A âBetter Thanâ Nyquist Pulse - CiteSeerX
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A “Better Than” Nyquist Pulse Norman C. Beaulieu1 , Fellow, IEEE, Christopher C. Tan2 , Member, IEEE, and Mohamed Oussama Damen1 , Member, IEEE Abstract— A novel ISI-free pulse is presented that has smaller maximum distortion, a more open receiver eye, and a smaller probability of error in the presence of symbol timing error than the Nyquist pulse for the same excess bandwidth.
T
I. Introduction HE Nyquist pulse with excess bandwidth, α, is specified by its frequency spectrum
1, S(f )= 0,1+cos 1 2
π (f−B(1 2Bα
0 ≤ f ≤ B(1 − α) , B(1 − α) ≤ f ≤ B(1 + α) (1)
− α))
B(1 + α) ≤ f,
where B is the bandwidth corresponding to symbol repetition rate T = 1/(2B) and its corresponding (scaled) time function [1] pNY (t)
=
sinc(t/T )
cos (παt/T ) . 1 − 4α2 t2 /T 2
(2)
Examination of (2) indicates that the tails of the Nyquist pulse for α > 0 decay asymptotically as t−3 , as is wellknown. The Nyquist pulse is widely used in modem design and is the benchmark pulse in communication theory [2]. A novel pulse is specified by the frequency spectrum
S(f ) =
1, ln2 [f − B(1 − α)]}, exp{ −αB 1 − exp{ −ln2 [B(1 + α) − f ]}, 0,
αB
0 ≤ f ≤ B(1 − α) B(1 − α) ≤ f ≤ B B ≤ f ≤ B(1 + α) B(1 + α) ≤ f,
(3)
with corresponding (scaled) time function p(t) = 2Bsinc(2Bt)
4βπt sin(2πBαt) + 2β 2 cos(2πBαt) − β 2 , (4) 4π 2 t2 + β 2
2 where β = ln αB . Fig. 1 shows the frequency spectrum and time functions, respectively, of the new pulse for α = 0.35. It can be proved using (4) that the tails of the new pulse for α > 0 decay asymptotically as t−2 . Note that an infinite sequence of pulse tails is absolutely summable for both pulses, though the asymptotic rate of decay is greater for the Nyquist pulse. Despite the latter fact, the new pulse is “better than” the Nyquist pulse as shown in the next section.
1 N. C. Beaulieu and M. O. Damen are with the Department of Electrical and Computer Engineering, University of Alberta, Edmonton, Alberta, Canada T6G 2G7. E-mails: {beaulieu, damen}@ee.ualberta.ca 2 C. C. Tan is with Calimetric Inc. Alameda, California USA 94501. E-mail: [email protected]
II. Transmission Properties Fig. 2 shows receiver eye diagrams for the new and Nyquist pulses. Observe that the eye for the new pulse is more open. This is highlighted by superimposing the inner boundary of the Nyquist pulse eye from Fig. 2 (b) on the eye of the new pulse in Fig. 2 (a). Interestingly, this is so despite the fact that the tails of the new and Nyquist pulses decay at t−2 and t−3 , respectively. The behavior is explained by examination of Fig. 1 (b) which shows that the magnitudes of the two largest sidelobes of the Nyquist pulse are larger than the magnitudes of the two largest sidelobes of the new pulse. Also observe that the maximum distortion (which occurs at t/T = 0.5) is less for the new pulse than the Nyquist pulse, being 1.53 for the former and 1.71 for the latter. In practical receivers, timing error increases the average symbol error probability as intersymbol interference (ISI) results when the receiver eye is sampled off center. Table I and Table II give the average symbol error probabilities of binary antipodal signaling [3] in the presence of symbol timing error for the same two pulses, computed using the method of [4]. Note that the error rates are smaller for all values of α and timing offset for the new pulse. This behavior is consistent with the wider (more open) eye of the new pulse. III. Conclusion A new pulse that has smaller maximum distortion, a more open receiver eye, and a smaller symbol error rate in the presence of symbol timing error than the Nyquist pulse with the same excess bandwidth has been presented. The pulse has theoretical and practical importance. References [1] H. Nyquist, “Certain topics in telegraph transmission theory,” AIEE Trans., vol. 47, pp. 617–644, 1928. [2] J. G. Proakis, Digital Communications, 4th edition. McGraw-Hill series in Electrical and Computer Engineering, 2000. [3] J. M. Wozencraft and I. M. Jacobs, Principles of Communication Engineering. John Wiley and sons, 1965. [4] N. C. Beaulieu, “The evaluation of error probabilities for intersymbol and cochannel interference,” IEEE Trans. on Communications, vol. 31, pp. 1740–1749, Dec. 1991.
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1 New pulse Nyquist pulse
1.5
0.9 0.8
1
0.7
0.5 Amplitude
S(f)
0.6 0.5 0.4
0
−0.5
0.3
−1
0.2 0.1 0 −1.5
−1.5
−1
−0.5
0 f/B
0.5
1
1.5
−1
−0.8
−0.6
−0.4
−0.2
(a)
0 t/T
0.2
0.4
0.6
0.8
1
0 t/T
0.2
0.4
0.6
0.8
1
(a)
1 New pulse Nyquist pulse
1.5
0.8
1
0.5 Amplitude
p(t)
0.6
0.4
0
−0.5 0.2
−1 0
−1.5 −0.2 −6
−4
−2
0 t/T
2
4
6
−1
−0.8
−0.6
−0.4
−0.2
(b)
(b)
Fig. 1. Frequency and time characteristics of the new and Nyquist pulses for an excess bandwidth α = 0.35, (a) frequency domain, (b) time domain.
Fig. 2. Eye diagrams of pulses sequences for an excess bandwidth α = 0.35, (a) new pulse, (b) Nyquist pulse. The dotted line in (a) is the inner boundary of the eye in (b).
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TABLE I ISI error probability of the new pulse for N = 210 interfering symbols and SNR=15 dB.
α
t/T = ±0.05
t/T = ±0.1
t/T = ±0.2
t/T = ±0.25
0.25 0.35 0.5 0.75 1
5.8117e − 08 3.9253e − 08 2.4134e − 08 1.3836e − 08 1.3150e − 08
1.2980e − 06 5.4021e − 07 1.8580e − 07 4.5668e − 08 3.5692e − 08
3.5678e − 04 1.0129e − 04 2.0878e − 05 3.2260e − 06 1.6144e − 06
2.9462e − 03 9.3536e − 04 2.0154e − 04 4.1433e − 05 2.2273e − 05
TABLE II ISI Error probability of Nyquist pulse for N = 210 interfering symbols and SNR=15 dB.
α
t/T = ±0.05
t/T = ±0.1
t/T = ±0.2
t/T = ±0.25
0.25 0.35 0.5 0.75 1
8.2189e − 08 5.9997e − 08 3.9723e − 08 2.2777e − 08 1.5281e − 08
2.8184e − 06 1.3896e − 06 5.4890e − 07 1.5761e − 07 5.8720e − 08
9.7462e − 04 3.9084e − 04 1.0217e − 04 1.5346e − 05 3.6543e − 06
6.7732e − 03 3.1988e − 03 9.4694e − 04 1.4907e − 04 3.9247e − 05
A “Better Than” Nyquist Pulse Norman C. Beaulieu1 , Fellow, IEEE, Christopher C. Tan2 , Member, IEEE, and Mohamed Oussama Damen1 , Member, IEEE Abstract— A novel ISI-free pulse is presented that has smaller maximum distortion, a more open receiver eye, and a smaller probability of error in the presence of symbol timing error than the Nyquist pulse for the same excess bandwidth.
T
I. Introduction HE Nyquist pulse with excess bandwidth, α, is specified by its frequency spectrum
1, S(f )= 0,1+cos 1 2
π (f−B(1 2Bα
0 ≤ f ≤ B(1 − α) , B(1 − α) ≤ f ≤ B(1 + α) (1)
− α))
B(1 + α) ≤ f,
where B is the bandwidth corresponding to symbol repetition rate T = 1/(2B) and its corresponding (scaled) time function [1] pNY (t)
=
sinc(t/T )
cos (παt/T ) . 1 − 4α2 t2 /T 2
(2)
Examination of (2) indicates that the tails of the Nyquist pulse for α > 0 decay asymptotically as t−3 , as is wellknown. The Nyquist pulse is widely used in modem design and is the benchmark pulse in communication theory [2]. A novel pulse is specified by the frequency spectrum
S(f ) =
1, ln2 [f − B(1 − α)]}, exp{ −αB 1 − exp{ −ln2 [B(1 + α) − f ]}, 0,
αB
0 ≤ f ≤ B(1 − α) B(1 − α) ≤ f ≤ B B ≤ f ≤ B(1 + α) B(1 + α) ≤ f,
(3)
with corresponding (scaled) time function p(t) = 2Bsinc(2Bt)
4βπt sin(2πBαt) + 2β 2 cos(2πBαt) − β 2 , (4) 4π 2 t2 + β 2
2 where β = ln αB . Fig. 1 shows the frequency spectrum and time functions, respectively, of the new pulse for α = 0.35. It can be proved using (4) that the tails of the new pulse for α > 0 decay asymptotically as t−2 . Note that an infinite sequence of pulse tails is absolutely summable for both pulses, though the asymptotic rate of decay is greater for the Nyquist pulse. Despite the latter fact, the new pulse is “better than” the Nyquist pulse as shown in the next section.
1 N. C. Beaulieu and M. O. Damen are with the Department of Electrical and Computer Engineering, University of Alberta, Edmonton, Alberta, Canada T6G 2G7. E-mails: {beaulieu, damen}@ee.ualberta.ca 2 C. C. Tan is with Calimetric Inc. Alameda, California USA 94501. E-mail: [email protected]
II. Transmission Properties Fig. 2 shows receiver eye diagrams for the new and Nyquist pulses. Observe that the eye for the new pulse is more open. This is highlighted by superimposing the inner boundary of the Nyquist pulse eye from Fig. 2 (b) on the eye of the new pulse in Fig. 2 (a). Interestingly, this is so despite the fact that the tails of the new and Nyquist pulses decay at t−2 and t−3 , respectively. The behavior is explained by examination of Fig. 1 (b) which shows that the magnitudes of the two largest sidelobes of the Nyquist pulse are larger than the magnitudes of the two largest sidelobes of the new pulse. Also observe that the maximum distortion (which occurs at t/T = 0.5) is less for the new pulse than the Nyquist pulse, being 1.53 for the former and 1.71 for the latter. In practical receivers, timing error increases the average symbol error probability as intersymbol interference (ISI) results when the receiver eye is sampled off center. Table I and Table II give the average symbol error probabilities of binary antipodal signaling [3] in the presence of symbol timing error for the same two pulses, computed using the method of [4]. Note that the error rates are smaller for all values of α and timing offset for the new pulse. This behavior is consistent with the wider (more open) eye of the new pulse. III. Conclusion A new pulse that has smaller maximum distortion, a more open receiver eye, and a smaller symbol error rate in the presence of symbol timing error than the Nyquist pulse with the same excess bandwidth has been presented. The pulse has theoretical and practical importance. References [1] H. Nyquist, “Certain topics in telegraph transmission theory,” AIEE Trans., vol. 47, pp. 617–644, 1928. [2] J. G. Proakis, Digital Communications, 4th edition. McGraw-Hill series in Electrical and Computer Engineering, 2000. [3] J. M. Wozencraft and I. M. Jacobs, Principles of Communication Engineering. John Wiley and sons, 1965. [4] N. C. Beaulieu, “The evaluation of error probabilities for intersymbol and cochannel interference,” IEEE Trans. on Communications, vol. 31, pp. 1740–1749, Dec. 1991.
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1 New pulse Nyquist pulse
1.5
0.9 0.8
1
0.7
0.5 Amplitude
S(f)
0.6 0.5 0.4
0
−0.5
0.3
−1
0.2 0.1 0 −1.5
−1.5
−1
−0.5
0 f/B
0.5
1
1.5
−1
−0.8
−0.6
−0.4
−0.2
(a)
0 t/T
0.2
0.4
0.6
0.8
1
0 t/T
0.2
0.4
0.6
0.8
1
(a)
1 New pulse Nyquist pulse
1.5
0.8
1
0.5 Amplitude
p(t)
0.6
0.4
0
−0.5 0.2
−1 0
−1.5 −0.2 −6
−4
−2
0 t/T
2
4
6
−1
−0.8
−0.6
−0.4
−0.2
(b)
(b)
Fig. 1. Frequency and time characteristics of the new and Nyquist pulses for an excess bandwidth α = 0.35, (a) frequency domain, (b) time domain.
Fig. 2. Eye diagrams of pulses sequences for an excess bandwidth α = 0.35, (a) new pulse, (b) Nyquist pulse. The dotted line in (a) is the inner boundary of the eye in (b).
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TABLE I ISI error probability of the new pulse for N = 210 interfering symbols and SNR=15 dB.
α
t/T = ±0.05
t/T = ±0.1
t/T = ±0.2
t/T = ±0.25
0.25 0.35 0.5 0.75 1
5.8117e − 08 3.9253e − 08 2.4134e − 08 1.3836e − 08 1.3150e − 08
1.2980e − 06 5.4021e − 07 1.8580e − 07 4.5668e − 08 3.5692e − 08
3.5678e − 04 1.0129e − 04 2.0878e − 05 3.2260e − 06 1.6144e − 06
2.9462e − 03 9.3536e − 04 2.0154e − 04 4.1433e − 05 2.2273e − 05
TABLE II ISI Error probability of Nyquist pulse for N = 210 interfering symbols and SNR=15 dB.
α
t/T = ±0.05
t/T = ±0.1
t/T = ±0.2
t/T = ±0.25
0.25 0.35 0.5 0.75 1
8.2189e − 08 5.9997e − 08 3.9723e − 08 2.2777e − 08 1.5281e − 08
2.8184e − 06 1.3896e − 06 5.4890e − 07 1.5761e − 07 5.8720e − 08
9.7462e − 04 3.9084e − 04 1.0217e − 04 1.5346e − 05 3.6543e − 06
6.7732e − 03 3.1988e − 03 9.4694e − 04 1.4907e − 04 3.9247e − 05
