IEEE COMMUNICATIONS LETTERS, VOL. 7, NO. 11, NOVEMBER 2003 517

**Totally** Blind **APP** Channel Estimation

**for** Mobile **OFDM** Systems

Frieder Sanzi and Marc C. Necker

Abstract—A new two-dimensional **blind** **channel** **estimation**

scheme **for** coherent detection of orthogonal frequency-division

multiplexing (**OFDM**) signals in a **mobile** environment is

presented. The **channel** **estimation** is based on the a posteriori

probability (**APP**) calculation algorithm. The time-variant **channel**

transfer function is completely recovered without phase ambiguity

with no need **for** any pilot or reference symbols, thus maximizing

the spectral efficiency of the underlying **OFDM** system. The phase

ambiguity problem is solved by using a 4-QAM (quadrature

amplitude modulation) scheme with asymmetrical arrangement.

The results clearly indicate that totally **blind** **channel** **estimation** is

possible **for** virtually any realistic time-variant **mobile** **channel**.

Index Terms—Blind **channel** **estimation**, iterative decoding, orthogonal

frequency-division multiplexing (**OFDM**).

Fig. 1.

Transmitter and **channel** model.

I. INTRODUCTION

IN MOBILE communication **systems**, **channel** **estimation**

(CE) is an important issue, since the receiver needs to

have knowledge of the time-varying **channel** transfer function

(CTF) in order to per**for**m coherent detection. A common way

to support **channel** **estimation** is to periodically insert pilot

symbols into the transmitted data stream and use finite-impulse

response (FIR) interpolation filters, as described in [1], **for**

example.

Another method to estimate the **channel** is based on the calculation

of the a posteriori probability (**APP**), as described in

[2]. The **estimation** of the two-dimensional CTF is per**for**med

by a concatenation of two one-dimensional **APP** estimators in

frequency and time direction, respectively. This method enables

a dramatical reduction of the amount of pilot symbols compared

to the FIR interpolation method. Furthermore, the **APP** **channel**

**estimation** stage can be embedded into an iterative decoding

loop with a soft in/soft out decoder.

Since pilot symbols reduce the spectral efficiency, a lot of

work has been done in the area of **blind** **channel** **estimation**,

which makes pilot symbols unnecessary. Most research has focused

on methods based on higher order statistics, which converge

slowly, making them unsuitable **for** **mobile** environments.

Moreover, a phase ambiguity is introduced in the **channel** estimate,

which makes at least one reference symbol necessary

to resolve. In [3] the authors present a fast converging **blind**

**channel** estimator **for** **OFDM**-**systems** based on the Maximum

Likelihood principle, which recovers the **channel**’s amplitude

Manuscript received March 20, 2003. The associate editor coordinating the

review of this letter and approving it **for** publication was Dr. O. Sunay.

F. Sanzi is with the Institute of Telecommunications, University of Stuttgart,

70569 Stuttgart, Germany (e-mail: sanzi@inue.uni-stuttgart.de).

M. C. Necker is with the Institute of Communication Networks and Computer

Engineering, University of Stuttgart, 70569 Stuttgart, Germany (e-mail:

necker@ikr.uni-stuttgart.de).

Digital Object Identifier 10.1109/LCOMM.2003.820085

Fig. 2.

4-QAM constellation diagram with asymmetrical arrangement.

and phase without the need **for** any reference symbols. This is

achieved by combining two modulation schemes, such as QPSK

and 3-PSK.

In this paper we combine the idea of totally **blind** and **APP**

**channel** **estimation** (**APP**-CE). We use a modulation scheme

with an asymmetrical arrangement to solve the phase ambiguity

problem. An iterative loop consisting of the **blind** **APP**-CE

and an **APP** decoder is applied at the receiver to further reduce

the BER [2]. Our approach is not limited to two-dimensional

**channel** **estimation** and can also be applied to higher order modulation

schemes. In addition to the BER-per**for**mance, the iterative

decoding loop is studied with the EXtrinsic In**for**mation

Transfer chart (EXIT chart) [4].

II. SYSTEM MODEL

The block diagram of the transmitter is given in Fig. 1. The

sequence from the binary source is encoded by a convolutional

encoder. Its output signal consists of the coded bits ,

which is fed into the interleaver with output signal . After interleaving,

two successive coded bits are grouped and mapped

onto a 4-QAM symbol with asymmetrical arrangement as

shown in Fig. 2.

The signal is modulated onto orthogonal sub-carriers

by an iFFT-block. The transmission is done on a block-by-block

basis, with blocks of sub-carriers in frequency and **OFDM**

symbols in time direction. After iFFT, the cyclic prefix (CP) of

length 1/4 is inserted and the output is fed into the **channel**.

1089-7798/03$17.00 © 2003 IEEE

518 IEEE COMMUNICATIONS LETTERS, VOL. 7, NO. 11, NOVEMBER 2003

For the **mobile** **channel** we use the wide-sense stationary uncorrelated

scattering (WSSUS) **channel** model introduced in [5].

The time varying CTF can be expressed as

(1)

The variable denotes the number of propagation paths;

the phase is uni**for**mly distributed, the delay spread exponentially

distributed with probability density function (PDF)

. The Doppler-shift is distributed according to Jakes’

power spectral density. The autocorrelation function in time is

, where is the duration of one

**OFDM** symbol (useful part plus CP), the discrete time index

and the maximal Doppler shift. The complex auto-correlation

function in frequency direction can be calculated as

(see [6])

where is the sub-carrier spacing and is the discrete

frequency index. is chosen such that

with being the maximal delay spread.

At the receiver, we obtain the received 4-QAM constellation

points after removal of the cyclic prefix and **OFDM** demodulation

with FFT:

whereby are statistically i. i. d. complex Gaussian noise

variables with componentwise noise power

. The

are sample values of the CTF.

The signal is fed to the two-dimensional **blind** **APP**-CE

stage as shown in Fig. 3. This stage outputs log-likelihood ratios

on the transmitted coded bits which are deinterleaved

and decoded in an **APP** decoder. Iterative **channel** **estimation**

and decoding is per**for**med by feeding back extrinsic in**for**mation

on the coded bits; after interleaving it becomes the a priori

knowledge to the **blind** **APP**-CE stage.

III. **APP** CHANNEL ESTIMATION

A. One-Dimensional **APP** Channel Estimation

The two-dimensional **blind** **APP** **channel** estimator consists

of two estimators **for** frequency and time direction, respectively

[2]. This **estimation** algorithm exploits the time and frequency

continuity of the CTF at the receiver.

For one-dimensional **APP** **estimation**, the symbol-by-symbol

MAP-algorithm is applied to an appropriately chosen metric.

To help understanding, the symbols at the transmitter in

Fig. 1 can be thought of being put into a virtual shift register at

the output of the mapper. Owing to this “artificial grouping” the

corresponding trellis exploits the time and frequency continuity

of the CTF at the receiver.

At frequency index , the **APP** **estimation** in frequency direction

is characterized **for** **OFDM** symbol with

by the metric increment

(2)

(3)

(4)

Fig. 3.

Receiver with iterative **blind** **APP** **channel** **estimation**.

with estimated **channel** coefficient

whereby the FIR filter coefficients are calculated with the

Wiener–Hopf equation based on the frequency auto-correlation

function [2]. is the prediction order. The denote

the hypothesized transmitted data symbol according to the trellis

structure and the are the L-values of the coded bits

which are fed to the **APP** estimator in frequency direction. The

bits and in the sum in (4) result from the hard demapping

of . The term is the variance of the **estimation**

error in frequency direction according to [7]. The **APP** **estimation**

in time direction is done in a similar way **for** each subcarrier

taking the time auto-correlation function of the CTF into

account [2]. The two one-dimensional **APP** estimators are concatenated

as shown in Fig. 3. The output of the **APP** estimator

in frequency direction becomes the a priori input

of the **APP** estimator in time direction. The prediction order **for**

**estimation** in frequency and time direction, respectively, is set

to two in the following.

B. **Totally** Blind **APP** Channel Estimation

The concept of totally **blind** **channel** **estimation** was first

introduced in [3]. It is founded on the idea of exploiting the

correlation of the CTF in frequency direction and using two

different PSK-modulation schemes on adjacent subcarriers

to resolve the phase ambiguity. If the CTF does not vary

fast in frequency direction, the receiver can determine symbols

sent on adjacent subcarriers without any ambiguity by

solving the following equation system **for** adjacent subcarriers

( , ):

(6)

(7)

If only QPSK is used,

has to be chosen,

introducing the well known phase ambiguity. This can be

resolved by using two different PSK-modulation schemes on

adjacent subcarriers. Let be a signal point of the first

PSK-modulation scheme and a signal point of the

second modulation scheme. If

denotes

the angle between both signal points in the complex plane,

the signal points must be chosen such that no two angles

are identical **for** all possible signal point combinations. For

example, QPSK and 3-PSK fulfill this condition. In this case,

the equation system (6) and (7) can be solved without any

phase ambiguity, since can no longer be chosen arbitrarily. In

(5)

SANZI AND NECKER: TOTALLY BLIND **APP** CHANNEL ESTIMATION FOR MOBILE **OFDM** SYSTEMS 519

Fig. 4.

(a)

(b)

EXIT charts, **blind** **APP**-CE stage and decoder with simulated

trajectory of the iterative decoding loop at E =N =8 dB. (a) f =

100 Hz and =20s. (b) f = 300 Hz and =40s.

the noisy case, an ML-approach has to be used [3]. Simulations

showed good BER-per**for**mance with COST207-**channel**s RA

and TU. For **channel**s with longer delay spreads, this concept

imposes problems as the condition of a slow varying CTF in

frequency direction only holds **for** some subcarriers.

The **APP** **channel** estimator calculates the most likely

transmitted symbol sequence conditioned on the received

symbol sequence in consideration of the time and

frequency continuity of the CTF. Using symmetrical constellations

diagrams the transmission of pilot symbols is mandatory

necessary [2], as any symbol sequence is a possible

solution **for** the **APP**-CE. Extending the concept of totally

**blind** **channel** **estimation** to **APP**-CE, we can use a modulation

scheme with an asymmetrical constellation diagram, e.g., as

shown in Fig. 2, to solve the phase ambiguity. Again, this is

possible because can no longer be chosen arbitrarily.

IV. SIMULATION RESULTS

We use a sub-carrier spacing of kHz and an **OFDM**symbol

duration of

s. The cyclic prefix of length

s adds redundancy, corresponding to an in**for**mation

rate of

For the block-wise transmission we use adjacent

sub-carriers and consecutive **OFDM** symbols.

was chosen **for** the 4-QAM modulation according to

Fig. 2, and Gray mapping with two bits per symbol was used.

Hence, the interleaving depth is .

As an example, the used convolutional code is recursive systematic

with feedback polynomial

, feed-**for**ward

polynomial , memory 4 and code rate .

Note, that in the following all -values are given with respect

to the overall in**for**mation rate

The simulations were done with different maximal Doppler

shifts and different maximal delay spreads .As

per**for**mance measure, the bit error ratio (BER) at the output

of the hard decision device of the receiver in Fig. 3 is used.

(8)

(9)

Fig. 5.

BER **for** iterative **blind** **APP**-CE after 0 and 2 iterations.

Furthermore, we apply the EXIT chart [4] to gain insight into

the convergence behavior of the iterative decoding loop.

Fig. 4 shows the EXIT charts **for** different **channel** parameters

and

dB. The trajectories show the exchange

of in**for**mation between the **blind** **APP**-CE stage and the **APP**

decoder. For

Hz, the trajectory ends at the intersection

of the characteristic curves after one iteration. For

Hz the characteristic curve of the **blind** **APP**-CE

stage starts at a lower mutual in**for**mation , resulting in a

higher iterative decoding gain. Hence, two iterations are necessary

**for** optimal per**for**mance.

In Fig. 5 the BER is shown **for** the proposed system. As can

be seen, totally **blind** CE with coherent demodulation delivers

excellent BER-per**for**mance after very few iterations.

V. CONCLUSION

The concept of totally **blind** **channel** **estimation** was successfully

applied to two-dimensional **APP** **channel** **estimation**. The

result is a true **blind** **channel** estimator, which is capable of

estimating the time-variant **channel** transfer function including

its absolute phase. This is achieved without the need **for** any

reference symbols, thus maximizing the spectral efficiency of

the underlying **OFDM** system. Our results clearly indicate that

totally **blind** **channel** **estimation** is possible **for** virtually any realistic

time-variant **mobile** **channel**.

REFERENCES

[1] P. Höher, S. Kaiser, and P. Robertson, “Two-dimensional pilot-symbolaided

**channel** **estimation** by Wiener filtering,” in Proc. ICASSP, Munich,

Germany, Apr. 1997, pp. 1845–1848.

[2] F. Sanzi and S. ten Brink, “Iterative **channel** **estimation** and decoding

with product codes in multicarrier **systems**,” in Proc. IEEE Vehicular

Technology Conf. (VTC), Boston, MA, Sept. 2000, pp. 1338–1344.

[3] M. C. Necker and G. L. Stüber, “**Totally** **blind** **channel** **estimation** fpr

**OFDM** over fast varying **mobile** **channel**s,” in Proc. IEEE Int. Conf. on

Communications (ICC), New York, Apr. 2002, pp. 421–425.

[4] S. ten Brink, “Convergence behavior of iteratively decoded parallel concatenated

codes,” IEEE Trans. Commun., vol. 49, pp. 1727–1737, Oct.

2001.

[5] P. Höher, “A statistical discrete-time model **for** the WSSUS multipath

**channel**,” IEEE Trans. Veh. Technol., vol. 41, pp. 461–468, Nov. 1992.

[6] O. Ed**for**s, M. Sandell, J.-J. van de Beek, S. K. Wilson, and P. O. Börjesson,

“**OFDM** **channel** **estimation** by singular value decomposition,”

IEEE Trans. Commun., vol. 46, pp. 931–939, July 1998.

[7] P. Höher and J. Lodge, “Iterative decoding/demodulation of coded

DPSK **systems**,” in Proc. IEEE Global Telecommun. Conf. (Globecom),

Sydney, Australia, Nov. 1998, pp. 598–603.