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Performance Analysis of DS-BPAM UWB System 225

performance of the coded TH-PPM UWB system over multipath fading channels waspresented in [5]. It is shown the coded scheme can achieve better performance. In ourknowledge, there are no much literatures about the analysis of coded and uncodedscheme of DS-BPAM UWB system. In this paper, the bit error rate (BER) of the two

systems is analyzed in detail over UWB multipath fading channels. For the codedsystem, we use convolutional encoder and the soft output Viterbi algorithm (SOVA)is considered here. The computer simulations show that the coded DS-BPAM UWBsystem can get better performance with the acceptable complexity compared with theuncoded scheme.

The organization of this paper is as follows. In Section II, an overview of the DS-BPAM UWB system model is provided and the performance analysis is given indetail over multipath fading channels and SRake receiver. Section III presents theproposed coded DS-UWB system and also gives the upper bound and lower bound of

the coded scheme. Section IV describes the simulation results obtained andinterpretations. Finally, conclusions are presented in Section V.

2 DS UWB System Description

The performance analysis of the direct sequence spread spectrum UWB system overmultipath fading channels is presented in this section. The transmitter of DS-BPAMUWB system is shown in Fig.1. This DS-BPAM UWB system is similar with [6]. Weassume that desired user has a unique pseudo-noise (PN) sequence with c N chips per

message symbol period f T such that f c cT N T = , where c N is the spread spectrum

processing gain. A typical transmitted signal can be expressed as

( ) ( )1

0

c N

tr j n tr f c j n

s t d c w t jT nT −∞

=−∞ == − −∑ ∑ , (1)

where ( )tr w t is the transmitted monocycle waveform, seen in [1], { } j D is the binary

information bit and { } jd is the modulated data symbols with 2 1s j j N d D

⎢ ⎥⎣ ⎦= − , N s is the

pulse repetition time, { }nc is the spread chips with duration cT . We show the

modulated waveforms of the DS-BPAM UWB modulation waveforms in Fig.2, wherethe DS sequence is { 1, 1, 1, 1+ − − + with 4c N = and 1s N = .

Bit Repetition Coder (Convolutional Encoder)

DirectSequenceGenerator

SpreadingPAM

Modulator Pulse

Shaper Information

bits

Fig. 1. Transmitter diagram of DS-BPAM-UWB (Uncoded/Coded) System



226 Z. Bai et al.

0 26 52 78 104 130 156 182 208 234 260 286 312

+1 +1 -1+1 -1 -1 +1 +1 -1 -1 +1 -1 +1 +1 -1

Fig. 2. DS-BPAM UWB modulation waveform ( N c=4, N s=1)

In this paper, we consider a simple L-tap multipath channel model instead of theclustered multipath channels in order to simplify the analysis and high light thebenefit of SRake receiver [7]. The channel mpulse response can be expressed by

( ) ( )1

0

L

l ll

h t a t δ τ −

== −∑ , (2)

where la and lτ are the amplitude attenuation and time delay of the lth path,

respectively, and L is the number of the corresponding branches in the SRakereceiver.

An ideal channel and antenna system is known to modify the shape of thetransmitted monocycle ( )tr w t to ( )recw t at the output of the receiver antenna. For the

purpose of analysis, we have assumed that the true transformed pulse shape ( )recw t is

known at the receiver.Through the UWB multipath fading channels, the received signal ( )r t can be

expressed as [8]( ) ( ) ( ) ( )recr t s t h t n t = ∗ + , (3)

where ( ) ( )1

0

c N

rec j n rec f c j n

s t d c w t jT nT −∞

=−∞ == − −∑ ∑ .

We can further write the expression of the received signal as

( ) ( ) ( )1 1

0 0

c N L

l j n tr f c l j n l

r t a d c w t jT nT n t τ −∞ −

=−∞ = == − − − +∑ ∑ ∑ , (4)

where ( )n t is the AWGN noise modeled as 00,2

N N ⎛ ⎞⎜ ⎟⎝ ⎠

.

SRake receiver with MRC (maximum ratio combination) is employed in thereceiver. The diagram of SRake receiver is shown in Fig.3. We can make the decision




Direct SequenceGenerator

Received Signal r (t )

Finger 1

Finger 2

Finger L

TemplateGenerator ChannelEstimator

Integrator

Integrator

Integrator

Branch

Combining

1v t t

2v t t

Lv t t

1a

2a

La

DecisionOutput

Recovered Signal

ConvolutionalDecoder

Recovered Signal

Fig. 3. Rake receiver diagram of DS-BPAM UWB (Uncoded and Coded) System

of the received information bit according to the output of branch combining. Since weassume that the receiver has achieved perfect synchronization for the signaltransmitted by the transmitter. According to the estimated channel impulse response,the template can be set as

( ) ( ) ( ) ( )1 11

0 0 0

*c c N N L

bit n rec f c l n rec f c ln l n

v t c w t jT nT h t a c w t jT nT τ − −−

= = == − − = − − −∑ ∑ ∑ . (5)

The SRake correlation output can be expressed as

( ) ( ) ( ) ( )1 1 11

0 00 0 0 0

s s c N N N LT T

bit l n rec f c l j j l n

r t v t dt r t a c w t jT nT dt α τ − − −−

= = = == = − − −∑ ∑ ∑ ∑∫ ∫ . (6)

For the quantity la , we assume the order 1 2 ... La a a> > > . [ ]0,T is the time interval

includes the selected L paths.The decision consists of

>0, 1; 0, 0 j jif D if Dα α = < = . (7)

In the following, we assume that the received waveform satisfies the

relation ( ) 0recw t dt +∞

−∞=∫ .

Let 1α ± and 21σ ± be the mean and variance of the correlation output conditioned on

1 j D = and 0 j D = , respectively. We calculate these two values as follows:

( )21 1 0

T

s c bit N N v t dt α α −= − = ∫ , (8)

( )2 2 201 1 02

T s c

bit

N N N v t dt σ σ −= = ∫ , (9)



228 Z. Bai et al.

with

( ) ( ) ( ) ( )2

1

1 1 1 1 12 2

00 0 0 0 0,

L L L L LT T

bit l rec l l rec k l l k l k T l k l l k l k

v t dt a w t a w t dt a a a Rτ τ τ τ − − − − −

= = = = = ≠= − − = + −∑ ∑ ∑ ∑ ∑∫ ∫ , (10)

where ( ) R τ denotes the autocorrelation of pulse ( )recw t , and that

is, ( )( ) ( )rec rec

w

w t w t dt R

E

τ τ

∞

−∞−

=∫

with ( )2w rec E w t dt

∞

−∞= ∫ .

We first evaluate signal to interference ratio ( SIR). For the BPAM scheme, the bit

error probability is equal to ( )bP Q SIR= , where the equation is taken on the fading

parameters.

Through the above analysis, we can find( ) ( )22

1 021 0

2T

s c bit N N v t dt SIR

N

α σ

±

±

= =∫

. (11)

The bit error probability of the system is given by

( ) ( )2

00

2 T s c

b bit

N N P Q SIR Q v t dt

N

⎛ ⎞= = ⎜ ⎟⎜ ⎟

⎝ ⎠∫

( )1 1 1

2

0 0 0,0

2L L L

s c wl l k l k

l l k l k

N N E Q a a a R N

τ τ − − −

= = = ≠

⎛ ⎞

⎡ ⎤⎜ ⎟= + −⎢ ⎥⎜ ⎟⎣ ⎦⎝ ⎠

∑ ∑ ∑ , (12)

where ( )Q is the Gaussian probability integral.

When the channel attenuation is assumed to be normalized to one,1

2

0

1 L

ll

a−

==∑ with ( )

1 1

0 0,

0 L L

l k l k l k l k

a a R τ τ − −

= = ≠− =∑ ∑ , then the BER reduces to be

02 s c w

b N N E P Q N

⎛ ⎞

= ⎜ ⎟⎜ ⎟⎝ ⎠

. (13)

3 Coded DS-UWB System Model

In this section, we consider the proposed coded DS-BPAM UWB system. We showthe proposed receiver diagram of the convolutional coded system in Fig.3. In order togive the comparison of these two systems, in the DS-UWB system, we will assume

that each data bit is transmitted by N s repetition pulses. We consider it as a simplerepetition block code with rate 1/ N s. In the coded DS-UWB system, we will use anorthogonal convolutional code with constraint length K . For each input bit, we can get2k -2 coded bits, here we set N s =2 k -2 in order to compare with the uncoded system. Inthe receiver end, we will apply soft output viterbi decoding algorithm.




For each encoded bit, the correlator output can be written as:

( ) ( )11

00 0

c N LT

j l n rec f c ll n

r t a c w t jT nT dt α τ −−

= == − − −∑ ∑∫ . (14)

In the proposed SOVA scheme, we will normalize the correlator output jα to the

input of the decoder. The metric of SOVA is generated according to the correlatoroutput jα . According to the characteristics of the correlator output, the SOVA is the

optimal decoding algorithm for this system. The decoding complexity grows linearlywith K . Since in the practical system, the value of K is relatively low, the codedsystem is completely practical. In this paper, we use a simple convolutional encoderwith code rate being 1/2.

According to the generating function of this kind of convolutional encoder given in

[4], [9]

( )( )

( )

2

3 2

1,

1 1 1 2

K

SO K K

W W T

W W W

β γ β

β

+

− −

−=

⎡ ⎤− + + −⎣ ⎦

, (15)

where γ and β in each term of the polynomial indicate the number of paths and

output-input path weights, respectively. We set-3 32 2K K

W Z γ −

= = and

( ) ( )0 1 Z p y p y dy+∞

−∞= ∫ , where ( )0 p y and ( )1 p y are the probability density

functions of pulse correlator output. The free distance of this code can be expressedas ( ) ( )-3

22 2 log 4 2K f s sd K N N = + = + . The bit error probability can be obtained

as [9],

0 0

0

2s s

f

f

E E k d

I I sb f k

k d

E P Q d b e e

I

⎛ ⎞ ⎛ ⎞∞ − ⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠

=

⎛ ⎞= ⎜ ⎟⎜ ⎟⎝ ⎠

∑ , (16)

where( )

1 1 12

0 0 0,0 0

L L Ls c w

l l k l k l l k l k

E N E a a a R

I N τ τ

− − −

= = = ≠

⎡ ⎤= + −

⎢ ⎥⎣ ⎦∑ ∑ ∑.

For a uniform power delay profile (PDP), we obtain the following upper bound P e on P b according to (16), it can be expressed as below [4], [9]:

( )0

00 1, ,

,2

s f

E s I

E d

I se f

Z e Z

dT Z E P e Q d

I d β γ

β β

⎛ ⎞−⎜ ⎟⎜ ⎟⎝ ⎠

⎛ ⎞⎜ ⎟⎝ ⎠

= = =

⎛ ⎞⎛ ⎞⎜ ⎟≤ ⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

( )( )

2

2

20

12

1 2 1

f d K s f

K

E W Q d W

I W W γ − +

−

⎛ ⎞ ⎛ ⎞⎛ ⎞ −⎜ ⎟ ⎜ ⎟= ⎜ ⎟

⎜ ⎟⎜ ⎟ − −⎝ ⎠ ⎝ ⎠⎝ ⎠

( )( )

2

20

12

1 2 1s

f K

E W Q d

I W W −

⎛ ⎞⎛ ⎞⎛ ⎞ −⎜ ⎟⎜ ⎟= ⎜ ⎟

⎜ ⎟⎜ ⎟ − −⎝ ⎠ ⎝ ⎠⎝ ⎠

(17)



230 Z. Bai et al.

( )( )( )

21 1 1

2

20 0 0,0

2 1

1 2 1

L L L f c w

e l l k l k K l l k l k

d N E W P Q a a a R

N W W τ τ

− − −

−= = = ≠

⎛ ⎞⎛ ⎞⎡ ⎤ −⎜ ⎟⎜ ⎟≤ + −⎢ ⎥

⎜ ⎟⎜ ⎟ − −⎣ ⎦ ⎝ ⎠⎝ ⎠

∑ ∑ ∑ . (18)

The lower bound in this case can be obtained as

( )1 1 1

2

0 0 0,0

2 L L L f c w

e l l k l k l l k l k

d N E P Q a a a R

N τ τ

− − −

= = = ≠

⎛ ⎞⎡ ⎤⎜ ⎟> + −⎢ ⎥⎜ ⎟⎣ ⎦⎝ ⎠

∑ ∑ ∑ . (19)

From the above equation, it can be realized that in the coded scheme, the effectiveorder of the processing gains is the product of L (the number of branches in SRakereceiver) and the free distance of the code. This product plays an important role indeciding the bit error performance and it makes the different performance of the

coded and uncoded system. In order to get better BER performance, we can increasethe SRake branches to gather more multipath components, but the system complexitywill be increased correspondingly.

4 Simulations Results

In this section, we present some computer simulation results about these twosystems. Like the same example in [1], the received pulse is modeled as

( )2 2

1 4 exp 2recm m

t t w t π π τ τ

⎡ ⎤ ⎡ ⎤⎛ ⎞ ⎛ ⎞⎢ ⎥ ⎢ ⎥= − −⎜ ⎟ ⎜ ⎟⎢ ⎥ ⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦ ⎣ ⎦

, shown in Fig.4, where 0.2mτ = ns,

0.5wT = ns. The gold code sequence used here has the characteristics with length

7c N = and chip duration 20cT = ns. For each information bit, we repeat it twice forthe comparison of the two systems. The information bit rate is

( )1 1b f s c c R T N N T = = bps.

The channel impulse response of the IEEE model [10] can be expressed as follows:

( ) ( )( )

1 1

K n N

nk n nk n k

h t X t T α δ τ = =

= − −∑ ∑ , (20)

Table 1. Parameters for IEEE UEB channel scenarios

Channel Model(CM1~4) Λ λ Γ γ σξ σζ σg

Case 1 LOS(0-4m) 0.0233 2.5 7.1 4.3 3.3941 3.3941 3

Case 2 NLOS(0-4m) 0.4 0.5 5.5 6.7 3.3941 3.3941 3

Case 3 NLOS(4-10m) 0.0667 2.1 14 7.9 3.3941 3.3941 3

Case 4 NLOSMultipath Channel 0.0667 2.1 24 12 3.3941 3.3941 3



232 Z. Bai et al.

0 3 6 9 12 15 1810

-4

10-3

10-2

10-1

100

SNR (dB)

B E R

DS-BPAM Uncoded Shceme (SRake=3)

CM3, DS-BPAM(Uncoded)CM3, DS-BPAM(Coded)CM4, DS-BPAM(Uncoded)CM4, DS-BPAM(Coded)

Fig. 5. BER performance of DS-UWB (SRake=3)

0 3 6 9 12 15 1810

-4

10-3

10-2

10-1

100

SNR (dB)

B E R

DS-BPAM Uncoded Shceme (SRake=5)

CM3,DS-BPAM(Uncoded)CM3,DS-BPAM(Coded)CM4,DS-BPAM(Uncoded)CM4,DS-BPAM(Coded)

Fig. 6. BER performance of Coded DS-UWB (SRake=5)

5 Conclusions

In this paper, the performance of coded and uncoded DS-UWB system is analyzedover UWB multipath fading channel. In order to minimize the power consumption




and get better performance, bipolar modulation is used here. A useful BER expressionis deduced for uncoded DS-UWB scheme; for coded DS-UWB scheme, an upperbound and lower bound of BER are achieved. The analysis and simulation resultsshow that the coded scheme is outperform the uncoded system significantly with

acceptable complexity improvement. From the bit error probability expression of thecoded scheme, we can see that the free distance f d and the number of SRake fingers

play an important role in deciding the system performance.

References

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