ2

k=

d(k)∈Ω

|d(k)|2Pr(d(k)|r;hq)

0 ≤ k ≤ N − 1.

(14)

Pr(d(k)|r;hq)

Assuming the a priori probability is the same for each

constellation point (i.e. Pr(d(k)) =

probability can be determined as

∝

p(r|d(k);hq)Pr(d(k)).

(15)

1

|Ω|), the symbol posteriori

Pr(d(k)|r;hq) = C · exp(−|r?(k) − cq(k)d(k)|2

where C is the normalized constant for symbol posteriori

probability such that?

subcarrier, the corresponding frequency response vector is

given by cq= Fhq, and r?(k) = r(k) − cq(k)p(k) is the

residual signal after being subtracted from the received pilot.At

the beginning of each iteration, the channel estimate of the

previous OFDM symbol is used for initialization.

The noise variance used in (16) is estimated by averaging

the square difference between the signal r?(k) and its closest

constellation point across the subcarriers

N0

),

(16)

d(k)∈ΩPr(d(k)|r;hq) = 1. cq(k),k =

0,··· ,N − 1, is the complex channel gain over the k-th

ˆ N0=

1

N

N−1

?

k=0

?r?(k) − ψ(r?(k))?2,

(17)

where ψ(·) is the slice function maps the received signal to

its closest discrete constellation point.

C. Sparse channel tap selection

Compared to single antenna OFDM systems, the equivalent

channel length in CDD systems is much longer. Therefore the

number of channel taps to be estimated becomes significantly

larger. This indicates more background noise is introduced due

to the long channel duration. On the other hand, the realistic

channel is only with a few non-zero taps while the others are

zero taps contaminated by noise.

To avoid over modeling and reduce noise level in channel

estimation, we exploit this inherent channel sparsity at each

iteration by choosing the most significant taps and suppressing

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Page 4

the rest. The procedure is applied as follows. Channel estimate

hqobtained in the q-th iteration with length N is first sorted in

descendent order according to each tap’s power. Then the Np

most significant taps are selected. The number of selected taps

Npcan be estimated using the preamble or are pre-determined

according to the considered channel environment as in [14].

IV. SIMULATION RESULTS

In this section, numerical results for the proposed CDD-

OFDM system are provided. In the system simulation, five

OFDM symbols are grouped together to form a data frame. For

each data symbol, standard compatible tail biting convolutional

code with generator (133,171,165)oct is used for channel

coding and the code rate is set to 1/2. The number of

subcarriers is N = 128 and the CP is set to G = N/16 = 8.

In addition, QPSK modulation scheme is used.

Chirp sequence P(k) = exp(jπ(k(k + 2))/N),0 ≤ k ≤

N − 1 is used as the superimposed pilot sequence. The

advantage of using chirp sequence is its constant envelope

property in both the time and frequency domains. Hence it

can keep the peak-to-average power ratio (PAPR) low, as is

essential in OFDM systems. The power allocation for both

pilot and data is set to ρ = 0.2 and 1 − ρ = 0.8 respectively,

with unit power in total. Therefore the pilot takes about 20%

of the total transmit power, which is a common pilot allocation

ratio for channel estimation in an OFDM system.

Channel model with three multipath components is used

for each branch individually. The taps are at sampling clock

normalized delays [0 2 4] with relative powers of [0 -

5 -10] dB. The OFDM sampling period is Ts = 0.2μs.

The time variation of each tap follows Jakes model and the

corresponding Doppler spread for all the taps is the same. The

cyclic delay for the first and second antenna is set to δ0= 0

and δ1 = 32 respectively, which means maximum channel

length of the equivalent channel is Leq= δ1− δ0+ L1= 37.

Fig.3 shows the normalized mean squared error (NMSE) of

the proposed iterative channel estimation scheme. The normal-

ized maximum Doppler frequency offset in this simulation is

fdNTs= 0.02. The initial estimate is obtained by averaging

within the frame and the lower bound is obtained based on

the known transmitted data and pilot (but still impacted by

the noise). As shown in the figure, the NMSE decreases as

the number of iterations increases. In the SNR region below

5dB, the iteration procedure does not improve estimation

performance. This is due to the error propagation at high noise

level. As the SNR increases, the iteration can improve the

estimate significantly and approach the lower bound.

Fig.4 shows the corresponding BER performance for the

proposed system. As indicated, the BER performance can be

improved as the iteration number increases, particularly in the

region where SNR is higher than 5dB. The reason for the

improvement is that more reliable channel information is used

in equalization and soft demodulation in each iteration.

Fig.5 compares BER performance between single antenna

OFDM and CDD-OFDM with two transmit antennas. In the

cases of ideal channel estimation, CDD-OFDM system shows

051015 202530

10

−4

10

−3

10

−2

10

−1

10

0

SNR(dB)

NMSE

Initial channel estimation

Channel estimation with 1 iteration

Channel estimation with 2 iterations

Channel estimation with 3 iterations

Lower bound

Fig. 3. NMSE of channel estimate for CDD-OFDM system with two transmit

antennas (fdNTs= 0.02).

02468 1012 1416 18 20

10

−5

10

−4

10

−3

10

−2

10

−1

10

0

SNR(dB)

Coded BER

Initial channel estimation

Channel estimation with 1 iteration

Channel estimation with 2 iterations

Channel estimation with 3 iterations

Ideal channel estimation

Fig. 4.

antennas (fdNTs= 0.02).

BER performance for coded CDD-OFDM system with two transmit

about 3dB gain at BER = 10−3. Even though channel esti-

mation error will lead to system performance degradation in a

practical scenario, the BER performance of CDD-OFDM with

the proposed algorithm is still better compared to the single

antenna OFDM with ideal channel estimation. This shows that

the diversity gain introduced by CDD can be retained and is

about 2dB at a BER = 10−3.

V. CONCLUSION

We have proposed an iterative channel estimation for CDD-

OFDM systems using superimposed pilots. As each pilot is

being added to each OFDM subcarrier individually, the system

can track the highly frequency selective channel caused by a

large delay spread. In the proposed iteration procedure, the

EM algorithm is performed based on the superimposed pilots

and temporarily soft demodulated symbols. To exploit channel

sparsity, a sorting based criterion is used to choose the non-

zero taps with the most significant power. This procedure can

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Page 5

02468101214161820

10

−5

10

−4

10

−3

10

−2

10

−1

10

0

SNR

BER

Single antenna OFDM with proposed C.E.

Single antenna OFDM with ideal C.E.

Two antenna CDD−OFDM with proposed C.E.

Two antenna CDD−OFDM with ideal C.E.

Fig. 5.

antenna and CDD-OFDM with two transmit antennas (fdNTs= 0.02).

BER performance comparison between OFDM with single transmit

avoid over modeling by reducing the linear model dimension,

therefore alleviate the noise impact and improve the estimation

performance. Numerical results show that the proposed system

achieves robust performance and saves pilot overhead in highly

frequency selective fading channels.

APPENDIX A

Q(h|hq) in the E-Step can be decomposed as

Q(h|hq) =

i

+ hHFH(P + Di)H(P + Di)Fh?.

?

Pr(di|r;hq)??r?2− 2?(rH(P + Di)Fh)

(18)

Since?

Q(h|hq) = ?r?2− 2?{rH(P + D)Fh}

+

Pr(di|r;hq)hHFH(P + Di)H(P + Di)Fh,

where D =?Pr(di|r;hq)Di. The third term in (19) can be

?

= hHFHPHPFh + 2hHFH?(DHP)Fh

+

Pr(di|r;hq)hHFHDH

iPr(di|r;hq) = 1, above formula can be simplified

as

(19)

?

i

decomposed as

i

Pr(di|r;hq)hHFH(P + Di)H(P + Di)Fh

?

i

iDiFh.

(20)

Denote Rd as the diagonal matrix with symbol variances,

which is given by

?

Substituting (21) into (20), we have

?

= hHFHPHPFh + 2hHFH?(DHP)Fh + hHFHRdFh.

Rd=

i

Pr(di|r;hq)DH

iDi.

(21)

i

Pr(di|r;hq)hHFH(P + Di)H(P + Di)Fh

(22)

Substituting (22) back into (19), Q(h|hq) is obtained as

Q(h|hq) = ?r?2− 2?{rH(P + D)Fh}

+ hHFH{PHP + Rd+ 2?(DHP)}Fh.

(23)

From (23), to minimize Q(h|hq), let

the property of DFT matrix FHF = N I, the estimateˆh is

given by

ˆh = [FH(PHP + Rd+ 2?(DHP))F]−1FH(P + D)Hr

=

NFH[PHP + Rd+ 2?(DHP)]−1(P + D)Hr.

Since P, Rd and D are all full rank diagonal matrices,

[PHP + Rd+ 2?(DHP)] is non-singular andˆh exists.

When constant envelope pilots and modulation (e.g. BPSK,

QPSK) are used, it yields PHP = ρI, where ρ denotes the

power allocation ratio for pilot symbols. As the transmission

power is normalized, DH

In this case, the CIR estimate can be further simplified as

1

NFH[I + 2?(DHP)]−1(P + D)Hr.

∂Q(h|hq)

∂ h

= 0. Using

1

(24)

iDi= (1−ρ)I, we have Rd= (1−ρ)I.

ˆh =

(25)

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This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE ICC 2011 proceedings