Low Complexity Time Synchronization Algorithm for OFDM Systems with Repetitive Preambles

Abstract

In this paper, a new time synchronization algorithm for OFDM systems with repetitive preamble is proposed. This algorithm makes use of coarse and fine time estimation; the fine time estimation is performed using a cross-correlation similar to previous proposals in the literature, whereas the coarse time estimation is made using a new metric and an iterative search of the last sample of the repetitive preamble. A complete analysis of the new metric is included, as well as a wide performance comparison, for multipath channel and carrier frequency offset, with the main time synchronization algorithms found in the literature. Finally, the complexity of the VLSI implementation of this proposal is discussed.

Appendices

Appendix A. Statitical Properties of the Timing Metric at the Repetitive Part of the Preamble

Each received sample is composed of an OFDM signal and a noise component $r(n) = {s_{{SS}}}(n) + w(n)$, where s _SS (n) are samples of the received preamble during the repetitive part (short symbols) and w(n) are samples of complex additive white Gaussian noise with variance $\sigma_w^2$ and zero mean. Using the common approach of calculating the mathematic expectation of ${R_{{SS}}}(n)$, the input signal power has a mean:

$$ E\left\{ {{R_{{SS}}}(n)} \right\} = \sum\limits_{{k = - L}}^{{L - 1}} {{{\left| {\,{s_{{SS}}}\left( {n + k} \right)\,} \right|}^2}} + E\left\{ {\sum\limits_{{k = - L}}^{{L - 1}} {{{\left| {\,w\left( {n + k} \right)\,} \right|}^2}} } \right\} = {\varepsilon_s}(n) + 2L\,\sigma_w^2{,} $$

where ${\varepsilon_s}(n)$ is the estimated local energy of the received preamble. As the short symbols are periodic with L, we can write: ${\varepsilon_s}(n) = 2{\varepsilon_{{SS}}}$, where ε _SS is the energy of a period of the received preamble. So, during the repetitive part of the preamble the mean of R is a constant due to the periodicity.

Another method to obtain this result is to take into account that r(n) is formed by noise (characterized as a complex Gaussian random variable) added to the received preamble samples (a deterministic signal). When the sum of the squared modulus of r(n) is obtained, we get a non-central chi-squared random variable represented by symbol $\chi_p^2$, where p represents the degrees of freedom (number of real random Gaussian variables, in this case p = 2 L). A central chi-squared random variable G obtained as $G = \sum\nolimits_{{i = 1}}^p {{{\left( {{{{{V_i}}} \left/ {{{\sigma_v}}} \right.}} \right)}^2}}$, where V _i are real random Gaussian variables with variance $\sigma_v^2$ and zero mean, has a mean of $p\sigma_v^2$ and a variance of $2p\sigma_v^4$ [27]. When the real Gaussian variables have mean μ _i , a non-central chi-squared random variable is generated with mean $(p + \lambda )\,\sigma_v^2$ and variance $2(p + 2\lambda )\,\sigma_v^4$, with $\lambda = {\sum\nolimits_{{i = 1}}^p {\left( {{{{{\mu_i}}} \left/ {{{\sigma_v}}} \right.}} \right)}^2}$. In our case, the real and imaginary components of the Gaussian noise have variance ${{{\sigma_w^2}} \left/ {2} \right.}$ and their mean is the value of the real or imaginary part of the received preamble: ${\mu_i} = {s_R}(i)$ or ${\mu_i} = {s_I}(i)$. Then, if we calculate parameter λ we obtain:

$$ \lambda = \sum\limits_{{i = - L}}^{{L - 1}} {\left( {\frac{{s_R^2(i)}}{{{{{\sigma_w^2}} \left/ {2} \right.}}} + \frac{{s_I^2(i)}}{{{{{\sigma_w^2}} \left/ {2} \right.}}}} \right)} = \frac{{2{\varepsilon_{{SS}}}}}{{{{{\sigma_w^2}} \left/ {2} \right.}}}. $$

So, the mean of R(n) can be obtained as:

$$ E\left\{ {{R_{{SS}}}} \right\} = (p + \lambda )\frac{{\sigma_w^2}}{2} = \left( {2\cdot (2L) + \frac{{{\varepsilon_s}(n)}}{{{{{\sigma_w^2}} \left/ {2} \right.}}}} \right)\frac{{\sigma_w^2}}{2} = 2{\varepsilon_{{SS}}} + 2L\,\sigma_w^2. $$

By this method the variance can be easily obtained as:

$$ {\rm var} \left\{ {{R_{{SS}}}} \right\} = 4{\varepsilon_{{SS}}}\,\sigma_w^2 + 2L\,\sigma_w^4. $$

Next, we study the numerator when it is applied to samples from the short symbols, we call this P _SS (n). In this case, M must be chosen in such a way that assures that: $s(n) = {s_{{SS}}}(n - L) = {s_{{SS}}}(n - 2L) = \cdots = {s_{{SS}}}(n - M \cdot L)$, then the signal component inside the squared modulus cancels out and this leaves only noise terms: $w'(n) = w(n) - {{{\left( {\sum\nolimits_{{m = 2}}^{{M + 1}} {w(n - L\cdot m} } \right)}} \left/ {M} \right.}$, where $w\prime (n)$ is a white Gaussian random variable with variance $\sigma_{{w'}}^2 = {{{(M + 1)\,\sigma_w^2}} \left/ {M} \right.}$. Using the same reasoning as before, the numerator is the sum of a squared modulus of Gaussian random variables and the obtained signal is a central chi-squared random variable with mean and variance:

$$ E\left\{ {{P_{{SS}}}} \right\} = 2L\,\frac{{\sigma_{{w'}}^2}}{2} = L\frac{{M + 1}}{M}\sigma_w^2{,} $$

$$ {\rm var} \left\{ {{P_{{SS}}}} \right\} = 2\left( {2L} \right){\left( {\frac{{\sigma_{{w'}}^2}}{2}} \right)^2} = L{\left( {\frac{{M + 1}}{M}} \right)^2}\sigma_w^4. $$

Finally, we need to calculate the covariance between P _SS and R _SS . The covariance is:

$$ {\rm cov} \left\{ {{P_{{SS}}},{R_{{SS}}}} \right\} = E\left\{ {{P_{{SS}}}\,{R_{{SS}}}} \right\} - E\left\{ {{P_{{SS}}}} \right\}\;E\left\{ {{R_{{SS}}}} \right\}. $$

After some mathematical manipulations, the first term is given by:

$$ E\left\{ {{P_{{SS}}}\,{R_{{SS}}}} \right\} = {\varepsilon_{{SS}}}L\,\sigma_w^2 + (2{L^2} + L)\,\sigma_w^4 + {\varepsilon_{{SS}}}\,L\,\frac{{\sigma_w^2}}{M} + 2{L^2}\frac{{\sigma_w^4}}{M}. $$

Using the mean of R and P calculated previously we obtain:

$$ {\rm cov} \left\{ {{P_{{SS}}},{R_{{SS}}}} \right\} = L\sigma_w^4. $$

Appendix B. Statistical Properties of the Timing Metric after the Repetitive Part of the Preamble

At the first sample of the GI the subtraction in (4) can be expressed as $u(n) = {r_{{GI}}}(n) - {r_{{SS}}}(n)$. We can decompose it in noise and a combination of samples from the preamble: $u(n) = z(n) + w'(n)$, where the noise is as in Appendix A:$w'(n) = w(n) - {{{\left( {\sum\nolimits_{{m = 1}}^M {w(n - L\cdot m} } \right)}} \left/ {M} \right.}$; and the other term is: $z(n) = {s_{{GI}}}(n) - {s_{{SS}}}(n)$, where ${s_{{SS}}}(n)$ is the received repetitive preamble and ${s_{{GI}}}(n)$ is the received guard interval. To obtain $E\left\{ {{P_{{GI}}}} \right\}$, that is, the mean of the numerator of the time metric at n = n _GI , we can use the same reasoning as in Appendix A, when we estimated $E\left\{ {{R_{{SS}}}} \right\}$, since signal u(n) is composed of a sum of a deterministic signal plus a white Gaussian noise with variance $\sigma_{{w'}}^2 = \sigma_w^2\left( {{{{M + 1}} \left/ {M} \right.}} \right)\,$. Therefore we obtain:

$$ E\left\{ {{P_{{GI}}}} \right\} = {\varepsilon_z} + L\left( {\frac{{M + 1}}{M}} \right)\,\sigma_w^2{,} $$

$$ {\rm var} \left\{ {{P_{{GI}}}} \right\} = 2{\varepsilon_z}\left( {\frac{{M + 1}}{M}} \right)\,\sigma_w^2 + L{\left( {\frac{{M + 1}}{M}} \right)^2}\,\sigma_w^4{,} $$

where ε _z is:

$$ \begin{gathered} {\varepsilon_z} = \sum\limits_{{n = 0}}^{{L - 1}} {{{\left| {z(n)} \right|}^2}} = \sum\limits_{{n = 0}}^{{L - 1}} {{{\left| {{s_{{GI}}}(n) - {s_{{SS}}}(n)} \right|}^2}} \\ = \sum\limits_{{n = 0}}^{{L - 1}} {{{\left| {{s_{{GI}}}(n)} \right|}^2}} + \sum\limits_{{n = 0}}^{{L - 1}} {{{\left| {{s_{{SS}}}(n)} \right|}^2}} - 2\sum\limits_{{n = 0}}^{{L - 1}} {{\rm Re} \left\{ {{s_{{GI}}}(n)\,s_{{SS}}^{*}(n)} \right\}} \\ = {\varepsilon_{{GI}}} + {\varepsilon_{{SS}}} - 2C. \\ \end{gathered} $$

And as the mean of R(n) includes part of SS and part of GI becomes:

$$ E\left\{ {{R_{{GI}}}} \right\} = {\varepsilon_{{SS}}} + {\varepsilon_{{GI}}} + 2L\,\sigma_w^2{,} $$

and the variance:

$$ {\rm var} \left\{ {{R_{{GI}}}} \right\} = 2({\varepsilon_{{SS}}} + {\varepsilon_{{GI}}})\,\sigma_w^2 + 2L\,\sigma_w^4. $$

The covariance between P _GI and R _GI can be obtained in a similar way to the one calculated in Appendix A giving:

$$ {\rm cov} \left\{ {{P_{{GI}}},{R_{{GI}}}} \right\} = L\sigma_w^4 + 2\sigma_w^2\left( {{\varepsilon_{{GI}}} - C} \right). $$

Appendix C. Effect of the Carrier Frequency Offset in the Time Metric

The signal model that includes the CFO was defined as: $r(n) = s(n)\;{e^{{j\omega \,n}}} + w(n)$. As the CFO affects the signal term, during the repetitive part of the preamble the mean of (4) can be written as:

$$ E\left\{ {{P_{{SS}}}(n)} \right\} = \sum\limits_{{k = 0}}^{{L - 1}} {{{\left| {\;{s_{{SS}}}(n + k)\;{e^{{j\omega (n + k)}}} - \frac{1}{M}\sum\limits_{{m = 1}}^M {{s_{{SS}}}\left( {n + k - L \cdot m} \right)} \;{e^{{j\omega (n + k - Lm)}}}\;} \right|}^2}} + L\frac{{M + 1}}{M}\sigma_w^2, $$

where the signal term does not cancel out as before. Now, the mean can be rewritten, due to the periodicity of s _SS (n), as:

$$ E\left\{ {{P_{{SS}}}(n)} \right\} = \sum\limits_{{k = 0}}^{{L - 1}} {{{\left| {\;{s_{{SS}}}(n + k)\;{e^{{j\omega (n + k)}}}\;\left( {1 - \frac{1}{M}\sum\limits_{{m = 1}}^M {{e^{{ - j\omega Lm}}}} \;} \right)\;} \right|}^2}} + L\frac{{M + 1}}{M}\sigma_w^2. $$

The sum of complex exponentials can be expressed as:

$$ \sum\limits_{{m = 1}}^M {{e^{{ - j\omega Lm}}}} = \frac{{\sin (M\omega L/2)}}{{\sin (\omega L/2)}}\;{e^{{ - j\omega L\left( {1 + \frac{M}{2}} \right)}}}{;} $$

then, the signal term is given by:

$$ \begin{gathered} \sum\limits_{{k = 0}}^{{L - 1}} {{{\left| {\;{s_{{SS}}}(n + k)\;{e^{{j\omega (n + k)}}}\;\left( {1 - \frac{1}{M}\;\frac{{\sin (M\omega L/2)}}{{\sin (\omega L/2)}}\;{e^{{ - j\omega L\left( {1 + \frac{M}{2}} \right)}}}} \right)\;} \right|}^2}} = \\ = \sum\limits_{{k = 0}}^{{L - 1}} {{{\left| {\;{s_{{SS}}}(n + k)\;\;} \right|}^2}} \;{\left| {{e^{{j\omega (n + k)}}}} \right|^2}\;{\left| {\;\;1 - \frac{1}{M}\;\frac{{\sin (M\omega L/2)}}{{\sin (\omega L/2)}}\;{e^{{ - j\omega L\left( {1 + \frac{M}{2}} \right)}}}\;} \right|^2} = \\ = \gamma (\omega, M,L)\;\,{\varepsilon_{{SS}}}\,\,, \\ \end{gathered} $$

where ε _SS is the energy of a period from the received preamble and $\gamma (\omega, L,M)$ is a CFO factor that is given by

$$ \gamma (\lambda, L,M) = {\left| {1 - \frac{1}{M}\;\frac{{\sin ({{{M\omega L}} \left/ {2} \right.})}}{{\sin ({{{\omega L}} \left/ {2} \right.})}}\,{e^{{ - j\omega L\left( {1 + \frac{M}{2}} \right)}}}} \right|^2}. $$

Finally, the mean of the estimator is:

$$ E\left\{ {{P_{{SS}}}} \right\} = \gamma (\lambda, L,M)\,\,{\varepsilon_{{SS}}} + L\frac{{M + 1}}{M}\sigma_w^2. $$

To evaluate the effect of the CFO on the mean of (4) at n _GI , we must calculate how the CFO affects the signal term:

$$ \hat{z}(n) = {s_{{GI}}}(n)\,{e^{{j\omega \,n}}} - \frac{1}{M}\sum\limits_{{k = 1}}^M {{s_{{SS}}}(n - kL)\,{e^{{j\omega \,(n - kL)}}}} . $$

Again we consider that all the terms from the repetitive preamble ${s_{{SS}}}(n - kL)$ are equal, then

$$ \hat{z}(n) = {s_{{GI}}}(n)\,{e^{{j\omega \,n}}} - {s_{{SS}}}(n)\,{e^{{j\omega \,n}}}\frac{1}{M}\sum\limits_{{k = 1}}^M {\,{e^{{ - j\omega \,kL}}}} {,} $$

where the sum of complex exponentials is:

$$ \kappa (\omega, M,L) = \frac{1}{M}\sum\limits_{{m = 1}}^M {{e^{{ - j\omega Lk}}}} = \frac{1}{M}\frac{{\sin (M\omega L/2)}}{{\sin (\omega L/2)}}{e^{{ - j\omega L\left( {1 + \frac{M}{2}} \right)}}}. $$

If we calculate the energy of the signal term:

$$ \begin{gathered} {\varepsilon_{{\hat{z}}}} = \sum\limits_{{n = 0}}^{{L - 1}} {{{\left| {\hat{z}(n)} \right|}^2}} = \sum\limits_{{n = 0}}^{{L - 1}} {{{\left| {{s_{{GI}}}(n)\,{e^{{j\omega \,n}}} - {s_{{SS}}}(n)\,{e^{{j\omega \,n}}}\kappa (\omega, M,L)} \right|}^2}} \, \\ = \sum\limits_{{n = 0}}^{{L - 1}} {{{\left| {{s_{{GI}}}(n)\,{e^{{j\omega \,n}}}} \right|}^2}} + {\kappa^2}(\omega, M,L)\sum\limits_{{n = 0}}^{{L - 1}} {{{\left| {{s_{{SS}}}(n)\,{e^{{j\omega \,n}}}} \right|}^2}} - 2\kappa (\omega, M,L)\sum\limits_{{n = 0}}^{{L - 1}} {{\rm Re} \left\{ {{s_{{GI}}}(n) - s_{{SS}}^{*}(n)} \right\}} \\ = {\varepsilon_{{GI}}} + {\kappa^2}(\omega, M,L)\,{\varepsilon_{{SS}}} - 2\kappa (\omega, M,L)\,C. \\ \end{gathered} $$

Then, the mean of (4) at n _GI is:

$$ E\left\{ {{P_{{GI}}}} \right\} = {\varepsilon_{{\hat{z}}}} + L\left( {\frac{{M + 1}}{M}} \right)\sigma_w^2. $$