Effects of Mult-Path Channel upon Constellation of OFDM Systems

Zhenyu Zhang1,2, Lijia Ge1,2, Gang Liu2, Fanxin Zeng2, Tao Shuang2, Jianyuan Luo2

1. The 63rd Research Institute of the General Stall, PLA University of Science and Technology, Nanjing, China 2. Chongqing Key Laboratory of Emergency Communications, Chongqing Communication Institute, Chongqing, China

{cqzhangzy, gelijiaydf, fzengx}@yahoo.com.cn

Abstract—Based on time shift properties of discrete Fourier transform, the constellation point distribution of orthogonal frequency division multiplexing (OFDM) system in multi-path channel is analyzed. Although inter-symbol interference (ISI) of OFDM can be mitigated by using cyclic prefix whose length is longer than the maximum multi-path delay of wireless channel, the constellation of OFDM symbol sharply deteriorates, which will have an important effect on performance of OFDM systems. For different sub-channels with different carrier frequencies, the effects on constellation are quite distinct, that is, there exists frequency selectivity. In this paper, the frequency selective properties of constellation caused by multi-path channel are discussed and the corresponding simulation results are provided, which can be used as a theoretical reference for channel estimation of OFDM systems.

Keywords-OFDM system; signal constellation; multi-path fading; channel estimation

I. INTRODUCTION As a main interference source of wireless communications,

multi-path fading will make an important effect on system performance due to different propagation paths with different amplitude and delay times. Compared with single carrier systems with a complicated equalizer, an orthogonal frequency division multiplex (OFDM) communication system can be seen as a more efficient way to deal with multi-path problem [1]. The OFDM symbol duration increases for the lower rate parallel subcarriers, and hence the relative amount of dispersion in time caused by multi-path delay spread is decreased. With the further introduction of cyclic prefix (CP) in every OFDM symbol, inter-symbol interference (ISI) can be eliminated as long as multi-path delay spread is smaller than the length of CP [2-4].

In spite of no ISI, multi-path propagation will still cause the scattering and distortion of the received signal constellation, which will lead to the degradation of OFDM performance. In order to mitigate the effects, channel estimation is necessary, especially for 4G communication systems with high transmission speed and high mobility, such as WiMAX [5] and LTE [6]. Generally, channel estimation approaches can be categorized into three classes, namely pilot-aided method [7-10], blind method [11] and semi-blind method [12]. Although they possess different characteristics, all of these methods are

very helpful to improve the performance of OFDM systems in multi-path channel.

In this paper, the effects of multi-path fading on the received signal constellation of OFDM systems are analyzed on the basis of time shift properties of discrete Fourier transform (DFT). The theoretical analyses and simulation results show that such influence possess frequency selectivity to those constellation points in different sub-carriers of OFDM systems. In the case of strong multi-path amplitudes and long delay time, some of original constellation points will drop into error decision regions even if there exists a high signal-to-noise ratio (SNR) in the OFDM receiver. By depicting the constellation patterns of several typical wireless channels and comparing them with the results after adopting suitable channel estimation methods, we illustrate the effects of multi-path propagation on signal constellations and provide a reference for channel estimation method of OFDM communication systems.

This paper can be organized as follows. We first introduce the preliminary knowledge in Section II. Then, frequency selectivity of OFDM constellation points in multi-path fading channel is presented in Section III. In order to mitigate the effect of frequency selectivity, the improved performance of OFDM systems on the basis of corresponding channel estimation method is provided in Section IV. Finally, Section V summarizes the results.

II. PRELIMINARY KNOWLEDGE In OFDM system, data symbols are transmitted in parallel

on a large number of subcarriers. The complex data symbols are modulated on multiple subcarriers by an inverse discrete Fourier transform (IDFT). Let ,i kX denote the transmission data symbol in the k-th subcarrier of the i-th OFDM symbol. Then the n-th time domain sample signal ,i nx of the i-th OFDM symbol can be given by

21

, , ,0

1IDFTnkN j

Ni n i k i k

kx X X e

N

π−

=

⎡ ⎤= =⎣ ⎦ ∑ , (1)

where N denotes the size of inverse fast Fourier transform (IFFT), and 0 1n N≤ ≤ − .

This work was supported in part by NSFC Grant #61002034, #61271251 and #61271003, the Scientific and Technological Project of CQ CSTC Grant #2011AB2044, the Natural Science Foundation Project of CQ CSTC Grant #2010BB2203, Open Research Fund of CQKL S&IP Grant #CQSIP-2010-01, Open Fund of Chongqing Key Lab of Mobile Communications Technology. ____________________________________978-1-4673-5699-2 /13/$31.00 ©2013 IEEE

Supposed that multi-path delays are sample spaced, so the impulse response of multi-path channel can be modeled as

1

0( ) ( )

L

l ll

h n h nδ τ−

=

= −∑ , (2)

where ( )nδ denotes a impulse function and L represents the number of multi-paths. The notations lh and lτ denote the channel tap and delay sample of the l-th path, respectively.

Due to the insertion of CP in OFDM system, the linear convolution between time domain signal ,i nx and channel impulse response ( )h n can be calculated by cyclic convolution as long as the length of CP is longer than the maximum multi-path delay time. Then, the received data ,i kY after multi-path channel and N-point fast Fourier transform (FFT) can be expressed as

, , , ,i k i k i k i kY Y H N= + , (3)

where ,i kN denotes additive white Gaussian noise (AWGN) and the channel transfer function ,i kH satisfies

[ ]1

2 /,

0DFT ( ) ( )

Nj nk N

i kn

H h n h n e π−

−

=

= = ∑ .

From Eq. (3), in the presence of high SNR, the degradation of OFDM system performance is mainly caused by multi-path fading.

III. FREQUENCY SELECTIVITY OF CONSTELLATION POINTS IN DIFFERENT SUB-CARRIERS

Due to the insertion of CP in OFDM systems, the multi-path delays can be considered as the sum of different cyclic shift signals of the first path signal as long as the maximum delay is not longer than the length of CP. For cyclic shift signals, DFT possess the time shift property as follows.

Lemma 1 [13]: Let the N-point DFT of sequence ( )x n be ( )X k . Then the DFT of cyclic l-shift sequence

( )( ) ( )NNx n l R n− of sequence ( )x n satisfies

( )( ) 2 /( ) ( )j lk NNN

DFT x n l R n e X kπ−⎡ ⎤− =⎣ ⎦ . (4)

According to the lemma, the received OFDM symbol in multi-path channel satisfies the following corollary.

Corollary 1: When the maximum delay of multi-path

channel with impulse response 1

0( ) ( )

L

l ll

h n h nδ τ−

== −∑ is not

longer than the length of CP of OFDM system, the received OFDM symbol ,i kY without AWGN can be expressed as

12 /

, ,0

lL

j k Ni k i k l

lY X h e πτ

−−

=

= ∑ , (5)

θ

0h 1h

I

Q

0

,i kY

Figure 1. Vector diagram of OFDM constellation in multi-path channel.

-1 0 1 2 3-1

-0.5

0

0.5

1

1.5

2

2.5

3

I

Q

The transmitted constellation pointsThe received constellation points

Figure 2. The scattering and distortion of OFDM constellation points in

vehicular test channel A without AWGN.

-1 0 1 2 3-1

-0.5

0

0.5

1

1.5

2

2.5

3

I

Q

The transmitted constellation pointsThe received constellation points

Figure 3. The scattering and distortion of OFDM constellation points in

pedestrian test channel B without AWGN.

where ,i kX is the transmitted symbol corresponding to ,i kY .

From Eq. (5), the amplitude and phase of OFDM symbol will be affected by multi-path fading. If we consider the received OFDM symbol ,i kY as a vector in complex plane with in-phase part and quadrature part, then ,i kY becomes the sum

of L vectors corresponding to L propagation paths. Fig. 1 shows the idea, where we suppose a two-path channel with

12 /k Nθ πτ= − . From Fig. 1, the received OFDM symbol ,i kY marked “o” in red is really the sum of the first path with channel tap 0h and the second path with channel tap 1h .

It should be noted that Eq. (5) is a function of variable k corresponding to the k-th sub-carrier. Therefore, the effects of multi-path fading on different sub-carriers are quietly distinct, that is, there exists frequency selectivity.

In order to investigate frequency selectivity of OFDM constellation in multi-path channel, we will employ vehicular test channel A and pedestrian test channel B of IEEE 802.16e [5] as examples. Fig. 2 and Fig.3 show the scattering and distortion of OFDM constellation points in vehicular test channel A and pedestrian test channel B, respectively. For two figures, 128N = and the number of data sub-carriers is equal to 64. To conveniently illustrate the effects of multi-path, we supposed that there is no AWGN and all of 64 symbols of data sub-carriers are equal to ( )1 / 2j+ , where 1j = − .

From Fig. 2 and Fig. 3, we can see that the multi-path fading will have an important effect on OFDM constellation even if there exists no ISI. In addition, it is obvious that the distortions of constellations are quietly distinct for different sub-carriers.

If a suitable channel estimation method can’t be used in OFDM system in multi-path channel, the degradation of OFDM system performance will be catastrophic. Especially for the strong multi-path fading, e.g., pedestrian test channel B of IEEE 802.16e, the deterioration of performance is quite severe even if employing low order modulation and high SNR.

IV. IMPROVEMENT OF OFDM SYSTEM PERFORMANCE IN MULTI-PATH CHANNEL BASED ON CHANNEL ESTIMATION From the analytical results of the above section, channel

estimation is necessary for OFDM systems in multi-path fading. In this section, we will employ block pilot signals and least square (LS) algorithm to mitigate the influence of multi-path channel.

The simulation parameters of OFDM system can be shown in Table I. For channel models, we combine AWGN channel with vehicular test channel A and pedestrian test channel B. The detail parameters of two channel models in IEEE 802.16e are listed in Table II, where both of vehicular test channel A and pedestrian test channel B are six-path channels. From Table II, pedestrian test channel B has longer delay time and higher average power than vehicular test channel A. As a result, pedestrian test channel B model will cause stronger multi-path fading for OFDM system.

By employing LS algorithm channel estimation, the scattering and distortion of OFDM system resulting from multi-path fading will be greatly improved, which can be illustrated in Fig.4 and Fig.5. Compared Fig.4 with Fig. 2, we can see that the distant errors between the estimated constellation points and original constellation points are much smaller than that of between the unestimated constellation

TABLE I. SIMULATION PARAMETERS

Parameter Value

Bandwidth 2MHz

FFT size 128

Number of data sub-carriers 64

Sub-carrier frequency spacing 25kHz

Duration of OFDM symbol 50μs

CP duration 10μs

Modulation QPSK

Channel model AWGN channel,

vehicular test channel A , pedestrian test channel B

TABLE II. CHANNEL MODELS

Vehicular test channel A Pedestrian test channel B Tap Delay

(ns) Average Power

(dB) Delay (ns)

Average Power (dB)

1 0 0 0 0

2 310 -1 200 -0.9

3 710 -9 800 -4.9

4 1090 -10 1200 -8

5 1730 -15 2300 -7.8

6 2510 -20 3700 -23.9

-1 0 1 2 3-1

-0.5

0

0.5

1

1.5

2

2.5

3

I

Q

The original constellation pointsThe estimated constellation points

Figure 4. The distribution of OFDM constellation points in vehicular test

channel A with LS algorithm channel estimation and SNR=15dB.

points and original constellation points, which shows that channel estimation is very helpful. The similar results can be obtained from the comparison between Fig.5 with Fig. 3.

In order to show the effects of the multi-path fading upon bit error rate (BER) performance of OFDM system, we depict the BER curves in Fig. 6. The three red dashed lines in Fig.6 provide the BER performance without any channel estimation.

It is obvious that the case of no multi-path (only AWGN) is clearly better than the cases of two multi-path models. Especially for pedestrian test channel B, the performance of OFDM system can’t be improved when SNR increases. In fact, such results can be explained by Fig. 3 where some of the received constellation points have scattered and dropped into adjacent decision regions for QPSK even if no AWGN.

When LS algorithm channel estimation is employed, the performance of OFDM system in multi-path fading is greatly improved while that in AWGN channel is worse than the case without channel estimation. The phenomenon results from the characteristics of LS algorithm, that is, LS algorithm only considers the influence of multi-path propagation and ignores AWGN. When more efficient algorithm, e.g., minimum mean square error (MMSE) method, is used in channel estimation, the problem can be solved.

V. CONCLUSION This paper investigates the effects of multi-path fading on

the received complex-valued signal constellation points of OFDM systems by using cyclic shift property of DFT. The obtained corollary shows that the scattering and distortion of constellation points in multi-path fading can be considered as the adding result of multiple vectors corresponding to multiple paths. In terms of the idea, constellation points of different sub-carriers possess distinct distortions and some points drop into adjacent decision regions, which will cause the severe degradation of OFDM system performance. Hence, channel estimation of OFDM system in multi-path fading is necessary. When suitable channel estimation algorithm is employed, the performance will be clearly improved.

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-1 0 1 2 3-1

-0.5

0

0.5

1

1.5

2

2.5

3

I

Q

The original constellation pointsThe estimated constellation points

Figure 5. The distribution of OFDM constellation points in pedestrian test

channel B with LS algorithm channel estimation and SNR=15dB.

0 5 10 15 20 2510-6

10-5

10-4

10-3

10-2

10-1

100

Eb/N0(dB)

BE

R

AWGNLS, AWGNAWGN+Vehicular ALS, AWGN+Vehicular AAWGN+Pedestrain BLS, AWGN+Pedestrain B

Figure 6. The BER performance comparison of OFDM system in different

cases.