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1、OFDM基础-外文资料翻译 附录A 外文资料OFDM BasicsINTRODUCTION The basic principle of OFDM is to split a high-rate data stream into a number of lowerrate streams that are transmitted simultaneously over a number of subcarriers. Because the symbol duration increases for the lower rate parallel subcarriers, the relati
2、ve amount of dispersion in time caused by multipath delay spread is decreased. Inter symbol interference is eliminated almost completely by introducing a guard time in every OFDM symbol. In the guard time, the OFDM symbol is cyclically extended to avoid inter carrier interference In OFDM system desi
3、gn, a number of parameters are up for consideration, such as the number of subcarriers, guard time, symbol duration, subcarrier spacing,modulation type per subcarrier, and the type of forward error correction coding. The choice of parameters is influenced by system requirements such as available ban
4、dwidth,required bit rate, tolerable delay spread, and Doppler values. Some requirements are conflicting. For instance, to get a good delay spread tolerance, a large number of subcarriers with a small subcarrier spacing is desirable, but the opposite is true for a good tolerance against Doppler sprea
5、d and phase noiseGENERATION OF SUBCARRIERS USING THE IFFT An OFDM signal consists of a sum of subcarriers that are modulated by using phase shift keying PSK or quadrature amplitude modulation QAM.If di are the complex QAM symbols, N is the number of subcarriers, T is the symbol duration, and f is th
6、e carrier frequency, then one OFDM symbol starting at t t, can be written as2.1 In the literature, often the equivalent complex baseband notation is used, which is given by 2.2. In this representation, the real and imaginary parts correspond to the in-phase and quadrature parts of the OFDM signal, w
7、hich have to be multiplied by a cosine and sine of the desired carrier frequency to produce the final OFDM signal.Figure 2.1 shows the operation of the OFDM modulator in a block diagram. 2.2 Figure 2.1 OFDM modulator As an example,Figure2.2 shows four subcarriers from one OFDM signal. In this exampl
8、e, all subcarriers have the same phase and amplitude, but in practice the amplitudes and phases may be modulated differently for each subcarrier. Note that each subcarrier has exactly an integer number of cycles in the interval T, and the number of cycles between adjacent subcarriers differs by exac
9、tly one. This property accounts forthe orthogonality between the subcarriers. For instance, if the jth subcarrier from 2.2 is demodulated by down converting the signal with a frequency of j/T and then integrating the signal over T seconds, the result is as written in 2.3. By looking at the intermedi
10、ate result, it can be seen that a complex carrier is integrated over T seconds.For the demodulated subcarrier j, this integration gives the desired output multiplied by a constant factor T, which is the QAM value for that particular subcarrier. For all other subcarriers, the integration is zero, bec
11、ause the frequency difference produces an integer number of cycles within the integration interval T,such that the integration result is always zero. 2.3 The orthogonality of the different OFDM subcarriers can also be demonstrated in another way. According to 2.1, each OFDM symbol contains subcarrie
12、rs that are nonzero over a T-second interval. Hence, the spectrum of a single symbol is a convolution of a group of Dirac pulses located at the subcarrier frequencies with the spectrum of a square pulse that is one for a T-second period and zero otherwise. The amplitude spectrum of the square pulse
13、is equal to sincnJT, which has zeros for all frequencies f that are an integer multiple of 1IT. This effect is shown in Figure 2.2,which shows the overlapping sinc spectra of individual subcarriers. At the imum of each subcarrier spectrum, all other subcarrier spectra are zero. Because an OFDM recei
14、ver essentially calculates the spectrum values at those points that correspond to the ima of individual subcarriers, it can demodulate each subcarrier free from any interference from the other subcarriers. Basically, Figure 2.3 shows that the OFDM spectrum fulfills Nyquists criterium for an intersym
15、bol interference free pulse shape.Notice that the pulse shape is present in the frequency domain and not in the time domain, for which the Nyquist criterium usually is applied. Therefore, instead of intersymbol interference ISI, it is intercarrier interference ICI that is avoided by havingthe imum o
16、f one subcarrier spectrum correspond to zero crossings of all the others. Figure 2.2 Example of four subcarriers within one OFDM symbol The complex baseband OFDM signal as defined by 2.2 is in fact nothing more than the inverse Fourier transform of N, QAM input symbols. The time discrete equivalent
17、is the inverse discrete Fourier transform IDFT, which is given by 2.4,where the time t is replaced by a sample number n. In practice, this transform can be implemented very efficiently by the inverse fast Fourier transform IFFT. An N point IDFT requires a total of N complex multiplications-which are
18、 actually only phase rotations. Of course, there are also additions necessary to do an IDFT, but since the hardware complexity of an adder is significantly lower than that of a multiplier or phase rotator, only the multiplications are used here for comparison. The IFFT drastically reduces the amount
19、 of calculations by exploiting the regularity of the operations in the IDFT. Using the radix-2 algorithm, an N-point IFFT requires only N/2.log2N complex multiplications I. For a 16-point transform, for instance, the difference is 256 multiplications for the IDFT versus 32 for the IFFT-a reduction b
20、y a factor of 8!This difference grows for larger numbers of subcarriers, as the IDFT complexity grows quadratically with N, while the IFFT complexity only grows slightly faster than linear. 2.4 The number of multiplications in the JFFT can be reduced even further by using a radix-4 algorithm. This t
21、echnique makes use of the fact that in a four-point IFFT,there are only multiplications by 1,-1 j,-j, which actually do not need to be implemented by a full multiplier, but rather by a simple add or subtract and a switch of real and imaginary parts in the case of multiplications by j or -j. In the r
22、adix-4 algorithm, the transform is split into a number of these trivial four-point transforms,and non-trivial multiplications only have to be performed between stages of these four-point transforms. In this way, an N-point FFT using the radix4 algorithm requires only 3/8Nlog2N-2 complex multiplicati
23、ons or phase rotations and Mog2N complex additions IGUARD TIME AND CYCLIC EXTENSION One of the most important reasons to do OFDM is the efficient way it deals with multipath delay spread. By dividing the input datastream in Ns subcarriers, the symbol duration is made Ns times smaller, which also red
24、uces the relative multipath delay spread, relative to the symbol time; by the same factor. To eliminate intersymbol interference almost completely, a guard time is introduced for each OFDM symbol. The guard time is chosen larger than the expected delay spread, such that multipath components from one
25、 symbol cannot interfere with the next symbol. The guard time could consist of no signal at all. In that case, however, the problem of intercarrier interference ICI would arise. ICI is crosstalk between different subcarriers, which means they are no longer orthogonal. This effect is illustrated in F
26、igure 2.6. In this example, a subcarrier 1 and a delayed subcarrier 2 are shown. When an OFDM receivertries to demodulate the first subcarrier, it will encounter some interference from the second subcarrier, because within the FFT interval, there is no integer number of cycles difference between sub
27、carrier 1 and 2. At the same time, there will be crosstalk from the first to the second subcarrier for the same reason. Figure 2.6 Effect of multipath with zero signal in the guard time; the delayed subcarrier 2 causes ICI on subcarrier 1 and vice versa.CHOICE OF OFDM PARAMETERS The choice of variou
28、s OFDM parameters is a trade off between various, often conflicting requirements. Usually, there are three main requirements to start with:bandwidth, bit rate, and delay spread. The delay spread directly dictates the guard time.As a rule, the guard time should be about two to four times the root-mea
29、n-squared delay spread. This value depends on the type of coding and QAM modulation. Higher order QAM like 64-QAM is more sensitive to ICI and IS1 than QPSK, while heavier coding obviously reduces the sensitivity to such interference. Now that the guard time has been set, the symbol duration can be
30、fixed. To minimize the signal-to-noise ratio SNR loss caused by the guard time, it is desirable to have the symbol duration much larger than the guard time. It cannot be arbitrarily large, however, because a larger symbol duration means more subcarriers with a smaller subcarrier spacing, a larger im
31、plementation complexity, and more sensitivity to phase noise and frequency offset 2, as well as an increased peak-to-average power ratio 3,4. Hence, a practical design choice is to make the symbol duration at least five times the guard time, which implies a 1-dB SNR loss because of the guard time. F
32、orm:OFDM for Wireless Multimedia CommunicationsOFDM基础介绍 OFDM的基本原理是将一串高速数据流变成同时传输在一些副载波的低速率数据流。由于低速率平行的副载波是符号持续时间增加,因而对多径效应引起的时延扩展有较强的抵抗力。符号间干扰可以通过在每个OFDM符号前引入一个保护间隔来完全消除。加入保护间隔后,OFDM符号通过周期性扩展来避免载波间干扰。 在OFDM系统设计中,大量的参数需要考虑,比如副载波的数量,保护间隔,持续时间,副载波间距,每一个副载波的调制类型,前项纠错编码的类型,参数的选择是受系统要求的如,可用带宽,需要的比特率,可容忍的延
33、时时间和多普勒扩散值。但是有些要求是相互矛盾的。例如,为了得到一个好的延迟扩展公差,大部分副载波用一个小副载波的间距是可取的,但事实恰恰相反,一个好的公差不利于多普勒扩散和相位噪声。用IFFT方法调制副载波 一个OFDM信号是利用相移键控调制相移键控或正交振幅调制组成的副载波之和,如果di是合成的QAM符号,N是子载波的数量,T是符号周期,f是载波频率,则一个OFDM符号从时刻开始,可以写成(2.1) 在文献中,等效的基带符号经常被写成式2.2。在这个式子中,实部与虚部分别对应于OFDM信号的同相与正交部分,需要乘以一个余弦和正弦所需的载波频率生成最终的OFDM信号。图2.1表示了OFDM调制
34、器的运算框图。 2.2图2.1 OFDM调制 举个例子来说,在图2.2中,显示了4个子载波的实例,在这个例子中,所有子载波都有相同的相位和振幅,但实际中,每个子载波调制振幅与相位可能不同。需要注意的是,每个子载波正好有一个整数数量的周期间隔T,周期数相邻的子载波刚好有一个不同。这个属性正好符合子载波之间的正交性方案。例如,假设式2.2的第j个子载波是通过频率的j / T将信号解调下来的,然后将信号在T秒内积分,结果就可以用式2.3表示。通过查看这个结果,可以看出,一个复杂的载体在T秒内积分。对于解调的子载波j,这种积分提供了所需的输出乘以一个常数因子T,这种特定的子载波就是QAM的优点。对于所
35、有其他的子载波,积分是零,因为频率差异在积分间隔T内产生了一个整数的周期数,所以积分结果总是零。 2.3 图2.2 一个OFDM信号的四个子载波例子 OFDM的不同子载波的正交性可以用另一种方法表示。根据式2.1,每个OFDM符号在其周期T内包括多个非零的子载波。因此其频谱可以看作是周期为T的矩形脉冲的频谱与一组位于各个子载波频率上的单位冲激函数的卷积。矩形脉冲的频谱幅值为函数,这种函数的零点出现在频率为1/T整数倍的位置上。这种现象可以参见图2.2,图中给出了相互覆盖的各个子信道内经过矩形波形成型得到的符号的函数频谱。在每个子载波频率最大值处,所有其他子信道的频谱值恰好为零。由于在对OFDM
36、符号进行解调的过程中,需要计算这些点上所对应的每个子载波频率的最大值,因此可以从多个相互重叠的子信道符号中提取每一个信道符号,而不会受到其它子信道的干扰。图2.3基本上就可以显示OFDM满足奈奎斯特的标准,即多个子信道频谱之间不存在相互干扰。需要注意的是,脉冲形状出现在频域而不是时域,但是奈奎斯特标准也是适用的。因此这种一个子信道频谱出现最大值而其它子信道频谱为零的特点可以避免载波间干扰(ICI),而不是符号间干扰(ISI)。 对于输入QAM信号N比较大的系统来说,式2.2中的OFDM等效基带信号可以采用傅立叶逆变换来实现。时间离散等效于逆离散傅里叶变换IDFT,它用式2.4表示,其中的时间t
37、替换为采样数n。然而在实际中,这种变换可以用非常有效的快速傅里叶逆变换来实现。一个N点IDFT需要共次复杂的乘法,但实际上只是相位旋转。当然,由于系统的需要,有必要增加IDFT点数,但是硬件复杂度一个加法器是明显低于一个乘数或相位旋转,这里只用复数乘法的运算量来比较。IFFT利用的规律性操作IDFT使计算量大大降低了。对于常用的基2 IFFT 算法来说,其复数乘法的次数仅为。对于一个涵盖面很广的16点变换,例如,不同的是IDFT需要256次乘法和而IFFT只需要32次乘法,是IDFT的,这种差异对于大量的子载波来说,IDFT的计算复杂度会随N的增加而呈现二次方增长,IFFT的计算复杂度的增加速
38、度只是稍稍快于线性变化。 2.4 对于子载波数量非常大的OFDM系统来说,可以进一步采用基4 IFFT算法。在4点的IFFT运算中,只存在与1,-1,j,-j的相乘运算,这实际上不需要采用完整的乘法器来实施这种乘法,而是只需要通过简单的加、减以及交换实部和虚部的运算(当与-j,j相乘时)来实现这种乘法。在基4算法中,IFFT变换可以被分为多个4点的IFFT变换,这样就只需要在两个级别之间执行完整的乘法操作。一个N点的基4 IFFT使用算法中只需要执行次复数乘法或相位旋转,以及次复数加法。保护间隔和循环扩展 应用OFDM的一个最重要是它可以有效的对抗多路延迟传播。通过把输入数据流串并变换到Ns个
39、并行的子信道中,符号持续时间较小,相对于符号时间这也相对降低了多路延迟传播的时间,为了最大限度的消除符号间干扰,还可以在每个OFDM符号间插入保护间隔。而且该保护间隔长度一般要大于无线信道中的最大时延扩展,这样一个符号的多径分量就不会对下一个符号造成干扰。在这段保护间隔内,可以不插入任何信号,即是一段空闲的传输时段,然而,在这种情况下,载波间干扰的问题ICI将会出现。即子载波之间的正交性遭到破坏,不同的子载波之间产生干扰,这种效果如图2.6。在这个例子中,给出了第一子载波和第二子载波的时延信号。由于在FFT运算时间长度内,第一子载波与带有时延的第二子载波之间的周期个数只差不再是整数,所以当接收
40、机试图对第一子载波进行解调时,第二子载波会对此造成干扰。同时,当接收机对第二子载波进行解调时,也会来自第一子载波的干扰。OFDM的参数选择 选择各种OFDM的参数是需要权衡各种相互冲突。通常,有3个主要的参数:带宽、比特率、保护间隔。延迟传播直接决定了保护间隔。一般来说,保护间隔的时间长度应该为应用移动环境信道的时延均方根值的24倍。这个值取决于编码类型和QAM调制。高阶QAM如64-QAM对ICI和ISI比QPSK更为敏感,而译码明显减少这种干扰的敏感性。 一旦确定了保护间隔,则OFDM符号周期长度就可以确定。为了最大限度的减少由于插入保护间隔所带来的信噪比的损失,理论上需要OFDM符号周期长度可以远远大于保护间隔长度。但是符号周期长度又不可能任意大,否则OFDM系统中包括更多的子载波数,从而导致子载波间隔相应减少,系统的实现复杂度增加,而且还加大了系统的峰值平均功率比,同时使系统对频率偏差更加敏感。因此在实际应用中,一般选择符号周期是保护间隔长度的5倍,这样由于插入保护比特所造成的信噪比损耗只有1dB左右。 源于:OFDM系统的无线多媒体通信
