Multiple access performance in UWB systems using time hopping vs ...
Multiple Access Performance in UWB Systems using Time Hopping vs. Direct Sequence Spreading V. Srinivasa Somayazulu Intel Labs. JF3-206, 2111 N.E. 25th Ave. Hillsboro, OR 97124
Abstract- Ultra-wideband radio systems operate using extremely short duration signaling pulses. Spread spectrum techniques for multiple access and interference suppression are commonly considered for such systems. We compare time-hopping spread spectrum with pulse position modulation (TH-PPM) and timehopping/direct-sequence spreading with antipodal signaling (TH/DS-BPSK) in terms of their multiple access performance, and present some arguments for the use of one versus the other in a given system.
analysis of PPM would be closer to that for orthogonal waveforms, with a corresponding loss in AWGN performance when compared to the antipodal PAM waveforms, or even the optimally spaced binary PPM waveforms of [1]. We also evaluate the performance of an MMSE multi-user detector with both kinds of waveforms (TH-PPM as well as TH/DS-BPSK), and show that for this more complex receiver, the performance difference is very small, especially at lower data rates.
I. INTRODUCTION Ultra-wideband (UWB) radio communications systems operate by using as signaling waveforms, baseband pulses of very short duration, typically on the order of a nanosecond (ns.) or less. The signal energy is thus spread over a band of frequencies upto a few GHz, and thus these signals would overlay many existing narrowband spectrum users. In addition, spread spectrum codes also provide a multiple access means for many users to share the available bandwidth. Spread spectrum techniques can also provide some immunity to interference from existing narrowband users overlaid by the UWB system, as well as mitigation of interference caused to these users. Time-hopping (TH) spreading codes are the inevitable choice of spread spectrum when using PPM data modulation of UWB pulses. When using antipodal signaling (e.g. BPSK), either direct sequence (DS) or TH spreading can be employed. In this paper, we consider the multiple access performance of these two kinds of spread spectrum codes/data modulations. In particular, [1],[4] presented results for the single user detector based multiple access performance for time hopping spread spectrum (THSS) with pulse position data modulation (TH-PPM). In this paper we compare this with the performance of a THSS or a DSSS system that employs bipolar modulation such as BPSK (TH-BPSK or DS-BPSK), in an environment consisting of only thermal noise and inter user interference. We quantify the improvement in performance obtained in the case of the simple matched filter based receiver. The results can be readily generalized to M-ary PAM. The results shown in [1] for PPM really apply to binary PPM, since for greater than two levels, it would be much harder to exploit the chosen pulse shape and obtain optimum correlation between the signaling waveforms. In the absence of such a choice, the
0-7803-7376-6/02/$17.00 (c) 2002 IEEE.
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Finally, we consider some other arguments for using THPPM vs. TH/DS-BPSK formats from a multiple access point of view. Reference [5] presents some more comparisons between the different UWB data modulation systems considering performance, complexity and spectral properties (from a point of view of optimizing performance within regulatory constraints). II. COMPARISON OF TH/DS-BPSK AND TH-PPM MULTIPLE ACCESS PERFORMANCE We consider a UWB system consisting of N u users, th
where the k user is assigned a unique time hopping spreading code c
k
of length N s , and with elements
c kj ^0, N h `. This spreading code is used to dither the transmitted pulse timing instants in units of time given by Tc and is designed such that N hTc T f . The pulse shape at the receiver antenna output for each user’s signal is taken to be the doubly differentiated gaussian pulse given below (also see Fig. 1):
w(t )
1 2
2SV p V p
2
e
t 2 / 2V p 2
§ t2 · ¨ ¸ 1 ¨V 2 ¸ © p ¹
(1)
For comparison with [1], we chose the same waveform, with
Vp
is given by
0.09ns. The signaling waveform of the k
th
user
N s 1
s k (t )
¦ w(t jT
f
c kj Tc )
Also, as in [1], in order to estimate the correlation properties of the spreading codes without choosing particular
(2)
k
codes, we assume that the elements c j for different j and k
j 0
and the transmitted signal for the k
th
user is given by
Ak ¦ d >i / N s @ s (t >i / N s @N sT f )
(3)
the bipolar modulated sequence
d ^ 1,1`
k
k
x (t )
i
where
k
>@
th
represents the data bits for the k user, and z the integer part of z. Further, the data rate (bit rate) is given by
Rs
1
st
. The received signal for the 1
N sT f
¦ x k (t W k ) V n n(t )
where n(t ) ~ N (0,1) is the thermal noise term.
The
th
user is
We form the vector of sufficient
statistics y obtained by collecting the outputs of the N u individual matched filters over 1 symbol (bit) with input r(t), written as below: y ARd V n n (5) Here,
A
diag[ A1 ,..., AN ] ,
d is the vector
u
containing the data bits for the current symbol period, n is a N (0, R ) random vector, and the cross-correlation matrix formed as:
R i, j
U i, j
uniformly distributed over the range
^
`
E s i (t W i ) s j (t W j )
(6)
spreading codes, i.e.,
Ui , j
i
j
iz j
being
[0, T f ) .
U
(7)
N s ³ w2 (t )dt Ns Tf
2 2
(8) 2
f
ªf º ( ) ( ) w t s w t dt « » ds ³ ³ f ¬ f ¼
(9)
This is valid for both TH and DS spreading codes of length N s . Then, as in [2], we define the leakage coefficients
E k as: Ek
Ak U 2 A1 U1
(10)
Finally,
Ai
assuming perfect power control, i.e., A i , the probability of error for the conventional
linear (single user matched filter) detector is given by [2]:
Pe
Next, we assume that the N u users all use equi-correlated
U1 ® ¯U2
are
From (2) and (6), we can derive the following:
(4)
k 1
given by s (t W k ) .
W k W1
i.i.d. random variables, with ( W k W 1 ) mod T f
and
Nu
k
assume that the transmission time differences
U1
matched filter for the single user detector for the k
@
(desired)
user’s signal is given by
r (t )
>
are i.i.d. random variables over the range 0, N h . We also
· § Nu V P¨¨ ¦ E k bk n n1 ! 1¸¸ AU1 ¹ ©k 2
(11)
(by using the assumptions of equal power and equicorrelated waveforms, we are able to exploit the symmetry to drop the user index from the probability of error). The Gaussian approximation, again following [2] gives us: 1 / 2 §§ V 2 · 2· n ¨ (12) Pe Q ¨¨ 2 ( N u 1) E ¸¸ ¸ ¨ © A U1 ¸ ¹ © ¹ where Q (.) is the complementary unit Gaussian cumulative
distribution function. Now, from (8)—(10), 2
f
E2
U 22 U12
ªf º ³f«¬³fw(t s)w(t )dt »¼ ds Rs 2 2 ³ w (t )dt
>
C x Rs Figure 1. Representation of received UWB monocycle pulse.
0-7803-7376-6/02/$17.00 (c) 2002 IEEE.
519 523
@
(13)
and
A2 U1
V n2
= SNR, i.e., the signal to noise ratio for the single
user case. For the pulse shape in (1), we have:
C
35 S Vp 24 2
1.82775V p
and so, we can write (given
Vp
(14)
0.09 ns):
Pe ,TH BPSK 1 / 2 §§ 1 · -10 · ¨ Q ¨ ( N u 1) Rs 1.6462 u 10 ¸ ¸ ¨ © SNR ¹ ¸¹ ©
(15)
It must be noted here that the Gaussian approximation in (12), following [2],[3] is quite loose for all but very low SNRs for the single user (matched filter) based detector. For comparison with the case of the “optimally spaced” PPM modulation waveform in [2], the probability of error there is computed as:
Pe ,TH PPM 1 / 2 §§ 1 · -10 · ¨ ( N u 1) Rs 2.1849 u 10 ¸ ¸ Q ¨ ¨ © SNR ¹ ¸¹ ©
(16)
In Fig. 2, we plot the log of the probability of error according to (15) and (16) for the case when the single user 3
SNR is fixed so as to ensure a probability of error of 10 . To make the same comparison in a different way, we next consider the number of users N u vs. the additional increase
Fig. 2. Comparison of the performance of TH/DS-BPSK and TH-PPM in the multiple access environment, with single user detector and MMSE detector. The data rate is 19.2 kbps and the single user SNR is chosen for a probability of bit error of 1e-3.
Along the same lines, we also compare the performance of a more complex receiver employing multi-user detection with both TH/DS-BPSK and TH-PPM. Specifically, we consider a linear multi-user detector such as the MMSE detector [2]. The advantage of the MMSE multi-user detector is that it admits adaptive implementations that only depend on the knowledge of the signaling waveform of the particular user under consideration. Thus, the MMSE detector can be adapted blindly, without training sequences or knowledge of other users’ signature sequences. The probability of error (under the Gaussian assumption) is given by [2]:
in power 'P needed to maintain the same SNR in multiple access interference, given by [4]:
1 1 § 1 · (17) ¨1 ¸ C x Rs SNR © 'P ¹ As 'P !! 1 , we get: 1 1 (18) N u ,max | C x Rs x SNR In Fig. 3 we plot N u vs. 'P as in (17) for Nu 1
Rs
19.2kbps data users, and the SNR fixed as before.
Furthermore, the maximum number of users for the case of TH/DS-BPSK is N u ,max | 33130, and for TH-PPM
N u ,max | 24962.
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520 524
Fig. 3. Total number of users vs. additional power 'P (in dB) for single user detector based receiver in multiple access interference environment. Data rate fixed at 19.2 kbps per user, and the single user SNR is chosen for a probability of bit error of 1e-3.
Pe, MMSE
1 / 2 §§ · § ·· ¨¨ ¸ ¸ ¨ ¸ 2 U ( N u 1) ¨¨ ¸ ¸ ¨ ¸ Q¨ SNR 1 ¸ 1 ¨ ¸ ¨ ¸ ( N u 2) U ¸ ¸ ¨¨ ¨ ¸¸ ¨ 1 SNR © ¹¹ ©© ¹
(19)
where SNR is the single user SNR that is fixed at a desired value based on a certain probability of error, and R u 1.6462 u 10 10 TH / DS BPSK (20) U2 ® s 10 R 2 . 1849 10 TH PPM u u ¯ s For moderately large values of SNR and for moderately large N u , it can be seen that the probability of error is
the natural choice for spread spectrum multiple access. For very low data rate systems on the other hand, this is no longer an important factor. In this case, the observation that the THBPSK UWB signal may be potentially less susceptible to the near-far effect than the DSSS UWB signal, as speculated upon in [1] may be a reason to prefer TH-BPSK over DSBPSK systems. Of course, further analysis of TH and DS codeset designs is needed to determine whether the above assumption of equal length spreading codes for a given level of performance is justified, or if TH codes of shorter length could be used for equivalent multiple access performance compared to a given DS codeset.
approximately given by:
1 / 2
Pe , MMSE | Q SNR 1 U
For low data rates Rs , since
(21)
U 1 for both TH/DS-
BPSK and TH-PPM, the probability of error for the MMSE detector will be little different for the two systems. In both cases, the MMSE detector will show very minimal loss in performance from the single user case as the number of users is increased – e.g., see the MMSE detector curve in Fig. 2. For higher data rates, once again, the TH/DS-BPSK system will have better properties than the TH-PPM system.
III. CONCLUSION For the simple matched filter based single user detector, either time-hopping or direct sequence spreading with bipolar modulation yields better performance than time-hopping with PPM. However, even with bipolar modulation, depending on the data rates there are other reasons to prefer time-hopping or direct sequence spreading. With more complex multi-user receivers such as the MMSE detector the difference between the TH-PPM and TH/DS-BPSK systems seems to be almost negligible, especially at lower data rates. REFERENCES
Even using BPSK modulation, however, there are other considerations that differentiate between TH and DS spreading. For a given pulse width and spreading code length, the TH spreading would force a smaller PRF than the DS spreading, and thus limit the system to lower data rates. This is because in order to have a TH spreading code with good correlation properties, the range of values of the spreading code c j ^0, N h ` must be reasonably large. k
Since N hTc T f , (where Tc | Td , the pulse duration, which is about 1ns) this forces the T f to be larger, i.e., the PRF to be smaller. For example, assuming a value of N h | 10 gives a minimum T f of approximately 10ns, or a PRF of
Abstract- Ultra-wideband radio systems operate using extremely short duration signaling pulses. Spread spectrum techniques for multiple access and interference suppression are commonly considered for such systems. We compare time-hopping spread spectrum with pulse position modulation (TH-PPM) and timehopping/direct-sequence spreading with antipodal signaling (TH/DS-BPSK) in terms of their multiple access performance, and present some arguments for the use of one versus the other in a given system.
analysis of PPM would be closer to that for orthogonal waveforms, with a corresponding loss in AWGN performance when compared to the antipodal PAM waveforms, or even the optimally spaced binary PPM waveforms of [1]. We also evaluate the performance of an MMSE multi-user detector with both kinds of waveforms (TH-PPM as well as TH/DS-BPSK), and show that for this more complex receiver, the performance difference is very small, especially at lower data rates.
I. INTRODUCTION Ultra-wideband (UWB) radio communications systems operate by using as signaling waveforms, baseband pulses of very short duration, typically on the order of a nanosecond (ns.) or less. The signal energy is thus spread over a band of frequencies upto a few GHz, and thus these signals would overlay many existing narrowband spectrum users. In addition, spread spectrum codes also provide a multiple access means for many users to share the available bandwidth. Spread spectrum techniques can also provide some immunity to interference from existing narrowband users overlaid by the UWB system, as well as mitigation of interference caused to these users. Time-hopping (TH) spreading codes are the inevitable choice of spread spectrum when using PPM data modulation of UWB pulses. When using antipodal signaling (e.g. BPSK), either direct sequence (DS) or TH spreading can be employed. In this paper, we consider the multiple access performance of these two kinds of spread spectrum codes/data modulations. In particular, [1],[4] presented results for the single user detector based multiple access performance for time hopping spread spectrum (THSS) with pulse position data modulation (TH-PPM). In this paper we compare this with the performance of a THSS or a DSSS system that employs bipolar modulation such as BPSK (TH-BPSK or DS-BPSK), in an environment consisting of only thermal noise and inter user interference. We quantify the improvement in performance obtained in the case of the simple matched filter based receiver. The results can be readily generalized to M-ary PAM. The results shown in [1] for PPM really apply to binary PPM, since for greater than two levels, it would be much harder to exploit the chosen pulse shape and obtain optimum correlation between the signaling waveforms. In the absence of such a choice, the
0-7803-7376-6/02/$17.00 (c) 2002 IEEE.
518 522
Finally, we consider some other arguments for using THPPM vs. TH/DS-BPSK formats from a multiple access point of view. Reference [5] presents some more comparisons between the different UWB data modulation systems considering performance, complexity and spectral properties (from a point of view of optimizing performance within regulatory constraints). II. COMPARISON OF TH/DS-BPSK AND TH-PPM MULTIPLE ACCESS PERFORMANCE We consider a UWB system consisting of N u users, th
where the k user is assigned a unique time hopping spreading code c
k
of length N s , and with elements
c kj ^0, N h `. This spreading code is used to dither the transmitted pulse timing instants in units of time given by Tc and is designed such that N hTc T f . The pulse shape at the receiver antenna output for each user’s signal is taken to be the doubly differentiated gaussian pulse given below (also see Fig. 1):
w(t )
1 2
2SV p V p
2
e
t 2 / 2V p 2
§ t2 · ¨ ¸ 1 ¨V 2 ¸ © p ¹
(1)
For comparison with [1], we chose the same waveform, with
Vp
is given by
0.09ns. The signaling waveform of the k
th
user
N s 1
s k (t )
¦ w(t jT
f
c kj Tc )
Also, as in [1], in order to estimate the correlation properties of the spreading codes without choosing particular
(2)
k
codes, we assume that the elements c j for different j and k
j 0
and the transmitted signal for the k
th
user is given by
Ak ¦ d >i / N s @ s (t >i / N s @N sT f )
(3)
the bipolar modulated sequence
d ^ 1,1`
k
k
x (t )
i
where
k
>@
th
represents the data bits for the k user, and z the integer part of z. Further, the data rate (bit rate) is given by
Rs
1
st
. The received signal for the 1
N sT f
¦ x k (t W k ) V n n(t )
where n(t ) ~ N (0,1) is the thermal noise term.
The
th
user is
We form the vector of sufficient
statistics y obtained by collecting the outputs of the N u individual matched filters over 1 symbol (bit) with input r(t), written as below: y ARd V n n (5) Here,
A
diag[ A1 ,..., AN ] ,
d is the vector
u
containing the data bits for the current symbol period, n is a N (0, R ) random vector, and the cross-correlation matrix formed as:
R i, j
U i, j
uniformly distributed over the range
^
`
E s i (t W i ) s j (t W j )
(6)
spreading codes, i.e.,
Ui , j
i
j
iz j
being
[0, T f ) .
U
(7)
N s ³ w2 (t )dt Ns Tf
2 2
(8) 2
f
ªf º ( ) ( ) w t s w t dt « » ds ³ ³ f ¬ f ¼
(9)
This is valid for both TH and DS spreading codes of length N s . Then, as in [2], we define the leakage coefficients
E k as: Ek
Ak U 2 A1 U1
(10)
Finally,
Ai
assuming perfect power control, i.e., A i , the probability of error for the conventional
linear (single user matched filter) detector is given by [2]:
Pe
Next, we assume that the N u users all use equi-correlated
U1 ® ¯U2
are
From (2) and (6), we can derive the following:
(4)
k 1
given by s (t W k ) .
W k W1
i.i.d. random variables, with ( W k W 1 ) mod T f
and
Nu
k
assume that the transmission time differences
U1
matched filter for the single user detector for the k
@
(desired)
user’s signal is given by
r (t )
>
are i.i.d. random variables over the range 0, N h . We also
· § Nu V P¨¨ ¦ E k bk n n1 ! 1¸¸ AU1 ¹ ©k 2
(11)
(by using the assumptions of equal power and equicorrelated waveforms, we are able to exploit the symmetry to drop the user index from the probability of error). The Gaussian approximation, again following [2] gives us: 1 / 2 §§ V 2 · 2· n ¨ (12) Pe Q ¨¨ 2 ( N u 1) E ¸¸ ¸ ¨ © A U1 ¸ ¹ © ¹ where Q (.) is the complementary unit Gaussian cumulative
distribution function. Now, from (8)—(10), 2
f
E2
U 22 U12
ªf º ³f«¬³fw(t s)w(t )dt »¼ ds Rs 2 2 ³ w (t )dt
>
C x Rs Figure 1. Representation of received UWB monocycle pulse.
0-7803-7376-6/02/$17.00 (c) 2002 IEEE.
519 523
@
(13)
and
A2 U1
V n2
= SNR, i.e., the signal to noise ratio for the single
user case. For the pulse shape in (1), we have:
C
35 S Vp 24 2
1.82775V p
and so, we can write (given
Vp
(14)
0.09 ns):
Pe ,TH BPSK 1 / 2 §§ 1 · -10 · ¨ Q ¨ ( N u 1) Rs 1.6462 u 10 ¸ ¸ ¨ © SNR ¹ ¸¹ ©
(15)
It must be noted here that the Gaussian approximation in (12), following [2],[3] is quite loose for all but very low SNRs for the single user (matched filter) based detector. For comparison with the case of the “optimally spaced” PPM modulation waveform in [2], the probability of error there is computed as:
Pe ,TH PPM 1 / 2 §§ 1 · -10 · ¨ ( N u 1) Rs 2.1849 u 10 ¸ ¸ Q ¨ ¨ © SNR ¹ ¸¹ ©
(16)
In Fig. 2, we plot the log of the probability of error according to (15) and (16) for the case when the single user 3
SNR is fixed so as to ensure a probability of error of 10 . To make the same comparison in a different way, we next consider the number of users N u vs. the additional increase
Fig. 2. Comparison of the performance of TH/DS-BPSK and TH-PPM in the multiple access environment, with single user detector and MMSE detector. The data rate is 19.2 kbps and the single user SNR is chosen for a probability of bit error of 1e-3.
Along the same lines, we also compare the performance of a more complex receiver employing multi-user detection with both TH/DS-BPSK and TH-PPM. Specifically, we consider a linear multi-user detector such as the MMSE detector [2]. The advantage of the MMSE multi-user detector is that it admits adaptive implementations that only depend on the knowledge of the signaling waveform of the particular user under consideration. Thus, the MMSE detector can be adapted blindly, without training sequences or knowledge of other users’ signature sequences. The probability of error (under the Gaussian assumption) is given by [2]:
in power 'P needed to maintain the same SNR in multiple access interference, given by [4]:
1 1 § 1 · (17) ¨1 ¸ C x Rs SNR © 'P ¹ As 'P !! 1 , we get: 1 1 (18) N u ,max | C x Rs x SNR In Fig. 3 we plot N u vs. 'P as in (17) for Nu 1
Rs
19.2kbps data users, and the SNR fixed as before.
Furthermore, the maximum number of users for the case of TH/DS-BPSK is N u ,max | 33130, and for TH-PPM
N u ,max | 24962.
0-7803-7376-6/02/$17.00 (c) 2002 IEEE.
520 524
Fig. 3. Total number of users vs. additional power 'P (in dB) for single user detector based receiver in multiple access interference environment. Data rate fixed at 19.2 kbps per user, and the single user SNR is chosen for a probability of bit error of 1e-3.
Pe, MMSE
1 / 2 §§ · § ·· ¨¨ ¸ ¸ ¨ ¸ 2 U ( N u 1) ¨¨ ¸ ¸ ¨ ¸ Q¨ SNR 1 ¸ 1 ¨ ¸ ¨ ¸ ( N u 2) U ¸ ¸ ¨¨ ¨ ¸¸ ¨ 1 SNR © ¹¹ ©© ¹
(19)
where SNR is the single user SNR that is fixed at a desired value based on a certain probability of error, and R u 1.6462 u 10 10 TH / DS BPSK (20) U2 ® s 10 R 2 . 1849 10 TH PPM u u ¯ s For moderately large values of SNR and for moderately large N u , it can be seen that the probability of error is
the natural choice for spread spectrum multiple access. For very low data rate systems on the other hand, this is no longer an important factor. In this case, the observation that the THBPSK UWB signal may be potentially less susceptible to the near-far effect than the DSSS UWB signal, as speculated upon in [1] may be a reason to prefer TH-BPSK over DSBPSK systems. Of course, further analysis of TH and DS codeset designs is needed to determine whether the above assumption of equal length spreading codes for a given level of performance is justified, or if TH codes of shorter length could be used for equivalent multiple access performance compared to a given DS codeset.
approximately given by:
1 / 2
Pe , MMSE | Q SNR 1 U
For low data rates Rs , since
(21)
U 1 for both TH/DS-
BPSK and TH-PPM, the probability of error for the MMSE detector will be little different for the two systems. In both cases, the MMSE detector will show very minimal loss in performance from the single user case as the number of users is increased – e.g., see the MMSE detector curve in Fig. 2. For higher data rates, once again, the TH/DS-BPSK system will have better properties than the TH-PPM system.
III. CONCLUSION For the simple matched filter based single user detector, either time-hopping or direct sequence spreading with bipolar modulation yields better performance than time-hopping with PPM. However, even with bipolar modulation, depending on the data rates there are other reasons to prefer time-hopping or direct sequence spreading. With more complex multi-user receivers such as the MMSE detector the difference between the TH-PPM and TH/DS-BPSK systems seems to be almost negligible, especially at lower data rates. REFERENCES
Even using BPSK modulation, however, there are other considerations that differentiate between TH and DS spreading. For a given pulse width and spreading code length, the TH spreading would force a smaller PRF than the DS spreading, and thus limit the system to lower data rates. This is because in order to have a TH spreading code with good correlation properties, the range of values of the spreading code c j ^0, N h ` must be reasonably large. k
Since N hTc T f , (where Tc | Td , the pulse duration, which is about 1ns) this forces the T f to be larger, i.e., the PRF to be smaller. For example, assuming a value of N h | 10 gives a minimum T f of approximately 10ns, or a PRF of
