Numerical Simulation
4.1.1 Peak to Average Power Ratio
There we will take (3.8) to compute the peak to average power ratio (PAPR) value, and the PAPR performance curves are shown in Figure 4.4. We use oversampling factors (L) = 1, 2, 4, 8, 16, and 32 respectively to simulate the PAPR performance for each filter. Then we also focus on which factor is the least oversampling factor that we need to use for computing the PAPR value.
4.1.2 Bit Error Rate
The bit error rate (BER) performance comes from the average of 105 random chan-nels. In this section, we show the average BER performance of Rayleigh fading channel and additive white Gaussian noise channel.
At first, we deal with the Rayleigh fading channel. We take one of the 105random continuous-time fading channels for example to show impulse response, magnitude re-sponse, and the BER performance of its equivalent discrete-time channel c1(n) in Figure 4.5. (θ1 : 5.6165, θ2 : 1.2512; α1 : 1.3951, α2 : 0.6319).
When the continuous-time channel is AWGN channel, its equivalent discrete-time channel is c2(n). The impulse response, magnitude response, and the BER performance of c2(n) is shown in Figure 4.6.
From these simulation results, we can know that:
1. Oversampling factor: We need to take oversampling factor L = 16 to get the PAPR closer to the continuous-time PAPR. L = 4 is not enough for used.
2. PAPR performance: When the stopband edge of filer is large (ws= 1.2f s), it has large useful band bandwidth. Its PAPR performance is worse.
3. BER performance: When the useful band bandwidth is large, the filter has better BER performance.
4 5 6 7 8 9 10 11 12 10-3
10-2 10-1 100
PAPRo(dB)
CCDF (PAPRo)
(a)
L=1L=2 L=4L=8 L=16L=32
4 5 6 7 8 9 10 11 12 13
10-3 10-2 10-1 100
PAPRo(dB)
CCDF (PAPRo)
(b)
L=1L=2 L=4L=8 L=16L=32
4 6 8 10 12 14 10-3
10-2 10-1 100
PAPRo(dB)
CCDF (PAPRo)
(c)
L=1L=2 L=4L=8 L=16L=32
4 6 8 10 12 14
10-3 10-2 10-1 100
PAPRo(dB)
CCDF (PAPRo)
(d)
Wp 0.5 fs-Ws 0.7 fs Wp 0.5 fs-Ws 1.0 fs Wp 0.5 fs-Ws 1.2 fs
Figure 4.4: The PAPR curves of elliptic analog filters with distinct stopband edges ws. (a) ws is 0.7 fs. (b)ws is 1.0 fs. (c) ws is 1.2 fs. (d) The PAPR curves of these three filters and the oversampling factor is 32.
0 0.5 1 1.5
0 5 10 15 20 25 30 35
Figure 4.5: (a) The impulse response (b) The magnitude response (c) BER performance of c1(n)
0 0.2 0.4 0.6 0.8 1 -200
-150 -100 -50 0 50 100
normalized frequency
magnitude response(dB)
(b)
Wp 0.5 fs-Ws 0.7 fs Wp 0.5 fs-Ws 1.0 fs Wp 0.5 fs-Wp 1.2 fs
0 2 4 6 8 10 12 14 16
10-4 10-3 10-2 10-1 100
SNR(dB)
BER
(c)
Wp 0.5 fs-Ws 0.7 fs Wp 0.5 fs-Ws 1.0 fs Wp 0.5 fs-Ws 1.2 fs
Figure 4.6: (a) The impulse response (b) The magnitude response (c) BER performance of c2(n)
case 2 Order ws(fs) Rp(dB) Rs(dB)
wp 0.2(fs) 2 1.8 1.0 -40
wp 0.5(fs) 2 1.5 1.0 -40
wp 0.8(fs) 3 1.2 1.0 -40
Table 4.3: Parameters of case 2
4.2 Case 2
The analog filters taken for simulation are elliptic analog filters. We want to see if the useful band bandwidth affects the PAPR performance and the BER performance.
Fix the passband ripple and stop attenuation, then using different passband edge and stopband edge to produce distinct useful band bandwidths. The transition bands center at 1.0 sampling frequency (fs). The transmitting and receiving pulses, p1(t) and p2(t), are chosen the same when we simulate the BER performance. In other words, we have three kinds of analog filters, and we have three BER performance curves in this case.
The parameters of filter design are given in table 4.3. The magnitude response of each filter in frequency domain and the impulse response of each filter in time domain are shown in Figure 4.7.
4.2.1 Peak to Average Power Ratio
There we will take (3.8) to compute the peak to average power ratio (PAPR) value, and the PAPR performance curves are shown in Figure 4.8. We use oversampling factors (L) = 1, 2, 4, 8, 16, and 32 respectively to simulate the PAPR performance for each filter. Then we also focus on which factor is the least oversampling factor that we need to use for computing the PAPR value.
0 0.5 1 1.5 2
Figure 4.7: (a)The magnitude responses (b)The impulse responses of elliptic analog filters with distinct transition band bandwidths which center at 1.0 fs.
4 5 6 7 8 9 10 11 12 10-3
10-2 10-1 100
PAPRo(dB)
CCDF (PAPRo)
(a)
L=1L=2 L=4L=8 L=16L=32
4.2.2 Bit Error Rate
The bit error rate (BER) performance comes from the average of 105 random chan-nels. In this section, we show the average BER performance of Rayleigh fading channel and additive white Gaussian noise channel.
At first, we deal with the Rayleigh fading channel. We take one of the 105random continuous-time fading channels for example to show impulse response, magnitude re-sponse, and the BER performance of its equivalent discrete-time channel c3(n) in Figure 4.9. (θ1 : 0.36374, θ2 : 2.21710; α1 : 1.5683, α2 : 0.8305).
When the continuous-time channel is AWGN channel, its equivalent discrete-time channel is c4(n). The impulse response, magnitude response, and the BER performance of c4(n) is shown in Figure 4.10.
From these simulation results, we can know that:
1. Oversampling factor: We need to take oversampling factor L = 16 to get the PAPR closer to the continuous-time PAPR. L = 4 is not enough for used.
2. PAPR performance: When the filter has large useful band bandwidth (wp = 0.8f s, ws = 1.2f s), its PAPR performance is better. The one has smaller
use-4 6 8 10 12 14 10-3
10-2 10-1 100
PAPRo(dB)
CCDF (PAPRo)
(c)
L=1L=2 L=4L=8 L=16L=32
4 6 8 10 12 14
10-3 10-2 10-1 100
PAPRo(dB)
CCDF (PAPRo)
(c)
L=1L=2 L=4L=8 L=16L=32
4 6 8 10 12 14
Figure 4.8: The PAPR curves of elliptic analog filters with distinct transition band bandwidths. (a) wp is 0.2 fs and ws is 0.8 fs. (b)wp is 0.5 fs and ws is 1.5 fs. (c) wp is 0.8 fs and ws is 1.2 fs. (d) The PAPR curves of these three filters and the oversampling factor is 32.
0 0.2 0.4 0.6 0.8 1
Figure 4.9: (a) The impulse response (b) The magnitude response (c) BER performance of c3(n)
0 0.5 1 1.5
0 5 10 15 20 25 10-4
10-3 10-2 10-1 100
SNR(dB)
BER
(c)
Wp 0.2 fs-Ws 1.8 fs Wp 0.5 fs-Ws 1.5 fs Wp 0.8 fs-Ws 1.2 fs
Figure 4.10: (a) The impulse response (b) The magnitude response (c) BER performance of c4(n)
ful band bandwidth, and its PAPR performance is better.
3. BER performance: When the useful band bandwidth is large, the filter has better BER performance.
4.3 Case 3
The analog filters taken for simulation are elliptic analog filters. We want to see if the useful band bandwidth affects the PAPR performance and the BER performance. Fix the stopband edge, passband ripple, and stopband attenuation. Then change passband edge to produce distinct useful band bandwidths. The transmitting and receiving pulses, p1(t) and p2(t), are chosen the same when we simulate the BER performance. In other words, we have three kinds of analog filters, and we have three BER performance curves in this case. The parameters of filter design are given in table 4.4. The magnitude response of each filter in frequency domain and the impulse response of each filter in time domain are shown in Figure 4.11.
case 3 Order ws(fs) Rp(dB) Rs(dB)
wp 0.2 (fs) 2 1.5 1.0 -40
wp 0.5 (fs) 3 1.5 1.0 -40
wp 0.8 (fs) 4 1.5 0.1 -40
Table 4.4: Parameters of case 3
0 0.5 1 1.5 2
Figure 4.11: (a)The magnitude responses (b)The impulse responses of elliptic analog filters with distinct passband edges.
4.3.1 Peak to Average Power Ratio
There we will take (3.8) to compute the peak to average power ratio (PAPR) value, and the PAPR performance curves are shown in Figure 4.12. We use oversampling factors (L) = 1, 2, 4, 8, 16, and 32 respectively to simulate the PAPR performance for each filter. Then we also focus on which factor is the least oversampling factor that we need to use for computing the PAPR value.
4 5 6 7 8 9 10
10-3 10-2 10-1 100
PAPRo(dB)
CCDF (PAPRo)
(a)
L=1L=2 L=4L=8 L=16L=32
4 5 6 7 8 9 10 11
10-3 10-2 10-1 100
PAPRo(dB)
CCDF (PAPRo)
(b)
L=1L=2 L=4L=8 L=16L=32
4 5 6 7 8 9 10 11 10-3
10-2 10-1 100
PAPRo(dB)
CCDF (PAPRo)
(c)
L=1L=2 L=4L=8 L=16L=32
4.3.2 Bit Error Rate
The bit error rate (BER) performance comes from the average of 105 random chan-nels. In this section, we show the average BER performance of Rayleigh fading channel and additive white Gaussian noise channel.
At first, we deal with the Rayleigh fading channel. We take one of the 105random continuous-time fading channels for example to show impulse response, magnitude re-sponse, and the BER performance of its equivalent discrete-time channel c5(n) in Figure 4.13. (θ1 : 4.994, θ2 : 6.012; α1 : 0.38437, α2 : 0.6352).
When the continuous-time channel is AWGN channel, its equivalent discrete-time channel is c6(n). The impulse response, magnitude response, and the BER performance of c6(n) is shown in Figure 4.14.
From these simulation results, we can know that:
1. Oversampling factor: We need to take oversampling factor L = 4 to get the PAPR closer to the continuous-time PAPR.
2. PAPR performance: When the passband edge of filer is large (wp = 0.8f s), it has large useful band bandwidth. Its PAPR performance is worse.
4 5 6 7 8 9 10 11
Figure 4.12: The PAPR curves of elliptic analog filters with distinct passband edges wp. (a) wp is 0.2 fs. (b)wp is 0.5 fs. (c) wp is 0.8 fs. (d) The PAPR curves of these three filters and the oversampling factor is 32.
0 0.5 1 1.5
0 0.2 0.4 0.6 0.8 1
Figure 4.13: (a) The impulse response (b) The magnitude response (c) BER performance of c5(n)
0 0.5 1 1.5
0 5 10 15 20 25 30 10-4
10-3 10-2 10-1 100
SNR(dB)
BER
(c)
Wp 0.2 fs-Ws 1.5 fs Wp 0.5 fs-Ws 1.5 fs Wp 0.8 fs-Ws 1.5 fs
Figure 4.14: (a) The impulse response (b) The magnitude response (c) BER performance of c6(n)
3. BER performance: When the useful band bandwidth is large, the filter has better BER performance.
4.4 Case 4
The analog filters taken for simulation are elliptic analog filters. We want to see if the useful band bandwidth affects the PAPR performance and the BER performance.
Fix the transition band bandwidth, passband ripple, and stopband attenuation. Then change passband edge and stopband edge to produce distinct useful band bandwidths.
The transmitting and receiving pulses, p1(t) and p2(t), are chosen the same when we simulate the BER performance. In other words, we have three kinds of analog filters, and we have three BER performance curves in this case. The parameters of filter design are given in table 4.5. The magnitude response of each filter in frequency domain and the impulse response of each filter in time domain are shown in Figure 4.15.
case 4 Order ws(fs) Rp(dB) Rs(dB)
wp 0.2 (fs) 4 0.9 0.01 -40
wp 0.5 (fs) 4 1.2 0.01 -40
wp 0.8 (fs) 4 1.5 0.01 -40
Table 4.5: Parameters of case 4
0 0.5 1 1.5 2
Figure 4.15: (a)The magnitude responses (b)The impulse responses of elliptic analog filters with the same transition band bandwidth.
4.4.1 Peak to Average Power Ratio
There we will take (3.8) to compute the peak to average power ratio (PAPR) value, and the PAPR performance curves are shown in Figure 4.12. We use oversampling factors (L) = 1, 2, 4, 8, 16, and 32 respectively to simulate the PAPR performance for each filter. Then we also focus on which factor is the least oversampling factor that we need to use for computing the PAPR value.
4.4.2 Bit Error Rate
The bit error rate (BER) performance comes from the average of 105 random chan-nels. In this section, we show the average BER performance of Rayleigh fading channel and additive white Gaussian noise channel.
At first, we deal with the Rayleigh fading channel. We take one of the 105random continuous-time fading channels for example to show impulse response, magnitude re-sponse, and the BER performance of its equivalent discrete-time channel c7(n) in Figure 4.17. (θ1 : 2.3274, θ2 : 4.41541; α1 : 0.70429, α2 : 0.69466).
When the continuous-time channel is AWGN channel, its equivalent discrete-time channel is c8(n). The impulse response, magnitude response, and the BER performance of c8(n) is shown in Figure 4.18.
From these simulation results, we can know that:
1. Oversampling factor: We need to take oversampling factor L = 4 to get the PAPR closer to the continuous-time PAPR.
2. PAPR performance: When the filer has large useful band bandwidth (wp = 0.8f sandws= 1.5f s), its PAPR performance is worse.
3. BER performance: When the useful band bandwidth is large, the filter has better BER performance.
4 5 6 7 8 9 10 10-3
10-2 10-1 100
PAPRo(dB)
CCDF (PAPRo)
(a)
L=1L=2 L=4L=8 L=16L=32
4 5 6 7 8 9 10 11
10-3 10-2 10-1 100
PAPRo(dB)
CCDF (PAPRo)
(b)
L=1L=2 L=4L=8 L=16L=32
4 5 6 7 8 9 10 11 10-3
10-2 10-1 100
PAPRo(dB)
CCDF (PAPRo)
(c)
L=1L=2 L=4L=8 L=16L=32
4 5 6 7 8 9 10 11
10-3 10-2 10-1 100
PAPRo(dB)
CCDF (PAPRo)
(d)
Wp 0.2 fs-Ws 0.9 fs Wp 0.5 fs-Ws 1.2 fs Wp 0.8 fs-Ws 1.5 fs
Figure 4.16: The PAPR curves of elliptic analog filters with the same transition band bandwidth and distinct useful band bandwidths. (a)wp is 0.2 fs and ws is 0.9 fs. (b)wp
is 0.5 fs and ws is 1.2 fs. (c) wp is 0.8 fs and ws is 1.5 fs. (d) The PAPR curves of these three filters and the oversampling factor is 32.
0 0.5 1 1.5
0 5 10 15 20 25 30 35 40
Figure 4.17: (a) The impulse response (b) The magnitude response (c) BER performance of c7(n)
0 0.2 0.4 0.6 0.8 1 -250
-200 -150 -100 -50 0 50
normalized frequency
magnitude response(dB)
(b)
Wp 0.5 fs-Ws 0.7 fs Wp 0.5 fs-Ws 1.0 fs Wp 0.5 fs-Wp 1.3 fs
0 2 4 6 8 10 12 14 16 18
10-4 10-3 10-2 10-1 100
SNR(dB)
BER
(c)
Wp 0.2 fs-Ws 0.9 fs Wp 0.5 fs-Ws 1.2 fs Wp 0.8 fs-Ws 1.5 fs
Figure 4.18: (a) The impulse response (b) The magnitude response (c) BER performance of c8(n)
4.5 Summary
These simulations are done to see if the transmitting and receiving pulses design affects the oversampling factor, peak to average power ratio (PAPR), and bit error rate (BER) performance. The results show that these quantities are indeed affected by the filter design.
1. Oversampling factor: The earlier studies that use analog framework OFDM system shown an oversampling factor of 4 gives a good estimate of continuous-time PAPR value. But the results of case 1 and case 2 show oversampling factor of 16 is needed to obtain the good estimate of actual PAPR value when we use such transmitting pulses.
2. PAPR performance: We estimate the PAPR value from xL, which can be viewed as linear combination of the sampled points of transmitting pulse impulse response.
The power is linear proportional to the absolute value. For convenience, we can sum all the absolute sampled points values, Vs, to observe the tendency of peak power. For case 1, the filter with large useful band bandwidth (wp is 0.5 fs, ws is 1.2 fs) has the large Vs than other two, and its PAPR performance is worse. The phenomenon can be observed in the rest cases. (case 2: wp is 0.8 fs, and ws is 1.2 fs; case 3: wp is 0.8 fs, and ws is 1.5 fs; case 4: wp is 0.8 fs, and ws is 1.5 fs.) The ratio of (
P|p1(nLT)|)2
P|p1(nLT)|2 is noted as γ. We can use γ to estimate the PAPR performance tendency. γ of the four cases are given in Table 4.6, Table 4.7, Table 4.8, and Table 4.8. In Table 4.6, we can find out that γ of filter with ws = 0.7f s is larger than that of filter with ws = 1.0f s by roughly 0.6 dB. γ of the filter with ws = 1.2f s is larger than that of filter with ws = 1.0f s by roughly 7 dB. The result and Figure 4.4(d) are matched. Furthermore, the result of Table 4.7 and Figure 4.8(d), Table 4.8 and Figure 4.12(d), and Table 4.9 and Figure 4.16(d) are respectively matched. Therefore, we can use the ratio γ to estimate the peak-to-average power
case 1 wp : 0.5f s − ws: 0.7f s wp : 0.5f s − ws: 1.0f s wp : 0.5f s − ws: 1.2f s
γ 1.99 2.32 2.75
Table 4.6: γ of case 1
case 2 wp : 0.2f s − ws: 1.8f s wp : 0.5f s − ws: 1.5f s wp : 0.8f s − ws: 1.2f s
γ 2.04 2.15 3.27
Table 4.7: γ of case 2
ratio (PAPR) performance.
3. BER performance: When the useful band bandwidth is large, the BER perfor-mance is better as expected.
case 3 wp : 0.2f s − ws: 1.5f s wp : 0.5f s − ws: 1.5f s wp : 0.8f s − ws: 1.5f s
γ 3.53 4.02 4.11
Table 4.8: γ of case 3
case 4 wp0.2f s − ws0.9f s wp0.5f s − ws1.2f s wp0.8f s − ws1.5f s
γ 1.69 2.38 3.08
Table 4.9: γ of case 4
Chapter 5 Conclusion
In this thesis, we use the digital framework representation OFDM system model to analyze the oversampling factor L, peak to average power ratio (PAPR), and bit error rate (BER) performance. Usually, we ignore the transmitting pulse and receiving pulse of the digital framework OFDM system model. There, we consider the effect of transmitting and receiving filters. We give four realizable elliptic analog filter design cases that based on IEEE 802.11a wireless LANs standard for simulation. Simulation results show that the oversampling factor, PAPR, and BER performance affected by transmitting and receiving the filter design.
For the oversammling factor, we can find out that oversampling factor of 4 is not always giving a good estimate of the actual PAPR. In case 1 and case 2, the oversapmling factor of 16 is needed to obtain a good estimate of the continuous-time PAPR value.
For the peak to average power ratio (PAPR), the simulation results show that the transmitting pulse design affects the PAPR performance. When the useful band bandwidth of the transmitting pulse is large, its PAPR performance is worse. We can see that from Figure 4.4(d) of case 1, Figure 4.8(d) of case 2, Figure 4.12(d) of case 3, and Figure 4.16 of case 4.
For the bit error rate (BER) performance, we can also find out that it affected by transmitting and receiving pulses. The different filter design will result in distinct
equivalent discrete-time channel responses. When the analog filters have large useful band bandwidth, the equivalent discrete-time channel has large magnitude response.
And large magnitude response results in better BER performance.
Bibliography
[1] J.A.C.Blingham, “Multicarrier Modulation For Data Transmission: An Idea Whose Time Has Come,” IEEE Commun. Mag, vol. 28, pp. 5 -14, May. 1990.
[2] Richard van Nee, “OFDM for Wireless Multimedia Communications.”
[3] Tellambura. C, “Computation of the continuous-time PAR of an OFDM signal with BPSK subcarriers,” IEEE Commun.Mag, vol. 5, pp. 185-187, May. 2001.
[4] Yuan-Pei Lin and See-May Phoong, “OFDM transmitters: analog representation and T-based implementation,” IEEE Sig.Proc, vol. 51, pp. 2450-2453, Sept. 2003.
[5] S. A. Hirst, B. Honary, G. Markarian, “Fast Chase algorithm with an application in turbo decoding,” IEEE Trans. Commun., vol. 49, pp. 1693-1699, Oct. 2001.
[6] R.O’Nell and L. B. Lopes, “Envelope Variations and Spectral Splatter in Clipped Multicarrier Signals,” Proc. IEEE PIMRC ’95, Toronto, Canada, pp.71-75, Sept.
1995.
[7] R. W. BAauml, R. F. H. Fisher, and J. B. Huber, “Reducing the Peak-to-Average Power Ratio of Multicarrier Modulation by Selected Mapping,” Elect, Lett, vol.22 no.32 pp.2056-2057, Oct. 1996.
[8] S. H. MAuller and J. B. Huber, “OFDM with Reduced Peak-to-Average Power Ratio by Optimum Combination of Partial Transmit Sequences,” Elect, Lett, vol.22 no.5 pp.368-369, Feb. 1997.
[9] Kun-Wah Yip, Tung-Sang Ng and Yik-Chung Wu, “Impacts of multipath fading on the timing synchronization of IEEE802.11a wireless LANs,” IEEE international Conference, vol 1, pp.517-521, 2002.
[10] Yik-Chung Wu, Kun-Wah Yip, and Tung-Sang Ng,“ML frame synchronization for IEEE 802.11a WLANs on multipath Rayleigh fading channels,” IEEE Circuits and Systems, vol. 2, no.9, pp.25-28, May. 2003.