Spectrum Characteristics of Ternary PSK Signals Amplified with Non-Linear Amplifiers HIDEYUKI TORII and MAKOTO NAKAMURA Department of Network Engineering Kanagawa Institute of Technology 100 Shimo-ogino, Atsugi-shi, Kanagawa, 4-09 JAPAN Abstract: - Phase shift keying (PSK) is one of the most important classes of digital modulation schemes. M-ary PSK of which the value of M is other than the n-th power of has not been focused on until now because it is not suitable for transmitting binary information. In a previous study, we reported that ternary PSK () has theoretically the best error rate characteristic and the minimum instantaneous amplitude fluctuation among all of the PSK schemes. From the latter feature, it is surmised that the scheme can suppress unnecessary spectra caused by no-linear amplifiers. However, in order to clarify detailed effectiveness of this feature, it is necessary to evaluate spectrum characteristics of signals. In the present paper, the spectrum characteristics of the signals which are amplified with non-linear amplifiers after being inputted into a square root cosine roll-off filter are evaluated by computer simulation. The results show that the spectrum characteristic of the signal is superior to that of signal and that of π /4 shift signal. Key-Words: - Digital modulation, Phase shift keying (PSK), Ternary PSK (), Spectrum characteristic, Non-linear amplifier, Square root cosine roll-off filter 1 Introduction Phase shift keying (PSK) is one of the most important classes of the digital modulation schemes because it can achieve excellent error rate characteristics [1]. Binary PSK (BPSK), which can transmit 1 bit per symbol, is the most fundamental scheme among the PSK schemes. If the number of phases is increased, the number of information bits which can be transmitted by one symbol also increases. This is a well-known method as M-ary PSK. It is possible to reduce occupied bandwidth by using the M-ary PSK. In the M-ary PSK schemes, the value of M is commonly the n-th power of. The M-ary PSK scheme of which M is other than n has not been hardly studied because it is not suitable for transmitting binary information. In addition, in the M-ary PSK schemes other than quadrature PSK (), the error rate characteristics to E b /N 0 have been considered to deteriorate from the BPSK. Therefore, the BPSK scheme and the scheme have been mainly used in mobile communication systems. Ternary PSK is a M-ary PSK scheme of which M equals. The number of bits which the can transmit per symbol is not an integer but log bits. In a previous study [7], we clarified the bit error rate characteristic of the by theoretical study and showed that the bit error rate characteristic of the excels that of the BPSK in the range where E b /N 0 exceeds 4.86 db. In the scheme, a complicated processing is needed in order to transfer a binary information stream to ternary signals because is not the n-th power of. In order to solve this problem, we proposed a new class of convolutional codes [7]. In addition, the proposed convolutional codes can provide an efficient error correcting ability for the ternary signals without expanding the occupied bandwidth. Generally, in mobile communication systems, the bandwidth of transmission signal is restricted by a filter and the signal is amplified by a non-linear amplifier after the restriction of the bandwidth. If the instantaneous amplitude of the signal is not constant, the radio wave envelopment also fluctuates by the restriction of the bandwidth and the unnecessary spectrum which was removed by the filter appears again after the non-linear amplification. This causes deterioration of communication quality. It is necessary to reduce the instantaneous amplitude fluctuation in order to avoid this problem. π /4 shift was
developed for this purpose. In the previous study [7], we showed that the instantaneous amplitude fluctuation of the is smaller than that of the π /4 shift. From this feature, it is surmised that the scheme can suppress the unnecessary spectrum caused by the no-linear amplification. However, in order to clarify detailed effectiveness of this improvement, it is necessary to evaluate the spectrum of signal. In the present paper, the spectrum characteristics of the signals which are amplified with non-linear amplifiers after being inputted into a square root cosine roll-off filter are evaluated by computer simulation. The results show that the spectrum characteristic of the signal is superior to that of signal and that of π /4 shift signal, that is, the scheme is effective in suppressing the unnecessary spectrum which appears after the non-linear amplification. S 1 D S S 1 D R D =A S 0 (A, 0) S 0 Bit Error Rate of In the previous study [7], we clarified the bit error rate of the. In this section, the bit error rate characteristic of the is briefly explained. The signal assignment of the is shown in Fig.1. Three signals are denoted as S i (i = 0,1,). In addition, A denotes the amplitude of the carrier wave. When the signal S 0 is transmitted in an additive white Gaussian noise channel, the probability density of the received signal becomes a two-dimensional Gaussian function [] with the mean value of (A, 0). In Fig.1, the received signal which is located in the oblique lined region R is decoded as the signal S 0, if the signals S i (i = 0,1,) have the same priori probability. Therefore, if S 0 is transmitted and is received at the region other than R, a symbol error occurs. The symbol error rate ε is given as: ( x A ) + y 1 σ 1 e πσ ( x, y) R ε = dx dy (1) The noise power is σ and the signal power is A /. Note that the symbol error rate agrees with the bit error rate in the. On the other hand, the signal assignment of the BPSK is shown in Fig.1. The error probability of the BPSK scheme is given as: 1 ε ( E ) λ b N e dλ E N π / 0 = () b 0 BPSK Fig.1 Constellations of PSK schemes As shown in Fig.1, the minimum distance from the signal of the to the decision threshold is expressed as: D = A () Similarly, as shown in Fig.1, the distance from the signal of the BPSK to the decision threshold is: D = A (4) One symbol of the BPSK can transmit 1 bit. On the other hand, one symbol of the can transmit log bits. If the transmission signal power is equalized per one information bit, the relation between A and A is represented as: A = log A (5) Then, the minimum distance from the signal of the to the decision threshold is expressed as: log D = D 1. 09D (6) Therefore, the distance D in the is longer than the distance D in the BPSK on the condition that the transmission signal powers per one information bit are equivalent to each other. From Equations (), (6) and Fig.1, the error rate ε is bounded as: ε 1.189 E N ) < ε < ε (1.189 E ) (7) ( b 0 b N0
Therefore, it is expected that the error rate characteristic of the is better than that of the BPSK in the range where E b /N 0 is sufficiently large. Fig. shows the error rate characteristics of the BPSK and the in the practical values of E b /N 0. The error rate of the scheme excels that of the BPSK scheme in the range where E b /N 0 is larger than 4.86 db. For detail descriptions, see [7]. BER 10-1 1.E-01 1.E-0 10-10 - 1.E-0 10-4 1.E-04 10-5 1.E-05 10-6 1.E-06 BPSK 1 4 5 6 7 8 9 10 11 Eb/No Fig. Bit error rate of Convolutional Codes for In the previous study [7], we proposed a new class of convolutional codes for the. In this section, the proposed convolutional codes are briefly explained. In the scheme, a particular processing is needed in order to transfer a binary information stream to ternary signals because is not n. In addition, it is indispensable for mobile communication systems to utilize error correction schemes. Therefore, convolutional codes which are suitable for the scheme are needed. The proposed convolutional codes can solve these problems simultaneously. An example of the generator polynomial of the proposed convolutional code is given as: 4 5 6 g ( D) = 1+ D + D + D + D + D (8) The constraint length of this convolutional code is 7. While the input signals are elements over GF(), the output signals are elements over GF(). In addition, all operations are calculated over GF(). The code rate of this convolutional code is 1, that is, one ternary signal is outputted every one binary input signal. Although the code rate is 1, the minimum free distance of this convolutional code is 6. This fact denotes that this convolutional code can provide an efficient error correcting ability without expanding the occupied bandwidth. This is an attractive feature which the conventional convolutional codes do not have. Note that this convolutional code can provide not only the error correcting ability but also the function of transforming from the binary signals to the ternary signals. In addition, Viterbi decoding can be applied to the proposed convolutional codes without extending the circuit size. We showed that the bit error rate characteristics of the with the proposed convolutional code and the BPSK with a conventional convolutional code are almost equivalent one another. For detail descriptions, see [7]. 4 Amplitude Fluctuation of It is clear that the BER of the with the proposed convolutional code is almost equivalent to that of the π /4 shift with the conventional convolutional code if both the schemes transmit the information bits at the same information rate. Note that the bit error rate characteristics of the and the π /4 shift agree with that of the BPSK. However, the has an advantage over theπ /4 shift. In the previous study [7], we showed that the instantaneous amplitude fluctuation of the is smaller than that of the π /4 shift. In this section, this feature is explained. The instantaneous amplitude of the BPSK and the becomes zero at the symbol transition. It causes a great fluctuation of the radio wave envelopment if it is transmitted via a bandwidth-limited channel. This is very serious for a mobile terminal that uses a C class power amplifier. If the signal is amplified by non-linear amplifiers after the restriction of bandwidth, the unnecessary spectrum which was removed by the filters appears again. If the instantaneous amplitude fluctuation is reduced, this problem will be improved. Therefore, the π /4 shift, which can reduce the amplitude fluctuation compared with the, is widely used for the digital cellular systems. In the π /4 shift, the minimum instantaneous amplitude is approximately 8 % of the maximum instantaneous amplitude. On the other hand, the instantaneous amplitude of the falls to only 50 % of the maximum value. Hence, the instantaneous amplitude fluctuation is improved by approximately 1. times by using the scheme instead of the conventional π /4 shift scheme. This feature is very attractive in practical mobile communication systems. 5 Spectrum Characteristic of As has been mentioned, the instantaneous amplitude fluctuation of the has the superior property to
that of the π /4 shift. Consequently, it is surmised that the can suppress the unnecessary spectrum caused by the no-linear amplification. However, in order to clarify detailed effectiveness of this improvement for the suppression of the unnecessary spectrum, it is necessary to evaluate the spectrum of the signal. In this section, the spectrum characteristics of the signals which are amplified with non-linear amplifiers after being inputted into a square root cosine roll-off filter are evaluated by computer simulation. Fig. shows a block diagram of our computer simulation. First, each base-band signal is inputted into a square root cosine roll-off filter. This filter is often used in mobile communication systems, in order to eliminate unnecessary spectra. This filter has the following impulse response: (1 α) πt 4αt ( 1+ α ) πt sin + cos g() t = T T T (9) πt 4αt 1 T T where α is a roll-off ratio and 1/T is the symbol rate of the signal. Next, the band-limited base-band signals are multiplied by Acos( π fct) or Asin( π fct). Here, A denotes amplitude and f c denotes a carrier frequency. Moreover, both the signals are added to each other. Note that the radio wave envelopment is fluctuating at this point. Finally, the signal is amplified by a non-linear amplifier. The unnecessary spectrum appears again at this point. It is significant to suppress the unnecessary spectrum in order to reduce the influence on the adjacent channels. Base-band signal (I-phase) Base-band signal (Q-phase) Acos Asin ( πf t) ( πf t) Non-linear amplifier c c Square root cosine roll-off filter Square root cosine roll-off filter FFT In this simulation, we used two non-linear amplifiers. Input-output characteristics of the two amplifiers are shown in Fig.4. -A -A -A/4 Output A Output A Input Fig.4 Input-output characteristics of amplifiers First, we calculated the spectra of the, the π /4 shift, and the. The results are shown in Fig.5. In Fig.5, 0000 symbols of random data are transmitted at each combination of the PSK schemes and the amplifiers. The vertical axis of Fig.5 denotes the power spectral density (db), and the horizontal axis denotes the frequency which is normalized by the carrier wave frequency and the symbol rate. The spectrum of the TPSk is smaller than that of the and that of the π /4 shift. For example, suppose that the amplifier is used, then the value of the power spectral density of the is smaller for approximately 15 db than that of the at the point in which the value of the normalized frequency is. Similarly, it is smaller for approximately 6 db than the value of the power spectral density of the π /4 shift at the same point. -A -A A/4 A Input A Fig. Block diagram of computer simulation
This fact shows that the spectrum of the does not spread compared with that of the π /4 shift and that of the. Normalized Bandwidth.5.5 1.5 1 0.5 90 95 99 Rario of Energy (%) Normalized Bandwidth.5 1.5 1 Fig.5 Spectrum characteristics of PSK schemes 0.5 90 95 99 Ratio of Energy (%) Fig.6 Occupied bandwidth Next, we examined a relation between the energy of each PSK scheme and the occupied bandwidth. The results are shown in Fig.6. The horizontal axis denotes a ratio of the energy. Note that the total energy from - to is 100%. The vertical axis denotes the normalized bandwidth occupied by the energy which is exhibited in the horizontal axis. For example, 95% of the total energy of the is focused on the central part whose bandwidth is 1.0 when the amplifier is used. Similarly, 95% of the total energy of the π /4 shift and that of the are focused on the central parts whose bandwidths are 1.08 and 1., respectively. In the both amplifiers, The occupied bandwidth of the is narrower than that of the π /4 shift and that of the, at every ratio. Finally, we calculated a ratio of desired signal power to undesired signal power from two neighboring channels under the assumption that a distance between the channel used by desired signal and the neighboring channel is.5. The results are shown in Table 1. The results show that the D/U ratio of the scheme is smaller than that of the π /4 shift scheme and that of the scheme. In addition, the BERs in consideration of the D/U ratio are shown in Fig.7. In the, the degree of the degradation of the BER by the undesired signals is smaller than other two schemes.
BER Amplifier PSK scheme D/U (db) 15.7 π /4 shift 1. 4.49 19.76 π /4 shift.48 5.05 1.0E-01 10-1 1.0E-0 10-1.0E-0 10-1.0E-04 10-4 1.0E-05 10-5 Table 1 D/U ratio the signals which are amplified with the non-linear amplifiers after being inputted into the square root cosine roll-off filter have been evaluated by computer simulation. The results show that the unnecessary spectrum of the is lower than that of the π /4 shift and that of the. Therefore, we can figure that the property that the instantaneous amplitude fluctuation of the is less than other PSKs is very effective to suppress the unnecessary spectrum which appears after the non-linear amplifications. 5 Acknowledgment This research was supported by a Grant-in-Aid for Young Scientists (B) (KAKENHI(1475018)) from the Ministry of Education, Culture, Sports, Science and Technology. BER 1.0E-06 10-6 1.0E-01 10-1 1.0E-0 10-1.0E-0 10-1.0E-04 10-4 1.0E-05 10-5 1.0E-06 10-6 1 4 5 6 7 8 9 10 Eb/N0 1 4 5 6 7 8 9 10 Eb/N0 Fig.7 BER in consideration of D/U From all of the results, we can verify that the property that the instantaneous amplitude fluctuation of the is less than other PSKs is effective to suppress the unnecessary spectrum which appears after the non-linear amplification. References: [1] V.K. Bhargavan, et al., Digital Communications by Satellite, Wiley, 1981 [] S. Lin, et al., Error Control Coding: Fundamentals and Applications, Prentice-Hall Inc., 198 [] J.M. Wozencraft, et al., Principles of Communication Engineering, Waveland Press Inc., 1990 [4] J. P. Odenwalder, Dual k Convolutional Codes for Noncoherently Demodulated Channels, Proc. Int. Telemetering Conf., vol.1, 1976, pp.165-174. [5] L.R. Bahl, et al., An Efficient Algorithm for Computing Free Distance, IEEE Trans. Inform. Theory, Vol.IT-18, 197, pp.47-49. [6] A.J. Viterbi, Error Bounds for Convolutional Codes and an Asymptotically Optimum Decoding Algorithm, IEEE Trans. Inform. Theory, Vol. IT-1, 1967, pp.60-69. [7] M. Nakamura, et al., Ternary Phase Shift Keying and Its Performance, Proc. Int. Symp. Wireless Personal Multimedia Communications, 00, pp.184-188. 6 Conclusions In the present paper, the spectrum characteristics of