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On the performance of a maximum likelihood method with delayed correlation for the coarse carrier frequency offset estimation of OFDM signals with multiple preamble symbols

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Abstract

Estimation of carrier frequency offset (CFO) is an important issue in the design of a wireless receiver that employs orthogonal frequency division multiplexing (OFDM) techniques. In this paper, using the ten short training symbols specified in the signal format of the IEEE 802.11a WLAN, we investigate the performance of a coarse CFO estimation scheme for OFDM signals with multiple preamble symbols. This scheme, which we call DC-ML, employs the maximum likelihood (ML) method with delayed correlation (DC). For AWGN channels under moderate signal to noise ratio (SNR) conditions, we develop an analysis to evaluate the variance of estimation error (VEER). The analysis is corroborated in light of simulations, and compared with the formulated Cramer-Rao lower bound (CRLB). VEER of the DC-ML in a multipath environment is studied via simulations. Numerical results show that a certain parameter combination can result in minimum VEER. Simulation results justify that the probability of estimation error (PEER) approximates Gaussian distribution in both AWGN and multipath scenarios. We also present a Two-Branch DC-ML (TBDC-ML) scheme, which comprises two correlation branches of DC-ML, and an associative ambiguity resolution algorithm. Numerical examples reveal that TBDC-ML outperforms DC-ML in both VEER and PEER. Assuming that the estimation error resulting from the two branches is jointly Gaussian, we derive the joint probability density function (pdf) and validate it via simulations.

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Acknowledgments

This work is supported in part by the National Science Council of Taiwan under Contract No. NSC 93-2213-E-216-027, and in part by the Chung Hua University under Contract No. CHU-93-TR-003.

Author information

Authors and Affiliations

  1. Department of Electrical Engineering, Chung Hua University, No. 707, Sec. 2, WuFu Rd., Hsinchu, Taiwan, ROC

    In-Hang Chung

  2. Institute of Engineering and Science, Chung Hua University, No. 707, Sec. 2, WuFu Rd., Hsinchu, Taiwan, ROC

    Ming-Ching Yen

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  1. In-Hang Chung
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  2. Ming-Ching Yen
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Correspondence to In-Hang Chung.

Additional information

This paper was presented in part at IEEE International Conference on Communications (ICC’2005), Seoul, Korea, May 16–20, 2005.

Appendices

Appendix 1

From (5) and using the inequality, \( \left| {a + b} \right| \le \left| a \right| + \left| b \right|, \) we can obtain

$$ \begin{aligned} \frac{1}{{GN_{s} }}\left| {r\left( {G,D} \right)} \right| &\le \frac{1}{{GN_{s} }}\sum\limits_{n = 0}^{G - 1} {} \sum\limits_{m = 0}^{{N_{s} - 1}} {\left[ {\;\left| {s_{{m + nN_{s} }}^{*} \exp \left\{ { - j\left( {\frac{{2\pi \left[ {m + \left( {n + D} \right)N_{s} } \right]\Updelta k}}{N} + \theta } \right)} \right\} n_{{m + \left( {n + D} \right)N_{s} }} } \right|} \right.} \\ & + \left| {s_{{m + nN_{s} }} \exp \left\{ {j\left( {\frac{{2\pi \left[ {m + nN_{s} } \right]\Updelta k}}{N} + \theta } \right)} \right\} n_{{m + nN_{s} }}^{*} } \right|\left. { + \left| {n_{{m + nN_{s} }}^{*} n_{{m + (n + D)N_{s} }} \exp \left\{ { - j\frac{{2\pi DN_{s} \Updelta k}}{N}} \right\}} \right|\;} \right] \\ &\le \left( {R_{1} + R_{2} + R_{3} } \right)/GN_{s} , \\ \end{aligned} $$
(31)

in which \( R_{1} = \sum\limits_{n = 0}^{G - 1} {\, \sum\limits_{m = 0}^{{N_{s} - 1}} {\left| {s_{{m + nN_{s} }} } \right|\,\left| {n_{{m + (n + D)N_{s} }} } \right|} } , \) \( R_{2} = \sum\limits_{n = 0}^{G - 1} { \sum\limits_{m = 0}^{{N_{s} - 1}} {\left| {s_{{m + nN_{s} }} } \right|\,\left| {n_{{m + nN_{s} }} } \right|} } , \) and \( R_{3} = \sum\limits_{n = 0}^{G - 1} { \sum\limits_{m = 0}^{{N_{s} - 1}} {\left| {n_{{m + nN_{s} }} } \right|\,\left| {n_{{m + (n + D)N_{s} }} } \right|} } . \) Rewrite R 1 as

$$ R_{1} = \sum\limits_{n = 0}^{G - 1} {\,\left[ {\,\begin{array}{*{20}c} {\left| {s_{{nN_{s} }} } \right|} & {\left| {s_{{1 + nN_{s} }} } \right|} & \cdots & {\left| {s_{{N_{s} - 1 + nN_{s} }} } \right|} \\ \end{array} \,} \right]\;\,\left[ {\,\begin{array}{*{20}c} {\left| {n_{{(n + D)N_{s} }} } \right|} & {\left| {n_{{1 + (n + D)N_{s} }} } \right|} & \cdots & {\left| {n_{{N_{s} - 1 + (n + D)N_{s} }} } \right|} \\ \end{array} \,} \right]^{\;T} } \, = \sum\limits_{n = 0}^{G - 1} \, x_{n} \bullet y_{n} , $$
(32)

where \( x_{n} = \left[ {\,\begin{array}{*{20}c} {\left| {s_{{nN_{s} }} } \right|} & {\left| {s_{{1 + nN_{s} }} } \right|} & \cdots & {\left| {s_{{N_{s} - 1 + nN_{s} }} } \right|} \\ \end{array} \,} \right]\,, \) \( y_{n} = \left[ {\,\begin{array}{*{20}c} {\left| {n_{{(n + D)N_{s} }} } \right|} & {\left| {n_{{1 + (n + D)N_{s} }} } \right|} & \cdots & {\left| {n_{{N_{s} - 1 + (n + D)N_{s} }} } \right|} \\ \end{array} \,} \right], \) and the symbol • denotes the inner product operator. Using the Cauchy-Schwartz inequality, \( u \bullet v \le \left\| u \right\|\;\left\| v \right\|, \) in which \( \left\| {} \right\| \) represents the norm of the specified vector, we have

$$ R_{1} \le \sum\limits_{n = 0}^{G - 1} {\,\left\| {x_{n} } \right\|} \;\left\| {y_{n} } \right\| = \left[ {\,\begin{array}{*{20}c} {\left\| {x_{0} } \right\|} & {\left\| {x_{1} } \right\|} & \cdots & {\left\| {x_{G - 1} } \right\|} \\ \end{array} \,} \right]\;\left[ {\,\begin{array}{*{20}c} {\left\| {y_{0} } \right\|} & {\left\| {y_{1} } \right\|} & \cdots & {\left\| {y_{G - 1} } \right\|} \\ \end{array} \,} \right]\,^{T} . $$
(33)

Applying the Cauchy-Schwartz inequality to (33) yields

$$ R_{1} \le \left( {\sum\limits_{n = 0}^{G - 1} {\,\left\| {x_{n} } \right\|}^{2} } \right)^{1/2} \left( {\sum\limits_{n = 0}^{G - 1} {\,\left\| {y_{n} } \right\|}^{2} } \right)^{1/2} = \left( {\sum\limits_{n = 0}^{G - 1} {\,\sum\limits_{m = 0}^{{N_{s} - 1}} {\left| {s_{{m + nN_{s} }} } \right|} }^{2} } \right)^{1/2} \left( {\sum\limits_{n = 0}^{G - 1} {\,\sum\limits_{m = 0}^{{N_{s} - 1}} {\left| {n_{{m + (n + D)N_{s} }} } \right|} }^{2} } \right)^{1/2} = GN_{s} \left( {\sigma_{s}^{2} } \right)^{1/2} \left( {\sigma_{n}^{2} } \right)^{1/2} . $$
(34)

Following above procedures, we can find that \( R_{2} \le GN_{s} \left( {\sigma_{s}^{2} } \right)^{1/2} \left( {\sigma_{n}^{2} } \right)^{1/2} \) and \( R_{3} \le GN_{s} \sigma_{n}^{2} . \) Consequently,

$$ \left| {r\left( {G,D} \right)} \right|/GN_{s} \le \left( {R_{1} + R_{2} + R_{3} } \right)/GN_{s} \le 2\left( {\sigma_{s}^{2} } \right)^{1/2} \left( {\sigma_{n}^{2} } \right)^{1/2} + \sigma_{n}^{2} . $$
(35)

One may note that the value of \( \left[ {r\left( {G,D} \right)/GN_{s} } \right] + \sigma_{s}^{2} \) can be interpreted as a point lying inside a circle with radius \( 2\left( {\sigma_{s}^{2} } \right)^{1/2} \left( {\sigma_{n}^{2} } \right)^{1/2} + \sigma_{n}^{2} \) and centered at (\( \sigma_{s}^{2} , \) 0) in the complex plane, as illustrated in Fig. 15. When SNR is high, the maximum value of the real part of \( \left[ {r\left( {G,D} \right)/GN_{s} } \right] + \sigma_{s}^{2} \) will be

$$ 2\left( {\sigma_{s}^{2} } \right)^{1/2} \left( {\sigma_{n}^{2} } \right)^{1/2} + \sigma_{n}^{2} + \sigma_{s}^{2} = \sigma_{s}^{2} \left[ {2\left( {\sigma_{n}^{2} /\sigma_{s}^{2} } \right)^{1/2} + \left( {\sigma_{n}^{2} /\sigma_{s}^{2} } \right) + 1} \right] \approx \sigma_{s}^{2} . $$
(36)

Therefore,

$$ {\text{RE}}\left\{ {\left[ {r(G, D )+ \sum\limits_{n = 0}^{G - 1} {} \sum\limits_{m = 0}^{{N_{s} - 1}} {s_{{m + nN_{s} }}^{*} s_{{m + nN_{s} }} } } \right]/GN_{s} } \right\} = {\text{RE }}\left\{ { \left[ {r(G,D)/GN_{s} } \right] + \sigma_{s}^{2} } \right\} \approx \sigma_{s}^{2} . $$
(37)
Fig. 15
figure 15

Geometric interpretation of (35)

Full size image

Appendix 2

To evaluate (11), we note that \( {\text{IM }}\left[ {r\left( {G,D} \right)} \right] = \left[ {r\left( {G,D} \right) - r^{*} \left( {G,D} \right)} \right]/(2j). \) Also note that the \( n_{{m + nN_{s} }}^{*} n_{{m + (n + D)N_{s} }} \) term in (5) will not take effect when expectation is taken. Consequently,

$$ \begin{aligned} {{\text {IM}}} \left[ {r\left( {G,D} \right)} \right] = \frac{1}{2j}\sum\limits_{n = 0}^{G - 1} {\sum\limits_{m = 0}^{{N_{s} - 1}} {\left[ {s_{{m + nN_{s} }}^{*} \exp \left\{ { - j\left[ {\frac{{2\pi \left( {m + nN_{s} } \right) \Updelta k}}{N} + \theta } \right]} \right\}\left( {n_{{m + \left( {n + D} \right)N_{s} }} \exp \left\{ { - j\frac{{2\pi DN_{s} \Updelta k}}{N}} \right\} - n_{{m + nN_{s} }} } \right)} \right.} } \\ + s_{{m + nN_{s} }} \exp \left\{ {j\left[ {\frac{{2\pi \left( {m + nN_{s} } \right) \Updelta k}}{N} + \theta } \right]} \right\}\left( {n_{{m + nN_{s} }}^{*} - n_{{m + \left( {n + D} \right)N_{s} }}^{*} \exp \left\{ {j\left( {\frac{{2\pi DN_{s} \Updelta k}}{N}} \right)} \right\}} \right) \\ \left. { + \left( {n_{{m + nN_{s} }}^{*} n_{{m + (n + D)N_{s} }} \exp \left\{ { - j\frac{{2\pi DN_{s} \Updelta k}}{N}} \right\} - n_{{m + nN_{s} }} n_{{m + \left( {n + D} \right)N_{s} }}^{*} \exp \left\{ {j\frac{{2\pi DN_{s} \Updelta k}}{N}} \right\}} \right)} \right]. \\ \end{aligned} $$
(38)

Using (38), we can obtain \( E\left\{ {\left[ {{\text{ IM }}\left\{ {r (G,D)} \right\} } \right]^{2} } \right\} \) as expressed in (39).

$$ \begin{aligned} E\left\{ {\left[ {{\text{ IM }}\left\{ {r \left( {G,D} \right)} \right\} } \right]^{2} } \right\} = \frac{1}{4}\left\{ {\sum\limits_{n = 0}^{G - 1} {\sum\limits_{m = 0}^{{N_{s} - 1}} {{\kern 1pt} \sum\limits_{i = 0}^{{G - 1}} {\sum\limits_{l = 0}^{{N_{s} - 1}} { \left[ {E \left\{ {s_{{m + nN_{s} }}^{*} s_{{l + iN_{s} }} } \right\} E \left( {n_{{m + \left( {n + D} \right)N_{s} }} n_{{l + \left( {i + D} \right)N_{s} }}^{*} - n_{{m + \left( {n + D} \right)N_{s} }} n_{{l + iN_{s} }}^{*} \exp \left\{ { - j\left( {\frac{{2\pi DN_{s} \Updelta k}}{N}} \right)} \right\}} \right.} \right.} } } } } \right. \\ \left. {\left. { + n_{{m + nN_{s} }} n_{{l + iN_{s} }}^{*} - n_{{m + nN_{s} }} n_{{l + \left( {i + D} \right)N_{s} }}^{*} \exp \left\{ {j\left( {\frac{{2\pi DN_{s} \Updelta k}}{N}} \right)} \right\}} \right)} \right] \\ + \left[ {E \left\{ {s_{{m + nN_{s} }} s_{{l + iN_{s} }}^{*} } \right\} E \left( {n_{{m + nN_{s} }}^{*} n_{{l + iN_{s} }} - n_{{m + nN_{s} }}^{*} n_{{l + \left( {i + D} \right)N_{s} }} \exp \left\{ { - j\left( {\frac{{2\pi DN_{s} \Updelta k}}{N}} \right)} \right\}} \right.} \right. \\ + n_{{m + \left( {n + D} \right)N_{s} }}^{*} n_{{l + \left( {i + D} \right)N_{s} }} \left. {\left. { - n_{{m + \left( {n + D} \right)N_{s} }}^{*} n_{{l + iN_{s} }} \exp \left\{ {j\left( {\frac{{2\pi DN_{s} \Updelta k}}{N}} \right)} \right\}} \right) } \right] \\ + \left[ {E \left\{ {s_{{m + nN_{s} }}^{*} s_{{l + iN_{s} }}^{*} } \right\} \exp \left\{ { - j\left( {\frac{{2\pi (m + nN_{s} + l + iN_{s} )\Updelta k}}{N} + 2\theta } \right)} \right\}E \left( {n_{{m + \left( {n + D} \right)N_{s} }} n_{{l + iN_{s} }} \exp \left\{ { - j\left( {\frac{{2\pi DN_{s} \Updelta k}}{N}} \right)} \right\}} \right.} \right. \\ \left. {\left. { + n_{{m + nN_{s} }} n_{{l + \left( {i + D} \right)N_{s} }} \exp \left\{ { - j\left( {\frac{{2\pi DN_{s} \Updelta k}}{N}} \right)} \right\} - n_{{m + \left( {n + D} \right)N_{s} }} n_{{l + \left( {i + D} \right)N_{s} }} \exp \left\{ { - j\left( {\frac{{4\pi DN_{s} \Updelta k}}{N}} \right)} \right\} - n_{{m + nN_{s} }} n_{{l + iN_{s} }} } \right)} \right] \\ + \left[ {E \left\{ {s_{{m + nN_{s} }} s_{{l + iN_{s} }} } \right\} \exp \left\{ {j\left( {\frac{{2\pi (m + nN_{s} + l + iN_{s} )\Updelta k}}{N} + 2\theta } \right)} \right\}E \left( {n_{{m + nN_{s} }}^{*} n_{{l + \left( {i + D} \right)N_{s} }}^{*} \exp \left\{ {j\left( {\frac{{2\pi DN_{s} \Updelta k}}{N}} \right)} \right\}} \right.} \right. \\ \left. {\left. { + n_{{m + \left( {n + D} \right)N_{s} }}^{*} n_{{l + iN_{s} }}^{*} \exp \left\{ {j\left( {\frac{{2\pi DN_{s} \Updelta k}}{N}} \right)} \right\} - n_{{m + \left( {n + D} \right)N_{s} }}^{*} n_{{l + \left( {i + D} \right)N_{s} }}^{*} \exp \left\{ {j\left( {\frac{{4\pi DN_{s} \Updelta k}}{N}} \right)} \right\} - n_{{m + nN_{s} }}^{*} n_{{l + iN_{s} }}^{*} } \right)} \right] \\ + \left[ {E \left( {n_{{m + nN_{s} }}^{*} n_{{m + \left( {n + D} \right)N_{s} }} n_{{l + iN_{s} }} n_{{l + \left( {i + D} \right)N_{s} }}^{*} - n_{{m + nN_{s} }}^{*} n_{{m + \left( {n + D} \right)N_{s} }} n_{{l + iN_{s} }}^{*} n_{{l + \left( {i + D} \right)N_{s} }} \exp \left\{ { - j\frac{{4\pi DN_{s} \Updelta k}}{N}} \right\}} \right.} \right. \\ + n_{{m + nN_{s} }} n_{{m + \left( {n + D} \right)N_{s} }}^{*} n_{{l + iN_{s} }}^{*} n_{{l + \left( {i + D} \right)N_{s} }} \left. {\left. { - n_{{m + nN_{s} }} n_{{m + \left( {n + D} \right)N_{s} }}^{*} n_{{l + iN_{s} }} n_{{l + \left( {i + D} \right)N_{s} }}^{*} \exp \left\{ {j\frac{{4\pi DN_{s} \Updelta k}}{N}} \right\}} \right)} \right] \\ \left. { + E \left[ {\text{terms which contain triple products of noise or its complex conjugate}} \right]} \right\} \\ \end{aligned}$$
(39)

For all values of m, n, l and i in (39), \( E \left\{ {s_{{m + nN_{s} }} s_{{l + iN_{s} }} } \right\}, \) \( E \left\{ {n_{{m + nN_{s} }}^{*} n_{{m + \left( {n + D} \right)N_{s} }} n_{{l + iN_{s} }}^{*} n_{{l + \left( {i + D} \right)N_{s} }} } \right\}, \) \( E\left\{ {s_{{m + nN_{s} }}^{*} s_{{l + iN_{s} }}^{*} } \right\}, \) and \( E\,\left\{ {n_{{m + nN_{s} }} n_{{m + \left( {n + D} \right)N_{s} }}^{*} n_{{l + iN_{s} }} n_{{l + \left( {i + D} \right)N_{s} }}^{*} } \right\} \)are zero. If \( m \ne l, \) \( E \left\{ {s_{{m + nN_{s} }}^{*} s_{{l + iN_{s} }} } \right\} = 0 \) and \( E \left\{ {s_{{m + nN_{s} }} s_{{l + iN_{s} }}^{*} } \right\} = 0 \) for all n and i. In addition, since an AWGN channel is considered, \( E \left\{ {n_{y}^{{}} n_{z}^{{}} } \right\} \)=\( E \left\{ {n_{y}^{ *} n_{z}^{*} } \right\} = 0 \) for \( y = m + nN_{s} \) or \( y = m + \left( {n + D} \right)N_{s} \), and \( z = l + iN_{s} \) or \( z = l + \left( {i + D} \right)N_{s} . \) Moreover, the last term in (39) is zero. Consequently, (39) can be simplified to

$$ \begin{aligned} E\left\{ {\left[ {{\text{ IM }}\left\{ {r \left( {G,D} \right)} \right\} } \right]^{2} } \right\} = \frac{1}{4}\left\{ {\sum\limits_{n = 0}^{G - 1} { \sum\limits_{{{\text{i}} = 0}}^{G - 1} {} \sum\limits_{m = 0}^{{N_{s} - 1}} { \left[ {E \left\{ {s_{{m + nN_{s} }}^{*} s_{{m + iN_{s} }} } \right\} E \left( {n_{{m + \left( {n + D} \right)N_{s} }} n_{{m + \left( {i + D} \right)N_{s} }}^{*} } \right.} \right.} } } \right. \\ \left. {\left. { - n_{{m + \left( {n + D} \right)N_{s} }} n_{{m + iN_{s} }}^{*} \exp \left\{ { - j\left( {\frac{{2\pi DN_{s} \Updelta k}}{N}} \right)} \right\} + n_{{m + nN_{s} }} n_{{m + iN_{s} }}^{*} - n_{{m + nN_{s} }} n_{{m + \left( {i + D} \right)N_{s} }}^{*} \exp \left\{ {j\left( {\frac{{2\pi DN_{s} \Updelta k}}{N}} \right)} \right\}} \right)} \right] \\ + \left[ {E \left\{ {s_{{m + nN_{s} }} s_{{m + iN_{s} }}^{*} } \right\} E \left( {n_{{m + nN_{s} }}^{*} n_{{m + iN_{s} }} - n_{{m + nN_{s} }}^{*} n_{{m + (i + D)N_{s} }} \exp \left\{ { - j(\frac{{2\pi DN_{s} \Updelta k}}{N})} \right\}} \right.} \right. \\ + n_{{m + \left( {n + D} \right)N_{s} }}^{*} n_{{m + \left( {i + D} \right)N_{s} }} \left. {\left. { - n_{{m + \left( {n + D} \right)N_{s} }}^{*} n_{{m + iN_{s} }} \exp \left\{ {j\left( {\frac{{2\pi DN_{s} \Updelta k}}{N}} \right)} \right\}} \right) } \right] \\ + \left[ {E \left( {n_{{m + nN_{s} }}^{*} n_{{m + \left( {n + D} \right)N_{s} }} n_{{m + iN_{s} }} n_{{m + \left( {i + D} \right)N_{s} }}^{*} } \right.} \right.\left. {\left. {\left. { + n_{{m + nN_{s} }} n_{{m + \left( {n + D} \right)N_{s} }}^{*} n_{{m + iN_{s} }}^{*} n_{{m + \left( {i + D} \right)N_{s} }} } \right){\kern 1pt} } \right]{\kern 1pt} {\kern 1pt} } \right\}. \\ \end{aligned} $$
(40)

In (40), expectation for products of the noise terms, \( E \left\{ {n_{{m + nN_{s} }}^{*} n_{{m + iN_{s} }} } \right\}, \) \( E \left\{ {n_{{m + \left( {n + D} \right)N_{s} }}^{*} n_{{m + \left( {i + D} \right)N_{s} }} } \right\}, \) \( E \left\{ {n_{{m + nN_{s} }}^{*} n_{{m\, + \,\left( {n + D} \right)N_{s} }} n_{{m + iN_{s} }} n_{{m + \left( {i + D} \right)N_{s} }}^{*} } \right\}, \) \( E \left\{ {n_{{m + nN_{s} }} n_{{m + \left( {n + D} \right)N_{s} }}^{*} n_{{m + iN_{s} }}^{*} n_{{m + \left( {i + D} \right)N_{s} }} } \right\}, \) \( E \left\{ {n_{{m + nN_{s} }} n_{{m + iN_{s} }}^{*} } \right\}, \) and \( E \left\{ {n_{{m + \left( {n + D} \right)N_{s} }} n_{{m + \left( {i + D} \right)N_{s} }}^{*} } \right\} \) will be nonzero only when \( n = i \). But values of the other terms, \( E \left\{ {n_{{m + \left( {n + D} \right)N_{s} }} n_{{m + iN_{s} }}^{*} } \right\} , \) \( E \left\{ {n_{{m + nN_{s} }} n_{{m + \left( {i + D} \right)N_{s} }}^{*} } \right\} , \) \( E \left\{ {n_{{m + nN_{s} }}^{*} n_{{m + \left( {i + D} \right)N_{s} }} } \right\} , \) and \( E \left\{ {n_{{m + \left( {n + D} \right)N_{s} }}^{*} n_{{m + iN_{s} }} } \right\} \) will depend on whether D ≥ G or D < G. In case D ≥ G, all these terms will be zero. Consequently,

$$ \begin{aligned} E\left\{ {\left[ {{\text{ IM }}\left\{ {r \left( {G,D} \right)} \right\} } \right]^{2} } \right\} &= \frac{1}{4}\left\{ {\sum\limits_{n = 0}^{G - 1} { \sum\limits_{m = 0}^{{N_{s} - 1}} { \left[ {E \left\{ {s_{{m + nN_{s} }}^{*} s_{{m + nN_{s} }} } \right\} E \left( {n_{{m + \left( {n + D} \right)N_{s} }} n_{{m + \left( {n + D} \right)N_{s} }}^{*} } \right.} \right.} } } \right.\left. {\left. { + n_{{m + nN_{s} }} n_{{m + nN_{s} }}^{*} } \right)} \right] \\ & \quad + \left[ {E \left\{ {s_{{m + nN_{s} }} s_{{m + iN_{s} }}^{*} } \right\} E \left( {n_{{m + nN_{s} }}^{*} n_{{m + nN_{s} }} } \right.} \right.\left. {\left. { + n_{{m + \left( {n + D} \right)N_{s} }}^{*} n_{{m + \left( {n + D} \right)N_{s} }} } \right) } \right] \\ & \quad + \left[ {E \left( {n_{{m + nN_{s} }}^{*} n_{{m + \left( {n + D} \right)N_{s} }} n_{{m + nN_{s} }} n_{{m + \left( {n + D} \right)N_{s} }}^{*} } \right.} \right.\left. {\left. {\left. { + n_{{m + nN_{s} }} n_{{m + \left( {n + D} \right)N_{s} }}^{*} n_{{m + nN_{s} }}^{*} n_{{m + \left( {n + D} \right)N_{s} }} } \right){\kern 1pt} } \right]\,} \right\} \\ & = GN_{s} \left( {\sigma_{s}^{2} \sigma_{n}^{2} + \sigma_{n}^{4} / 2} \right) .\\ \end{aligned} $$
(41)

On the other hand, as D < G, some of the above terms will not be zero for n + D = i or i + D = n. Thus,

$$ \begin{aligned} E\left\{ {\left[ { IM \left\{ {r \left( {G,D} \right)} \right\} } \right]^{{{\kern 1pt} 2}} } \right\} & = \frac{1}{4}\left\{ {\sum\limits_{n = 0}^{G - 1} { \sum\limits_{m = 0}^{{N_{s} - 1}} { \left[ {E \left\{ {s_{{m + nN_{s} }}^{*} s_{{m + nN_{s} }} } \right\} E \left( {n_{{m + (n + D)N_{s} }} n_{{m + (n + D)N_{s} }}^{*} } \right.} \right.} } } \right.\left. {\left. { + n_{{m + nN_{s} }} n_{{m + nN_{s} }}^{*} } \right)} \right] \\ & + \left[ {E \left\{ {s_{{m + nN_{s} }} s_{{m + iN_{s} }}^{*} } \right\} E \left( {n_{{m + nN_{s} }}^{*} n_{{m + nN_{s} }} } \right.} \right.\left. {\left. { + n_{{m + (n + D)N_{s} }}^{*} n_{{m + (n + D)N_{s} }} } \right) } \right] \\ & + \left[ {E \left( {n_{{m + nN_{s} }}^{*} n_{{m + (n + D)N_{s} }} n_{{m + nN_{s} }} n_{{m + (n + D)N_{s} }}^{*} } \right.} \right.\left. {\left. {\left. { + n_{{m + nN_{s} }} n_{{m + (n + D)N_{s} }}^{*} n_{{m + nN_{s} }}^{*} n_{{m + (n + D)N_{s} }} } \right)} \right]{\kern 1pt} } \right\} \\ & - \frac{1}{4}\left\{ {\sum\limits_{n = 0}^{G - D - 1} {\sum\limits_{m = 0}^{{N_{s} - 1}} {} \left[ {E\left\{ {s_{{m + nN_{s} }}^{*} s_{{m + (n + D)N_{s} }} } \right\}E\left\{ {n_{{m + (n + D)N_{s} }} n_{{m + (n + D)N_{s} }}^{*} } \right\} \exp \left\{ { - j\left( {\frac{{2\pi DN_{s} \Updelta k}}{N}} \right)} \right\}} \right.} } \right. \\ & + \left. {E\left\{ {s_{{m + nN_{s} }} s_{{m + (n + D)N_{s} }}^{*} } \right\} E\left\{ {n_{{m + (n + D)N_{s} }}^{*} n_{{m + (n + D)N_{s} }} } \right\} \exp \left\{ {j\left( {\frac{{2\pi DN_{s} \Updelta k}}{N}} \right)} \right\}} \right] \\ & + \sum\limits_{i = 0}^{G - D - 1} {\sum\limits_{m = 0}^{{N_{s} - 1}} {} \left[ {E \left\{ {s_{{m + (i + D)N_{s} }}^{*} s_{{m + iN_{s} }} } \right\} E \left\{ {n_{{m + (i + D)N_{s} }} n_{{m + (i + D)N_{s} }}^{*} } \right\}\exp \left\{ {j\left( {\frac{{2\pi DN_{s} \Updelta k}}{N}} \right)} \right\}} \right.} \\ & \left. {\left. { + E \left\{ {s_{{m + (i + D)N_{s} }} s_{{m + iN_{s} }}^{*} } \right\}E\left\{ {n_{{m + (i + D)N_{s} }}^{*} n_{{m + (i + D)N_{s} }} } \right\} \exp \left\{ { - j\left( {\frac{{2\pi DN_{s} \Updelta k}}{N}} \right)} \right\}} \right] } \right\} \\ & = GN_{s} \sigma_{s}^{2} \sigma_{n}^{2} + GN_{s} \sigma_{n}^{4} /2 - (G - D)N_{s} \sigma_{s}^{2} \sigma_{n}^{2} \cos \left( {2\pi DN_{s} \Updelta k/N} \right) \\ & \approx DN_{s} \sigma_{s}^{2} \sigma_{n}^{2} + GN_{s} \sigma_{n}^{4} /2,\,if\, 2\pi DN_{s} \Updelta k/N \ll 1. \\ \end{aligned} $$
(42)

Appendix 3

Note that Δk 1 = Δk 2 = Δk. Using (11) and (38) with (D, G) replaced by \( \left( {D_{1} , G_{1} } \right) \) or \( (D_{2} , G_{2} ), \) we find

$$ \begin{aligned} E\left\{ {\Updelta \tilde{k}_{1} \Updelta \tilde{k}_{2} } \right\} & = \frac{{ - N^{2} }}{{16\pi^{2} D_{1} G_{1} D_{2} G_{2} N_{s}^{4} \sigma_{s}^{4} }}E\left\{ {\sum\limits_{n = 0}^{{G_{1} - 1}} {} \sum\limits_{m = 0}^{{N_{s} - 1}} {\left[ {s_{{m + nN_{s} }}^{*} \exp \left\{ { - j\left[ {\frac{{2\pi (m + nN_{s} ) \Updelta k}}{N} + \theta } \right]} \right\}^{ } } \right.} } \right. \\ & \quad \times \left( {n_{{m + (n + D_{1} )N_{s} }} \exp \left\{ { - j\frac{{2\pi D_{1} N_{s} \Updelta k}}{N}} \right\} - n_{{m + nN_{s} }} } \right) \\ \,\; & \quad + s_{{m + nN_{s} }} \exp \left\{ {j\left[ {\frac{{2\pi (m + nN_{s} ) \Updelta k}}{N} + \theta } \right]} \right\}\left( {n_{{m + nN_{s} }}^{*} - n_{{m + (n + D_{1} )N_{s} }}^{*} \exp \left\{ {j\frac{{2\pi D_{1} N_{s} \Updelta k}}{N}} \right\}} \right) \\ & & \;\;{\kern 1pt} \;\left. { + \left( {n_{{m + nN_{s} }}^{*} n_{{m + (n + D_{1} )N_{s} }} \exp \left\{ { - j\frac{{2\pi D_{1} N_{s} \Updelta k}}{N}} \right\} - n_{{m + nN_{s} }} n_{{m + (n + D_{1} )N_{s} }}^{*} \exp \left\{ {j\frac{{2\pi D_{1} N_{s} \Updelta k}}{N}} \right\}} \right)} \right] \\ & \quad \times \sum\limits_{i = 0}^{{G_{2} - 1}} {} \sum\limits_{l = 0}^{{N_{s} - 1}} {\left[ {s_{{l + iN_{s} }}^{*} \exp \left\{ { - j\left[\frac{{2\pi (l + iN_{s} ) \Updelta k}}{N} + \theta \right]} \right\}\left( {n_{{l + (i + D_{2} )N_{s} }} \exp \left\{ { - j(\frac{{2\pi D_{2} N_{s} \Updelta k}}{N})} \right\} - n_{{l + iN_{s} }} } \right)} \right.} \\ & \;\;\,{\kern 1pt} + s_{{l + iN_{s} }} \exp \left\{ {j\left[ {\frac{{2\pi (l + iN_{s} ) \Updelta k}}{N} + \theta } \right]} \right\}\left( {n_{{l + iN_{s} }}^{*} - n_{{l + (i + D_{2} )N_{s} }}^{*} \exp \left\{ {j\left( {\frac{{2\pi D_{2} N_{s} \Updelta k}}{N}} \right)} \right\}} \right) \\ & \;\;\,\left. {\left. { + \left( {n_{{l + iN_{s} }}^{*} n_{{l + (i + D_{2} )N_{s} }} \exp \left\{ { - j\frac{{2\pi D_{2} N_{s} \Updelta k}}{N}} \right\} - n_{{l + iN_{s} }} n_{{l + (i + D_{2} )N_{s} }}^{*} \exp \left\{ {j\frac{{2\pi D_{2} N_{s} \Updelta k}}{N}} \right\}} \right)} \right] } \right\}. \\ \end{aligned} $$
(43)

Let p denote the value of the expectation \( E\left\{ \cdot \right\} \) on the right hand side of (43). Note that if \( m \ne l, \) \( E \left\{ {s_{{m + nN_{s} }}^{*} s_{{l + iN_{s} }} } \right\} = 0 \) and \( E \left\{ {s_{{m + nN_{s} }} s_{{l + iN_{s} }}^{*} } \right\} = 0 \) for all n and i. Also note that \( D_{1} < D_{2} \). After simplification and some mathematical manipulations, p and \( E\left\{ {\Updelta \tilde{k}_{1} \Updelta \tilde{k}_{2} } \right\} \)can be obtained as

$$ p = N_{s} \left[ {\sum\limits_{n = 0}^{{G_{2} - 1}} { \left( { - 2\sigma_{s}^{2} \sigma_{n}^{2} } \right) + } \sum\limits_{n = 0}^{{G_{2} - D_{1} - 1}} { \left( {2\sigma_{s}^{2} \sigma_{n}^{2} } \right) + } \sum\limits_{i = 0}^{{G_{1} - D_{2} - 1}} { \left( {2\sigma_{s}^{2} \sigma_{n}^{2} } \right) + } \sum\limits_{i = 0}^{{G_{2} - 1}} { \left( { - 2\sigma_{s}^{2} \sigma_{n}^{2} } \right) } } \right] = - 4D_{1} N_{s} \sigma_{s}^{2} \sigma_{n}^{2} . $$
(44)
$$ E\left\{ {\Updelta \tilde{k}_{1} \Updelta \tilde{k}_{2} } \right\} = \frac{{N^{2} \sigma_{n}^{2} }}{{4\pi^{2} D_{2} G_{1} G_{2} N_{s}^{3} \sigma_{s}^{2} }} = \frac{{N^{2} }}{{4\pi^{2} D_{2} G_{1} G_{2} N_{s}^{3} }} \cdot \frac{1}{\text{SNR}}. $$
(45)

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Chung, IH., Yen, MC. On the performance of a maximum likelihood method with delayed correlation for the coarse carrier frequency offset estimation of OFDM signals with multiple preamble symbols. Wireless Netw 16, 641–657 (2010). https://doi.org/10.1007/s11276-008-0159-5

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