A new noncoherent UWB impulse radio receiver - Communications ...
IEEE COMMUNICATIONS LETTERS, VOL. 9, NO. 2, FEBRUARY 2005
151
A New Noncoherent UWB Impulse Radio Receiver Mi-Kyung Oh, Byunghoo Jung, Student Member, IEEE, Ramesh Harjani, Senior Member, IEEE, and Dong-Jo Park, Member, IEEE Abstract— This letter analyzes the implementation issues related to coherent receivers for UWB impulse radio with a special emphasis on timing and jitter problems. We propose a new jitter tolerant receiver design that is easy to implement. Analytical BER analysis and simulations verify that the performance of the proposed receiver is comparable to that of a correlator-based receiver that includes jitter. The new design is a promising candidate for low-cost low-power UWB IR receiver implementations. Index Terms— Ultra wideband (UWB), noncoherent, receiver.
I. I NTRODUCTION
R
ECENTLY, Ultra Wideband (UWB) Impulse Radio (IR) systems have received significant attention from industry, media and academia due to the benefits that primarily accrue from having the energy spread over a wide spectrum. These benefits include: a) the possibility of low-cost low-power transceivers due to the absence of a carrier b) better immunity to multipath propagation, and c) low-interference [1]. However, there are still many technical challenges that remain to be solved for the widespread deployment of UWB IR systems. The reduced complexity and other implementation advantages offered by IR are somewhat offset by the stringent timing requirements [1]. The majority of the research on IR receiver design has been focused on correlator-based receivers [2]. It has recently demonstrated that the throughput of the correlator-based receiver degrades markedly for relatively modest increase in timing jitter and tracking error [3]. This mandates that a significant portion of the design process be focused on low jitter and timing error analysis and optimization. It is these findings that has triggered and motivated the noncoherent IR receiver design proposed in this letter. In this letter we verify the performance of the proposed noncoherent receiver. This receiver uses a square law device as a detector in combination with an integration/dump filter, which allows for a low-cost low-power implementation of a UWB IR system. II. S YSTEM M ODELS We develop a system model to analyze the impact of timing jitter on the proposed architecture. Let the transmitted pulse t be represented as pTX (t) = −∞ p(u)du. The received signal can then be modeled as r(t) = p(t) + n(t), if we ignore signal attenuation and delay due to propagation. The effect
Manuscript received May 10, 2004. The associate editor coordinating the review of this letter and approving it for publication was Prof. Gianluca Mazzini. M.-K. Oh and D.-J. Park are with the Dept. of EECS, KAIST, 373-1 Guseong-dong, Yuseong-gu, Daejon 305-701, Republic of Korea (e-mail: [email protected], [email protected]). B. Jung and R. Harjani are with the Dept. of ECE, Univ. of Minnesota, Minneapolis, MN, USA (e-mail: {jbh, harjani}@ece.umn.edu). Digital Object Identifier 10.1109/LCOMM.2005.02010.
Fig. 1.
Optimal correlator receiver.
of an antenna on the transmitted pulse can be modeled as a differentiation operation. The noise n(t) is assumed to be AWGN with power density σ 2 := No /2. In this letter we assume binary pulse position modulation (PPM) as the modulation scheme so that the transmitted signal consists of Np time-shifted low duty cycle pulses that can be defined as: Np −1
si (t) =
pTX (t − jTf − ci,j τp ), 0 ≤ t ≤ Np Tf
(1)
j=0
where each signal {si (t)}1i=0 is completely identified by the sequence of time shifts ci,j τp ∈ {0, τp } and Tf is the frame duration that is chosen to be sufficiently large, i.e., Tf Tp + τp where Tp denotes the duration of pulse p(t). On the receiver side, the signals can then be given by Np −1
ri (t) =
p(t − jTf − ci,j τp ) + n(t).
(2)
j=0
To simplify the ensuing analysis, we consider only one frame (Np = 1), defined within the time-interval 0 ≤ t ≤ Tf . Thus,(2) can be rewritten as: ri (t) = p(t − iτp ) + n(t), i = 0, 1.
(3)
In the absence of multipath interference, the optimum receiver for this system is the correlator receiver shown in Fig. 1, where the receiver generates a local template signal that is synchronized with the incoming pulse p(t). We note that we obtain the decision statistic Z0 at sampling time t = Tf /2 and Z1 at t = Tf . So only one correlator is required. Comparing Z0 − Z1 with the decision threshold for 0 accomplishes detection process of the transmitted symbol. However, this optimum receiver is subject to timing error and jitter, as shown in Fig. 1. Our goal in this letter is to provide details for a noncoherent receiver that does not require exact timing, and compare it to the optimal correlator receiver in terms of error performance and ease of implementation. III. N ONCOHERENT UWB IR R ECEIVER Instead of a correlator employing a pulse generator and a timing-control clock as shown in Fig. 1, we use a linear filter
c 2005 IEEE 1089-7798/05$20.00
152
Fig. 2.
IEEE COMMUNICATIONS LETTERS, VOL. 9, NO. 2, FEBRUARY 2005
Block diagram of the proposed receiver.
known as a matched filter whose sampled outputs are exactly the values obtained from the correlator [4]. The frequency response of the matched filter is designed to match the frequency spectrum of the signal, and is given by HMF (f ) = P ∗ (f )e−j2πf Tf /2 , where P ∗ (f ) is the complex conjugate of the pulse spectrum. We note that the frequency spectrum of the pulse used in UWB system is limited so as to meet the FCC spectral mask (see Fig. 3). In the absence of noise, the output of the matched filter has a spectrum |P (f )|2 ·e−j2πf Tf /2 which corresponds to the autocorrelation of the pulse in the time domain. For optimal receiver operation, exact sampling at the maximum peak of the autocorrelation of the pulse is required, which makes it difficult to implement a UWB IR. The “Square-law device and Integrator”, shown in Fig. 2, is a method that does not require exact timing. Because the exact pulse position is not known, the proposed receiver obtains the approximate peak based on the average by using the “Squarelaw device and Integrator” instead of sampling the signal at the exact peak. The benefits of the proposed receiver are twofold: a) the template signal generator and timing-control block, that consume a significant portion of the power, are eliminated and b) the degradation in performance is limited in comparison to a coherent receiver that includes the effect of jitter. A. Performance Analysis Let us consider the BER performance of the proposed receiver shown in Fig. 2 in comparison to that of the optimal correlator receiver shown in Fig. 1. If the receiver is assumed to be perfectly synchronized with the transmitter, the union bound can be given by Pe = √ on the bit error probability √ ∞ SNR , where Q(x) = 1/ 2π x exp(−t2 /2)dt, and Q T SNR:=Ep /No with energy Ep := 0 p [p(t)]2 dt [1], [2]. We evaluate the BER of the proposed receiver. Assuming that s0 (t) is transmitted, we define the random variable Z∆ := Z0 − Z1 , where the conditional probability of error is [2]: E[Z∆ |s0 ]2 Pe (s0 ) = P [Z∆ < 0|s0 (t) sent] = Q . (4) Var[Z∆ |s0 ] In the rest of this section, we obtain the conditional mean and variance of Z∆ and derive expressions for the BER. To keep its analysis simple, the matched filter is replaced by a brick wall band pass filter (BPF) because most energy of the UWB pulse can be captured within the FCC mask, which frequency response is given by: 1 , 3.1 GHz ≤ f ≤ 10.6 GHz (5) HBPF (f ) = 0, elsewhere With this in mind, let us consider a random process y0 (t) := |x0 (t)|2 = |r0 (t) ∗ hBPF (t)|2 , which is obtained from the
square-law device. We note that when the time is fixed at t0 , y0 (t0 ) is a noncentral chi-square random variable with 2 degrees of freedom where 0 ≤ t0 ≤ Tf /2, whereas y0 (t1 ) becomes a central chi-square random variable with 2 degrees of freedom where Tf /2 ≤ t1 ≤ Tf . Their pdfs can be expressed as:
1 (y0 (t0 ) + s2 (t0 )) exp − P (y0 (t0 )) = 2σ 2 2σ 2 y0 (t0 )s(t0 ) (6) · I0 ( )U (y0 (t0 )) 2 σ
1 y0 (t1 ) exp − P (y0 (t1 )) = U (y0 (t1 )), 2σ 2 2σ 2 where I0 (·) is the 0th -order modified Bessel function of the first kind [4], s2 (t0 ) can easily be shown to be p2 (t0 ), and U (·) is the unit step function. Considering an integrator as a linear t filter, let us define a new random process z0 (t) as z0 (t) = t−Tf /2 y0 (u)du. Notice that the decision statistics Z0 and Z1 are obtained at time t = Tf /2 and t = Tf , respectively. z0 (t) can be reformulated in terms of the convolution of y0 (t) and the rectangular impulseresponse function hw (t) defined as 1 for 0 ≤ t ≤ Tf /2 and 0 ∞ for otherwise. The area of hw (t) is −∞ hw (t)dt = Tf /2, and the finite autocorrelation of hw (t) is the triangular function: | , |τ | ≤ Tf /2 1 − T|τ f /2 . (7) rhw (τ ) = 0, otherwise Now we can use the fact that a random process through a time-invariant linear continuous-time filter has the following mean and autocorrelation [5]:
∞ Tf hw (t)dt = E[y0 (t)] , E[z0 (t)] = E[y0 (t)] 2 −∞ (8)
Tf /2 Rz0 (τ ) = rhw (u)Ry0 (τ − u)du. −Tf /2
We note that the variance is σz20 = E[z02 (t)]−E[z0 (t)]2 , where E[z02 (t)] = Rz0 (0). If we approximate Ry0 (u) E[y02 (t)] which is its maximum, i.e., Ry0 (0), we have: Tf , (9) 2 where Tf /2 is the area of the triangular function. Since a continuous time integral can be thought of as discrete sums, it can be shown that the decision statistics {Zi }1i=0 are Gaussian random variables, thanks to the central limit theorem. As a result, the random variable Z∆ is also conditionally Gaussian. We now derive the mean and the variance of y0 (t), whose magnitude has the pdf given in (6). It is known that the first two moments of a noncentral chi-square random variable with 2 degrees of freedom at time t are [4]: Rz0 (0) ≤ E[y02 (t)]
E[y0 (t)] = 2σ 2 + s2 (t),
2 E[y02 (t)] = 4σ 4 + 4σ 2 s2 (t) + 2σ 2 + s2 (t) .
(10)
Notice that if s2 (t) = 0, i.e., there is no pulse, (10) is equivalent to that of a central chi-square random variable. Based on equations (8), (9), and (10), we can calculate
OH et al.: A NEW NONCOHERENT UWB IMPULSE RADIO RECEIVER 0.4
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the conditional mean and variance of {Zi }1i=0 and Z∆ as summarized in the following table. 2σ 2 Tf /2 + Ep 2σ 2 Tf /2
4σ 4 Tf /2 + 4σ 2 Ep 4σ 4 Tf /2
Z∆
Ep
4σ 4 Tf + 4σ 2 Ep
Using the expressions in the table and (4), the conditional probability of error for high SNR can be calculated as: Ep Pe (s0 ) ≥ Q . (11) 2No Note that we used the maximum value of Rz0 (0) in (9), which leads to the inequality in (11). Since the signals s0 (t) and s1 (t) are equally likely to be transmitted, the average probability of error Pe is equal to (11). As can be seen, the proposed receiver requires approximately 3 dB additional Ep /N0 than an ideal correlator based receiver with no jitter for the same BER performance. IV. S IMULATION R ESULTS We conducted BER simulations with and without finite jitter. In all experiments, we used a UWB pulse obtained from [6], for which the pulse duration Tp was 1ns. The spectrum of the pulse meets the power spectral constraint of the FCC mask, as shown in Fig. 3. Fig. 4 shows the simulated BER performance for the correlator receiver and for the proposed noncoherent receiver in the face of a different timing jitter. As expected, the ideal correlator receiver shows the best performance. The proposed receiver with a matched filter requires 3.8 dB more SNR than the ideal correlator receiver for a BER of 10−3 . This coincides well with the analytical performance analysis done earlier. But the performance of the correlator based receiver drops rapidly with increasing jitter. For a BER of 10−3 , the correlator receiver requires a higher SNR than the proposed receiver when the RMS jitter reaches 18 ps. This performance degradation due to jitter is easily explained while considering the pulse shape in Fig. 3 where the zero crossing spacing is less than 100 ps. We also note that there is about 1 dB gap
6 SNR [dB]
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Rake correlator (ideal) Rake correlator (w/10ps jitter) Rake correlator (w/20ps jitter) Rake correlator (w/30ps jitter) Proposed (Matched filter)
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BER performance in multipath environments.
between the proposed receiver with a matched filter and the one with a brick wall BPF for a BER of 10−3 . Fig. 5 shows the BER performance in a multipath environment. The multipath channel is modeled as a linear, random time-varying filter with a maximum delay spread of Tm . The assumption of Tf Tm + Tp + τp eliminates intersymbol and intrasymbol interference effects. A rake receiver consisting of correlators exploits diversity by combining the separable UWB pulses from distinguishable propagation paths [2]. The results show very similar behavior to that of an AWGN channel for both the rake and the proposed receiver. It has been observed that the complexity of a rake receiver rapidly increases with increasing number of correlators in the receiver, and reinforces the idea that the proposed receiver is a good candidate as a UWB IR receiver. R EFERENCES [1] W. Zhuang, X. Shen, and Q. Bi, “Ultra-wide wireless communications,” Wireless Commun. & Mobile Computing, vol. 3, pp. 663-685, Sept. 2003. [2] J. D. Choi and W. E. Stark, “Performance of ultra-wideband communications with suboptimal receivers in multipath channels,” IEEE J. Select. Areas Commun., vol. 20, pp. 1754-1766, Dec. 2002. [3] W. M. Lovelace and J. K. Townsend, “The effects of timing jitter and tracking on the performance of impulse radio,” IEEE J. Select. Areas Commun., vol. 20, pp. 1646-1651, Dec. 2002. [4] J. G. Proakis, Digital Communications. McGraw-Hill, 2000. [5] H. Stark and J. W. Wood, Probability and Random Processes with Applications to Signal Processing. Prentice Hall, 3rd edition, 2002. [6] B. Parr et al., “A novel ultra-wideband pulse design algorithm,” IEEE Commun. Lett., vol. 7, pp. 219-221, May 2003.
151
A New Noncoherent UWB Impulse Radio Receiver Mi-Kyung Oh, Byunghoo Jung, Student Member, IEEE, Ramesh Harjani, Senior Member, IEEE, and Dong-Jo Park, Member, IEEE Abstract— This letter analyzes the implementation issues related to coherent receivers for UWB impulse radio with a special emphasis on timing and jitter problems. We propose a new jitter tolerant receiver design that is easy to implement. Analytical BER analysis and simulations verify that the performance of the proposed receiver is comparable to that of a correlator-based receiver that includes jitter. The new design is a promising candidate for low-cost low-power UWB IR receiver implementations. Index Terms— Ultra wideband (UWB), noncoherent, receiver.
I. I NTRODUCTION
R
ECENTLY, Ultra Wideband (UWB) Impulse Radio (IR) systems have received significant attention from industry, media and academia due to the benefits that primarily accrue from having the energy spread over a wide spectrum. These benefits include: a) the possibility of low-cost low-power transceivers due to the absence of a carrier b) better immunity to multipath propagation, and c) low-interference [1]. However, there are still many technical challenges that remain to be solved for the widespread deployment of UWB IR systems. The reduced complexity and other implementation advantages offered by IR are somewhat offset by the stringent timing requirements [1]. The majority of the research on IR receiver design has been focused on correlator-based receivers [2]. It has recently demonstrated that the throughput of the correlator-based receiver degrades markedly for relatively modest increase in timing jitter and tracking error [3]. This mandates that a significant portion of the design process be focused on low jitter and timing error analysis and optimization. It is these findings that has triggered and motivated the noncoherent IR receiver design proposed in this letter. In this letter we verify the performance of the proposed noncoherent receiver. This receiver uses a square law device as a detector in combination with an integration/dump filter, which allows for a low-cost low-power implementation of a UWB IR system. II. S YSTEM M ODELS We develop a system model to analyze the impact of timing jitter on the proposed architecture. Let the transmitted pulse t be represented as pTX (t) = −∞ p(u)du. The received signal can then be modeled as r(t) = p(t) + n(t), if we ignore signal attenuation and delay due to propagation. The effect
Manuscript received May 10, 2004. The associate editor coordinating the review of this letter and approving it for publication was Prof. Gianluca Mazzini. M.-K. Oh and D.-J. Park are with the Dept. of EECS, KAIST, 373-1 Guseong-dong, Yuseong-gu, Daejon 305-701, Republic of Korea (e-mail: [email protected], [email protected]). B. Jung and R. Harjani are with the Dept. of ECE, Univ. of Minnesota, Minneapolis, MN, USA (e-mail: {jbh, harjani}@ece.umn.edu). Digital Object Identifier 10.1109/LCOMM.2005.02010.
Fig. 1.
Optimal correlator receiver.
of an antenna on the transmitted pulse can be modeled as a differentiation operation. The noise n(t) is assumed to be AWGN with power density σ 2 := No /2. In this letter we assume binary pulse position modulation (PPM) as the modulation scheme so that the transmitted signal consists of Np time-shifted low duty cycle pulses that can be defined as: Np −1
si (t) =
pTX (t − jTf − ci,j τp ), 0 ≤ t ≤ Np Tf
(1)
j=0
where each signal {si (t)}1i=0 is completely identified by the sequence of time shifts ci,j τp ∈ {0, τp } and Tf is the frame duration that is chosen to be sufficiently large, i.e., Tf Tp + τp where Tp denotes the duration of pulse p(t). On the receiver side, the signals can then be given by Np −1
ri (t) =
p(t − jTf − ci,j τp ) + n(t).
(2)
j=0
To simplify the ensuing analysis, we consider only one frame (Np = 1), defined within the time-interval 0 ≤ t ≤ Tf . Thus,(2) can be rewritten as: ri (t) = p(t − iτp ) + n(t), i = 0, 1.
(3)
In the absence of multipath interference, the optimum receiver for this system is the correlator receiver shown in Fig. 1, where the receiver generates a local template signal that is synchronized with the incoming pulse p(t). We note that we obtain the decision statistic Z0 at sampling time t = Tf /2 and Z1 at t = Tf . So only one correlator is required. Comparing Z0 − Z1 with the decision threshold for 0 accomplishes detection process of the transmitted symbol. However, this optimum receiver is subject to timing error and jitter, as shown in Fig. 1. Our goal in this letter is to provide details for a noncoherent receiver that does not require exact timing, and compare it to the optimal correlator receiver in terms of error performance and ease of implementation. III. N ONCOHERENT UWB IR R ECEIVER Instead of a correlator employing a pulse generator and a timing-control clock as shown in Fig. 1, we use a linear filter
c 2005 IEEE 1089-7798/05$20.00
152
Fig. 2.
IEEE COMMUNICATIONS LETTERS, VOL. 9, NO. 2, FEBRUARY 2005
Block diagram of the proposed receiver.
known as a matched filter whose sampled outputs are exactly the values obtained from the correlator [4]. The frequency response of the matched filter is designed to match the frequency spectrum of the signal, and is given by HMF (f ) = P ∗ (f )e−j2πf Tf /2 , where P ∗ (f ) is the complex conjugate of the pulse spectrum. We note that the frequency spectrum of the pulse used in UWB system is limited so as to meet the FCC spectral mask (see Fig. 3). In the absence of noise, the output of the matched filter has a spectrum |P (f )|2 ·e−j2πf Tf /2 which corresponds to the autocorrelation of the pulse in the time domain. For optimal receiver operation, exact sampling at the maximum peak of the autocorrelation of the pulse is required, which makes it difficult to implement a UWB IR. The “Square-law device and Integrator”, shown in Fig. 2, is a method that does not require exact timing. Because the exact pulse position is not known, the proposed receiver obtains the approximate peak based on the average by using the “Squarelaw device and Integrator” instead of sampling the signal at the exact peak. The benefits of the proposed receiver are twofold: a) the template signal generator and timing-control block, that consume a significant portion of the power, are eliminated and b) the degradation in performance is limited in comparison to a coherent receiver that includes the effect of jitter. A. Performance Analysis Let us consider the BER performance of the proposed receiver shown in Fig. 2 in comparison to that of the optimal correlator receiver shown in Fig. 1. If the receiver is assumed to be perfectly synchronized with the transmitter, the union bound can be given by Pe = √ on the bit error probability √ ∞ SNR , where Q(x) = 1/ 2π x exp(−t2 /2)dt, and Q T SNR:=Ep /No with energy Ep := 0 p [p(t)]2 dt [1], [2]. We evaluate the BER of the proposed receiver. Assuming that s0 (t) is transmitted, we define the random variable Z∆ := Z0 − Z1 , where the conditional probability of error is [2]: E[Z∆ |s0 ]2 Pe (s0 ) = P [Z∆ < 0|s0 (t) sent] = Q . (4) Var[Z∆ |s0 ] In the rest of this section, we obtain the conditional mean and variance of Z∆ and derive expressions for the BER. To keep its analysis simple, the matched filter is replaced by a brick wall band pass filter (BPF) because most energy of the UWB pulse can be captured within the FCC mask, which frequency response is given by: 1 , 3.1 GHz ≤ f ≤ 10.6 GHz (5) HBPF (f ) = 0, elsewhere With this in mind, let us consider a random process y0 (t) := |x0 (t)|2 = |r0 (t) ∗ hBPF (t)|2 , which is obtained from the
square-law device. We note that when the time is fixed at t0 , y0 (t0 ) is a noncentral chi-square random variable with 2 degrees of freedom where 0 ≤ t0 ≤ Tf /2, whereas y0 (t1 ) becomes a central chi-square random variable with 2 degrees of freedom where Tf /2 ≤ t1 ≤ Tf . Their pdfs can be expressed as:
1 (y0 (t0 ) + s2 (t0 )) exp − P (y0 (t0 )) = 2σ 2 2σ 2 y0 (t0 )s(t0 ) (6) · I0 ( )U (y0 (t0 )) 2 σ
1 y0 (t1 ) exp − P (y0 (t1 )) = U (y0 (t1 )), 2σ 2 2σ 2 where I0 (·) is the 0th -order modified Bessel function of the first kind [4], s2 (t0 ) can easily be shown to be p2 (t0 ), and U (·) is the unit step function. Considering an integrator as a linear t filter, let us define a new random process z0 (t) as z0 (t) = t−Tf /2 y0 (u)du. Notice that the decision statistics Z0 and Z1 are obtained at time t = Tf /2 and t = Tf , respectively. z0 (t) can be reformulated in terms of the convolution of y0 (t) and the rectangular impulseresponse function hw (t) defined as 1 for 0 ≤ t ≤ Tf /2 and 0 ∞ for otherwise. The area of hw (t) is −∞ hw (t)dt = Tf /2, and the finite autocorrelation of hw (t) is the triangular function: | , |τ | ≤ Tf /2 1 − T|τ f /2 . (7) rhw (τ ) = 0, otherwise Now we can use the fact that a random process through a time-invariant linear continuous-time filter has the following mean and autocorrelation [5]:
∞ Tf hw (t)dt = E[y0 (t)] , E[z0 (t)] = E[y0 (t)] 2 −∞ (8)
Tf /2 Rz0 (τ ) = rhw (u)Ry0 (τ − u)du. −Tf /2
We note that the variance is σz20 = E[z02 (t)]−E[z0 (t)]2 , where E[z02 (t)] = Rz0 (0). If we approximate Ry0 (u) E[y02 (t)] which is its maximum, i.e., Ry0 (0), we have: Tf , (9) 2 where Tf /2 is the area of the triangular function. Since a continuous time integral can be thought of as discrete sums, it can be shown that the decision statistics {Zi }1i=0 are Gaussian random variables, thanks to the central limit theorem. As a result, the random variable Z∆ is also conditionally Gaussian. We now derive the mean and the variance of y0 (t), whose magnitude has the pdf given in (6). It is known that the first two moments of a noncentral chi-square random variable with 2 degrees of freedom at time t are [4]: Rz0 (0) ≤ E[y02 (t)]
E[y0 (t)] = 2σ 2 + s2 (t),
2 E[y02 (t)] = 4σ 4 + 4σ 2 s2 (t) + 2σ 2 + s2 (t) .
(10)
Notice that if s2 (t) = 0, i.e., there is no pulse, (10) is equivalent to that of a central chi-square random variable. Based on equations (8), (9), and (10), we can calculate
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the conditional mean and variance of {Zi }1i=0 and Z∆ as summarized in the following table. 2σ 2 Tf /2 + Ep 2σ 2 Tf /2
4σ 4 Tf /2 + 4σ 2 Ep 4σ 4 Tf /2
Z∆
Ep
4σ 4 Tf + 4σ 2 Ep
Using the expressions in the table and (4), the conditional probability of error for high SNR can be calculated as: Ep Pe (s0 ) ≥ Q . (11) 2No Note that we used the maximum value of Rz0 (0) in (9), which leads to the inequality in (11). Since the signals s0 (t) and s1 (t) are equally likely to be transmitted, the average probability of error Pe is equal to (11). As can be seen, the proposed receiver requires approximately 3 dB additional Ep /N0 than an ideal correlator based receiver with no jitter for the same BER performance. IV. S IMULATION R ESULTS We conducted BER simulations with and without finite jitter. In all experiments, we used a UWB pulse obtained from [6], for which the pulse duration Tp was 1ns. The spectrum of the pulse meets the power spectral constraint of the FCC mask, as shown in Fig. 3. Fig. 4 shows the simulated BER performance for the correlator receiver and for the proposed noncoherent receiver in the face of a different timing jitter. As expected, the ideal correlator receiver shows the best performance. The proposed receiver with a matched filter requires 3.8 dB more SNR than the ideal correlator receiver for a BER of 10−3 . This coincides well with the analytical performance analysis done earlier. But the performance of the correlator based receiver drops rapidly with increasing jitter. For a BER of 10−3 , the correlator receiver requires a higher SNR than the proposed receiver when the RMS jitter reaches 18 ps. This performance degradation due to jitter is easily explained while considering the pulse shape in Fig. 3 where the zero crossing spacing is less than 100 ps. We also note that there is about 1 dB gap
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BER performance in multipath environments.
between the proposed receiver with a matched filter and the one with a brick wall BPF for a BER of 10−3 . Fig. 5 shows the BER performance in a multipath environment. The multipath channel is modeled as a linear, random time-varying filter with a maximum delay spread of Tm . The assumption of Tf Tm + Tp + τp eliminates intersymbol and intrasymbol interference effects. A rake receiver consisting of correlators exploits diversity by combining the separable UWB pulses from distinguishable propagation paths [2]. The results show very similar behavior to that of an AWGN channel for both the rake and the proposed receiver. It has been observed that the complexity of a rake receiver rapidly increases with increasing number of correlators in the receiver, and reinforces the idea that the proposed receiver is a good candidate as a UWB IR receiver. R EFERENCES [1] W. Zhuang, X. Shen, and Q. Bi, “Ultra-wide wireless communications,” Wireless Commun. & Mobile Computing, vol. 3, pp. 663-685, Sept. 2003. [2] J. D. Choi and W. E. Stark, “Performance of ultra-wideband communications with suboptimal receivers in multipath channels,” IEEE J. Select. Areas Commun., vol. 20, pp. 1754-1766, Dec. 2002. [3] W. M. Lovelace and J. K. Townsend, “The effects of timing jitter and tracking on the performance of impulse radio,” IEEE J. Select. Areas Commun., vol. 20, pp. 1646-1651, Dec. 2002. [4] J. G. Proakis, Digital Communications. McGraw-Hill, 2000. [5] H. Stark and J. W. Wood, Probability and Random Processes with Applications to Signal Processing. Prentice Hall, 3rd edition, 2002. [6] B. Parr et al., “A novel ultra-wideband pulse design algorithm,” IEEE Commun. Lett., vol. 7, pp. 219-221, May 2003.
