Improved Nyquist filters using interpolation with sine functions

SETIT 2009 th

5 International Conference: Sciences of Electronic, Technologies of Information and Telecommunications March 22-26, 2009 – TUNISIA

Improved Nyquist filters using interpolation with sine functions Nicolae Dumitru ALEXANDRU *and Sorin POHOATA ** *

”Gh. Asachi” Technical University of Iaşi Department of Telecommunications Bd. Carol I, no.11, RO-700506 IAŞI-ROMANIA [email protected] **

”Ştefan cel Mare” University of Suceava str. Universităţii, no.13, RO-720225 SUCEAVA-ROMANIA [email protected] Abstract: This paper investigates a novel approach for constructing a family of ISI-free pulses that equal or ouperform some recently proposed pulses in terms of ISI performance in the presence of sampling errors. We propose a new family of improved Nyquist filters derived from an ideal staircase frequency characteristic using interpolation with sine functions. The performances of new ISI-free pulses are studied with respect to the ISI error probability. This family provides flexibility in designing an appropriate pulse even after the roll-off factor has been chosen. The results for error probability outperform the 4th degree polynomial pulse for a reasonable number of interpolation intervals. The proposed pulses for were also investigated for OFDM use including DVB systems in order to reduce their sensitivity to frequency offset. The results presented in this paper equal those of recently found pulses in terms of intercarrier interference (ICI) power. Key words: DVB, Nyquist filter, improved impulse response.

frequency domain by equation (1):

INTRODUCTION In order to deal with ISI (Inter-Symbol Interference), the common solution is to use a Nyquist filter with a frequency characteristic showing odd symmetry around cut-off frequency.

1, f  B1  α       G f B 1 α , B   1  α   f  B  S(f)   1  G B1  α   f , B  f  B1  α  0, B1  α   f 

Nyquist proved that one can use any type of lowpass filter with odd symmetry about the corresponding ideally band-limited cut-off frequency to replace the theoretically ideal characteristic, physically unrealizable. Recent works [ASA 04], [SAN 05], [CHA 05] have reported and examined new families of pulses which are inter-symbol interference (ISI)free and that asymptotically decay as t-3, t-2 and t-k for any integer value of k, respectively [CHA 05].

(1)

where G(f) is a function having G(0) = 1. In [ASA 04], [SAN 05] and [CHA 05], G(f) was chosen to be described by a particular waveform in the frequency interval B(1-α) ≤ |f| ≤ B in order to transfer spectral energy to the higher frequencies. This results in a pulse that decreases asymptotically as t-2, as compared with t-3 for the RC pulse.

Recently, new improved Nyquist pulses were introduced [BEA 01], [ASA 04], [SAN 05], [CHA 05] [ONO 08], [BEA 04] and [PEN 04], showing smaller maximal distortion, a more open eye diagram at the receiver and a smaller error rate in the presence of synchronization errors, i.e. sampling the received signal with an offset with regard to ideal sampling instants.

The resulted time domain pulses have a decreased size first side lobe at the expense of a slight increase in the size of remaining lobes. In a real transmission the received signal is sampled with an offset with regard to ideal sampling instants and due to the decrease of the first side lobe this results in a decrease of the intersymbol interference and a smaller value for the

The improved Nyquist pulses are defined in the

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SETIT2009

X k 1  α B   1 and X k (B)  1/2

probability of error. The constraints on transfer function Si f  such that the asymptotic decay rate ADR of the impulse response is t-k. For this, the first (k−2) derivatives of Si f  should be continuous and the (k−1)th derivative should have one or more finite amplitude discontinuities [BEA 04]. For k = 2 , this implies the the first derivative should have one or more finite amplitude discontinuities.

(3)

The rectangular functions used for this example appear to be desirable in minimizing the bandwidth and the results [ONO 08] evaluated in terms of error probability for a large number of parameters outperform those in most recent works.

2. Staircase characteristic 1. Proposed method

We shall consider as an example a staircase transfer function of the improved Nyquist filter depending on one parameter (k = 1) in equation (4):

In a previous paper [ONO 08] a low-pass filter with ideal piece-wise rectangular transfer characteristic showing odd symmetry about the corresponding ideally band-limited cut-off frequency was proposed and investigated. In order to derive the Nyquist filter characteristic one started from an ideal one composed of rectangles of equal width and obeying an odd-symmetry law. Its frequency characteristic was denoted as staircase characteristic. This frequency characteristic is defined for positive frequencies by equation (2) and is illustrated in Figure 1. The characteristic was built so that the shaded surfaces obeying odd-symmetry in Figure 1 are equal.

1,  a1,  ...  a k ,  X k (f)   1/2,  1  a ,  k ...  1  a1,  0

X1 (f)  F(α(a 1 ))

The resulting filter characteristic is defined by relation (5) and is dependent on the parameters α and a1.

1, a ,  1 X1 (f)  0.5, 1  a , 1  0,

f  (1  α)B

 

(1  α)B  f   1 

(4)

2k  1  α B 2k  1 

1  3 α B  f  1  1 α B  2k  1   2k  1      1  1 α B  f  1  1 α B  2k  1   2k  1     

0  f  1 α 1  α  f  1  α/3 1  α/3  f  1  α/3 1  α/3  f  1  α 1 α  f

(5)

The transfer function X 1 f  is constant over five intervals in the filter bandwidth for positive frequency, as shown in Figure 2. The interpolation points are marked with circles.

1  2k  1 α B  f  (1  α)B  2k  1    f  (1  α)B

Because the ideal piece-wise rectangular characteristic composed of rectangles of equal width is physicaly unrealizable, several interpolation techniques can be used in order that the filter frequency characteristic should be physically implemented.

(2)

Xk(f) 1 a1 a2 ak 1/2 1-ak 1-a2 1-a1 0

1- 

1

1+ 

f/B

Figure 2 Particular filter frequency characteristic (k = 1)

Figure 1 Proposed filter frequency characteristic The function X k f  was chosen in order to satisfy:

3. Interpolation method The -2-

method

for

constructing

the

filter

SETIT2009 characteristics proposed and investigated here uses sine functions approximation. Instead of approximating the staircase characteristic by a single polynomial on the entire interval [x 0 , x n ] as in [CHA 05] (polynomial pulse), we approximated the staircase characteristic by n sine functions that cross the rectangles that compose the ideal staircase transfer characteristic at the midpoint of their upper sides (see Figure 3 for k = 4).

f(x 2 )  y 2

(8)

Imposing that the sine function should be maximal flat in the interpolation point, so that the limit to the right of the derivative of sine function is zero, we get:

f(x 2 )  a  b  y 2  b  a  y 2

The idea is to link two points of coordinates (x1 , y1 ) and (x 2 , y 2 ) by a piece of a sine function, as shown in Figure 2 for a piece-wise rectangular transfer characteristic composed of 10 rectangles (k = 4), or using another arrangement.

(9)

Also,

2π m x 2    3π /2    3π /2 - 2 π mx 2

(10)

This condition was imposed in order for the interpolated characteristic to present discontinuities of the first derivative at the interpolation points. As the pieces of sine functions used for interpolations do not encompass an integer number of half-periods, the value of the first derivative evaluated at the other end of interpolation interval will not be zero. This results in an impulse response that decays as t-2, as shown in [BEA 04]. Figure 3 Interpolated filter frequency characteristic (k = 4) using sine functions

f(x)  a  b  sin 2π m (x  x 2 )  3π /2  y1

In a previous paper [ONO 08] it was shown that if the frequency characteristic is flat around Nyquist frequency, this results in improved performance regarding error probability, when the impulse response is sampled with a timing offset, as it happens in real life. So, we chose that the sine function should have a phase of 3π/2 at the point defined by the coordinates (x 2 , y 2 ) . There are 3 parameters that can be used in order to adjust the sine function to the imposed task, namely frequency, amplitude and phase.

or

f(x)  a  (a  y 2 )  sin 2π m (x  x 2 )  3π /2

(12)

Then evaluating f(x) at x  x1 , we get eq. (13):

Let us denote by m the frequency of sine function that passes through the points defined by (x1 , y1 ) ,





f(x1 )  a  (a  y 2 )  sin 2π m (x1  x 2 )  3π /2  y1

and (x 2 , y 2 ) where m is not restricted to be an integer, by b the amplitude of sine function, by a the offset on vertical axis and by φ the phase. So, equation (6) is:

f(x)  a  b  sin(2π m x   )

(11)

(13)

Solving eq. (6) for a, we get relation (14) and (15):

(6)

a

y1  y 2  sin 2π m (x1  x 2 )  3π /2 1  sin 2π m (x 1  x 2 )  3π /2

(14)

So, it is obvious that

f(x1 )  y1

An example of sine functions that pass through two particular points of coordinates (0.4, 0.8) and (0.7, 0.1) for three values of m, namely 0.7, 1 and 2 is illustrated in Figure 4.

(7)

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SETIT2009









(1  α /3, 1  α/3) (1  α /3, 1  2α /3)



α/3

(1 α /3, 0.5) , (1 α /3, 0.5) (1 α /3, 0.5) , (1  2α /3, 1  a 1 ) (1  2α/3, 1-a1 ) , (1 α, 0)

As an example we shall write the equation of a sine function of phase 3π /2 at (1  2α /3, a 1 ) that

(15)



α/3

(1  2α /3, 1  α)

 y1  y 2  sin2π m (x1  x 2 )  3π /2     1  sin2π m (x  x )  3π /2  y 2     1 2  sin 2π m (x  x 2 )  3π /2

2α /3

No interpolation

y  y 2  sin 2π m (x1  x 2 )  3π /2 f(x)  1  1  sin 2π m (x1  x 2 )  3π /2

passes through the points of coordinates (1 α, 1) and (1  2α /3, a 1 ) .

y1  1, y 2  a 1 x1  1  α, x 2  1  2α /3

f1 (x) 

(16)

1  a 1  sin 2π mα /3  3π /2  1  sin  2π mα /3  3π /2

  1  a1  sin  2π mα /3  3π /2  a1       1  sin  2π mα /3  3π /2  sin2π m (x  1  2α /3)  3π /2

The impulse response p(t) is obtained by performing the inverse Fourier transform of several pieces of sine function that pass through successive interpolation points. For instance, in the interval (1  α, 1  2α /3) , the associated piece of sine function is given by relation (17), and its contribution to the impulse response p(t) is eq. (18):

Figure 4 Sine functions that pass through the points (0.4, 0.8) and (0.7, 0.1) for three values of m The interpolation will result in a new characteristic that is close to the staircase characteristic. This behaviour will be reflected in results obtained in terms of error probability. Increasing the number k of interpolation points will also determine the shape of the new characteristic to be closer to the ideal staircase characteristic. The same is valid for decreasing m.

1 2α /3

p1 (t) 

Also,

α/3

(1  2α/3, 1  α/3)

α/3

j 2π xt

dx

for

the

interval (1  α /3, 1  α/3) ,

f 3 (x)  0.5 we get eq. (20).

Table 1 Coordinates of interpolation points (k = 1)

(1  α, 1  2α /3)

1

(18)

Performing the inverse Fourier transform results in eq. (19).

For k = 1, the pair of interpolation points for sine functions are presented in Table 1.

Interval width

 f (x)e

1 α

The impulse response for the improved Nyquist filter with a frequency characteristic that is derived from an ideal one with a piece-wise rectangular characteristic known as staircase, is obtained applying the inverse Fourier transform to several pieces of sine functions that pass through successive points, as shown in Figure 3. We denote by-n the number of interpolation points (x0, …, xn), which are set in the middle of constant intervals).

Ends of interpolation interval

(17)

Coordinates of interpolation points (1 α, 1) ,

(1  2α /3, a 1 ) (1  2α /3, a 1 ) , (1 α /3, 0.5)

-4-

as

SETIT2009 p1 (t) 

1 2



2

8π t (t  m )





Table 2 ISI error probability of the proposed Nyquist pulses for N = 210 interfering symbols and SNR = 15 dB

 2π m α  2 2   2   m  a 1t   3 

  cos  

X1(f) k=1 New (k=1) poly[4] α

 2π m α    2   a1 (m  t )  cos  3    sin  3  (3  2α ) π t       2

2

 2π mα   2 2   2(m  t )   a1  cos  3   1  sin2(α  1) π t      

t/TB = 0.05 a1 0.62

0.25

0,005

0.58

  2π (α m  3t  3at)    3

 (a1  1)  t  (m  t)  sin 

 sin

 2α m π   3  2(α  1) π t   

  α  2  sin  (3  α ) π t   sin 21   π t  3    3  p 3 (t)  4 πt

(19)

a2

a3

a4

40

-100

85

0.35

The impulse response p(t) results from a sum of six Fourier transforms and suffers a time scaling, t  t / 2 taking into account the constraints imposed by Nyquist I criterion for ISI-free signaling. The impulse response p(t) is illustrated in Figure 5 for α  0.5 together with impulse response h(t) defined in [4] by a2 = 25, a3 = -64, a4 = 55 for poly pulse.

0.64

m

0.62

0.005

a2

a3

a4

31

-80

69

0.68

m

0.666

0.005

0.5

(20)

Pe m

a2

a3

a4

25

-64

55

X1(f) k=1 New (k=1) poly[4] α

X1(f) k =1 new (k = 1) poly[6] X1(f) k=1 new (k = 1)

4.555 ×10-8 4.868 ×10-8 4.734 ×10-8 3.092 ×10-8 3.337 ×10-8 3.290 ×10-8

poly[6] X1(f) k=1 new (k = 1)

2.008 ×10-8 2.163 ×10-8 2.057 ×10-8

poly[6]

t/TB = 0.1 a1

Pe

0.64

m

0.628

0.005

0.25

a2

a3

a4

40

-100

85

0.35

0.68

m

0.665

0.005

a2

a3

a4

31

-80

69

0.5

0.68

m

0.69

0.005

Figure 5 Impulse responses of new filter (k = 1) and poly filter [4] for α  0.5

a2

a3

a4

25

-64

55

The optimal value of parameters a1 were determined for the piece-wise rectangular improved Nyquist characteristic in a previous paper [ONO 08] for several combinations (α , a 1 ) and are presented in Table 2, together with the values of the error probability. They are compared with the values for interpolated characteristic and poly [CHA 05] characteristic defined by a2, a3 and a4.

4. OFDM use

X1(f) k=1 new (k = 1) poly[6] X1(f) k=1 new (k = 1) poly[6] X1(f) k=1 new (k = 1) poly[6]

8.423 ×10-7 9.722 ×10-7 8.834 ×10-7 3.788 ×10-7 4.467 ×10-7 3.839 ×10-7 1.44 ×10-7 1.728 ×10-7 1.354 ×10-7

This section is focused on the problem of reducing the ICI power in transmission over OFDM systems, in particular DVB. OFDM is very sensitive to carrier frequency offset caused by the jitter of carrier wave and phase errors between the transmitter and receiver. Next we present and examine the use of the new ISIfree pulses defined above in an OFDM system. -5-

SETIT2009 The complex envelope of one radio frequency (RF) N-subcarrier OFDM block with pulse-shaping [PEN 04] is expressed in equation (21):

x t   e j 2 π f c t

N 1

a

kp

t e j 2 π f

k

t

(21)

k0

where: j   1 , f c is the carrier frequency, f k is the subcarrier frequency of the k-th subcarrier, pt  is the time-limited pulse shaping function and a k is the data symbol transmitted on the k-th subcarrier and has mean zero and normalized average symbol energy; data symbols are uncorrelated.

Figure 6 SIR for proposed pulse and polynomial pulse in a 64-subcarrier OFDM system The error probability measures the performances of the pulses regarding inter-symbol interferences and includes the effects of noise, synchronization error and distortion. The error probability Pe was evaluated as in [BEA 91] using Fourier series by equation (24).

Frequency offset, Δf Δf  0 , and phase error θ , are introduced during transmission because of channel distortion or receiver crystal oscillator inaccuracy. The average ICI power, averaged across different sequences [PEN 04] is equation (22):

Pe  σm ICI 

N 1



km k0

km  P   Δf  T 

1 2  2 π

2





 exp  m 2 ω 2 /2 sin mωg 0       m  m 1 

M



M odd

(22)



N2

 cos mωg

k



k  N1

The average ICI power depends not only on the desired symbol location m , and the transmitted symbol sequence, but also on the pulse-shaping function at the frequencies    k  m  /T   Δf  , k  m,

Here M represents the number of coefficients considered in the approximate Fourier series of noise complementary distribution function; ω  2 π /Tf angular frequency; Tf is the period used in the series; N1 and N2 represent the number of interfering symbols before and after the transmitted symbol; g k  p(t  kT) where p(t) is the pulse shape used and T is the bit interval. The results are computed using Tf  40 and M = 61 for N = 210 interfering symbols and a signal to noise ratio SNR = 15 dB, for all the cases.

k = 0, 1, ..., N-1 and the number of subcarriers. The ratio of average signal power to average ICI power is denoted as SIR and expressed in equation (23).

SIR  P ΔF  / 2

N 1



P k  m /T   Δf

km k 0

(24)

2

(23)

It is desirable to determine the values of the parameters a 1 , ...., a k so that the error probability should be minimized.

Relation (22) was evaluated for a number N = 64 subcarriers both for the poly pulse [CHA 05] and the new pulse produced by sine interpolation, defined by k = 1 and. Perfect overlapping of the SIR characteristics was found, as inferred from Figure 6, where SIR is represented as a function of frequency offset.

The ISI error probability is calculated using the method in [BEA 91] in the presence of time sampling errors for pulses produced by X1 (f) for different values of parameter a 1 and roll-off factor α . One can observe that in the presence of samplingtime error the results obtained for error probability if k = 1 are comparable, but do not outperform the reported reference results [CHA 05], which are reported for polynomial pulse.

5. Simulation results In this section, the pulses produced by the lowpass filters previously mentioned specified are studied in terms of ISI error probability.

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SETIT2009

6. Improved pulse In order to get better results, both the number of rectangles in the staircase characteristic, as well as the number of interpolation intervals was increased. We worked with k = 2 and a number of 8 interpolation intervals, as illustrated in Figure 7 and Table 3.

Figure 8 Impulse responses of poly and new filter (α = 0.5) This is illustrated in Figures 8, 9 and 10 together with poly pulse, as a reference.

Figure 7 Improved filter frequency characteristic (k=2) Table 3 Coordinates of interpolation points (k = 2) Ends of interpolation interval

Interval width

(1  α, 1  4α/5)

α/5

(1  4α/5, 1  3α/5)

α/5

(1  3α/5, 1  2α/5)

α/5

(1  2α/5, 1  α/5)

α/5

No interpolation

2 α/5

(1  α/5,1  α/5) (1  α/5,1  2α/5)

α/5

(1  2α/5,1  3α/5)

α/5

(1  3α/5,1  4α/5)

α/5

(1  4α/5,1  α/5)

α/5

where b 

a1 +a 2 2

c

Coordinates of interpolation points (1 α, 1) ,

(1  4α/5, a1 ) (1  4α/5, a1 ) , (1  3α/5, b)

Figure 9 Impulse responses of poly and new filter (α = 0.35)

(1  3α/5, b) . (1  2α/5, a2 ) (1  2α/5, a2 ) (1  α/5, 0.5) (1  α/5,0.5) , (1  α/5, 0.5) (1  α/5, 0.5) , (1  2α/5,1  a 2 ) (1  2α/5,1  a 2 ) , (1  3α/5, c) (1  3α/5, c) , (1  4α/5,1  a1 ) (1  4α/5,1  a1 ) , (1 α, 0)

Figure 10 Impulse responses of poly and new filter (α = 0.25)

2  a1  a2 . 2

Also, the ISI error probability in the presence of time sampling errors for pulses produced by interpolating the X 2 (f) staircase characteristic for

Performing inverse Fourier transforms on the pieces of sine functions defined on 8 intervals plus 2 rectangular functions defined in Table 3 and summing all the contributions resulted in the impulse response p(t).

different values of parameters a 1 , a2 and roll-off factor α was calculated using the method in [BEA 91]. The results are reported in Table 4 together with those for X 2 (f) and poly [CHA 05] pulse, taken as a reference. -7-

SETIT2009 Table 4 ISI error probability of the proposed Nyquist pulses (k = 2) and poly for N = 210 interfering symbols and SNR = 15 dB

We proposed and evaluated a new type of a Nyquist transfer function that approximates an ideal staircase frequency characteristic with 2k  2 levels using pieces of sine functions that pass through interpolation points in terms of inter-symbol interference. For the associated functions we started with the values of parameters a k (k = 1, 2) found for minimum error probability obtained with the pulses produced by filter with X k f  characteristics (k = 1, 2).

t/TB = 0.05

X 2 (f)

X 2 (f) α  0.25

new (k = 2) poly

X 2 (f) α  0.35

new (k = 2) poly

X 2 (f) new

α  0.5

(k = 2) poly

a1

a2

0.62

0.58

4.53042 ×10-8

0.62

0.58

4.688 ×10-8

(40, -100, 85) 0.64

0.58

0.69

0.6

( 32, -80, 69) 0.68

0.6

0.69

0.62

(25, -64, 55)

Pe

The best results for error probability are obtained when we used a larger number of levels for the staircase characteristic, which are evenly distributed and placed near the level 0.5. Adding a new level is equivalent with introducing new points for function interpolation.

4.734 ×10-8 3.05974 ×10-8 3.180 ×10-8

The results for error probability in the presence of symbol timing error for the same excess bandwidth are comparable with 4th degree polynomial pulse [CHA 04] for k = 1 and n = 6 and outperform it for k = 2 and n = 10. The new pulse and poly and are equal in terms of SIR (ratio of average signal power to average ICI power) for OFDM use in the presence of frequency offset.

3.290 ×10-8 1.9494 ×10-8 1.985 ×10-8

The performance of the improved Nyquist filters presented here could be further improved by increasing the number of k of intervals and of interpolation points n.

2.057 ×10-8

t/TB = 0.1

X 2 (f)

X 2 (f) α  0.25

new (k = 2) poly

X 2 (f) α  0.35

new (k = 2) poly

X 2 (f) α  0.5

new (k = 2) poly

a1

a2

0.64

0.58

8.26096 ×10-7

0.58

8.735 ×10-7

0.65

(40, -100, 85) 0.68

0.6

0.69

0.62

( 32, -80, 69) 0.68

0.6

0.71

0.63

(25, -64, 55)

Pe

REFERENCES [ASA, 04] A. Assalini, and A. M. Tonello, “Improved Nyquist pulses”, IEEE Communications Letters, vol. 8, pp. 87 - 89, Febr.2004. [BEA, 91] N. C. Beaulieu, “The evaluation of error probabilities for intersymbol and cochannel interference”, IEEE Trans. Commun., vol. 31, pp.1740–1749, Dec.1991. [BEA, 01] N. C. Beaulieu, C. C. Tan, and M. O. Damen, “A better than Nyquist pulse”, IEEE Commun. Lett., vol. 5, pp. 367–368, Sept.2001. [BEA, 04] C. Beaulieu, and M. O. Damen, “Parametric construction of Nyquist-I pulses”, IEEE Trans. Communications, vol. COM-52, pp. 2134 - 2142, Dec.2004. [CHA, 05] S. Chandan, P. Sandeep, and A.K. Chaturvedi, “A family of ISI-free polynomial pulses”, IEEE Commun. Lett., vol. 9, No.6, pp. 496-498, June 2005. [DEM, 98] T. Demeechai, “Pulse-shaping filters with ISI-free matched and unmatched filter properties”, IEEE Trans. Commun., vol. 46, pp. 992, Aug.1998. [HEL, 86] C.W. Helstrom “Calculating error probabilities for intersymbol and cochannel interference”, IEEE Trans. Commun., vol. 34, pp. 430-435, May 1986. [KUM, 07] V. Kumbasar, O. Kucur, “ICI reduction in OFDM systems by using improved sinc power pulse”, Digital Signal Processing, 17, pp.997–1006, 2007 [MOU, 06] H.M. Mourad, “Reducing ICI in OFDM systems using a proposed pulse shape”, Wireless Personal Communications, 40, pp.41–48, 2006.

8.834 ×10-7 3.56887 ×10-7 3.735 ×10-7 3.839 ×10-7 1.33245 *10-7 1.318 ×10-7 1.354 ×10-7

7. Conclusions We found the minimal values for error probability that are listed in Table 2. As expected, the error probability is reduced if the roll-off factor α increases. Also, better values of the error probability can be obtained if k and the number n of interpolation points are increased, as shown for k = 2 and n = 10. -8-

SETIT2009 [ONO, 08] Onofrei, L.A., Alexandru, N.D., “Optimization of the Improved Nyquist Filter with a Piece-Wise Rectangular Characteristic”, Proc. of 9 th Int. Conference Development and Application Systems, Suceava, 23-25 May, pp.128-137, 2008. [PEN, 04] Peng Tan, N.C. Beaulieu, “Reduced ICI in OFDM Systems Using the Better Than Raised-Cosine Pulse”, IEEE Commun. Lett., vol. 8, No. 3, pp.135-137, March 2004. [POL, 95] Pollet T., Bladel M.V., Moeneclaey M., “BER sensitivity of OFDM systems to carrier frequency offset and Wiener phase noise” IEEE Trans. Commun., vol. 4, pp. 191-193, Feb./Mar./Apr1995. [SAN, 05] P. Sandeep, S. Chandan, and A.K. Chaturvedi “ISI-Free pulses with reduced sensitivity to timing errors”, IEEE Commun. Lett., vol. 9, No.4, pp. 292 - 294, April 2005.

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