Comparison Between Several Methods Of PPM ... - Semantic Scholar
Comparison Between Several Methods Of PPM Demodulation Based On Iterative Techniques M. Shariat, M. Ferdosizade Naeiny, M. J. Abdoli, B. Makouei, A.Yazdanpanah, F. Marvasti Multimedia Laboratory, Iran Telecommunication Research Center Department of Electrical Engineering, Sharif University of Technology Abstract – In this paper we examine demodulation methods for nonuniform Pulse Position Modulation (PPM), which is generated by the crossings of a modulating signal with a sawtooth wave. At high sampling rates (several times the Nyquist), demodulation can be achieved by an appropriate low pass filter of the PPM pulses. However, at sampling rates near the Nyquist rate, this approach creates unacceptable distortion. Here, we are interested in PPM at the Nyquist rate and illustrate the efficiency of iterative methods. Lowering the sampling rate we essentially reduce the bandwidth requirement of the transmitted PPM pulses. Several iterative methods such as Wiley/Marvasti, Time-varying, adaptive weight, Zero-order-hold, Voronoi, linear interpolation algorithms are discussed, where the PPM demodulation is converted as a nonuniform sampling reconstruction problem. Also another approach called the inverse system approach is introduced and the performance of the two algorithms is assessed.
But this approximation is acceptable only when the average sampling rate (1/T) is at least 3-4 times greater than the Nyquist rate. In the following sections we will show that iterative methods work well even at sampling rates close to the Nyquist rate. The block diagram of a generic iterative algorithm is shown in Figure 1. The block G is the ‘distortion block’. In fact the iterative method is a general method for recovery of signals distorted by any kind of distortion. It can be shown that if SNR at the output of the block is greater than one, then the iterative process converges to the original signal at infinity [1]. But because of limitations on computational precision, in practice, the obtained SNR saturates after some n iterations. However, this saturation level and also the convergence speed can to the block G in be improved by adding a gain each iteration. In the PPM case, the block G can be replaced by various systems. We have classified these systems into 2 groups. The first group treats the PPM demodulation as a nonuniform sampling reconstruction problem while the second group takes conventional PPM modulator for the G block. In the next sections we will present different methods for each group and then compare the results.
I. INTRODUCTION The idea of the PPM modulation was developed around 50 years ago but it is only recently that a revival of interest has been experienced with the development of Impulse Radio and Fiber-Optic transmission systems [4]. The special properties of the PPM modulation make it a suitable candidate for fiber-optic communications. Specially, because there is no information in the amplitude of PPM signals, this modulation technique is immune to the effects of nonlinearity of the fiber-optic devices. Also the PPM modulation is a good choice for the short-range communications in dense multi-path environments [67] using the Impulse Radio communications. Theoretically it has been shown that PPM systems are effective when the signals are power-limited rather than bandlimited [5]-[6].
Figure 1. Block diagram of the proposed iterative method II. ITERATIVE APPROACHES In this part, we introduce several approaches for PPM demodulation using iterative methods: the nonuniform sampling approach and the PPM inverse system approach. In the first approach, PPM pulses are multiplied by a synchronized sawtooth signal yielding a set of nonuniform samples of the original signal. Thus we can use the nonuniform sampling recovery techniques. We will introduce six different techniques in this approach.
The main idea in the PPM modulation is to send pulses with equal amplitude at the times of intersection between the message signal and a sawtoothwave r(t). Traditionally, for demodulation, the PPM pulses are first passed through a low pass filter. Then the DC offset of the output is removed and the resulting signal is passed though an integrator to produce an approximation of the message signal.
In the second approach, we consider the PPM modulator and demodulator as the G distortion in Fig. 2. In this approach, we can present different iterative
1
N x(t ) − x(ti ) (t − ti ) + x(ti )I (ti , ti +1 ) (8) Sx(t ) = ∑ i +1 ti +1 − ti i =1
methods depending on the type of the demodulation technique. Note that the demodulator used in each iteration is a low performance one and we improve its performance in the iteration. We will examine two systems using different demodulators in this approach. In the next sections we will explain these approaches more precisely.
Adaptive weight method (ADPW) is the next technique that we apply it to our problem. In this technique we multiply each sample by the weight equal to the corresponding Voronoi interval:
II.1. NON-UNIFORM SAMPLING APPROACH In this approach, we consider the G distortion in Fig. 2 as a sampling process S followed by a low pass filtering P [2]. Thus we have:
xk +1 (t ) = λ (PSx(t ) − PSxk (t )) + xk (t )
(1)
x(t i )δ (t − t i )
∞ 1 d x (t ) )) y (t ) = f s (1 − ( ) x (t ))1 + 2∑ cos( nω s (t − µ dt µ i =1
(2)
N P ∑ x(t i )δ (t − t i ) i =1 Sx(t ) = P{comb(t )}
t k = kTs +
(3)
p
N
∑ δ (t − t ) . i
In the Zero-Order-Hold method we have:
Sx(t ) = ∑ x(t i ) Φ i (t )
x(t )
µ
(11)
Algorithm II: Since the signal samples at the times t k are equal to the value of the saw tooth signal at these times, we can multiply the PPM pulses (which have unity amplitudes) by the saw tooth to obtain nonuniform samples of the signal. These nonuniform samples are then low pass filtered as shown in Fig5.
N
(4)
i =1
where (5)
and (6)
The Voronoi method have sampling process similar to Eq. (4) but with different Φ i (t ) . In this method we have:
t +t t +t Φ i (t ) = I i −1 i , i i +1 2 2
(10)
and µ is the saw tooth slope. Interpreting Eq.(10) we see that PPM with non-uniform sampling is a combination of linear and exponential carrier modulation. At each harmonic the signal is phase modulated and its derivative is amplitude modulated at the output. We can therefore retrieve the original message by low-pass filtering and integrating with the incorporation of a DC block as shown in Fig4.
where P{y (t )} is the low pass filtered version
1 t1 ≤ t ≤ t 2 I (t1 , t 2 ) = 0 otherwise
(9)
Where y(t) is the PPM wave of constant amplitude with pulses positioned at non-uniformly sampled times of:
and in the time-varying method as follow:
Φ i (t ) = I (t i , t i +1 )
2
Algorithm I: Analyzing the spectrum of a PPM signal, it can be shown [1] that the modulation output has the form of:
i =1
i =1
i =1
In this approach the block G (in Figure 2) consists of PPM modulator followed by PPM demodulator. Both methods use the same PPM modulator shown in Fig3, but different in PPM demodulation. This approach is divided into two methods:
N
of y (t ) and comb(t ) =
(t i +1 − t i −1 )
II.2. PPM INVERSE SYSTEM APPROACH
The main difference between the various forms of iterative methods is due to the difference of sampling processes [3]. We examine six techniques in this category: Wiley/Marvasti, Time-varying, adaptive weight method, Zero-order-hold, Voronoi and linear interpolation. In the Wiley/Marvasti method the operator S is as follow:
Sx(t ) = ∑ x(t i )δ (t − t i )
N
Sx(t ) = ∑
Fig 1.1: PPM Modulation Figure 2. PPM modulator
(7)
In the linear interpolation method we can write:
2
Figure 3. Demodulator of the algorithm I
Figure 4. Demodulator of the algorithm II
Figure 5. The mean square error of the first approach methods
Fig 5 shows the MSE versus the number of iterations using different techniques of the second approach (inverse system).
III.SIMULATION RESULTS The following results have been obtained for the demodulation of a lowpass signal using different iterative techniques described before. The period of the sawtooth signal is chosen so that the average number of PPM pulses in unit time satisfies the Nyquist rate. In order to test performance of these methods, we implemented them for some various s and found its optimal value for each of the methods. We considered the optimality in the sense of minimum mean-square error in the tenth iteration. Fig6 shows the mean square error of demodulator (in dB) using different techniques of the first approach (nonuniform sampling) versus the number of iterations. The parameter λ in each method is chosen optimally (Table 1). In this figure we see that the Voronoi method (symmetric hold) has the highest convergence of all, but its complexity is relatively high and some delay is necessary in its implementation. The mean square error of all methods finally converges to approximately -21.5dB except the simple zero-order hold method (non symmetric). Another important result is approximately the same performance of the Wiley/Marvasti and ADPW method. The Table 2 represents the MSE of this methods after 3 ,5 , 10 and 20th iteration.
Figure 6. The mean square error of the second approach methods
The complexities of the first approach methods are approximately the same except in ADPW that we have the additional complexity of the multiplication for each sample. For the algorithms I and II there is an additional complexity because of PPM modulation block in each iteration. In fact this block involves N additions (plus N comparisons). Furthermore, in the algorithm I the demodulation stage requires 2N additions and a filtering (the same as others). But the algorithm II involves N additions and a filtering at the demodulation stage.
We can observe from the Figure that the method I has better results than the method II. The MSE obtained from the method I in the tenth iteration is about 3dB larger than that is obtained from the method II. Although the method II has better performance than the method I in the sense of MSE, the synchronization is a serious problem for this method. In fact the saw tooth signal required at receiver for the method II must be exactly synchronous with that of transmitter. This critical parameter causes to use the first method widely than the second.
As final note, we present a modification of the method I for bandpass signals. The modified method has two differences with the original method. First, it uses a bandpass filter instead of lowpass filter at each iteration. Second, instead of removing a specific DC value in the DC block, we force the DC component of signal to be zero (as it is for a bandpass signal). Fig8 ( =0.85). We represents the result for the optimal can see that the modified method involves better MSE and faster convergence for bandpass signals than the original method. λ
3
Table 2 : The optimum convergence factor λopt for different iterative methods technique
λopt
technique
λopt
Wiley/Marvasti
0.8
Zero-order hold
1.3
Time varying
0.6
Voronoi
0.5
ADPW
0.95
Algorithm I
0.45
Linear interpolation
1.6
Algorithm II
0.85
Figure 7. The mean square error of the modified algorithm I
III.CONCLUSION
REFERENCES
We introduced several techniques for PPM demodulation and assessed their performances. We can compare these techniques from two points of view: the Mean Square Error (MSE) and the convergence rate. The MSE is in the saturation region of the performance curves depicted in Figs 68, which we calculate at the 20th iteration. The convergence rate is a criterion for comparing the MSE in the transition region that is in the lower number of iterations. For convenience, we have collected the MSE of the two approaches in Table 1 for certain number of iterations.
[1] F.Marvasti, Nonuniform Sampling Theory and Practice,Kluwer Academic/Plenum Publisher,NewYork,2001. [2] F.Marvasti, M.Analouei and M.Gamshadzahi, Recovery of Signals from Samples Using Iterative Methods., IEEE Trans on Signal Processing, vol.39 ,no.4,pp.872-878, April 1991 [3] P. Azmi, and F. Marvasti, “Comparison between Several Iterative Methods of Recovering Signals from Nonuniformly Spaced Samples”, in Proc. IEEE Conference on Sampling Theory and Applications SampTA2001, Florida-U.S.A., pp., 49-54, 13-17 May 2001.
As shown in Table 1, Algorithm II yields better performance after twenty iterations but the convergence rate is slow, however the convergence rate of the Voronoi technique is the best between one and three iterations.
[4] M. Z. Win, and R. A. Scholtz, “Ultra-Wide Bandwidth Time-Hopping Spread-Spectrum Impulse Radio for Wireless Multiple Access Communications” IEEE Trans on Com., vol. 48, no. 4, pp. 679-689, April 2000.
Table 1 : The MSE of the first approach techniques for some iterations.
Iteration
3
5
10
20
Wiley/Marvasti
-16.53
-19.15
-21.28
-21.55
Time varying
-18.68
-20.9
-21.55
-21.55
ADPW
-16.82
-19.64
-21.44
-21.57
Linear interpolation
-20.41
-21.34
-21.54
-21.54
Zero-order hold
-1.311
-2.616
-6.239
-9.652
[6] B. Wilson, and Z. Ghassemlooy, “Pulse Time Modulation Techniques for Optical Communications: A Review”, IEE Proceeding-J, vol. 140, no. 6, pp. 346-357, Dec. 1993.
Voronoi
-21
-21.46
-21.55
-21.55
[7] J. R. Pierce, and E. C. Posner, Introduction to Communication Science and Systems, Plenum, 1976.
Algorithm I
-13.15
-14.92
-17.23
-17.06
Algorithm II
-13.31
-15.98
-21.22
-26.63
[5] L. Zhao, and A. Haimovich, “Multi-User Capacity of M-ary PPM Ultra-Wideband Communications” in Proc.IEEE Conference on Ultrawideband Systems and Communications (UWBST) 2002, Baltimore, Maryland, pp. 175-179, May 2002.
4
But this approximation is acceptable only when the average sampling rate (1/T) is at least 3-4 times greater than the Nyquist rate. In the following sections we will show that iterative methods work well even at sampling rates close to the Nyquist rate. The block diagram of a generic iterative algorithm is shown in Figure 1. The block G is the ‘distortion block’. In fact the iterative method is a general method for recovery of signals distorted by any kind of distortion. It can be shown that if SNR at the output of the block is greater than one, then the iterative process converges to the original signal at infinity [1]. But because of limitations on computational precision, in practice, the obtained SNR saturates after some n iterations. However, this saturation level and also the convergence speed can to the block G in be improved by adding a gain each iteration. In the PPM case, the block G can be replaced by various systems. We have classified these systems into 2 groups. The first group treats the PPM demodulation as a nonuniform sampling reconstruction problem while the second group takes conventional PPM modulator for the G block. In the next sections we will present different methods for each group and then compare the results.
I. INTRODUCTION The idea of the PPM modulation was developed around 50 years ago but it is only recently that a revival of interest has been experienced with the development of Impulse Radio and Fiber-Optic transmission systems [4]. The special properties of the PPM modulation make it a suitable candidate for fiber-optic communications. Specially, because there is no information in the amplitude of PPM signals, this modulation technique is immune to the effects of nonlinearity of the fiber-optic devices. Also the PPM modulation is a good choice for the short-range communications in dense multi-path environments [67] using the Impulse Radio communications. Theoretically it has been shown that PPM systems are effective when the signals are power-limited rather than bandlimited [5]-[6].
Figure 1. Block diagram of the proposed iterative method II. ITERATIVE APPROACHES In this part, we introduce several approaches for PPM demodulation using iterative methods: the nonuniform sampling approach and the PPM inverse system approach. In the first approach, PPM pulses are multiplied by a synchronized sawtooth signal yielding a set of nonuniform samples of the original signal. Thus we can use the nonuniform sampling recovery techniques. We will introduce six different techniques in this approach.
The main idea in the PPM modulation is to send pulses with equal amplitude at the times of intersection between the message signal and a sawtoothwave r(t). Traditionally, for demodulation, the PPM pulses are first passed through a low pass filter. Then the DC offset of the output is removed and the resulting signal is passed though an integrator to produce an approximation of the message signal.
In the second approach, we consider the PPM modulator and demodulator as the G distortion in Fig. 2. In this approach, we can present different iterative
1
N x(t ) − x(ti ) (t − ti ) + x(ti )I (ti , ti +1 ) (8) Sx(t ) = ∑ i +1 ti +1 − ti i =1
methods depending on the type of the demodulation technique. Note that the demodulator used in each iteration is a low performance one and we improve its performance in the iteration. We will examine two systems using different demodulators in this approach. In the next sections we will explain these approaches more precisely.
Adaptive weight method (ADPW) is the next technique that we apply it to our problem. In this technique we multiply each sample by the weight equal to the corresponding Voronoi interval:
II.1. NON-UNIFORM SAMPLING APPROACH In this approach, we consider the G distortion in Fig. 2 as a sampling process S followed by a low pass filtering P [2]. Thus we have:
xk +1 (t ) = λ (PSx(t ) − PSxk (t )) + xk (t )
(1)
x(t i )δ (t − t i )
∞ 1 d x (t ) )) y (t ) = f s (1 − ( ) x (t ))1 + 2∑ cos( nω s (t − µ dt µ i =1
(2)
N P ∑ x(t i )δ (t − t i ) i =1 Sx(t ) = P{comb(t )}
t k = kTs +
(3)
p
N
∑ δ (t − t ) . i
In the Zero-Order-Hold method we have:
Sx(t ) = ∑ x(t i ) Φ i (t )
x(t )
µ
(11)
Algorithm II: Since the signal samples at the times t k are equal to the value of the saw tooth signal at these times, we can multiply the PPM pulses (which have unity amplitudes) by the saw tooth to obtain nonuniform samples of the signal. These nonuniform samples are then low pass filtered as shown in Fig5.
N
(4)
i =1
where (5)
and (6)
The Voronoi method have sampling process similar to Eq. (4) but with different Φ i (t ) . In this method we have:
t +t t +t Φ i (t ) = I i −1 i , i i +1 2 2
(10)
and µ is the saw tooth slope. Interpreting Eq.(10) we see that PPM with non-uniform sampling is a combination of linear and exponential carrier modulation. At each harmonic the signal is phase modulated and its derivative is amplitude modulated at the output. We can therefore retrieve the original message by low-pass filtering and integrating with the incorporation of a DC block as shown in Fig4.
where P{y (t )} is the low pass filtered version
1 t1 ≤ t ≤ t 2 I (t1 , t 2 ) = 0 otherwise
(9)
Where y(t) is the PPM wave of constant amplitude with pulses positioned at non-uniformly sampled times of:
and in the time-varying method as follow:
Φ i (t ) = I (t i , t i +1 )
2
Algorithm I: Analyzing the spectrum of a PPM signal, it can be shown [1] that the modulation output has the form of:
i =1
i =1
i =1
In this approach the block G (in Figure 2) consists of PPM modulator followed by PPM demodulator. Both methods use the same PPM modulator shown in Fig3, but different in PPM demodulation. This approach is divided into two methods:
N
of y (t ) and comb(t ) =
(t i +1 − t i −1 )
II.2. PPM INVERSE SYSTEM APPROACH
The main difference between the various forms of iterative methods is due to the difference of sampling processes [3]. We examine six techniques in this category: Wiley/Marvasti, Time-varying, adaptive weight method, Zero-order-hold, Voronoi and linear interpolation. In the Wiley/Marvasti method the operator S is as follow:
Sx(t ) = ∑ x(t i )δ (t − t i )
N
Sx(t ) = ∑
Fig 1.1: PPM Modulation Figure 2. PPM modulator
(7)
In the linear interpolation method we can write:
2
Figure 3. Demodulator of the algorithm I
Figure 4. Demodulator of the algorithm II
Figure 5. The mean square error of the first approach methods
Fig 5 shows the MSE versus the number of iterations using different techniques of the second approach (inverse system).
III.SIMULATION RESULTS The following results have been obtained for the demodulation of a lowpass signal using different iterative techniques described before. The period of the sawtooth signal is chosen so that the average number of PPM pulses in unit time satisfies the Nyquist rate. In order to test performance of these methods, we implemented them for some various s and found its optimal value for each of the methods. We considered the optimality in the sense of minimum mean-square error in the tenth iteration. Fig6 shows the mean square error of demodulator (in dB) using different techniques of the first approach (nonuniform sampling) versus the number of iterations. The parameter λ in each method is chosen optimally (Table 1). In this figure we see that the Voronoi method (symmetric hold) has the highest convergence of all, but its complexity is relatively high and some delay is necessary in its implementation. The mean square error of all methods finally converges to approximately -21.5dB except the simple zero-order hold method (non symmetric). Another important result is approximately the same performance of the Wiley/Marvasti and ADPW method. The Table 2 represents the MSE of this methods after 3 ,5 , 10 and 20th iteration.
Figure 6. The mean square error of the second approach methods
The complexities of the first approach methods are approximately the same except in ADPW that we have the additional complexity of the multiplication for each sample. For the algorithms I and II there is an additional complexity because of PPM modulation block in each iteration. In fact this block involves N additions (plus N comparisons). Furthermore, in the algorithm I the demodulation stage requires 2N additions and a filtering (the same as others). But the algorithm II involves N additions and a filtering at the demodulation stage.
We can observe from the Figure that the method I has better results than the method II. The MSE obtained from the method I in the tenth iteration is about 3dB larger than that is obtained from the method II. Although the method II has better performance than the method I in the sense of MSE, the synchronization is a serious problem for this method. In fact the saw tooth signal required at receiver for the method II must be exactly synchronous with that of transmitter. This critical parameter causes to use the first method widely than the second.
As final note, we present a modification of the method I for bandpass signals. The modified method has two differences with the original method. First, it uses a bandpass filter instead of lowpass filter at each iteration. Second, instead of removing a specific DC value in the DC block, we force the DC component of signal to be zero (as it is for a bandpass signal). Fig8 ( =0.85). We represents the result for the optimal can see that the modified method involves better MSE and faster convergence for bandpass signals than the original method. λ
3
Table 2 : The optimum convergence factor λopt for different iterative methods technique
λopt
technique
λopt
Wiley/Marvasti
0.8
Zero-order hold
1.3
Time varying
0.6
Voronoi
0.5
ADPW
0.95
Algorithm I
0.45
Linear interpolation
1.6
Algorithm II
0.85
Figure 7. The mean square error of the modified algorithm I
III.CONCLUSION
REFERENCES
We introduced several techniques for PPM demodulation and assessed their performances. We can compare these techniques from two points of view: the Mean Square Error (MSE) and the convergence rate. The MSE is in the saturation region of the performance curves depicted in Figs 68, which we calculate at the 20th iteration. The convergence rate is a criterion for comparing the MSE in the transition region that is in the lower number of iterations. For convenience, we have collected the MSE of the two approaches in Table 1 for certain number of iterations.
[1] F.Marvasti, Nonuniform Sampling Theory and Practice,Kluwer Academic/Plenum Publisher,NewYork,2001. [2] F.Marvasti, M.Analouei and M.Gamshadzahi, Recovery of Signals from Samples Using Iterative Methods., IEEE Trans on Signal Processing, vol.39 ,no.4,pp.872-878, April 1991 [3] P. Azmi, and F. Marvasti, “Comparison between Several Iterative Methods of Recovering Signals from Nonuniformly Spaced Samples”, in Proc. IEEE Conference on Sampling Theory and Applications SampTA2001, Florida-U.S.A., pp., 49-54, 13-17 May 2001.
As shown in Table 1, Algorithm II yields better performance after twenty iterations but the convergence rate is slow, however the convergence rate of the Voronoi technique is the best between one and three iterations.
[4] M. Z. Win, and R. A. Scholtz, “Ultra-Wide Bandwidth Time-Hopping Spread-Spectrum Impulse Radio for Wireless Multiple Access Communications” IEEE Trans on Com., vol. 48, no. 4, pp. 679-689, April 2000.
Table 1 : The MSE of the first approach techniques for some iterations.
Iteration
3
5
10
20
Wiley/Marvasti
-16.53
-19.15
-21.28
-21.55
Time varying
-18.68
-20.9
-21.55
-21.55
ADPW
-16.82
-19.64
-21.44
-21.57
Linear interpolation
-20.41
-21.34
-21.54
-21.54
Zero-order hold
-1.311
-2.616
-6.239
-9.652
[6] B. Wilson, and Z. Ghassemlooy, “Pulse Time Modulation Techniques for Optical Communications: A Review”, IEE Proceeding-J, vol. 140, no. 6, pp. 346-357, Dec. 1993.
Voronoi
-21
-21.46
-21.55
-21.55
[7] J. R. Pierce, and E. C. Posner, Introduction to Communication Science and Systems, Plenum, 1976.
Algorithm I
-13.15
-14.92
-17.23
-17.06
Algorithm II
-13.31
-15.98
-21.22
-26.63
[5] L. Zhao, and A. Haimovich, “Multi-User Capacity of M-ary PPM Ultra-Wideband Communications” in Proc.IEEE Conference on Ultrawideband Systems and Communications (UWBST) 2002, Baltimore, Maryland, pp. 175-179, May 2002.
4
