Eﬀect of Pulse Shaping Filters on a Fractional Sampling ... · tiplexing (FS OFDM) system is...

1484IEICE TRANS. COMMUN., VOL.E92–B, NO.5 MAY 2009

PAPER Special Section on Radio Access Techniques for 3G Evolution

Effect of Pulse Shaping Filters on a Fractional Sampling OFDMSystem with Subcarrier-Based Maximal Ratio Combining

Mamiko INAMORI†a), Student Member, Takashi KAWAI†b), Tatsuya KOBAYASHI†c), Nonmembers,Haruki NISHIMURA†d), Student Member, and Yukitoshi SANADA†e), Member

SUMMARY In this paper, the effect of the impulse response of pulseshaping filters on a fractional sampling orthogonal frequency division mul-tiplexing (FS OFDM) system is investigated. FS achieves path diversitywith a single antenna through oversampling and subcarrier-based maxi-mal ratio combining (MRC). Though the oversampling increases diversityorder, correlation among noise components may deteriorate bit error rate(BER) performance. To clarify the relationship between the impulse re-sponse of the pulse shaping filter and the BER performance, five differentpulse shaping filters are evaluated in the FS OFDM system. Numerical re-sults of computer simulations show that the Frobenius norm of a whiteningmatrix corresponding to the pulse shaping filter has significant effect on theBER performance especially with a small numbers of subcarriers. It is alsoshown that metric adjustment based on the Frobenius norm improves BERperformance of the coded FS OFDM system.key words: OFDM, fractional sampling, diversity, pulse shaping filter

1. Introduction

Orthogonal frequency division multiplexing (OFDM) andorthogonal frequency code division multiplexing (OFCDM)have received large attention as modulation schemes to real-ize broadband transmission. OFDM based systems achievehigh frequency utilization efficiency due to orthogonalitybetween subcarriers. The primary advantage of OFDMbased schemes over single carrier schemes is its robustnessto severe multipath channels. OFDM based schemes havebeen implemented in various wireless standards such asthe IEEE 802.11 standards, digital terrestrial broadcasting,mobile worldwide interoperability for microwave access(WiMAX), or international mobile telecommunications-advanced (IMT-Advanced).

Many diversity schemes have been actively investi-gated for OFDM based systems [2]–[4]. One of the typi-cal diversity schemes is antenna diversity in which multipleantenna elements are implemented in a receiver [2]. How-ever, it may be difficult to implement multiple antenna ele-ments in a small mobile terminal. Therefore, a new diver-sity scheme called fractional sampling (FS) has been pro-posed in [5]. This scheme tries to acquire diversity gain

Manuscript received August 25, 2008.Manuscript revised December 25, 2008.†The authors are with the Dept. of Electronics and Electrical

Engineering, Keio University, Yokohama-shi, 223-8522 Japan.a) E-mail: [email protected]) E-mail: [email protected]) E-mail: [email protected]) E-mail: [email protected]) E-mail: [email protected]

DOI: 10.1587/transcom.E92.B.1484

through the signal sampled faster than the Nyquist rate inthe receiver. FS is known to covert a single-input single-output channel into a single-input multiple-output channel.In [5], subcarrier-based noise whitening and maximal ratiocombining (MRC) have been investigated because of its lowcomplexity.

Though the oversampling increases diversity order,correlation among noise components may deteriorate bit er-ror rate (BER) performance. In order to solve this prob-lem, a frequency spreading scheme for OFCDM has beenproposed [6]. This scheme cancels the correlated noisecomponents among adjacent subcarriers and improves theBER performance. However, the proposed scheme reducesthe number of available spreading codes. Moreover, thisscheme is not applicable to OFDM systems. Since the noisepasses through the pulse shaping filter (baseband filter) inthe receiver, the impulse response of the filter determinesthe correlation among the noise components. In order to pre-vent the BER degradation due to the correlated noise com-ponents without spreading codes, impulse responses of thepulse shaping filter are evaluated for the FS OFDM systemin this paper. The effect of the impulse response of the filteron the BER performance is then clarified.

This paper is organized as follows. Firstly, a systemmodel is described briefly in Sect. 2. The correlation amongthe noise components is then discussed in Sect. 3. Numer-ical results are shown in Sect. 4. Finally, conclusions arepresented in Sect. 5.

2. Receiver Structure with Fractional Sampling

At the receiver side, FS and MRC are used to achieve diver-sity over a multipath channel [5]. The block diagram of anOFDM receiver with FS is shown in Fig. 1. The transmitted

Fig. 1 Block diagram of a receiver.

Copyright c© 2009 The Institute of Electronics, Information and Communication Engineers

INAMORI et al.: EFFECT OF PULSE SHAPING FILTERS ON A FS OFDM SYSTEM WITH SUBCARRIER-BASED MRC1485

signal with the guard interval (GI), u[l], is given as

u[l] =1√N

N−1∑k=0

s[k]e− j2πkl/N , l = 0, . . . , P − 1, (1)

where N is the inverse discrete Fourier transform (IDFT)length, s[k] is the symbol transmitted on the k-th subcarrier,P is the sum of the IDFT length and the length of GI. Thereceived signal, y(t), is expressed as follows,

y(t) =P−1∑l=0

u[l]h(t − lTs) + v(t), (2)

where 1/Ts is the baud rate, h(t) is the impulse response ofthe composite channel and is given by h(t) = (p� c� p′)(t),� denotes convolution, p(t) is the impulse response of thepulse shaping filter (=Tx or Rx baseband filter), p′(t) =p(−t), c(t) is the impulse response of the physical channel,and v(t) is the additive white Gaussian noise [5]. The re-ceived signal which is sampled at a rate of G/Ts is expressedas follows,

yg[n] =P−1∑l=0

u[l]hg[n − l] + vg[n], g = 0, · · · ,G − 1,

(3)

where n is the time index, yg[n] = y(nTs + gTs/G), hg[n] =h(nTs + gTs/G), and vg[n] = v(nTs + gTs/G). The demod-ulated signal received on the k-th subcarrier, z[k], is derivedafter removal of the GI and demodulation by the DFT at thereceiver for each g. z[k] is expressed as

z[k] = H[k]s[k] + w[k], k = 0, · · · ,N − 1, (4)

where

z[k] = [z0[k], · · · , zG−1[k]]T , (5)

zg[k] =1√N

N−1∑n=0

yg[n]e− j2πkn/N , (6)

H[k] = [H0[k], · · · ,HG−1[k]]T , (7)

Hg[k] =L−1∑n=0

hg[n]e− j2πkn/N , (8)

w[k] = [w0[k], · · · , wG−1[k]]T , (9)

wg[k] =N−1∑n=0

vg[n]e− j2πkn/N , (10)

and L is the number of multipath.When sampling at the receiver is carried out at the baud

rate of 1/Ts, we have a usual OFDM input/output relation-ship with white noise. However, when sampling is per-formed at the multiple of the baud rate, the noise is colored.Noise whitening is necessary because MRC maximizes thesignal-to-noise ratio (SNR) when the noise is white. Inorder to take subcarrier-based MRC combining approach,subcarrier-by-subcarrier noise whitening is carried out. Thecovariance matrix of the noise on the k-th subcarrier is given

as

Rw[k] = E[w[k]wH[k]], (11)

where E[ ] denotes expectation and H represents Hermitiantranspose. After noise whitening, Eq. (4) is converted as

R− 1

2w [k]z[k] = R

− 12w [k]H[k]s[k] + R

− 12w [k]w[k]. (12)

This equation turns to the following expression.

z′[k] = H′[k]s[k] + w′[k], (13)

where R− 1

2w [k]z[k] = z′[k], R

− 12w [k]H[k] = H′[k], and

R− 1

2w [k]w[k] = w′[k]. The estimate of s[k], s[k], through

MRC is then given as

s[k] =H′H[k]z′[k]

H′H[k]H′[k]

=(R− 1

2w [k]H[k])HR

− 12w [k]z[k]

(R− 1

2w [k]H[k])HR

− 12w [k]H[k]

. (14)

3. Noise Correlation among Samples

In order to derive the effect of the noise whitening, the re-ceived signal is expressed in the vector form in this section.From Eq. (4), the received signal for all N subcarriers is ex-pressed as

z = Hs + w, (15)

where

z = [zT [0], · · · , zT [N − 1]]T , (16)

H = diag[H[0], · · · ,H[N − 1]], (17)

s = [s[0], · · · , s[N − 1]]T , (18)

w = [wT [0], · · · ,wT [N − 1]]T . (19)

The noise vector w is colored and can be expressed as

w = R12wω, (20)

where Rw is the correlation matrix of the noise, ω is thewhite noise in the vector form and it is given as

ω = [ωT [0], · · · ,ωT [N − 1]]T , (21)

ω[k] = [ω0[k], · · · , ωG−1[k]]T , (22)

and ωg[k] is the white noise of the g-th sample componenton the k-th subcarrier. The noise covariance matrix is Rw :=E[wwH] whose (k1G + g1, k2G + g2)-th element is given by

E[wg1[k1]w∗g2[k2]]

= σ2v

1N

N−1∑n1=0

N−1∑n2=0

p2((n2 − n1 + (g2 − g1)/G)Ts)

× e j 2πN (k2n2−k1n1) (23)


where p2(t) is the composite response of the filters givenas p2(t) = (p � p)(t), σ2

v is the variance of v(t), {k1, k2} =0, · · · ,N − 1, and {g1, g2} = 0, · · · ,G − 1. After subcarrier-based noise whitening, Eq. (15) is converted as

Rwwz = RwwHs + Rwww, (24)

where Rww = diag[R− 1

2w [0], · · · ,R− 1

2w [N−1]]. Eq. (24) results

in the following equation.

z′ = H′s + w′, (25)

where

z′ = Rwwz

= [z′T [0], · · · , z′T [N − 1]]T , (26)

H′ = RwwH

= diag[H′[0], · · · ,H′[N − 1]], (27)

and

w′ = [w′T [0], · · · ,w′T [N − 1]]T

= Rwww

= RwwR12wω

=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

IG Rn[0, 1] · · · Rn[0,N − 1]

Rn[1, 0] IG. . .

......

. . .. . .

...Rn[N − 1, 0] · · · · · · IG

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

×

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣ω[0]ω[1]...

ω[N − 1]

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ , (28)

where Rn[k1, k2] is the G ×G matrix, which corresponds to

the (k1, k2)-th subblock of the NG×NG matrix, RwwR12w . The

g1-th element of w′[k1] is expressed as

w′g1 [k1] =N−1∑k2=0

G−1∑g2=0

[Rn[k1, k2]]g1 ,g2ωg2 [k2]

= ωg1 [k1] +N−1∑k2=0k2�k1

G−1∑g2=0

[Rn[k1, k2]]g1,g2ωg2 [k2],

(29)

where [Rn[k1, k2]]g1,g2 is the (g1, g2)-th element of Rn[k1, k2].The second term of the right side of this equation gives thecorrelation between the noise components after subcarrierbased noise whitening. These components may deterioratethe BER performance of the receiver.

An example of the correlation among the noise com-

ponents, R12w , over the subcarriers and oversampling indexes

(k1G + g1 ≤ 100, k2G + g2 ≤ 100) is shown in Fig. 2. A sincpulse is assume as the impulse response of the pulse shap-ing filter. From Eq. (23), this is the function of the impulse

Fig. 2 Correlation of the noise components (logarithm representation ofabsolute value).

response of the pulse shaping filter in the receiver. In orderto improve the BER performance, the Frobenius norm ofRn[k1, k2] should be kept small. Here, the Frobenius normof a matrix A is given as

||A||F =√√√G−1∑g1=0

G−1∑g2=0

([A]g1,g2 )2 (30)

where || · ||F denotes the Frobenius norm [7].From Eq. (28), Rn[k1, k2] is given as follows.

Rn[k1, k2] = R− 1

2w [k1]R

12w [k1, k2], (31)

where R12w[k1, k2] is the (k1, k2)-th subblock of R

12w . The dis-

tribution of the eigenvalues for particular types of matriceshas been investigated, especially for MIMO transmission[8], [9]. However, the distribution is not given for general

matricies. Since R12w [k1, k2] does not have a specific matrix

structure, the eigenvalues of R12w [k1, k2] are not able to be

analysed. On the other hand, since R12w [k1] is an Hermitian

matrix and positive semidefinite, the Frobenius norm of the

whitening matrix, R− 1

2w [k1], is given by

||R− 12w [k1]||2F =

G−1∑g=0

(1/λg[k1])2, (32)

where λg[k1] is the g-th eigenvalue of R12w[k1] [7]. Also,

||Rw[k1]||2F =∑G−1g=0 (λg[k1])4. From Eq. (23), the (g1, g2)-th

element of Rw[k1] is given as

[Rw[k1]]g1 ,g2

= σ2v

1N

N−1∑nd=−N+1

(N − |nd |)p2((nd + (g2 − g1)/G)Ts)

× e− j2πknd

N . (33)

From Eqs. (30), (32) and (33), the inverse of the Frobenius


norm of R− 1

2w [k1] is related to the specturm of the com-

posite response, p2(t), because of the term, p2((nd + (g2 −g1)/G)Ts)e− j

2πkndN , which is the same as the one in the dis-

crete Fourier transform. Therefore, in this paper, the Frobe-nius norm of the whitening matrix for five different impulseresponses of the pulse shaping filter is investigated throughcomputer simulation.

4. Numerical Results

4.1 Simulation Conditions

Simulation conditions are shown in Table 1. The data ismodulated with QPSK, 16QAM, and 64QAM, and multi-plexed with OFDM. The bandwidth of the OFDM system is80 MHz. The DFT size is 64 and 1024 while the number ofdata subcarriers is 48 and 768. The received signal is sam-pled at the rates of 1/Ts, 2/Ts, and 4/Ts (G = 1, 2, 4). Aschannel coding, a convolutional code with a coding rate of1/2 is used. Soft decision Viterbi decoding is employed inthe receiver.

4.2 Channel Models

In this paper, three channel models are assumed in the sim-ulation. One is 6-ray GSM Typical Urban model. The 6-ray GSM Typical Urban model is one of the propagationmodels that are mentioned in the main body of 3GPP TS45.005. The parameters of the 6-ray GSM Typical Urbanmodel (TU6) is defined in Table 2. Figure 3 shows the delayprofile of 6-ray GSM Typical Urban model. The amplitudeof the path follows Rayleigh distribution. The others areRayleigh fading channel models. A 16 path Rayleigh fadingmodel with a uniformed delay profile is shown in Fig. 4(a)[5]. The interval between the path delays in this model isTs/4. A 24 path Rayleigh fading model with an exponentialdelay profile is shown in Fig. 4(b) [1]. The interval betweenthe path delays is 5Ts. The channel response is assumed tobe constant during one OFDM symbol interval. The GSMmodel and 24 path Rayleigh fading model are applied to theOFDM system with 1024 subcarriers.

Table 1 Simulation conditions.

Bandwidth 80 MHzNumber of data subcarriers 48/768

Guard interval 0.2/3.2 [μsec]Subcarrier spacing Δ f 1250/78.1 [kHz]

Number of IDFT points 64/1024DFT sampling speed Ts 12.5 [nsec]

Data modulation QPSK, 16QAM, 64QAM/OFDMChannel estimation Ideal

Fractional sampling ratio G 1,2,4Channel model Rayleigh fading

(16path uniform/24path exponential)GSM Typical urban model

Channel coding Convolutional code(R = 1/2, K = 7)

Channel decoding Soft decision Viterbi decoder

4.3 Pulse Shaping Filters

To clarify the effect of the spectrum of the filter on the BERperformance, the pulse shaping filter with the following 5different impulse responses are employed both at the trans-mitter and the receiver [5], [10]. These impulse responseshave different bandwidths and are classified into two differ-ent types, which are a sinc-based pulse shaping filter and acosine-based pulse shaping filter [11].

Table 2 6-ray GSM Typical Urban model parameters.

Tap Relative Average relative Delaynumber time (μs) power (dB) in samples

1 0.0 −3.0 02 0.2 0.0 33 0.5 −2.0 84 1.6 −6.0 255 2.3 −8.0 356 5.0 −10.0 77

Fig. 3 6-ray GSM Typical Urban model.

(a) 16 path Rayleigh fading model with uniform delay spread.

(b) 24 path Rayleigh fading model with exponential delay spread.

Fig. 4 Multipath Rayleigh fading channel models.


(a) Impulse responses of the pulse shaping filter.

(b) Frequency spectrums of the pulse shaping filter.

Fig. 5 Graphical illustration of the pulse shaping filters.

(1) Sinc-based pulse shaping filter

1. Sinc pulse truncated to [−Ts,Ts] β = 0.64 is given by

g(t) = sinc

(t

Ts

)rect

(t

2Ts

). (34)

2. Fourth-power sinc pulse over [−Ts,Ts], β = 0.164 isgiven by

g(t) = sinc4

(t

Ts

)rect

(t

2Ts

). (35)

3. Fifth-power sinc pulse truncated to [−Ts,Ts], β =0.1046 is given by

g(t) = sinc5

(t

Ts

)rect

(t

2Ts

). (36)

(2) Cosine-based pulse shaping filter

1. Quadrature overlapped cubed raised cosine (QOCRC)

pulse truncated to [−Ts,Ts], β = 0.125 is given by

g(t) = cos4

(πt

2Ts

)rect

(t

2Ts

). (37)

2. Quadrature overlapped squared raised cosine (QOSRC)pulse truncated to [−Ts,Ts], β=0.25 is given by

g(t) = cos6

(πt

2Ts

)rect

(πt

2Ts

). (38)

Here, β represents the scaling effect on the received signalsamples due to the pulse shape at the offset sampling in-stants of ±Ts/2. The impulse responses and the frequencyresponses of the pulse shaping filter are shown in Fig. 5(a)and 5(b). As a reference, a root cosine roll-off filter (roll-offfactor α = 0) with the duration of ±4Ts is shown in Fig. 5(b).In this case, as suggested in [5], no diversity gain is obtainedbecause of the sharp frequency response of the filter.

4.4 Frequency Spectrum of the Filter and Frobenius Normof the Whitening Matrix

Figures 6 and 7 show the Frobenius norm of the whiten-

ing matrix, R− 1

2w [k1], with different impulse responses of the

pulse shaping filter. The number of subcarriers is 64 andthe oversampling ratio is set to G = {2, 4}, here. As itis suggested with Eq. (33), there is a relation between thespectrum of the filter and the Frobenius norm (though it isnot exactly the same as the spectrum of the filter due to theweighting term (N − |nd |)). The number of dip points inthe spectrum is proportional to the number of peak pointsin the Frobenius norm. If the oversampling ratio increases,the interval of the samples, Ts/G, reduces in Eq. (33). Thus,Eq. (33) covers larger spectrum. Both the number of dippoints in the spectrum and the number of peak points in theFrobenius norm then increase. If the number of subcarri-ers increases, the resolution of the spectrum in Eq. (33) im-proves, the depth of the dip points in the spectrum becomeslarger, and the peak value of the Frobenius norm grows.

4.5 Uncoded FS OFDM

4.5.1 Effect of Pulse Shaping Filter with 64 Subcarriers

In Figs. 9, 10 and 11, the BER curves with different pulseshaping filters for G = {1, 2, 4} on the 16 path Rayleigh fad-ing channel are presented. From Fig. 9, when G = 1, all ofthe BERs except the one with the sinc pulse filter are almostthe same. The reason is that the bandwidth of the sinc pulsefilter is smaller than those of the other filters. Thus, the SNRof the subcarriers in the band edges are smaller and more biterrors are observed. When G = 2, the whitening matrix forthe sinc pulse filter shows the large amount of the norm onthe specific subcarriers in Fig. 6. These subcarriers gener-ate more bit errors due to the correlated noise. As a result,the BER with the sinc pulse filter is larger than those withthe other filters. The norm of the whitening matrix with the


Fig. 6 Frobenius norm of the whitening filter for different impulseresponses (Number of subcarriers = 64, G = 2).


QOSRC pulse filter is slightly larger than the rests of thefilters except the sinc pulse filter. Therefore, the BER per-formance is also slightly worse.

When G = 4, BER performances of QOSRC pulse fil-ter and the fourth-power sinc pulse filter are deteriorated.From Fig. 7, the norm with the QOSRC pulse filter showsthe largest and the fourth-power sinc pulse filter shows thesecond largest on the specific subcarriers. On those subcar-riers more bit errors are produced. Even though Eb/N0 in-creases, the BER does not reduce as those with the QOCRCpulse filter or the fifth-power sinc pulse filter. Therefore, theBER curves for those pulse filters in Fig. 11 are worse thanthose with the other filters except the sinc pulse filter. Forthe case of the sinc pulse filter, the bandwidth of the filteris smaller than the others as shown in Fig. 6. It is suggested


Fig. 9 BER performance vs. Eb/N0 on the 16 path Rayleigh fading chan-nel with the uniform delay profile (QPSK, Number of subcarriers = 64,G = 1).

in [5] that the excess bandwidth of the filter allows diversitygain in FS. The BER with the sinc pulse filter is then largerthan those with the QOCRC pulse filter or fifth-power sincpulse filter that has larger bandwidth.

Through Figs. 9–11, no diversity gain can be obtainedwith the root cosine roll-off filter. It has been suggested in[5] that the excessive bandwidth of the filter gives diversitygain with FS. Thus, the sharp frequency response of the rootcosine filter limits diversity gain.

Figures 12 and 13 show the BER curves with the differ-ent pulse shaping filters for G = 4 when 16QAM or 64QAMmodulation is employed. In those figures, the same tendencyon the BER performance can be observed as the case withQPSK modulation for G = 4. As far as we have investi-gated, when G = {1, 2}, the BER performance with 16QAMor 64QAM also show the same tendency as the case withQPSK.


Fig. 10 BER performance vs. Eb/N0 on the 16 path Rayleigh fadingchannel with the uniform delay profile (QPSK, Number of subcarriers =64, G = 2).


4.5.2 Effect of Pulse Shaping Filter with 1024 Subcarriers

The effect of the pulse shaping filters with larger numbers ofsubcarriers are also investigated. In Fig. 14, the BER curveswith different pulse shaping filters for G = 4 are presented.The channel model we assume here is 16 path Rayleigh fad-ing channel with the uniform delay profile. When G = 4, thecurves of the Frobenius norm of the whitening filters shownin Fig. 8 are different as compared to those in Fig. 7. In thiscase, the norm with the fourth-power sinc pulse filter showsthe largest and the one with QOSRC pulse filter shows thesecond largest on the specific subcarriers. On those sub-carriers more bit errors are produced. However, the BERsdo not increase significantly in Fig. 14 as compared to thosein Fig. 11. For the cases of the fourth-power sinc pulse fil-ter and the QOSRC pulse filter, though the bit error rates

Fig. 12 BER performance vs. Eb/N0 on the 16 path Rayleigh fadingchannel with the uniform delay profile (16QAM, Number of subcarriers =64, G = 4).


on some particular subcarriers are larger, it is averaged overthe BERs of the large number of subcarriers. On the otherhand, the BER with the sinc filter is larger than those withthe other filters because of the smaller bandwidth and lim-ited diversity gain. When G = {1, 2}, the same tendency onthe BER performance can be observed as the cases with 64subcarriers.

Figures 15 and 16 show the BER curves with differentpulse shaping filters on the 16 path Rayleigh fading channelmodel when 16QAM or 64QAM modulation is applied. Inthose figures, the same tendency on the BER performancecan be observed as the case with QPSK modulation for G =4.

Figures 17 and 18 show the BER performance forQPSK modulation with different pulse shaping filters on the24 path Rayleigh fading channel model and the GSM Typi-




cal Urban model. In those figures, the same as the case withthe 16 path Rayleigh fading channel model, the BER degra-dation due to the different pulse shaping filters is smallerthan that of the OFDM system with 64 subcarriers. Sincethe number of multipath is smaller for the GSM model thanthe Rayleigh fading channel model, diversity gain throughFS is smaller in Fig. 18. Thus, the BER curves for all theimpulse responses of the filter are a little worse than thosein Fig. 17.

4.6 Coded FS OFDM

Figure 19 shows the BER curves of the coded FS OFDMwith different impulse responses of the pulse shaping filter.The number of subcarriers is 64 and the oversampling ratiois set to G = 4, here. A rate 1/2 convolutional code with its


Fig. 17 BER performance vs. Eb/N0 on the 24 path Rayleigh fadingchannel with the exponential delay profile (QPSK, Number of subcarriers= 1024, G = 4).

generating matrix G = [1338, 1718] and interleaving speci-fied in the IEEE 802.11a standard are employed [12]. Softdecision Viterbi decoding is performed in the receiver. Ascompared to Fig. 11, the improvement of the BER curvesfor the QOSRC and fourth-power sinc pulse filters are lim-ited. This is because the effect of the large Frobenius normis spread over the subcarriers due to the channel coding andthe interleaving. To reduce the effect of the large Frobeniusnorm, the metric in the Viterbi decoder is adjusted accordingto the Frobenius norm on each subcarrier. Figure 20 showsthe BER curves of the coded FS OFDM with the adjustedmetric. In this figure, the BER curves with the QOSRC andfourth-power sinc pulse filters are improved as compared tothose in Fig. 19. The metric adjustment can mitigates theeffect of the large Frobenius norm.

Figure 21 shows the BER curves of the coded FS


Fig. 18 BER performance vs. Eb/N0 on the GSM Typical Urban model(QPSK, Number of subcarriers = 1024, G = 4).

Fig. 19 BER performance vs. Eb/N0 of Coded OFDM (QPSK, Numberof subcarriers = 64, G = 4).

OFDM with different impulse responses of the pulse shap-ing filter when the number of subcarriers is 1024. Fig-ure 22 shows the BER curves of the coded FS OFDMwith the adjusted metric according to the Frobenius norm.A rate 1/2 convolutional code with its generating matrixG = [1338, 1718] and interleaving specified in the IEEE802.16 standard are employed [13]. In those figures, thesame as the coded system, the BER is averaged over thelarge number of subcarriers and the difference due to thepulse shaping filters is smaller than that of the system with64 subcarriers.

5. Conclusions

In this paper, the effect of the pulse shaping filters on the FSOFDM system with subcarrier-based MRC has been inves-tigated. The Frobenius norm of the whitening filter closelycorresponds to the frequency spectrum of the pulse shap-

Fig. 20 BER performance vs. Eb/N0 of Coded OFDM with AdjustedMetric (QPSK, Number of subcarriers = 64, G = 4).

Fig. 21 BER performance vs. Eb/N0 of Coded OFDM (QPSK, Numberof subcarriers = 1024, G = 4).

ing filter. It has been shown that the Frobenius norm ofthe whitening matrix has significant effect on the BER per-formance irrespective of modulation schemes and channelmodels. If the Frobenius norm is large, the power of thecorrelated noise components increases and the BER on thecorresponding subcarrier is deteriorated. If the number ofsubcarriers is 64, the average BER also increases. Whenthe number of subcarriers is 1024, the large amount of theFrobenius norm is concentrated on the specific subcarriers.Although the more number of bit errors can be observed onthose subcarriers, it is less significant to the average BER ifthe number of subcarriers is large.

When the channel coding is employed, the large Frobe-nius norm deteriorates the total BER performance of the sys-tem with 64 subcarriers. It has also been shown that theViterbi decoder with the adjusted metric according to theFrobenius norm improves the BER performance.

As a conclusion, it is required to design the pulse shap-


Fig. 22 BER performance vs. Eb/N0 of Coded OFDM with AdjustedMetric (QPSK, Number of subcarriers = 1024, G = 4).

ing filters in order to reduce the amount of the correlatednoise caused by the Frobenius norm and to obtain diversitygain. The Frobenius norm can be calculated at the stageof designing the pulse shaping filters. Moreover, when thenumber of subcarrier is small and the pulse shaping filter in-troduces the large amount of the Frobeinus norm on the spe-cific subcarriers, the adjusted metric based on the Frobeniusnorm can mitigate the effect of the correlated noise compo-nents in the coded FS OFDM system.

Acknowledgments

This work is supported in part by a Grant-in-Aid for theGlobal Center of Excellence for high-Level Global Cooper-ation for Leading-Edge Platform on Access Spaces from theMinistry of Education, Culture, Sport, Science, and Tech-nology in Japan.

References

[1] H. Atarashi, S. Abeta, and M. Sawahashi, “Variable spreadingfactor-othogonal frequency and code division multiplexing (VSF-OFCDM) for broadband packet wireless access,” IEICE Trans.Commun., vol.E86-B, no.1, pp.291–299, Jan. 2003.

[2] N. Maeda, H. Atarashi, S. Abeta, and M. Sawahashi, “Antenna di-versity reception appropriate for MMSE combining in frequencydomain for forward link OFCDM packet wireless access,” IEICETrans. Commun., vol.E85-B, no.10, pp.1966–1977, Oct. 2002.

[3] K. Suto and T. Otsuki, “Space-time-frequency block codes over fre-quency selective fading channels,” IEICE Trans. Commun., vol.E87-B, no.7, pp.1939–1945, July 2004.

[4] N. Miki, H. Atarashi, and M. Sawahashi, “Effect of time diver-sity in hybrid ARQ considering space and path diversity for VSF-OFCDM downlink broadband wireless access,” Proc. PIMRC’04,vol.1, pp.604–608, Barcelona, Spain, Sept. 2004.

[5] C. Tepedelenlioglu and R. Challagulla, “Low-complexity multipathdiversity through fractional sampling in OFDM,” IEEE Trans. SignalProcess., vol.52, no.11, pp.3104–3116, Nov. 2004.

[6] M. Inamori, H. Nishimura, Y. Sanada, and M. Ghavami, “Fractionalsampling OFCDM with alternative spreading code,” Proc. Eleventh

IEEE International Conference on Communications Systems, Nov.2008.

[7] G. Golub and C.V. Loan, Matrix Computations, 3rd edition, TheJohns Hopkins University Press, 1996.

[8] S.L. Loyka, “Channel capacity of MIMO architecture using the ex-ponential correlation matrix,” IEEE Commun. Lett., vol.5, no.9,pp.369–371, Sept. 2001.

[9] M. Chiani, M.Z. Win, and A. Zanella, “On the capacity of spatiallycorrelated MIMO Rayleigh-fading channels,” IEEE Trans. Inf. The-ory, vol.49, no.10, pp.2363–2371, Oct. 2003.

[10] K. Zhang, Y.L. Guan, and B.C. Ng, “Full-rate orthogonal spacetime block code with pulse-shaped offset QAM for four transmit an-tennas,” IEEE Trans. Wirel. Commun., vol.6, no.4, pp.1551–1559,April 2007.

[11] I. Sasase, R. Nagayama, and S. Mori, “Bandwidth efficient quadra-ture overlapped squared raised-cosine modulation,” IEEE Trans.Wirel. Commun., vol.33, no.1, pp.101–103, Jan. 1985.

[12] Part 11: Wireless LAN Medium Access Control (MAC) and Phys-ical Layer (PHY) Specifications; Highspeed Physical Layer in the5GHz Band, 1999, IEEE.802.11a.

[13] Part 16: Air Interface for Fixed Broadband Wireless Access Sys-tems, 2004, IEEE Std. 802.16.

Mamiko Inamori was born in Japan in1982. She received her B.E. and M.E. degreesin electronics engineering from Keio University,Japan in 2005 and 2007, respectively. SinceApril 2007, she has been a Ph.D. candidate at theSchool of Integrated Design Engineering, Grad-uate School of Science and Technology, KeioUniversity. Her research interests are mainlyconcentrated on software-defined radio.

Takashi Kawai was born in Tokyo, Japan in1985. He received his B.E. degree in electron-ics engineering from Keio University, Japan in2009. His research interests are mainly concen-trated on OFDM.

Tatsuya Kobayashi was born in Yokohama,Japan in 1985. He received his B.E. degreein electronics engineering from Keio University,Japan in 2008. Since April 2008, he has been agraduate student in School of Integrated DesignEngineering, Graduate School of Science andTechnology, Keio University. His research in-terests are mainly concentrated on software de-fined radio.


Haruki Nishimura was born in Chiba,Japan in 1983. He received his B.E. degreein electronics engineering from Keio University,Japan in 2007. Since April 2007, he has been agraduate student in School of Integrated DesignEngineering, Graduate School of Science andTechnology, Keio University. His research in-terests are mainly concentrated on software de-fined radio.

Yukitoshi Sanada was born in Tokyo in1969. He received his B.E. degree in electricalengineering from Keio University, Yokohama,Japan, his M.A.Sc. degree in electrical engi-neering from the University of Victoria, B.C.,Canada, and his Ph.D. degree in electrical engi-neering from Keio University, Yokohama, Japanin 1992, 1995, and 1997, respectively. In 1997he joined the Faculty of Engineering, Tokyo In-stitute of Technology as a Research Associate.In 2000 he joined the Advanced Telecommuni-

cation Laboratory, Sony Computer Science Laboratories, Inc., as an asso-ciate researcher. In 2001 he joined the Faculty of Science and Engineering,Keio University, where he is now an associate professor. He received theYoung Engineer Award from IEICE Japan in 1997. His current researchinterests are in software-defined radio and ultra wideband systems.

Eﬀect of Pulse Shaping Filters on a Fractional Sampling ... · tiplexing (FS OFDM) system is...

Documents

Transcript of Eﬀect of Pulse Shaping Filters on a Fractional Sampling ... · tiplexing (FS OFDM) system is...