OFDM同步算法之SC算法_算法_matlab_其他

<html xmlns="http://www.w3.org/1999/xhtml"><head><meta charset="utf-8"><meta name="generator" content="pdf2htmlEX"><meta http-equiv="X-UA-Compatible" content="IE=edge,chrome=1"><link rel="stylesheet" href="https://csdnimg.cn/release/download_crawler_static/css/base.min.css"><link rel="stylesheet" href="https://csdnimg.cn/release/download_crawler_static/css/fancy.min.css"><link rel="stylesheet" href="https://csdnimg.cn/release/download_crawler_static/12190445/raw.css"><script src="https://csdnimg.cn/release/download_crawler_static/js/compatibility.min.js"></script><script src="https://csdnimg.cn/release/download_crawler_static/js/pdf2htmlEX.min.js"></script><script>try{pdf2htmlEX.defaultViewer = new pdf2htmlEX.Viewer({});}catch(e){}</script><title></title></head><body><div id="sidebar" style="display: none"><div id="outline"></div></div><div id="pf1" class="pf w0 h0" data-page-no="1"><div class="pc pc1 w0 h0"><img class="bi x0 y0 w1 h1" alt="" src="https://csdnimg.cn/release/download_crawler_static/12190445/bg1.jpg"><div class="t m0 x1 h2 y1 ff1 fs0 fc0 sc0 ls0 ws0">IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 45, NO. 12, DECEMBER 1997 1613</div><div class="t m0 x2 h3 y2 ff1 fs1 fc0 sc0 ls0 ws0">Robust Frequency and Timing</div><div class="t m0 x3 h3 y3 ff1 fs1 fc0 sc0 ls0 ws0">Synchronization for OFDM</div><div class="t m0 x4 h4 y4 ff1 fs2 fc0 sc0 ls0 ws0">Timothy M. Schmidl and Donald C. Cox, Fellow, IEEE</div><div class="t m0 x5 h5 y5 ff3 fs4 fc0 sc0 ls0 ws0">Abstract— A rapid synchronization method is presented for</div><div class="t m0 x1 h5 y6 ff4 fs4 fc0 sc0 ls0 ws0">an orthogonal frequency-division multiplexing (OFDM) system</div><div class="t m0 x1 h5 y7 ff4 fs4 fc0 sc0 ls0 ws0">using either a continuous transmission or a burst operation over</div><div class="t m0 x1 h5 y8 ff4 fs4 fc0 sc0 ls0 ws0">a frequency-selective channel. The presence of a signal can be</div><div class="t m0 x1 h5 y9 ff4 fs4 fc0 sc0 ls0 ws0">detected upon the receipt of just one training sequence of two</div><div class="t m0 x1 h5 ya ff4 fs4 fc0 sc0 ls0 ws0">symbols. The start of the frame and the beginning of the symbol</div><div class="t m0 x1 h5 yb ff4 fs4 fc0 sc0 ls0 ws0">can be found, and carrier frequency offsets of many subchannels</div><div class="t m0 x1 h5 yc ff4 fs4 fc0 sc0 ls0 ws0">spacings can be corrected. The algorithms operate near the</div><div class="t m0 x1 h5 yd ff4 fs4 fc0 sc0 ls0 ws0">Cram</div><div class="t m0 x6 h5 ye ff4 fs4 fc0 sc0 ls0 ws0">´</div><div class="t m0 x6 h5 yd ff4 fs4 fc0 sc0 ls0 ws0">er–Rao lower bound for the variance of the frequency</div><div class="t m0 x1 h5 yf ff4 fs4 fc0 sc0 ls0 ws0">offset estimate, and the inherent averaging over many subcarriers</div><div class="t m0 x1 h5 y10 ff4 fs4 fc0 sc0 ls0 ws0">allows acquisition at very low signal-to-noise ratios (SNR’s).</div><div class="t m0 x5 h5 y11 ff3 fs4 fc0 sc0 ls0 ws0">Index Terms— Carrier frequency, orthogonal frequency-</div><div class="t m0 x1 h5 y12 ff4 fs4 fc0 sc0 ls0 ws0">division multiplexing, symbol timing estimation.</div><div class="t m0 x7 h6 y13 ff1 fs3 fc0 sc0 ls0 ws0">I. INTRODUCTION</div><div class="t m0 x1 h7 y14 ff4 fs6 fc0 sc0 ls0 ws0">I</div><div class="t m0 x8 h6 y15 ff1 fs3 fc0 sc0 ls0 ws0">N AN orthogonal frequency-division multiplexing (OFDM)</div><div class="t m0 x8 h6 y16 ff1 fs3 fc0 sc0 ls0 ws0">system, synchronization at the receiver is one important</div><div class="t m0 x1 h6 y17 ff1 fs3 fc0 sc0 ls0 ws0">step that must be performed. This paper describes a method</div><div class="t m0 x1 h6 y18 ff1 fs3 fc0 sc0 ls0 ws0">to acquire synchronization for either a continuous stream of</div><div class="t m0 x1 h6 y19 ff1 fs3 fc0 sc0 ls0 ws0">data as in a broadcast application or for bursty data as in</div><div class="t m0 x1 h6 y1a ff1 fs3 fc0 sc0 ls0 ws0">a wireless local area network (WLAN). In both cases the</div><div class="t m0 x1 h6 y1b ff1 fs3 fc0 sc0 ls0 ws0">receiver must continuously scan for incoming data, and rapid</div><div class="t m0 x1 h6 y1c ff1 fs3 fc0 sc0 ls0 ws0">acquisition is needed. The ratio of the number of overhead bits</div><div class="t m0 x1 h6 y1d ff1 fs3 fc0 sc0 ls0 ws0">for synchronization to the number of message bits must be kept</div><div class="t m0 x1 h6 y1e ff1 fs3 fc0 sc0 ls0 ws0">to a minimum, and low-complexity algorithms are needed.</div><div class="t m0 x9 h6 y1f ff1 fs3 fc0 sc0 ls0 ws0">Synchronization of an OFDM signal requires ﬁnding the</div><div class="t m0 x1 h6 y20 ff1 fs3 fc0 sc0 ls0 ws0">symbol timing and carrier frequency offset. Symbol timing</div><div class="t m0 x1 h6 y21 ff1 fs3 fc0 sc0 ls0 ws0">for an OFDM signal is signiﬁcantly different than for a single</div><div class="t m0 x1 h6 y22 ff1 fs3 fc0 sc0 ls0 ws0">carrier signal since there is not an “eye opening” where a</div><div class="t m0 x1 h6 y23 ff1 fs3 fc0 sc0 ls0 ws0">best sampling time can be found. Rather there are hundreds or</div><div class="t m0 x1 h6 y24 ff1 fs3 fc0 sc0 ls0 ws0">thousands of samples per OFDM symbol since the number of</div><div class="t m0 x1 h6 y25 ff1 fs3 fc0 sc0 ls0 ws0">samples necessary is proportional to the number of subcarriers.</div><div class="t m0 x1 h6 y26 ff1 fs3 fc0 sc0 ls0 ws0">Finding the symbol timing for OFDM means ﬁnding an</div><div class="t m0 x1 h6 y27 ff1 fs3 fc0 sc0 ls0 ws0">estimate of where the symbol starts. There is usually some</div><div class="t m0 x1 h6 y28 ff1 fs3 fc0 sc0 ls0 ws0">tolerance for symbol timing errors when a cyclic preﬁx is</div><div class="t m0 x1 h6 y29 ff1 fs3 fc0 sc0 ls0 ws0">used to extend the symbol. Synchronization of the carrier</div><div class="t m0 x1 h6 y2a ff1 fs3 fc0 sc0 ls0 ws0">frequency at the receiver must be performed very accurately,</div><div class="t m0 x1 h6 y2b ff1 fs3 fc0 sc0 ls0 ws0">or there will be loss of orthogonality between the subsymbols.</div><div class="t m0 x1 h6 y2c ff1 fs3 fc0 sc0 ls0 ws0">OFDM systems are very sensitive to carrier frequency offsets</div><div class="t m0 x1 h6 y2d ff1 fs3 fc0 sc0 ls0 ws0">since they can only tolerate offsets which are a fraction of the</div><div class="t m0 xa h8 y2e ff1 fs5 fc0 sc0 ls0 ws0">Paper approved by M. Luise, the Editor for Synchronization of the</div><div class="t m0 x1 h8 y2f ff1 fs5 fc0 sc0 ls0 ws0">IEEE Communications Society. Manuscript received April 16, 1996; revised</div><div class="t m0 x1 h8 y30 ff1 fs5 fc0 sc0 ls0 ws0">February 11, 1997. This work was supported in part by a National Science</div><div class="t m0 x1 h8 y31 ff1 fs5 fc0 sc0 ls0 ws0">Foundation Graduate Fellowship. This work was presented in part at the IEEE</div><div class="t m0 x1 h8 y32 ff1 fs5 fc0 sc0 ls0 ws0">International Conference on Communications (ICC), Dallas, TX, June 1996.</div><div class="t m0 xa h8 y33 ff1 fs5 fc0 sc0 ls0 ws0">T. M. Schmidl is with DSP Research and Development Center at Texas</div><div class="t m0 x1 h8 y34 ff1 fs5 fc0 sc0 ls0 ws0">Instruments Incorporated, Dallas, TX 75243 USA (e-mail: schmidl@ti.com).</div><div class="t m0 xa h8 y35 ff1 fs5 fc0 sc0 ls0 ws0">D. C. Cox is with the STAR Laboratory, Department of Electrical</div><div class="t m0 x1 h8 y36 ff1 fs5 fc0 sc0 ls0 ws0">Engineering, Stanford University, Stanford, CA 94305-4055 USA (e-mail:</div><div class="t m0 x1 h8 y37 ff1 fs5 fc0 sc0 ls0 ws0">dcox@nova.stanford.edu).</div><div class="t m0 xa h8 y38 ff1 fs5 fc0 sc0 ls0 ws0">Publisher Item Identiﬁer S 0090-6778(97)09083-1.</div><div class="t m0 xb h6 y5 ff1 fs3 fc0 sc0 ls0 ws0">spacing between the subcarriers without a large degradation in</div><div class="t m0 xb h6 y39 ff1 fs3 fc0 sc0 ls0 ws0">system performance [1].</div><div class="t m0 xc h6 y3a ff1 fs3 fc0 sc0 ls0 ws0">There have been several papers on the subject of synchro-</div><div class="t m0 xb h6 y3b ff1 fs3 fc0 sc0 ls0 ws0">nization for OFDM in recent years. Moose gives the maximum</div><div class="t m0 xb h6 y3c ff1 fs3 fc0 sc0 ls0 ws0">likelihood estimator for the carrier frequency offset which is</div><div class="t m0 xb h6 y3d ff1 fs3 fc0 sc0 ls0 ws0">calculated in the frequency domain after taking the FFT [2].</div><div class="t m0 xb h6 y3e ff1 fs3 fc0 sc0 ls0 ws0">He assumes that the symbol timing is known, so he just has to</div><div class="t m0 xb h6 y3f ff1 fs3 fc0 sc0 ls0 ws0">ﬁnd the carrier frequency offset. The limit of the acquisition</div><div class="t m0 xb h6 y40 ff1 fs3 fc0 sc0 ls0 ws0">range for the carrier frequency offset is</div><div class="t m0 xd h6 y41 ff1 fs3 fc0 sc0 ls0 ws0">the subcarrier</div><div class="t m0 xb h6 y42 ff1 fs3 fc0 sc0 ls0 ws0">spacing. He also describes how to increase this range by using</div><div class="t m0 xb h6 y43 ff1 fs3 fc0 sc0 ls0 ws0">shorter training symbols to ﬁnd the carrier frequency offset.</div><div class="t m0 xb h6 y44 ff1 fs3 fc0 sc0 ls0 ws0">For example shortening the training symbols by a factor of</div><div class="t m0 xb h6 y45 ff1 fs3 fc0 sc0 ls0 ws0">two would double the range of carrier frequency acquisition.</div><div class="t m0 xb h6 y46 ff1 fs3 fc0 sc0 ls0 ws0">This approach will work to a point, but the estimates get</div><div class="t m0 xb h6 y47 ff1 fs3 fc0 sc0 ls0 ws0">worse as the symbols get shorter because there are fewer</div><div class="t m0 xb h6 y48 ff1 fs3 fc0 sc0 ls0 ws0">samples over which to average, and the training symbols need</div><div class="t m0 xb h6 y49 ff1 fs3 fc0 sc0 ls0 ws0">to be kept longer than the guard interval so that the channel</div><div class="t m0 xb h6 y4a ff1 fs3 fc0 sc0 ls0 ws0">impulse response does not cause distortion when estimating</div><div class="t m0 xb h6 y4b ff1 fs3 fc0 sc0 ls0 ws0">the frequency offset.</div><div class="t m0 xc h6 y4c ff1 fs3 fc0 sc0 ls0 ws0">Nogami and Nagashima [4] present algorithms to ﬁnd the</div><div class="t m0 xb h6 y4d ff1 fs3 fc0 sc0 ls0 ws0">carrier frequency offset and sampling rate offset. They use</div><div class="t m0 xb h6 y4e ff1 fs3 fc0 sc0 ls0 ws0">a null symbol where nothing is transmitted for one symbol</div><div class="t m0 xb h6 y4f ff1 fs3 fc0 sc0 ls0 ws0">period so that the drop in received power can be detected to</div><div class="t m0 xb h6 y50 ff1 fs3 fc0 sc0 ls0 ws0">ﬁnd the beginning of the frame. The carrier frequency offset</div><div class="t m0 xb h6 y51 ff1 fs3 fc0 sc0 ls0 ws0">is found in the frequency domain after applying a Hanning</div><div class="t m0 xb h6 y52 ff1 fs3 fc0 sc0 ls0 ws0">window and taking the FFT. The null symbol is also used in</div><div class="t m0 xb h6 y53 ff1 fs3 fc0 sc0 ls0 ws0">[11]. This extra overhead of using a null symbol is avoided by</div><div class="t m0 xb h6 y54 ff1 fs3 fc0 sc0 ls0 ws0">using the technique described in this paper. If instead of a con-</div><div class="t m0 xb h6 y55 ff1 fs3 fc0 sc0 ls0 ws0">tinuous transmission mode, a burst mode is used, it would be</div><div class="t m0 xb h6 y56 ff1 fs3 fc0 sc0 ls0 ws0">difﬁcult to use a null symbol since there would be no difference</div><div class="t m0 xb h6 y57 ff1 fs3 fc0 sc0 ls0 ws0">between the null symbol and the idle period between bursts.</div><div class="t m0 xc h6 y58 ff1 fs3 fc0 sc0 ls0 ws0">Van de Beek [3] describes a method of using a correlation</div><div class="t m0 xb h6 y59 ff1 fs3 fc0 sc0 ls0 ws0">with the cyclic preﬁx to ﬁnd the symbol timing. If this method</div><div class="t m0 xb h6 y5a ff1 fs3 fc0 sc0 ls0 ws0">were used to ﬁnd the symbol timing, while using one of the</div><div class="t m0 xb h6 y5b ff1 fs3 fc0 sc0 ls0 ws0">previous methods to ﬁnd the carrier frequency offset, there</div><div class="t m0 xb h6 y5c ff1 fs3 fc0 sc0 ls0 ws0">would still be a problem of ﬁnding the start of the frame to</div><div class="t m0 xb h6 y5d ff1 fs3 fc0 sc0 ls0 ws0">know where the training symbols are located.</div><div class="t m0 xc h6 y5e ff1 fs3 fc0 sc0 ls0 ws0">Classen introduces a method which jointly ﬁnds both the</div><div class="t m0 xb h6 y5f ff1 fs3 fc0 sc0 ls0 ws0">symbol timing and carrier frequency offset [5]. However, it</div><div class="t m0 xb h6 y60 ff1 fs3 fc0 sc0 ls0 ws0">is very computationally complex because it uses a trial and</div><div class="t m0 xb h6 y61 ff1 fs3 fc0 sc0 ls0 ws0">error method where the carrier frequency is incremented in</div><div class="t m0 xb h6 y62 ff1 fs3 fc0 sc0 ls0 ws0">small steps over the entire acquisition range until the correct</div><div class="t m0 xb h6 y63 ff1 fs3 fc0 sc0 ls0 ws0">carrier frequency is found. It is impractical to do the exhaustive</div><div class="t m0 xb h6 y64 ff1 fs3 fc0 sc0 ls0 ws0">search and go through a large amount of computation at each</div><div class="t m0 xb h6 y65 ff1 fs3 fc0 sc0 ls0 ws0">possible carrier frequency offset.</div><div class="t m0 xc h6 y66 ff1 fs3 fc0 sc0 ls0 ws0">This paper introduces some modiﬁcations of Classen’s</div><div class="t m0 xb h6 y67 ff1 fs3 fc0 sc0 ls0 ws0">method which both greatly simplify the computation necessary</div><div class="t m0 xb h6 y68 ff1 fs3 fc0 sc0 ls0 ws0">for synchronization and extend the range for the acquisition</div><div class="t m0 xe h9 y69 ff1 fs5 fc0 sc0 ls0 ws0">0090–6778/97$10.00  1997 IEEE</div></div><div class="pi" data-data='{"ctm":[1.568627,0.000000,0.000000,1.568627,0.000000,0.000000]}'></div></div></body></html>

<div id="pf2" class="pf w0 h0" data-page-no="2"><div class="pc pc2 w0 h0"><img class="bi x0 y0 w1 h1" alt="" src="https://csdnimg.cn/release/download_crawler_static/12190445/bg2.jpg"><div class="t m0 x1 h2 y1 ff1 fs0 fc0 sc0 ls0 ws0">1614 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 45, NO. 12, DECEMBER 1997</div><div class="t m0 x1 h8 y6a ff1 fs5 fc0 sc0 ls0 ws0">Fig. 1. Block diagram of OFDM transmitter.</div><div class="t m0 x1 h8 y6b ff1 fs5 fc0 sc0 ls0 ws0">Fig. 2. Block diagram of OFDM receiver.</div><div class="t m0 x1 h6 y6c ff1 fs3 fc0 sc0 ls0 ws0">of carrier frequency offset. The method in this paper avoids</div><div class="t m0 x1 h6 y6d ff1 fs3 fc0 sc0 ls0 ws0">the extra overhead of using a null symbol, while allowing</div><div class="t m0 x1 h6 y6e ff1 fs3 fc0 sc0 ls0 ws0">a large acquisition range for the carrier frequency offset. By</div><div class="t m0 x1 h6 y6f ff1 fs3 fc0 sc0 ls0 ws0">using one unique symbol which has a repetition within half a</div><div class="t m0 x1 h6 y70 ff1 fs3 fc0 sc0 ls0 ws0">symbol period, this method can be used for bursts of data to</div><div class="t m0 x1 h6 y71 ff1 fs3 fc0 sc0 ls0 ws0">ﬁnd whether a burst is present and to ﬁnd the start of the burst.</div><div class="t m0 x9 h6 y72 ff1 fs3 fc0 sc0 ls0 ws0">Acquisition is achieved in two separate steps through the</div><div class="t m0 x1 h6 y73 ff1 fs3 fc0 sc0 ls0 ws0">use of a two-symbol training sequence, which will usually</div><div class="t m0 x1 h6 y74 ff1 fs3 fc0 sc0 ls0 ws0">be placed at the start of the frame. First the symbol/frame</div><div class="t m0 x1 h6 y75 ff1 fs3 fc0 sc0 ls0 ws0">timing is found by searching for a symbol in which the</div><div class="t m0 x1 h6 y76 ff1 fs3 fc0 sc0 ls0 ws0">ﬁrst half is identical to the second half in the time domain.</div><div class="t m0 x1 h6 y77 ff1 fs3 fc0 sc0 ls0 ws0">Then the carrier frequency offset is partially corrected, and</div><div class="t m0 x1 h6 y78 ff1 fs3 fc0 sc0 ls0 ws0">a correlation with a second symbol is performed to ﬁnd</div><div class="t m0 x1 h6 y79 ff1 fs3 fc0 sc0 ls0 ws0">the carrier frequency offset.</div><div class="t m0 xf h6 y7a ff1 fs3 fc0 sc0 ls0 ws0">II. OFDM P</div><div class="t m0 x10 h8 y7b ff1 fs5 fc0 sc0 ls0 ws0">RINCIPLES</div><div class="t m0 x9 h6 y7c ff1 fs3 fc0 sc0 ls0 ws0">The OFDM signal is generated at baseband by taking the</div><div class="t m0 x1 h6 y7d ff1 fs3 fc0 sc0 ls0 ws0">inverse fast Fourier transform (IFFT) of quadrature amplitude</div><div class="t m0 x1 h6 y7e ff1 fs3 fc0 sc0 ls0 ws0">modulated (QAM) or phase-shift keyed (PSK) subsymbols</div><div class="t m0 x11 h6 y7f ff1 fs3 fc0 sc0 ls0 ws0">(Fig. 1). In the ﬁgure, the block P/S represents</div><div class="t m0 x1 h6 y80 ff1 fs3 fc0 sc0 ls0 ws0">a parallel-to-serial converter. An OFDM symbol has a useful</div><div class="t m0 x1 h6 y81 ff1 fs3 fc0 sc0 ls0 ws0">period</div><div class="t m0 x12 h6 y82 ff1 fs3 fc0 sc0 ls0 ws0">and preceding each symbol is a cyclic preﬁx of length</div><div class="t m0 x9 h6 y83 ff1 fs3 fc0 sc0 ls0 ws0">, which is longer than the channel impulse response so</div><div class="t m0 x1 h6 y84 ff1 fs3 fc0 sc0 ls0 ws0">that there will be no intersymbol interference (ISI) [6]. The</div><div class="t m0 x1 h6 y85 ff1 fs3 fc0 sc0 ls0 ws0">frequencies of the complex exponentials are</div><div class="t m0 x13 h6 y86 ff1 fs3 fc0 sc0 ls0 ws0">, and</div><div class="t m0 x1 h6 y87 ff1 fs3 fc0 sc0 ls0 ws0">the useful part for</div><div class="t m0 x14 h6 y88 ff1 fs3 fc0 sc0 ls0 ws0">subcarriers is given by</div><div class="t m0 x15 h6 y89 ff1 fs3 fc0 sc0 ls0 ws0">(1)</div><div class="t m0 x1 h6 y8a ff1 fs3 fc0 sc0 ls0 ws0">The baseband signal is quadrature modulated, up-converted to</div><div class="t m0 x1 h6 y8b ff1 fs3 fc0 sc0 ls0 ws0">the radio frequency (RF) and transmitted through the channel.</div><div class="t m0 x9 h6 y8c ff1 fs3 fc0 sc0 ls0 ws0">At the receiver (Fig. 2), the signal is down-converted to an</div><div class="t m0 x1 h6 y8d ff1 fs3 fc0 sc0 ls0 ws0">intermediate frequency (IF), and quadrature demodulated. The</div><div class="t m0 x1 h6 y8e ff1 fs3 fc0 sc0 ls0 ws0">block S/P represents a serial-to-parallel converter. A carrier</div><div class="t m0 x1 h6 y8f ff1 fs3 fc0 sc0 ls0 ws0">frequency offset of</div><div class="t m0 x16 h6 y90 ff1 fs3 fc0 sc0 ls0 ws0">causes a phase rotation of .</div><div class="t m0 x1 h6 y91 ff1 fs3 fc0 sc0 ls0 ws0">If uncorrected this causes both a rotation of the constellation</div><div class="t m0 x1 h6 y92 ff1 fs3 fc0 sc0 ls0 ws0">and a spread of the constellation points similar to additive</div><div class="t m0 x1 h6 y93 ff1 fs3 fc0 sc0 ls0 ws0">white Gaussian noise (AWGN). A symbol-timing error will</div><div class="t m0 x1 h6 y94 ff1 fs3 fc0 sc0 ls0 ws0">have little effect as long as all the samples taken are within</div><div class="t m0 x1 h6 y95 ff1 fs3 fc0 sc0 ls0 ws0">the length of the cyclically-extended OFDM symbol.</div><div class="t m0 x17 h6 y96 ff1 fs3 fc0 sc0 ls0 ws0">III. E</div><div class="t m0 x18 h6 y97 ff1 fs5 fc0 sc0 ls0 ws0">STIMATION OF SYMBOL TIMING</div><div class="t m0 x1 ha y98 ff2 fs3 fc0 sc0 ls0 ws0">A. Symbol Timing Estimation Algorithm</div><div class="t m0 x9 h6 y99 ff1 fs3 fc0 sc0 ls0 ws0">The symbol timing recovery relies on searching for a</div><div class="t m0 x1 h6 y9a ff1 fs3 fc0 sc0 ls0 ws0">training symbol with two identical halves in the time do-</div><div class="t m0 xb h6 y9b ff1 fs3 fc0 sc0 ls0 ws0">main, which will remain identical after passing through the</div><div class="t m0 xb h6 y9c ff1 fs3 fc0 sc0 ls0 ws0">channel, except that there will be a phase difference be-</div><div class="t m0 xb h6 y9d ff1 fs3 fc0 sc0 ls0 ws0">tween them caused by the carrier frequency offset. The two</div><div class="t m0 xb h6 y9e ff1 fs3 fc0 sc0 ls0 ws0">halves of the training symbol are made identical (in time</div><div class="t m0 xb h6 y9f ff1 fs3 fc0 sc0 ls0 ws0">order) by transmitting a pseudonoise (PN) sequence on the</div><div class="t m0 xb h6 ya0 ff1 fs3 fc0 sc0 ls0 ws0">even frequencies, while zeros are used on the odd frequen-</div><div class="t m0 xb h6 ya1 ff1 fs3 fc0 sc0 ls0 ws0">cies. This means that at each even frequency one of the</div><div class="t m0 xb h6 ya2 ff1 fs3 fc0 sc0 ls0 ws0">points of a QPSK constellation is transmitted. In order to</div><div class="t m0 xb h6 ya3 ff1 fs3 fc0 sc0 ls0 ws0">maintain an approximately constant signal energy for each</div><div class="t m0 xb h6 ya4 ff1 fs3 fc0 sc0 ls0 ws0">symbol the frequency components of this training symbol are</div><div class="t m0 xb h6 ya5 ff1 fs3 fc0 sc0 ls0 ws0">multiplied by</div><div class="t m0 x19 h6 ya6 ff1 fs3 fc0 sc0 ls0 ws0">at the transmitter, or the four points of</div><div class="t m0 xb h6 ya7 ff1 fs3 fc0 sc0 ls0 ws0">the QPSK constellation are selected from a larger constel-</div><div class="t m0 xb h6 ya8 ff1 fs3 fc0 sc0 ls0 ws0">lation, such as 64-QAM, so that points with higher energy</div><div class="t m0 xb h6 ya9 ff1 fs3 fc0 sc0 ls0 ws0">can be used. Transmitted data will not be mistaken as the</div><div class="t m0 xb h6 yaa ff1 fs3 fc0 sc0 ls0 ws0">start of the frame since any actual data must contain odd</div><div class="t m0 xb h6 yab ff1 fs3 fc0 sc0 ls0 ws0">frequencies. Note that an equivalent method of generating</div><div class="t m0 xb h6 yac ff1 fs3 fc0 sc0 ls0 ws0">this training symbol is to use an IFFT of half the normal</div><div class="t m0 xb h6 yad ff1 fs3 fc0 sc0 ls0 ws0">size to generate the time domain samples. The repetition is</div><div class="t m0 xb h6 yae ff1 fs3 fc0 sc0 ls0 ws0">not generated using the IFFT, so instead of just using the</div><div class="t m0 xb h6 yaf ff1 fs3 fc0 sc0 ls0 ws0">even frequencies, a PN sequence would be transmitted on</div><div class="t m0 xb h6 yb0 ff1 fs3 fc0 sc0 ls0 ws0">all of the subcarriers to generate the time domain samples</div><div class="t m0 xb h6 yb1 ff1 fs3 fc0 sc0 ls0 ws0">which are half a symbol in duration. These time-domain</div><div class="t m0 xb h6 yb2 ff1 fs3 fc0 sc0 ls0 ws0">samples are repeated (and properly scaled) to form the ﬁrst</div><div class="t m0 xb h6 yb3 ff1 fs3 fc0 sc0 ls0 ws0">training symbol.</div><div class="t m0 xc h6 yb4 ff1 fs3 fc0 sc0 ls0 ws0">The second training symbol contains a PN sequence on the</div><div class="t m0 xb h6 yb5 ff1 fs3 fc0 sc0 ls0 ws0">odd frequencies to measure these subchannels, and another</div><div class="t m0 xb h6 yb6 ff1 fs3 fc0 sc0 ls0 ws0">PN sequence on the even frequencies to help determine</div><div class="t m0 xb h6 yb7 ff1 fs3 fc0 sc0 ls0 ws0">frequency offset. Table I illustrates the use of PN sequences</div><div class="t m0 xb h6 yb8 ff1 fs3 fc0 sc0 ls0 ws0">in the training sequence for an OFDM signal with nine</div><div class="t m0 xb h6 yb9 ff1 fs3 fc0 sc0 ls0 ws0">subcarriers with the points chosen from a subset of a 64-</div><div class="t m0 xb h6 yba ff1 fs3 fc0 sc0 ls0 ws0">QAM constellation. The selection of a particular PN sequence</div><div class="t m0 xb h6 ybb ff1 fs3 fc0 sc0 ls0 ws0">should not have much effect on the performance of the</div><div class="t m0 xb h6 ybc ff1 fs3 fc0 sc0 ls0 ws0">synchronization algorithms. Instead the PN sequence can be</div><div class="t m0 xb h6 ybd ff1 fs3 fc0 sc0 ls0 ws0">chosen on the basis of being easy to implement or having a low</div><div class="t m0 xb h6 ybe ff1 fs3 fc0 sc0 ls0 ws0">peak-to-average power ratio so that there is little distortion in</div><div class="t m0 xb h6 ybf ff1 fs3 fc0 sc0 ls0 ws0">the transmitter ampliﬁer.</div><div class="t m0 xc h6 yc0 ff1 fs3 fc0 sc0 ls0 ws0">Complex samples</div><div class="t m0 x1a h6 yc1 ff1 fs3 fc0 sc0 ls0 ws0">are taken by mixing the received</div><div class="t m0 xb h6 yc2 ff1 fs3 fc0 sc0 ls0 ws0">signal down to the IF, splitting the signal into two branches,</div><div class="t m0 xb h6 yc3 ff1 fs3 fc0 sc0 ls0 ws0">multiplying by both the in-phase and quadrature local oscil-</div><div class="t m0 xb h6 yc4 ff1 fs3 fc0 sc0 ls0 ws0">lators, and low-pass ﬁltering and sampling to get baseband</div><div class="t m0 xb h6 yc5 ff1 fs3 fc0 sc0 ls0 ws0">in-phase and quadrature components (Fig. 2). This can be</div><div class="t m0 xb h6 yc6 ff1 fs3 fc0 sc0 ls0 ws0">expressed mathematically by writing the IF local oscillator</div><div class="t m0 xb h6 yc7 ff1 fs3 fc0 sc0 ls0 ws0">for the in-phase branch as</div><div class="t m0 x1b h6 yc8 ff1 fs3 fc0 sc0 ls0 ws0">(2)</div></div><div class="pi" data-data='{"ctm":[1.568627,0.000000,0.000000,1.568627,0.000000,0.000000]}'></div></div>

<div id="pf3" class="pf w0 h0" data-page-no="3"><div class="pc pc3 w0 h0"><img class="bi x0 y0 w1 h1" alt="" src="https://csdnimg.cn/release/download_crawler_static/12190445/bg3.jpg"><div class="t m0 x1 h2 y1 ff1 fs0 fc0 sc0 ls0 ws0">SCHMIDL AND COX: ROBUST FREQUENCY AND TIMING SYNCHRONIZATION FOR OFDM 1615</div><div class="t m0 x1c h8 yc9 ff1 fs5 fc0 sc0 ls0 ws0">TABLE I</div><div class="t m0 x1d h8 yca ff1 fs5 fc0 sc0 ls0 ws0">I</div><div class="t m0 x6 h8 ycb ff1 fs7 fc0 sc0 ls0 ws0">LLUSTRATION OF USE OF PN SEQUENCES FOR TRAINING SYMBOLS</div><div class="t m0 x1 h6 ycc ff1 fs3 fc0 sc0 ls0 ws0">and the IF local oscillator for the quadrature branch at the</div><div class="t m0 x1 h6 ycd ff1 fs3 fc0 sc0 ls0 ws0">receiver as</div><div class="t m0 x15 h6 yce ff1 fs3 fc0 sc0 ls0 ws0">(3)</div><div class="t m0 x1 h6 ycf ff1 fs3 fc0 sc0 ls0 ws0">Let the output of the mixer after down-conversion be</div><div class="t m0 x13 h6 yd0 ff1 fs3 fc0 sc0 ls0 ws0">. The</div><div class="t m0 x1 h6 yd1 ff1 fs3 fc0 sc0 ls0 ws0">demodulated signal before the sampler can be expressed as</div><div class="t m0 x15 h6 yd2 ff1 fs3 fc0 sc0 ls0 ws0">(4)</div><div class="t m0 x1 h6 yd3 ff1 fs3 fc0 sc0 ls0 ws0">where</div><div class="t m0 x1e h6 yd4 ff1 fs3 fc0 sc0 ls0 ws0">means to low-pass ﬁlter the terms in the</div><div class="t m0 x1 h6 yd5 ff1 fs3 fc0 sc0 ls0 ws0">argument. The output of the in-phase branch is considered to</div><div class="t m0 x1 h6 yd6 ff1 fs3 fc0 sc0 ls0 ws0">be real and the output of the quadrature branch is considered to</div><div class="t m0 x1 h6 yd7 ff1 fs3 fc0 sc0 ls0 ws0">be imaginary. This is a mathematical convention to represent</div><div class="t m0 x1 h6 yd8 ff1 fs3 fc0 sc0 ls0 ws0">the in-phase and quadrature components as a complex number.</div><div class="t m0 x1 h6 yd9 ff1 fs3 fc0 sc0 ls0 ws0">After sampling, the complex samples are denoted as</div><div class="t m0 x13 h6 yda ff1 fs3 fc0 sc0 ls0 ws0">.</div><div class="t m0 x9 h6 ydb ff1 fs3 fc0 sc0 ls0 ws0">Consider the ﬁrst training symbol where the ﬁrst half is</div><div class="t m0 x1 h6 ydc ff1 fs3 fc0 sc0 ls0 ws0">identical to the second half (in time order), except for a phase</div><div class="t m0 x1 h6 ydd ff1 fs3 fc0 sc0 ls0 ws0">shift caused by the carrier frequency offset. If the conjugate of</div><div class="t m0 x1 h6 yde ff1 fs3 fc0 sc0 ls0 ws0">a sample from the ﬁrst half is multiplied by the corresponding</div><div class="t m0 x1 h6 ydf ff1 fs3 fc0 sc0 ls0 ws0">sample from the second half (</div><div class="t m0 x1f h6 ye0 ff1 fs3 fc0 sc0 ls0 ws0">seconds later), the effect of</div><div class="t m0 x1 h6 ye1 ff1 fs3 fc0 sc0 ls0 ws0">the channel should cancel, and the result will have a phase</div><div class="t m0 x1 h6 ye2 ff1 fs3 fc0 sc0 ls0 ws0">of approximately</div><div class="t m0 x4 h6 ye3 ff1 fs3 fc0 sc0 ls0 ws0">. At the start of the frame,</div><div class="t m0 x1 h6 ye4 ff1 fs3 fc0 sc0 ls0 ws0">the products of each of these pairs of samples will have</div><div class="t m0 x1 h6 ye5 ff1 fs3 fc0 sc0 ls0 ws0">approximately the same phase, so the magnitude of the sum</div><div class="t m0 x1 h6 ye6 ff1 fs3 fc0 sc0 ls0 ws0">will be a large value.</div><div class="t m0 x9 h6 ye7 ff1 fs3 fc0 sc0 ls0 ws0">Let there be</div><div class="t m0 xf h6 ye8 ff1 fs3 fc0 sc0 ls0 ws0">complex samples in one-half of the ﬁrst</div><div class="t m0 x1 h6 ye9 ff1 fs3 fc0 sc0 ls0 ws0">training symbol (excluding the cyclic preﬁx), and let the sum</div><div class="t m0 x1 h6 yea ff1 fs3 fc0 sc0 ls0 ws0">of the pairs of products be</div><div class="t m0 x15 h6 yeb ff1 fs3 fc0 sc0 ls0 ws0">(5)</div><div class="t m0 x1 h6 yec ff1 fs3 fc0 sc0 ls0 ws0">which can be implemented with the iterative formula</div><div class="t m0 x15 h6 yed ff1 fs3 fc0 sc0 ls0 ws0">(6)</div><div class="t m0 x1 h6 yee ff1 fs3 fc0 sc0 ls0 ws0">Note that</div><div class="t m0 x20 h6 yef ff1 fs3 fc0 sc0 ls0 ws0">is a time index corresponding to the ﬁrst sample in</div><div class="t m0 x1 h6 yf0 ff1 fs3 fc0 sc0 ls0 ws0">a window of</div><div class="t m0 x1e h6 yf1 ff1 fs3 fc0 sc0 ls0 ws0">samples. This window slides along in time as</div><div class="t m0 x1 h6 yf2 ff1 fs3 fc0 sc0 ls0 ws0">the receiver searches for the ﬁrst training symbol. The received</div><div class="t m0 x1 h6 yf3 ff1 fs3 fc0 sc0 ls0 ws0">energy for the second half-symbol is deﬁned by</div><div class="t m0 x15 h6 yf4 ff1 fs3 fc0 sc0 ls0 ws0">(7)</div><div class="t m0 xb h8 yf5 ff1 fs5 fc0 sc0 ls0 ws0">Fig. 3. Example of the timing metric for the AWGN channel (</div><div class="t m0 x21 hb yf5 ff6 fs8 fc0 sc0 ls1 ws0">SNR = 10</div><div class="t m0 xb h8 yf6 ff1 fs5 fc0 sc0 ls0 ws0">dB).</div><div class="t m0 xb h6 yf7 ff1 fs3 fc0 sc0 ls0 ws0">which can also be calculated iteratively. may be used</div><div class="t m0 xb h6 yf8 ff1 fs3 fc0 sc0 ls0 ws0">as part of an automatic gain control (AGC) loop. A timing</div><div class="t m0 xb h6 yf9 ff1 fs3 fc0 sc0 ls0 ws0">metric can be deﬁned as</div><div class="t m0 x1b h6 yfa ff1 fs3 fc0 sc0 ls0 ws0">(8)</div><div class="t m0 xc h6 yfb ff1 fs3 fc0 sc0 ls0 ws0">Fig. 3 shows an example of the timing metric as a window</div><div class="t m0 xb h6 yfc ff1 fs3 fc0 sc0 ls0 ws0">slides past coincidence for the AWGN channel for an OFDM</div><div class="t m0 xb h6 yfd ff1 fs3 fc0 sc0 ls0 ws0">signal with 1000 subcarriers, a carrier frequency offset of 12.4</div><div class="t m0 xb h6 yfe ff1 fs3 fc0 sc0 ls0 ws0">subcarrier spacings, and an signal-to-noise ratio (SNR) of 10</div><div class="t m0 xb h6 yff ff1 fs3 fc0 sc0 ls0 ws0">dB, where the SNR is the total signal (all the subcarriers) to</div><div class="t m0 xb h6 y100 ff1 fs3 fc0 sc0 ls0 ws0">noise power ratio. The timing metric reaches a plateau which</div><div class="t m0 xb h6 y101 ff1 fs3 fc0 sc0 ls0 ws0">has a length equal to the length of the guard interval minus</div><div class="t m0 xb h6 y102 ff1 fs3 fc0 sc0 ls0 ws0">the length of the channel impulse response since there is no</div><div class="t m0 xb h6 y103 ff1 fs3 fc0 sc0 ls0 ws0">ISI within this plateau to distort the signal. For the AWGN</div><div class="t m0 xb h6 y104 ff1 fs3 fc0 sc0 ls0 ws0">channel, there is a window with a length of the guard interval</div><div class="t m0 xb h6 y105 ff1 fs3 fc0 sc0 ls0 ws0">where the metric reaches a maximum, and the start of the frame</div><div class="t m0 xb h6 y106 ff1 fs3 fc0 sc0 ls0 ws0">can be taken to be anywhere within this window without a loss</div><div class="t m0 xb h6 y107 ff1 fs3 fc0 sc0 ls0 ws0">in the received SNR. For the frequency selective channels, the</div><div class="t m0 xb h6 y108 ff1 fs3 fc0 sc0 ls0 ws0">length of the impulse response of the channel is shorter than</div><div class="t m0 xb h6 y109 ff1 fs3 fc0 sc0 ls0 ws0">the guard interval by design choice of the guard interval, so</div><div class="t m0 xb h6 y10a ff1 fs3 fc0 sc0 ls0 ws0">the plateau in the maximum of the timing metric is shorter</div><div class="t m0 xb h6 y10b ff1 fs3 fc0 sc0 ls0 ws0">than for the AWGN channel.</div><div class="t m0 xc h6 y10c ff1 fs3 fc0 sc0 ls0 ws0">This plateau leads to some uncertainty as to the start of</div><div class="t m0 xb h6 y10d ff1 fs3 fc0 sc0 ls0 ws0">the frame. For the simulations in this paper, OFDM symbols</div><div class="t m0 xb h6 y10e ff1 fs3 fc0 sc0 ls0 ws0">are generated with 1000 frequencies,</div><div class="t m0 x22 h6 y10f ff1 fs3 fc0 sc0 ls0 ws0">500 to 499. They are</div><div class="t m0 xb h6 y110 ff1 fs3 fc0 sc0 ls0 ws0">slightly oversampled at a rate of 1024 samples for the useful</div><div class="t m0 xb h6 y111 ff1 fs3 fc0 sc0 ls0 ws0">part of each symbol. In an actual hardware implementation, the</div><div class="t m0 xb h6 y112 ff1 fs3 fc0 sc0 ls0 ws0">ratio of the sampling rate to the number of frequencies would</div><div class="t m0 xb h6 y113 ff1 fs3 fc0 sc0 ls0 ws0">be higher to ease ﬁltering requirements. The guard interval is</div><div class="t m0 xb h6 y114 ff1 fs3 fc0 sc0 ls0 ws0">set to about 10% of the useful part, which is 102 samples.</div><div class="t m0 xb ha y115 ff2 fs3 fc0 sc0 ls0 ws0">B. Performance of Symbol Timing Estimator</div><div class="t m0 xc h6 y116 ff1 fs3 fc0 sc0 ls0 ws0">There are two issues to consider when evaluating the</div><div class="t m0 xb h6 y117 ff1 fs3 fc0 sc0 ls0 ws0">performance of the symbol timing estimator. First, since the</div><div class="t m0 xb h6 y118 ff1 fs3 fc0 sc0 ls0 ws0">timing metric is also used to determine whether the training</div><div class="t m0 xb h6 y119 ff1 fs3 fc0 sc0 ls0 ws0">sequence has been received, there is a probability of either</div><div class="t m0 xb h6 y11a ff1 fs3 fc0 sc0 ls0 ws0">missing a training sequence and not detecting the signal or</div><div class="t m0 xb h6 y11b ff1 fs3 fc0 sc0 ls0 ws0">falsely detecting a training sequence when none is there. In</div><div class="t m0 xb h6 y11c ff1 fs3 fc0 sc0 ls0 ws0">this paper the distribution of the timing metric at the correct</div><div class="t m0 xb h6 y11d ff1 fs3 fc0 sc0 ls0 ws0">start of the frame is calculated. Using this distribution, the</div><div class="t m0 xb h6 y11e ff1 fs3 fc0 sc0 ls0 ws0">number of samples that need to be processed during the</div><div class="t m0 xb h6 y11f ff1 fs3 fc0 sc0 ls0 ws0">detection phase can be determined and a threshold can be</div></div><div class="pi" data-data='{"ctm":[1.568627,0.000000,0.000000,1.568627,0.000000,0.000000]}'></div></div>