Performance Analysis of Impulse Radio Under Timing Jitter Using M-ary Bipolar Pulse Waveform and Position Modulation

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Performance Analysis of Impulse Radio Under Timing Jitter

Using M-ary Bipolar Pulse Waveform and Position Modulation

Shih-Chun Lin and Tzi-Dar Chiueh Graduate Institute of Electronics Engineering

and Department of Electrical Engineering National Taiwan University, Taipei, Taiwan, 10617 E-mail sclin@analog.ee.ntu.edu.tw, chiueh@cc.ee.ntu.edu.tw

Abstract-In this paper, we derive analytical bit error rate (BER) performance of impulse radio using spread-spectrum technique under correlated random timing jitters. The result is applied to the bipolar pulse waveform and position modulation (BPWF'M) to study its degradation in performance. High-order derivatives of the Gaussian pulse with about 500 ps pulse width are chosen to comply the FCC part 15 limit. The analysis concludes that both jitter mot-mean square (RMS) value and correlation between jitters are important facton that inhence system performance. When the jitters are highly correlated, different signal sets will have different levels of performance degradation even with the same jitter RMS value, which needs be taken into consideration in impulse radio system design.

I. INTRODUCTION

By transmitting nano-second duration, low duty-cycle baseband pulses, ultra-wide hand (UWB) impulse radio is a prominent candidate for low-cost short-range wireless communication application However, as indicated in [l], impulse radio will induce large interference to global position system

(GPS) under the original FCC part 15 level. Thus FCC has further restricted the part 15 mask for communication usage

to 3.1 - 10.6 GHz [2]. Without frequency-translation mixer in the transmitter, the original Gaussian monocycle pulse used in [I], [3] has too wide a bandwidth that violates the revised FCC mask. This motivates us to explore the impact of pulse waveform and consider waveform as a parameter in system design.

The other issue that, affects UWB system performance significantly and strongly depends on pulse waveform is pulse timing precision. The impact of independent jitter on impulse radio systems was simulated in [3] and

the performance degradation is more serious than that in narrowband systems. In practice, correlated jitter sources such as substrate and supply noise are commonly seen in transmitterlreceiver circuits [4], [5]. Thus we take the correlation between jitters into consideration and study the impact of correlated jitters on the BER performance in additive white Gaussian noise (AWGN) channel. The results

can he used as a theoretical upper bound for evaluating system performances in multipath channels.

This paper is organized as follows. In Section I1 we will introduce the impulse radio system and signal model. Next, signal set design is presented in Section

III.

Section N pro- vides the analysis of the impact caused by correlated random jitters on error performance with numerical examples given in Section V. Finally, Section VI concludes this paper.

mihis work was supponed in part by MediaTek Inc.

11. SYSTEM AND SIGNAL MODEL

A. Channel Model and Pulse Waveform

Consider a single-user system and assume that the time- domain characteristic of the antenna is a differentiator. The pulse propagated in the channel is P&) = Jf_P(Qd<, and the received pulse in the baseband processor is AP(t - 7)

+

n ( t ) , where P ( t ) , A, and T represent the pulse waveform, the link path loss, and the path delay, respectively. The noise process n ( t ) is an additive white Gaussian process with two- sided power spectral density No/2.

As in [l], we select those pulse waveforms derived from the Gaussian pulse for detailed study. These waveforms are widely used in image and spatial vision processing [61. The

kth derivative of the Gaussian pulse is expressed as

where the pulse width Tp is about 2.3. and the pulse energy Ep = J ? ~ [ P ( < ) ] 2 d ~ is normalized to unity through amplitude Ax. As indicated in [l], increasing the order of the derivative while keeping the pulse width unchanged will shift the power spectral density toward higher frequency and induce less interference below 3.1 GHz. To this end, we choose higher-order G&) and G7(t) with f = 210 ps and they both comply the FCC part 15 limit.

B. Spread Spectrum Signal F o m t

To combat the narrowband interference from existing systems, conventionally spread-spectrum techniques are used in UWB signaling. In this paper, we adopt the DSSS-UWB signaling format as described in [7] . We first define & ( t ) as a possible symbol, and consider the bipolar DSSS-UWB signal, which can be represented as

N,-l

"=O

s i ( t ) = ai - nTf -&) (2)

where ai is the polarity;

N,

is the number of pulses per symbol (spreading factor);

e

is the nth chip of the spreading sequence; T' is the pulse repetition (frame) time and Si is the pulse offset. Note that each chip in the spread-spectrum signal is waveform-mapped to a single pulse. One feature of impulse radio is that we can let

Tf

>>

Tp to obtain the extra processing gain 1OloglO(Tf/Tp) in addition to the spreading factor. The spreading sequence is a pseudo- random (PN) code to provide resistance against interference.

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If the receiver hardware complexity is the main system consideration, Golay sequence is suggested over conventional PN codes [8].

111. SIGNAL SET DESIGN

In this section, we will introduce the signal set design cri- teria and resultant design. The signal set design parameters corresponding to the data modulation are

ai E {-1,+1} (3)

pi(?) E {wO(t),wl (t), .

.

, wdw-l

(?)I

6i E (70 = 0

<

71

<

' ' _

<

7 d s - ] }

where d, 2 1 and d, 2 1 are integers that represent the dimension of waveforms Pi(t) and timing offset 6i, respec- tively. The size of the signal set is given hy

M

= 2 . d ,

.

d,.

The error performance of the signal set depends on the normalized correlation value between symbols Si(?) and

B

p

S j ( f ) . Assuming that the receiver is synchronized and the spreading sequence is known, the correlation between two symbols is

Fig. 1.

signals using the Golay sequence with code length 256.

Power spechal densities of the (a) BPWPMI and (b) B P W P M Z

(4)

"

= a i a j . R i j ( 6 j - S i ) ,

where the normalized pulse cross-correlation function Rij(r)

With the defined correlation value, we then study the signal set performance using union bound for the symbol error probability (SER) [9]. With perfectly synchronized spread spectrum signals in the optimal coherent detector under free- space channel and equally likely signals assumption, the union bound gives

where P,(Si + S j ) is the pairwise symbol error probability

of transmitting S,(t) and deciding erroneously to S,(t);

e(.)

is the Gaussian tail integral; b = log,M and the received

signal to noise ratio (SNR) per bit "fb is AzEs/(Nob).

When M = 2, it is well known that antipodal signal set achieves minimum symbol error rate performance. It can be shown that for the case M

>

2 with parameters in (3), the minimum UBP, occurs when the signal set is biorthogonal. From (4), this can he done by choosing disjoint time slots,

i.e. letting 8, - 6i be sufficiently larger than T,. Here we

define a design parameter TOR as the minimum 6j -6j that satisfies this condition. When two overlapping pulse waveforms, Pi(?) E { W o ( t ) , W ~ ( t ) } , are used, we can also make p j j = 0 by letting S j -61 = 0 and ::J W & ) W ~ ( t ) d t =

0. For example, if WO(?) = G&) and W I ( ? ) = & ( t ) , this condition can be satisfied.

The resultant signal sets are summarized in Table I. Fig. 1 shows the transmitted signal power spectral densities of these signals. As the design in [7] that intends to ease the sensitivity loss in other narrowband systems from impulse radio signals, we set signal PSD to he 6 dB lower than the FCC part 15 mask. Since the BER performance is more relevant in digital communication, we use the union bound

on BER given by

where dH(i, j ) is the Hamming distance between bit patterns assigned to S i ( t ) and S j ( t ) .

IV. PERFORMANCE ANALYSIS UNDER CORRELATED

RANDOM TIMING JITTERS

In this section, we will derive formulas that describe the impact of correlated random timing jitters on the BER performance of different modulation schemes listed in Ta- ble l. To simplify the theoretical analysis and notation, the following additional assumptions are made:

1) The receiver is synchronized and there is

no

inter-chip interference (ICI) caused by small timing jitter. The spreading sequence

&'

will then have no effects in the analysis, so we set

2

= 1,vn for simplicity.

,...,

E" ,...

are samples of a zero-mean wide-sense stationary Gaussian process with variance crz for all E. [31, [51.

To model the hand-limited phenomenon of this pro- cess, we assume this process is non-white with autocorrelation function which has essentially zero values when the relative time difference between samples are longer than a certain threshold, i.e. E [ ( E ~ ) ( E I ) ] = 0, when II - nl 2 in. In other words, induced jitters will he correlated over several adjacent pulses, and they are independent for pulses farther apart. Furthermore, 2) The sequence of random timing jitters

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TABLE I

SUMMARY OF M-ARY BIPOLAR PULSE WAVEFORM POSITION MODULATION FAMILY.

~~~

Tvoe of Signal oattem Size of Union bound Correlators

*

All ui = - 1 ' h ' in both signal sets

the spreading factor N, is larger than the threshold,

N , > m > l .

The objective now is to compute the general US% in (7) in the case of such timing jitters. We first compute the individual pairwise SER, P,{Si + S j } . The received signal r ( t ) at the correlator input is

and the correlator template waveform is

where

lJij(t) = u i 4 ( t - 6 j ) - u j P j ( f - 6 j ) .

The decision metric yij at the output of the correlator is

yij =

1

r(t)v(t)dt = dij + n ; j ,

I E T

where T is the symbol time, and d i j and nij are the data component and the noise component, respectively. An error is made when the decision metric is less than zero, i.e. yij

<

0.

The receiver is confronted with an M-ary hypothesis testing prohlem with hypothesis Hi : A . Si(? - 5 )

+

n ( t ) , 0

5 i

<

M

- 1. To compute the pairwise SER Pe{Si i S j } = P(yij

<

OlHi), we need the statistics of both dij and njj. The

noise component nij is J E T n ( t ) v ( r ) d t , and it can be shown that this random variable is independent of random jitter

and is normally distributed with distribution

" i j - N ( O , N o N , ( I - p i j ) ) (8)

where Z

-

N ( p , o z ) means that random variable Z is a

normal distribution with mean p and variance

&.

The statistics of data component d i j is far more complex, we first write it as

dij = A l s r ~ a i P ; ( r - . r - n T f - G j ) v i j ( t - . r - n T f - & , ) d r

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N,-1

where R i ( 7 ) is the autocorrelation function of Pi(?) obtained from setting P j ( t ) = Pi@) in (5) and S j i = 6 j

-

6i is the relative timing offset between & ( t ) and S j ( t ) . Here we define a new function

hj(.)

to simplify the notation. Since the waveforms used in impulse radio system are not simple rectangular pulses, f i j ( 5 ) is usually a nonlinear function in its argument.

From the above derivations, we h o w that the data com- ponent dij is summation of nonlinear functions of correlated

random variables E,, typically it is hard to identify the statistics of such complex combination of random variables. Fomnately, f i j ( . ) is a zero-memory nonlinear function, and the sequence of random variables fij(&"),O

5

n

<

Ns has the following properties:

1) Sequence f i j ( c n ) is stationary since the sequence of jitters E . is stationary.

2 ) It can be shown that sequence f i j ( c ) is m-dependent sequence of random variables since sequence &" can

be modelled as m-dependent. The definition of m- dependent sequence of random variables can be found in [lo, p. 2151.

3) Let E[.] be the expectation function, then E [ [ f i j ( ~ . ) 1 ~ ]

exists and is bounded in the range from 0 to Z3 since Based on these three properties, we can invoke the central limit theorem for m-dependent sequence of random variables in [lo] to compute the statistics of dij, and

Ifij(&n)l

i

2.

d i j - N ( A N s p i j , A 2 N s a f , ) , (10)

where

pij = E [ h j ( ~ g ) l

4

= v ~ r V i j ( ~ g ) I + 2 E ~ o v [ f i j ( e ~ ) f i j ( & n ) I >

where Vur[.] denotes the variance of a random variable and Cov[.] the covariance of two random variables.

From the statistics of data component djj in

(IO)

and noise component nij in (8) and the fact that they are independent Gaussian random variables, the resulting error probability

Pe{Si i S j } = P(dij +nij

<

OIH;) is

m

(4)

where we consider only the case when p;j

>

0. Basically, this formula is sufficient to illustrate the pairwise SER behavior since parameters pij and ab can be computed using numerical integral with known probability density function (PDF) of EO and joint PDF of EO and &". However, this formula carries less insight and requires long computation time on the double integral. To this end, we use the Taylor series expansion on the nonlinear

fil(.)

with moment functions of

random jitters to estimate p,, and

at..

The estimation of mean in (10) using &order approximation is

= ( l - p i j ) + L i j ,

where

$fil"'(.)

is the kth derivative of

&(.)

and Ljj represents variance can also be estimated with K,th Taylor polynomial

in two variables

the effect of jitter on pjj. Note that f , (0) (0) = 1 - p i , . Co-

COV[fij(Eg).fij(En)I (13)

Now the problem becomes finding the joint moment of random values

eo

and E", E[(Q)'(&.)*]. Since they are jointly Gaussian this joint moment can he found in [ I l , p. 2121, which is a polynomial function of aE and pS," =

E [ ( E ~ ) ( E . ) ] / $ . Moment function of

a,

E [ 4 ] in (12) is a

special case of E [ ( Q ) ~ ( & ~ ) * ] , which can be obtained by setting E. =

ea

and q = 0. Replacing (12) and (13) into (IO), we conclude that the effect of Gaussian random jitter sequence on pij and

:

,

a

can be approximated by polynomial functions of jitter variance a; and correlation pE,". The influences

of

pulse waveform are reflected on the polynomial coefficients through nonlinearity

A,(.).

Finally, the painvise SER is given by

where Vjj denotes the approximation of a2- When no jitter exists,

as

= 0 then Lij and Vj, will be e y u J i o zero, thus the painvise

SER reduces to

the

case in

(6).

As shown

in

(14),

the effect of random jitters can be treated as another additive

Gaussian noise source with non-zero mean and SNR related variance.

The performance for signal sets listed in Table I using

(14) will now be discussed. In these modulation schemes, three different cases of painvise SER are considered. We can observe that in these cases with Pj(t) = Gk(t) and

P j ( r ) = G q ( l ) , all

fi,(.r)

in (9)

can

be reduced to or expressed in a common form, i.e. R ~ ( T )

-

0 . R G ~ G ~ ( T ) , where 0 = 0 or *l, R c , ( r ) is Gaussian pulse autocorrelation function and RckcP ( 5 ) is Gaussian pulse cross-correlation function, respectively. Thus we can define two new variables, Lkq(0) and Vkq(0) to respectively denote L;j and Vij when f:j(.r)

equals to the above form.

The first case is Si(t) = - S j ( t ) , i.e. the two signals are antipodal with

E ( t )

= G&) and fjj(.r) equals to 2Rck(7)

(k=7 in BPWPMl and k=6,7 in BPWPM2 signal set). According to (14) with pi, = -1, Lij = 2Lkk(0) and V, = 4Vkk(O), the resulting SNR; for the antipodal signal set is

Note that &(O) is always less than zero since signal autocorrelation functions are always maximum and concave at 7 = 0, i.e. SNR), is always smaller than the perfect timing case. In addition, the autocorrelation functions reach global extremum and

f!!)(O)

is always zero at 7 = 0, and Vkk(0) is related only to ;he higher power (larger than one) of jitter variance

a,'.

The second case is orthogonal signal pair obtained by choosing disjoint time slots between two signals. Since signal correlation values of Gaussian pulses are almost zero when the relative timing offset is larger than or equal to TOR,

thus the term Ri,(S,i + E ~ ) in (9) can be neglected for small

jitter E.. When P;(t) =

G&),

we can set fi,(r) to RG (7).

this kind of orthogonal signal set is

Using this fact together with p j j = 0 in (14), the SNRij

d

for

Note that SNPDoF is almost the same as S N R L except that

the noise power IS twice as large.

The thiid case is the overlapped orthogonal signal sets. From S j j = 0 together with Pi(t) = Gk(r), P,(t) = Gq(t), and

aiaj = +1, Ljj and

V ,

in (14) are Lkq(+l) and Vkq(+l),

respectively. Then the

S N R

becomes

where k = 6,q = 1 or k = 7,q = 6 in BPWPM2 system. When ajaj = -1, the S N R is SNR&, which is the same

as SNRA$R except the arguments in Lkq and V q are both -1. Substituting (15) through (17) in

(7),

we then summarize the union bound on BER in the presence of jitter for the two

signal sets in Tahle

II. v.

NUMERICAL EXAMPLE

In this section we compute the BER under 20 ps RMS jitter using Table

II.

To make a fair comparison, we fix the

symbol time T = N,Tf and

M

= 16 for both signal sets. As can be seen in Table I, BPWPM2 signals can have half Tf as BPWPMl at the same

M

thus having twice Ns. So we set N, = 128 for BPWPMl and N, = 256 for BPWPM2 signals. The correlated time window

m

is affected by Tf and the bandwidth of the jitter process, and typically it is on the order of tens. This research covers

a

selection of

m which is small compared to N,. Since BPWPM2 signals have half T' as BPWF'Ml and thus twice m, we set m = 10 for BPWPMl and m = 20 for BPWPM2. The bandlimited effect is modelled as first-order low-pass filtering and pE,"

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TABLE ll

UNION BOUND BER UNDER JITTERS FOR THE TWO SIGNAL SETS

Type Union bound BER

of argnal BPWPMl

under random timing jitters

Q

(m)

+

v .

Q

(&)

f ,

[Q(-) + Q

(m)]

+

h,

[Q

(m)

+ Q

I

(-)

+ BPWPM2

$2.

[n

(m)

+

B

(&Gi)]

+

V ,

[Q

(-)

+ a

)I

(&

SNR p s r M rh

Fig. 2. Union bound BER of two signal se8 inflicted with correlated jiner of 20 ps R M S value and different m.

expansion and joint moments of Gaussian random variables. Two bipolar signal sets derived from the Gaussian pulses are used in this study. Both have the advantage of low receiver

complexity. It is shown that not only the jitter RMS value but

also the correlation between jitters influence the error rate performance. Inflicted with independent jitters of 20 ps RMS

value, about 4

dB

additional signal power must be added to achieve BER. Even more power, which can be more than 1 dB, must be added if the jitters are correlated. The effect of correlation is closely related to the pulse repetition time and the spreading factor. All these factors must be considered in impulse radio signal set design.

R E F E R E N C E S

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I31

141

1700, Dec 2002.

“Revision of P m 15 the Commissions rules regarding ultra-wideband transmission systems:’ FCC ET Docket 98.153, 2002.

W. M. Lovelace and I. K. Townsend, ‘The effects of timing jitter and tracking on the performance of impulse radio:’ IEEE J. Select. Areas

Commun., vol. 20, no. 9, pp. 1616 1651, Dec 2002.

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[IO] D. A. S. Fraser, Nonpormetric methods in smtisrics. John Wiley, 1957.

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is exponentially decaying in chip time index n. The order of Taylor approximation is set to 44 which guarantees the estimation error to be less than 0.001%. The BER curves are shown in Fig. 2, we find that BPWPM2 signals have smaller degradation than BPWF’Ml signals at low y, when jitters are correlated. It results from the values of R G ~ ( E ) decay less than R G ~ ( E ) with small timing offset E . Nevertheless, the advantage of BPWPM2 signals will disappear at high yb

due to the large equivalent additive variance in (17). When the jitter is independent (m = 0), the advantage of BPWPMZ signals will he more significant. The reason can be observed from (14). Since BPWPMZ have larger

N,

than BPWPMI, it should have smaller degradation. However, this advantage will be cancelled by larger m if m

#

0.

VI. CONCLUSION

In this paper, we investigate the impact of correlated random timing jitters on impulse radio using 3 to 10 GHz hand. The analysis shows that the influence of jitter can be

treated as another additive noise source, whose mean and variance can be efficiently approximated by Taylor series