In this chapter we will see more simulation results. The block diagram of the OFDM system using SPR method is shown in Fig. 6.1. The IDFT block at the transmitter is put in the SPR method block in Fig. 4.2 and Fig. 4.5. The input symbols have the same variance εs because there is usually no bit/power allocation in the OFDM system. The symbols are assumed to be zero mean and uncorrelated, which is usually a reasonable assumption after proper interleaving of the input bit stream. The autocorrelation matrix of the input vector s is thus Rs = εsIM, where IM is a M × M identity matrix. Then the output vector of IDFT matrix x = W†s has autocorrelation matrix Rx = E[xx†] given by:
Rx = W†RsW = εsIM. (6.1) Therefore x are uncorrelated and their variances are the same, equal to εs.
Fig. 6.2 shows the clipping with a maximum permissible amplitude A. The operation of clipping a real number x can be written as:
y = f (x) =
½ x, |x| ≤ A
A, |x| > A (6.2)
16QAM SPR P/S clipping CP
channel
remove S/P CP
demod DFT
noise bit
stream
s x y
Figure 6.1: The OFDM system with SPR method.
- A A
Figure 6.2: A clipping with maximum amplitude A.
In Fig. 6.1, there is a clipping block before the channel. The n-th element of the transmitted signal y can be written as
yn = yr,n+ jyi,n, (6.3)
where yr,n and yi,n are the real and imaginary part of yn respectively. The am-plitude clipping operates on yr,n and yi,n individually. The output of clipper becomes f (yr,n) + jf (yi,n). The clipping ratio γ is defined as
γ = Amax
√P0, (6.4)
where Amax =√
A2 + A2 =√ 2A.
Here we use the commonly decision rule which called nearest neighbor decision
constellation points become smaller after modulo-D process and hence the bit error rate (BER) may be increased slightly.
The BER depends on the signal power and the noise power. The BER curves are plotted against the signal-to-noise (SNR). But here we will plot the BER curves of average-transmission-power-to-noise ratio (P0/N0), where P0 is the av-erage transmission power of y before clipping and N0 be the noise variance of AWGN channel.
We will see some examples in the following. In our simulations, the IDFT size is 64 and total 105 random OFDM blocks are used in the simulation. The input symbols are 16 QAM. P0 is the same as εs. The SPR method uses 8 PR subchannels: {0, 7, 15, 23, 31, 39, 47, 56} and the rotation factors are selected from {+1, −1}. a = 1 in the tone injection method. We selected 8 PR subchannels and SPR-ES will lose 8 bits for a block transmission.
Fixed noise variance (N0 = 1). The noise variance N0 is fixed to 1. The BER of SPR-ES and tone injection are in Fig. 6.3. We can see under the same noise variance, SPR-ES has better BER performance. When BER is fixed to 10−3, P0 are 16.5 dB and 17.3 dB for SPR-ES and tone injection. This is meant that the average transmission power are 44.67 and 53.7 respectively. The average transmission power of SPR-ES is 20% smaller than tone injection. However, SPR-ES used 8 PR subchannels and lost 8 bits when 16 QAM constellation is used. This is equivalent to a loss of 3% in transmission rate. Under the same noise variance and BER, SPR-ES needs smaller average transmission power than tone injection. The CCDF of PAPR of both methods are in Fig. 6.4. For large PAPR0 SPR-ES has better performance than tone injection.
5 10 15 20 10−3
10−2 10−1 100
P0/N0 (dB)
Probability of error
SPR−ES
Tone Injection [6]
Figure 6.3: BER. N0=1. SPR-ES uses 8 PR subchannels and rotation factors are selected from {+1, −1} and tone injection.
4 6 8 10 12
10−3 10−2 10−1 100
PAPR0 (dB)
Pr(PAPR > PAPR0)
SPR−ES
Tone Injection [6]
Figure 6.4: PAPR. N0=1, BER=10−3. SPR-ES uses 8 PR subchannels and rotation factors are selected from {+1, −1} and tone injection.
Fixed average transmission power. (P0 = 10). In this example, we fixed the average transmission power P0 to 10. When fixed P0, the minimum distance of the constellation used the tone injection method is smaller than that of SPR-ES. The minimum distance of SPR-ES and tone injection are 2 and 1.88 respectively. We can see SPR-ES has better BER performance in Fig. 6.5. For a BER of 10−3, SPR-ES is 1 dB less than tone injection.
5 10 15 20
10−3 10−2 10−1 100
P0/N
0 (dB)
Probability of error
SPR−ES
Tone Injection [6]
Figure 6.5: BER. Fixed average transmission power. P0 = 10. SPR-ES uses 8 PR subchannels and rotation factors are selected from {+1, −1} and tone injection.
Fixed average transmission power and the clipping ratio. (P0 = 10, γ = 2). In the following two examples we add a clipping operator before the signal passing through the channel. The BER performance after adding the clipping is worse, especially for the smaller clipping ratio γ. If P0 and γ are fixed, the peak value can be decided. When γ = 2 and P0 = 10 we can get Amax = 6.32 and the corresponding A = 4.47. The peak value of the two methods are all equal to (4.47)2 + (4.47)2 = 39.96. Under the same peak value and P0, SPR-ES has slightly better BER performance.
5 10 15 20
10−3 10−2 10−1 100
P0/N
0 (dB)
Probability of error
SPR−ES
Tone Injection [6]
Figure 6.6: BER. Fixed average transmission power and the clipping ratio. (P0 = 10, γ = 2). SPR-ES uses 8 PR subchannels and rotation factors are selected from {+1, −1} and tone injection.
Fixed average transmission power and the clipping ratio. (P0 = 10, γ = 3). In this example, γ = 3 and P0 = 10. Amax = 9.49 and the corresponding A = 6.71. The peak value of the two methods are all equal to (6.71)2 + (6.71)2 = 90.05. SPR-ES in this example still has better performance than the tone injection method.
5 10 15 20
10−3 10−2 10−1 100
P0/N
0 (dB)
Probability of error
SPR−ES
Tone Injection [6]
Figure 6.7: BER. Fixed average transmission power and the clipping ratio. (P0 = 10, γ = 3). SPR-ES uses 8 PR subchannels and rotation factors are selected from {+1, −1} and tone injection.
Chapter 7 Conclusion
In this thesis, we proposed a new method called SPR. SPR method reduces PAPR by multiplying rotations to the symbols of the PR subchannels individu-ally. Two schemes of SPR method were introduced. Simulation results showed both shcemes can reduce at least half dB from the original system with no PAPR reduction method when only two PR subchannels were selected. The average transmission power of SPR method remains the same. A few bits were lost on the PR subchannels but no side information is needed. Although the computa-tional complexity increases, the numbers of complex multiplications and complex additions are few compared with tone injection method.
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