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Joint Frequency and Timing Recovery for Pulse

Shaped 4-CPFSK with h = 0.25

Zhijian Yu, Minjian Zhao, Lifeng Liu

Department of Information Science & Electronic EngineeringZhejiang University, Hangzhou, Zhejiang Prov. China, 310027

Email: yuzhijian@263.net

Zhiyong Luo

Haige Communication Co. Ltd.Guangzhou, Guangdong Prov. China, 510655

Abstract A data-aided synchronization method for jointlyestimating the symbol timing and carrier frequency offset hasbeen proposed for 4-ary CPFSK modulation with h = 0.25. Theproposed algorithm is based on a special preamble and has afeedforward structure that is suitable for digital realizations.Simulation results indicate the timing and frequency recoveryalgorithm can be employed with short preamble of 16 symbolsand is well suited for burst mode transmissions.

I. INTRODUCTION

Due to their superior bandwidth efficiency and constant

envelope properties, M-ary continuous phase frequency shiftkeying (M-CPFSK) signals with modulation index 1/M arevery attractive for data transmission over nonlinear chan-

nels. Demodulation in digital communication systems requires

knowledge of the symbol timing and carrier frequency offset.

Mistiming and frequency drifts arise due to propagation,

Doppler effects and mismatch between transmitter and receiver

oscillators.Some symbol timing and frequency recovery have been

proposed in the literature [6]-[10]. In [10] carrier recovery

scheme for 4-ary CPFSK with h = 0.25 is presented, wherethe CPFSK signal is shaped with a rectangle frequency pulse.

A raised cosine frequency pulse is preferred for lowering

the adjacent interfere and improving the error rate perfor-

mance [13]. In [6] a nondata-aided algorithm is proposed to

recover the symbol timing and carrier frequency offset for

MSK signals. The general case of MSK-type modulation is

discussed in [7]. However in burst mode transmissions, rapid

timing and frequency synchronization is essential as receivers

must be able to correctly synchronize on short burst of data.

Data-aided synchronization techniques are preferred for these

applications.

In this paper we propose a data-aided timing and frequency

recovery scheme for 4-ary CPFSK (4-CPFSK) with modu-

lation index h = 0.25. The pulse shaped CPFSK signal isconsidered along with a raised cosine pulse.

The remainder of the paper is organized as follows. The

signal model and some basic notations are introduced in

Section II. In Section III, the algorithm is described. Numerical

results are provided in Section IV. Section V contains our

conclusions are.

II . SIGNAL MODEL

The complex envelope of an M-CPFSK signal may bewritten as

s(t) = ej(t;) (1)

where

(t; ) = 2h

+k=

kq(t kT) (2)

is the information bearing phase. In the above equation, h isthe modulation index, T is the symbol interval, q(t) is thephase pulse, and = {k} are independent data symbolstaking on the values in the set R = {1, 3, , (M1)}.The phase pulse q(t) is related to the frequency pulse h(t) bythe relation

q(t) =

t

h()d. (3)

The pulse h(t) is time limited to the interval (0, LT) and isnormalized so that

q(LT) = 1/2. (4)

A raised cosine (LRC) frequency pulse with

h(t) =

1

2LT

1 cos

2 tLT

, 0 t LT

0, otherwise.(5)

is preferred because the error rate performance of M-aryCPFSK signals with modulation index 1/M can be signifi-cantly improved by employing a raised cosine baseband pulse

[13].

We assume that s(t) is transmitted over an AWGN channel.The complex envelope of the received signal is modelled as

x(t) = ej2ft+s(t ) + n(t) (6)

where f and represent the frequency offset and the carrierphase, respectively, is the timing epoch, and n(t) is thechannel noise which is assumed to be white and Gaussian

with a one side spectral density N0 = 2n. Then the signal-to-

noise rate (SNR) per symbol is given as SNR= Es/N0, whereEs represents the received signal energy per symbol.

In a digital implementation of the receiver, the waveform

x(t) is sampled at some rate Ts = T /N, where N is theoversampling factor. In the study, we take N large enough toavoid aliasing.

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Denoting xk(i) the sample of x(t) taken at t = kT + iTs,we have

xk(i) = ej[(kT+iTs;)+2f(kT+iTs)+] + nk(i) (7)

with 0 i N 1. In the above equation, the index kcounts the symbol intervals while i counts the samples withina symbol interval.

III . TIMING AND FREQUENCY ESTIMATION

In this section, we describe the synchronization algorithm

for 2RC pulse shaped 4-CPFSK modulation. This discussion

is also suitable for LRC pulse shaped 4-CPFSK signals with

any other L.For 2RC pulse shaped 4-CPFSK modulation with h = 0.25,

(2) can be written as

(t; ) =

2

+k=

kq(t kT) (8)

where k {1, 3}.

We set the preamble to the following structure

0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 .

Then the signal in the preamble interval is periodic with a

period 4T. We can describe s(t) as

s(t) = exp

j

2

+k=

q4T

(t 4kT)

(9)

with

q4T

(t) =

14

12

sin 2t2T

, 0 < t T

14 +t2T , T < t 2T

3

4 +1

2 sin2t

2T , 2T < t 3T94

t2T

, 3T < t 4T.

(10)

To recover the symbol timing and carrier frequency offset,

consider the one-lag autocorrelation of x4(t)

c(t; , f) = E

[x(t)x(t T)]4

(11)

where E{} denotes the expectation operation. Inserting (6)and (9) into (11) yields

c(t; , f) = E

exp

j2

+k=

p(t 4kT)

ej8fT + N(t) (12)

where N(t) is a noise term and

p(t) = q4T

(t) q4T

(t T). (13)

For the convenience, we write c(t; , f) as

c(t; , f) = c(t )ej8fT + N(t) (14)

with

c(t) = E

exp

j2

+k=

p(t 4kT)

. (15)

2 1.5 1 0.5 0 0.5 1 1.5 2

1

0.8

0.6

0.4

0.2

0

0.2

Normalized time, t/T

c(t)

L=1L=2L=3

Fig. 1. Shapes ofc(t).

Its shown from (10) that (13) can be written as

p(t) =

12 +t2T

12

sin 2t2T , 0 < t T

12 + t2T 12 sin 2t2T , T < t 2T32

t2T

+ 12

sin 2t2T

, 2T < t 3T32

t2T +

12 sin

2t2T , 3T < t 4T.

(16)

Its obvious that p(t) is also a periodic signal with a periodof 4T. The signal snapshots of (0, T], (T, 2T], (2T, 3T]and (3T, 4T] appear with equal probability 1/4. Then theexpectation in (15) can be got

c(t)

=1

4exp

j2

1

2+

t

2T

1

2sin

2t

2T

+ 14

exp

j2

12

+ t + T2T

12

sin 2(t + T)2T

+

1

4exp

j2

3

2

t + 2T

2T+

1

2sin

2(t + 2T)

2T

+1

4exp

j2

3

2

t + 3T

2T+

1

2sin

2(t + 3T)

2T

=1

4exp

j

t

T

exp

j sin

2t

2T

+ exp

j sin

2t

2T

1

4exp

j

t

T

exp

j sin

2t

2T

exp

j sin

2t

2T

=j

2

sinsin 2t2T

expj tT expj t

T

= sin

sin

t

T

sin

t

T

. (17)

The line with a legend L = 2 in Fig. 1 illustrates the shapeof c(t). Its shown that c(t) is periodic with a period of T.Substituting (17) into (14) yields

c(t; , f) = sin

sin

(t )

T

sin

(t )

T

ej8fT + N(t). (18)

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Its clear that c(t; , f) provides information about theparameter and f. Assuming for simplicity that the noiseterm is negligible, we get

(t) = sin

sin

(t )

T

sin

(t )

T

ej8fT. (19)

|(t)| is even with = 0 and the location of the maximum

of |(t)| is T2 . Let us denote by (i) the samples of (t)

taken at the time t = iT/N. Then from the above equationwe have

(i) = sin

sin

i

N

T

sin

i

N

T

ej8fT. (20)

For |(i)|, taking the Fourier transform and rearranging yields

= T

2arg

N1

i=0|(i)| ej2i/N

. (21)

gives an estimation of the location of the maximum of |(t)|.As is explained above, the location of the maximum of |(t)|is T2 . Then the estimation of is given as

= +T

2. (22)

and 0 < T.Let imax denote the index of the maximum of |(i)|, which

can be taken as the round of . Then the estimation offrequency offset is given as

f T =arg{(imax)}

8. (23)

In a digital implementation the computation of the expecta-

tion is performed by an averaging filter of length L0 over thesequence of samples [xk(i)x

k1(i)]4 in the preamble interval

where i is fixed. Then (i) is given by

(i) =1

L0

L01k=0

xk(i)x

k1(i)4

(24)

where 0 i N 1 and L0 should be an integer multipleof 4 because of the preamble structure.

c(t) for other L can be derived as the discussion above. Fig.1 shows the shape of c(t) for other L such as L =1 and 3.

IV. NUMERICAL RESULTS

In this section, we provide some numerical results about the

performance of the timing and frequency recovery algorithm

on the AWGN channel. We assume the receiver filter band-

width be large enough not to distort the signal components.

The oversampling factor has been set to 8 and the averaging

filter length L0 to 16, 32, 64 and 128.Fig. 2 illustrates the average frequency estimations

E{f T} as function of f T for Es/N0 =10dB. The perfecttiming is assumed at the receiver. From (23), the frequency

offset estimation f T range is (18 ,18

] for < arg{} .

0.2 0.15 0.1 0.05 0 0.05 0.1 0.15 0.20.2

0.15

0.1

0.05

0

0.05

0.1

0.15

0.2

Ave

ragefreqeuncyestimations

Normalized freqency, fT

idealL

0=128

L0=16

Fig. 2. Average frequency estimations for 2RC pulse shaped 4-CPFSK(SNR=10dB).

5 10 15 20 2510

7

106

105

104

103

102

FrequencyMSE

Es/No(dB)

L0=128

L0=64

L0=32

L0=16

Fig. 3. Frequency MSE for 2RC pulse shaped 4-CPFSK (fT = 0.05).

If the normalized frequency offset f T is out of the range(18 ,18

], the average frequency estimation will be

E{f T} = f T +k

4(25)

where k is chosen to satisfy 18

< f T + k8

18

. Then the

maximum unbiased estimation range is |f T| < 18 . Fig. 2appears that the estimations are unbiased over range |f T|