Comparison of zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA - - PDF document

SLIDE 1

Comparison of zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA

GMSK

and linear approximated

GMSK

for use in Software Radio zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA

• A. Wiesler, R. Machauer, F. Jondral

Institut fur Nachrichtentechnik, Universitat Karlsruhe D-76128 Karlsruhe, Germany e-mail: zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA

wiesler@inssl.etec.uni-karlsruhe.de

phone: +49 721 608 37 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA

48

Abstract- In 1 1 1 a common Software Radio

structure for second generation mobile systems has been introduced. This SWRADIO combi- nes different standards of mobile communica- tion systems like GSM, DECT, IS-54 and PDC. All baseband functions like channel coding, mo- dulation and equalisation are implemented in a general, parametrized way, so that all of them can be used for the selected standards. This structure has several advantages like a redu- ced size of the hardware platform, fast perfor- mance by changing the air interface for a system handover and the possibility of global roaming. The linear approximated GMSK is used in the

SWRADIO because this enables a common I/Q-

modulator for all second generation

• systems. In

this paper it is proved, that with a usual recei- ver (Viterbi equalizer with least square channel estimate) there is no performance loss by using the approximated GMSK instead of the original GMSK.

• I. INTRODUCTION

Third generation systems like the European UMTS will perform a very flexible communication technology [2]. As the new standards can not replace established systems like GSM at once, a seamless change to the third generation is aimed. That means future hand- helds must be able to perform different communication technologies and also seamless system handover. This can only be realized with a transmitter structure which is totally software programmable, a so called Software Radio [3]. In [l] a solution for different second generation systems (GSM, DECT, IS-54, PDC) is introduced to show that a common software structure is possible for different

• standards. These are all TDMA systems so the main

difference between them lies in the modulation. The European systems use nonlinear GMSK, the American and Japanese systems use a/4-DQPSK. For a common, parametrized modulator structure a linear approxima- tion of the GMSK is necessary, which is reviewed in Section 11. In the SWRADIO a common MLSE equa- lizer using the Viterbi algorithm is implemented. This algorithm is based on a discrete channel model which assumes a linear modulation. Also the estimate of the

fax: f49 721 608 60 71

system impulse response assumes this linear channel

• model. The consequence of the usage of this estima-

tes for the nonlinear GMSK is described in Section 111. Some simulation results are discussed in Section IV.

• 11. LINEAR

APPROXIMATION

OF GMSK

GMSK is a special kind of a 2-level FSK with mo- dulation index zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA h = 0.5 [4]. The complex envelope of a GMSK modulated signal is zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA

g(r - zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA

nT)dT

(1)

• 03

1

with the NRZ-stream zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA

dn E {-1,l) and the frequency

impulse zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA

g(t). For MSK a rectangular frequency impulse

is used, which causes hard phase changes and thus a broad spectrum. To reduce the bandwidth for GMSK the following frequency impulse is used. h ~ ~ ~ ~ ~ ( t ) ist the known Gaussian impulse with the time bandwidth product BT. The reduction of bandwidth is achieved with the trade of a controlled in- tersymbol interference (ISI). For the GSM-system the factor BT = 0.3 was chosen, that results in a IS1 over about 2 symbols but a small bandwidth. With DECT a BT = 0.5 is used, which causes minor ISI, because the frequency impulse is shorter and so this GMSK is more a kind of MSK. The theoretical infinite long Gaussian impulse h ~ ~ ~ ~ ~ ( t ) is cut to the length LT, with L 2 3. Thus at time t =

nT the phase response is determined to

t

(3)

• 03
• Fig. 1

shows the phase response for MSK and GMSK. With q(t) the GMSK signal s(t) can be described as follows

03

s ( t ) =

exp j27rh

dnQ(t

• nT)

] [

n = O

(4)

The nth NRZ-bit dn causes a phase change of zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA

nq(t) or

• .lrq(t), which is added to the changes of the previous

557

0-7803-4281-X/97/$10.00 01998 IEEE

SLIDE 2

, 2 5 U G M S K , zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA

, t/T

That means zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA

s(t)

consists of a linear part slin(t) and a nonlinear part snl(t). In Fig.2 the two first impulses zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA CO and C

1 for GMSK are displayed.

0 0

1 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA

2 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA 3 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA

4 1 r

• f zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA

N, = P1 impulses zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA

CK:

Since CO has 99% of the signal energy, the following

n

linear approximation of sft) is obvious:

L-1

A K , ~

= 4 -

dn-1. Q K , ~ (6) with

01)

s(t)

M srin(t) =

ZnCo(t -

nT)

(14)

n = O zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA

1=1

with the symbols from (10). That means the symbols zn which are determinated from the accumulated bit stream di are only formed by one impulse CO. So this approximated GMSK has the advantage, that it can be realized with a usual I/&-Modulator for PSK or &AM and is therefore easy to integrate into a Software Radio Now the question whether the approximation of the This representation can be used for all CPM-signals. For example, the complex envelope of a MSK-signal (here is L = 1) can be written as follows

01)

S M S K (t)

=

exp

COMSK (t -

nT) (8)

structure.

n = O

with

{ sin(r;/2T)

O<t 5 2T

(9)

0 6 -

• therwise

COMSK

=

0 0 -

This is the well known representation of MSK as 0- QPSK modulation. The symbols

• 06-

E {-I, 1, - j , j ) . (10)

1

b

;

• il

BT =

. 3

BT =

0.5

are alternately real and imaginary.

• Fig. 3. Complex envelope of the approximated GMSK

For GMSK with L = 4 the exact superposition (5) is made by eight impulses. So the following represen- tation o

f the complex envelope of an exact GMSK-

modulated signal s(t) can be used: GMSK causes it performance loss will be discussed. First the approximation of the GMSK causes a change

• f the eye diagram, the scatter diagram, and the power

density spectrum. In Fig.3 the sca.tter diagram of the approximated GMSK with different values for BT is

• shown. The approximation causes fluctuations of the

complex envelope din(t). For BT = 0.5 the fluctua- tions are very small. As it was mentioned before the GMSK with BT = 0.5 is very similar to the linear MSK and therefore the linear approximation does not change the signal remarkably. So the main emphasis of the dis- cussion lies on the GMSK which is used in the GSM

• system. The eye diagram of the linear approximated

558

SLIDE 3

1 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA

.o

0.5 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA

0.0

• 0.5
• 1

. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA

• Fig. 4. Eye diagram of the exact GMSK zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA

1.0 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA

• Fig. 5. Eye diagram of the linear approximated GMSK

GMSK and exact GMSK are shown in Fig.4 and Fig.5. Also the power spectral density (PSD) has to be inve-

• stigated. In Fig6 the PSD for both GMSK schemes

are plotted. The approximated GMSK fits the requi- rements of the standardization at the PSD as good as the exact GMSK. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA

• 20 -

exact GMSK

E l

• approx. GMSK
• 40
• 60 '
• 80 - zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA

L

• m
• 3.0 -2.0
• 1

. 1 .

2.0 3.0

• Fig. 6.

Power spectrum density of approximated and exact GMSK signal

• 111. CHANNEL

ESTIMATE

AND EQUALIZATION

At the receiver a usual equalizer and demodulator structure for GSM systems are used. That means a Viterbi algorithm for MLSE equalisation is implemen-

• ted. For a MLSE decision the impulse response of the

whole transmitter system has to be known. It can be estimated by correlation with the training sequence in the middle of the GSM-burst. The Viterbi equalizer and the least square estimate of the channel both as- sume a linear modulation scheme. That means that the symbol stream z, is formed by an impulse shaper

g ~ ( t ) .

The received signal y(t) is therefore assumed as

00

y(t) = zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA

z, h(t -

nT, t )

+

r(t) (15)

n = O

with the impulse response which is the convolution of the impulse shaper gr(t), the time varia.nt impulse response of the mobile channel

k(r,

t ) and the receiver filter g ~ ( t ) . Additionally, there is the filtered AWGN ~ ( t )

= n(t) * g ~ ( t ) .

The mobile channels impulse response is finite and approximately time invariant for the lengt,h of one GSM burst. So the channel estimate must be performed only once for the equalization of one burst. The sampling of (15) with symbol timing leads to the received symbols of

• ne GSM burst, zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA

N

y ,

= z Zn-phfi + P ,

n = 0,. . . ,147 (17)

fi=O

with the system impulse response h, of length zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA N + 1. This discrete model is shown in Fig.7. For the GSM

• Fig. 7.

Discrete channel model which is used by the Viterbi equalizer

system N is usually 5 . The channel estimate uses the I< known training symbols c

, , n = 0,.

.

. ,

K and the

received training symbols 2

1 = y60+1, n = 0, .

.

. , K to

estimate h,, n = 0,. . .

,

N . With the matrix and the vector the relation between the received and transmitted trai-. ning symbols can be described by a linear equation sy- stem, if the linear channel model (17) is assumed: S . h + r = x

(22)

The best estimate in the sense of the least square cri- teria, which means that

559

SLIDE 4

has to be minimized, is the following solution In the case of the GSM training sequence the normali- zed AKF matrix 1/16. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA

S*TS is the unity matrix. The-

rewith the channel estimate can be performed with the correlation zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA i ; zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA

=

S * T X . zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA

(25) The error power of all coefficients

• f k

for this estimate is zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA n = 0,.

.

.,

N

E ( I A h n 1 2 ) = zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA

K+I

(26) zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA

Uf with the average power zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA

c,”

• f r. The estimate & of the

system impulse is used for hhe MLSE equalizer which determines the most likely symbol sequence {ii}

• ut of

all 2w possible sequences of length W . The sequence

{ i ; }

with minimal cost compared to all other possible sequences is chosen. The received signal g(t) of a GMSK-modulated si- gnal in a mobile environment is zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA

G(t) =

s ( t ) * JE(7-9 t )

* g R ( t ) +

n(t)

* g R ( t )

(28) with the time variant impuise response of the mobile channel k ( ~ , t ) and the receiver filter gR(t). Eq. (13) leads to a(t)

(Pn(t)

+

P’(t))

* k ( ~ ,

t )

* gR(t) + (29)

+r(tf

00

=

zn * h(t -

nT)

+

(30)

n = O

+sfi4((t)

* L(7-J) + zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA

T ( t )

with h(T,t) = C

O ( . ) * k(r,t) * g R ( T )

(31)

k(T,t) = k ( 7 , t )

*!JR(T)

(32) Sampling $(t) with symbol timing leads to

N N

ijm

=

z,-phl* +

s:!-,&~ +

r,

(33)

p=O p=O

with the sampled nonlinear part of the exact GMSK

SE^. Due to the assumption of a linear channel the

nonlinear part of the exact GMSK means an additive disturbance to the receiver. The resulting disturbance process has a higher average power than rn which is the additive disturbance in the case of the linear GMSK. So it is obvious that a lower performance of the exact GMSK has to be expected.

• IV. SIMULATION

RESULTS AND CONCLUSION

Some simulations with different mobile channels have been conducted to compare the performance of the two modulation schemes. In Fig.8 the bit error rates (BER) for the exact and linear approximated GMSK are plotted. The simulated mobile channels are the typical channels published by COST 207 [7] with dif- ferent speed (in km/h) of the mobile participant. The

0.0 5.0 10.0 15.0

20.0

SNR (dB]

• Fig. 8. Bit error rates for exact and approximated GMSK with

different mobile channels

results show that the BER performance of the linear approximated GMSK is as good as or for good SNR even better than the performance of the exact GMSK. Another advantage has to bee mentioned here: In the UTRA proposal for the TD/CDMA transmission [2] the main impulse CO

• f the linear GMSK is used as

impulse shaper. By using the linear GMSK for GSM,

• nly one filter has to be implemented in a software

radio for UMTS. REFERENCES

[I] A. Wiesler and F. Jondral, “Software radio structure for se- cond generation mobile communication systems,” in Procee-

dings oj IEEE Vehicular Technology Conference, May 1998, [2] ETSI, Universal Mobile Communications Systems (UMTS), [3] “Software radios,” IEEE Communications Magazine, 1995. [4] K. Murota and M. Hirade, “GMSK modulation for digital mobile radio telephony,” IEEE Trans. Communzcations, vol. 29, no. 7, pp. 1044-1050, July 1981. [5] P.A. Laurent, “Exact and approximateconstruction of digital phase modulations by superposition of amplitude modulated pulses (amp),” IEEE Trans. Commun., vol. COM-34, pp. 150-160, February 1986. [6] P. Jung, “Laurent’s representation of binary digital conti- nuous phase modulated signals with modulation index 1/2 revisited,” IEEE Trans. Comrnuncations, vol. 42, pp. 221-

224, 1994.

[7] COST 207, Digital land mobile radio communications, Com- mission of the European Communities, 1988.

• vol. 48, pp. 2363-2367.
• vol. version 3.0.0, Dec. 1997.

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