Carrier Phase Synchronization through the Correlation Principle

Observation window that determines the start of correlation

In another article on correlation, we said that correlation is the Master Algorithm for a digital communication system, not only for data detection but for parameter estimation of various kinds as well. We applied it to derive the matched filter for optimal detection. Here, we apply the principle of maximum correlation for solving our phase synchronization problem. The discussion in this context usually considers QPSK modulation but the extension to higher-order modulation schemes is a straightforward task. A similar framework is used for finding and compensating for other distortions as well such as carrier frequency offset and symbol timing offset, the examples of which we will see later.

Correlation with what?


In order to implement correlation, one signal is obviously the Rx signal r(nT_S). It is correlated with a clean version of what is expected at the Rx, the expected signal template. That is the most logical solution to detect the signal and estimate the desired parameters. For example, in case of a carrier phase offset, the Rx signal r(nT_S) will be correlated with a perfect Tx signal but rotated in phase by an unknown offset. This procedure then generates an algorithm to estimate that phase offset.

In the presence of noise, the received and sampled signal is

(1)   \begin{equation*}     \begin{aligned}         r(nT_S) &= s(nT_S) + \textmd{noise} \\                 &= v_I(nT_S) \sqrt{2}\cos \left(2\pi \frac{k_C}{N}n + \theta_\Delta\right) - \\&\hspace{1in} v_Q(nT_S) \sqrt{2}\sin \left(2\pi \frac{k_C}{N}n + \theta_\Delta\right) + \textmd{noise}     \end{aligned}     \end{equation*}

where k_C (in a DFT size of N) corresponds to carrier frequency F_C and v_I(nT_S) and v_Q(nT_S) are the inphase and quadrature waveforms (each time we define the inphase and quadrature waveforms, there is a reason behind choosing index i or index m that becomes clear as we progress through that particular derivation).

    \begin{equation*}       \begin{aligned}         v_I(nT_S)\: &= \sum _{m} a_I[m] p(nT_S-mT_M) \\         v_Q(nT_S) &= \sum _{m} a_Q[m] p(nT_S-mT_M)       \end{aligned}     \end{equation*}

The correlation between r(nT_S) and its expected template s(nT_S) is defined as

(2)   \begin{align*}         \textmd{corr}[j] &= r(nT_S) ~\heartsuit~ s(nT_S)\nonumber \\                   &= \sum \limits _{n = -\infty} ^{\infty} r(nT_S) s(nT_S-j)      \end{align*}

We face three problems in implementing the above relation.

  • [1. Observation interval] The above sum is computed from -\infty to +\infty, which implies that we could theoretically continue computing this correlation with every received sample r(nT_S). However, the useful component of the received signal r(nT_S) is given by a span of N_{0} symbols while every sample outside this interval is noise only. The time duration of such N_0 symbols is

    (3)   \begin{equation*}                 T_{0} = N_{0} T_M             \end{equation*}

    Thus, the finite observation interval is given by

        \begin{equation*}               0 \quad \rightarrow \quad \underbrace{T_{0}}_{\textmd{seconds}} = \underbrace{N_0}_{\textmd{symbols}} \cdot T_M = \underbrace{N_{0}\cdot L}_{\textmd{samples}} \cdot T_S             \end{equation*}

    where L is samples/symbol, or T_M/T_S. Therefore, the observation window consists of n=0,1,\cdots,LN_0-1 samples. It is understood in this calculation that the group delay arising from pulse shaping and matched filtering on either side of the transmission sequence has been taken care of by the Rx.

  • [2. Start of frame] The second problem is to determine the starting point of the useful component in a sampled signal, as many of its initial samples are noise only. It is the job of a frame synchronization unit to determine the symbol level boundaries of the Rx signal, while the symbol timing synchronization block establishes the optimal sampling instant within a symbol. This will be explained in detail in another article but Figure below illustrates this alignment problem for a visual clarification.
  • [3. Distortion at edges] Due to the pulse shape spreading beyond a single symbol and actually extending to G symbols in each direction, there is a distortion introduced at the edges of the observation interval T_0 in the correlation result. However, it can be ignored if T_0 is sufficiently long.

Observation window that determines the start of correlation

We also assume downconversion with perfect frequency recovery, i.e., Tx and Rx oscillators have the same frequency F_C, to emphasize on phase recovery problem. After perfect downconversion, our expected signal template is the complex sequence x(nT_S), with which the correlation of r(nT_S) needs to be implemented. The overall setup is drawn in Figure below.

Block diagram for phase recovery

Continuing from the article on effect of a phase offset and ignoring the double frequency terms (the higher frequency terms as a product of downconversion process are either filtered by an analog filter in the RF frontend or a digital filter in the digital frontend and furthermore, the matched filter also suppresses such terms, as it is also a lowpass filter due to the pulse shape originally designed to restrict the spectral contents of the wireless signal), the downconverted complex signal is expressed as

(4)   \begin{equation*}       \begin{aligned}         x_I(nT_S)\: &= v_I(nT_S) \cos \theta_\Delta - v_Q(nT_S) \sin\theta_\Delta       \\         x_Q(nT_S) &= v_Q(nT_S) \cos \theta_\Delta + v_I(nT_S) \sin\theta_\Delta       \end{aligned}     \end{equation*}

Next, the correlation between the Rx signal r(nT_S) and our expected template signal x(nT_S) is

(5)   \begin{align*}         \textmd{corr}[j] &= r(nT_S) ~\heartsuit~ x(nT_S)\nonumber \\                   &= \sum \limits _{n = 0} ^{LN_0-1} r(nT_S) x^*(nT_S-j)      \end{align*}

where the conjugate arises due to the signals being complex. Having determined j=0 through frame synchronization block, we can write Eq (5) as

    \begin{equation*}         \textmd{corr}[0] = \sum \limits _{n = 0} ^{LN_0-1} r(nT_S) x^*(nT_S)     \end{equation*}

Using the complex multiplication rule, the actual computations can be written as

(6)   \begin{equation*}       \begin{aligned}         \textmd{corr}_I[0]\: &= \sum \limits _{n = 0} ^{LN_0-1} r_I(nT_S) x_I(nT_S) + \sum \limits _{n=0} ^{LN_0-1} r_Q(nT_S) x_Q(nT_S) \\         \textmd{corr}_Q[0] &= \sum \limits _{n = 0} ^{LN_0-1} r_Q(nT_S) x_I(nT_S) - \sum \limits _{n=0} ^{LN_0-1} r_I(nT_S) x_Q(nT_S)       \end{aligned}     \end{equation*}

Recall that two complex signals are most similar when the inphase part of their correlation at time zero is maximum. Consequently, we ignore the Q component above and plug the definition of x(nT_S) from Eq (4) above.

    \begin{align*}         \begin{aligned}             \textmd{corr}_I[0]\: &= \sum \limits _{n = 0} ^{LN_0-1} r_I(nT_S) \Big(v_I(nT_S) \cos \theta_\Delta - v_Q(nT_S) \sin\theta_\Delta\Big) + \\&\hspace{1in}             \sum \limits _{n=0} ^{LN_0-1} r_Q(nT_S) \Big(v_Q(nT_S) \cos \theta_\Delta + v_I(nT_S) \sin\theta_\Delta\Big)         \end{aligned}     \end{align*}

Substituting the expressions for v_I(nT_S) and v_Q(nT_S) yields

(7)   \begin{equation*}         \begin{aligned}             \textmd{corr}_I[0]\: &= \sum \limits _{n = 0} ^{LN_0-1} r_I(nT_S) \Big(\sum _{m} a_I[m] p(nT_S-mT_M) \cos \theta_\Delta -  \\&\hspace{1.2in}\sum _{m} a_Q[m] p(nT_S-mT_M) \sin\theta_\Delta\Big) \nonumber\\&\hspace{0.2in}+             \sum \limits _{n=0} ^{LN_0-1} r_Q(nT_S) \Big(\sum _{m} a_Q[m] p(nT_S-mT_M) \cos \theta_\Delta +\\&\hspace{1.2in} \sum _{m} a_I[m] p(nT_S-mT_M) \sin\theta_\Delta\Big)         \end{aligned}     \end{equation*}

Ignoring the distortion at the edges of the summation, the above relation can be simplified recalling the fact that p(nT_S) are the samples of a square-root Nyquist pulse with support -LG \le n \le LG samples.

    \begin{equation*}         \begin{aligned}             \textmd{corr}_I[0]\: &= \sum \limits _{m = 0} ^{N_0-1}\Bigg\{ a_I[m] \underbrace{\sum \limits _{n=(m-G)L} ^{(m+G)L} r_I(nT_S) p(nT_S-mT_M)}_{\textmd{Inphase matched filter output}} \cos \theta_\Delta -  \\&\hspace{1.2in} a_Q[m] \underbrace{\sum \limits _{n=(m-G)L} ^{(m+G)L} r_I(nT_S) p(nT_S-mT_M)}_{\textmd{Inphase matched filter output}} \sin\theta_\Delta \\&\hspace{0.2in}+             a_Q[m]\underbrace{\sum \limits _{n=(m-G)L} ^{(m+G)L} r_Q(nT_S) p(nT_S-mT_M)}_{\textmd{Quadrature matched filter output}} \cos \theta_\Delta +\\&\hspace{1.1in} a_I[m] \underbrace{\sum \limits _{n=(m-G)L} ^{(m+G)L} r_Q(nT_S) p(nT_S-mT_M)}_{\textmd{Quadrature matched filter output}} \sin\theta_\Delta\Bigg\}         \end{aligned}     \end{equation*}

Notice that after the received signal r(nT_S) is downconverted to baseband, it is correlated with a similar pulse shape to the Tx, p(nT_S). As we saw in the details of matched filter, another way of implementing this correlation is through filtering the signal with its flipped version, p(-nT_S) (convolution operation in a filter flips p(-nT_S) again back to p(nT_S) and the operation becomes correlation). This is known as matched filtering. Clearly, after summing over the sample index n, we are left with a signal with symbol rate spaced samples at times mT_M.

Owing to the above description, we can identify the following terms as matched filtered outputs, one in the inphase and other in the quadrature arm.

(8)   \begin{equation*}           \begin{aligned}                 z_I(mT_M)\: &= \sum \limits _{n=(m-G)L} ^{(m+G)L} r_I(nT_S) p(nT_S-mT_M)\\                 z_Q(mT_M) &= \sum \limits _{n=(m-G)L} ^{(m+G)L} r_Q(nT_S) p(nT_S-mT_M)         \end{aligned}     \end{equation*}

Now the expression for inphase part of the correlation is simplified as

(9)   \begin{equation*}     \begin{aligned}          \textmd{corr}_I[0] &= \sum \limits _{m = 0} ^{N_0-1} a_I[m] \Bigg\{z_I(mT_M) \cos \theta_\Delta +z_Q(mT_M) \sin \theta_\Delta\Bigg\} \\ &\hspace{.1in}+\sum \limits _{m = 0} ^{N_0-1} a_Q[m] \Bigg\{z_Q(mT_M) \cos \theta_\Delta - z_I(mT_M) \sin \theta_\Delta\Bigg\}     \end{aligned}     \end{equation*}

The above expression is important. As we proceed, this Eq (9) will form the basis of many types of phase estimators, both in feedforward and feedback schemes.

A question at this stage is: starting from correlation, what have we achieved so far? While not clear as of now, we are on our way to solve the fundamental problem of the synchronization process. In a few other articles, we will discuss data-aided, decision-directed and non-data-aided techniques for phase synchronization.

2 Comments

  1. Graham Cottew

    I don’t remember this from our talks – perhaps it’s coming up @zenmountainman

    Reply
    1. Qasim Chaudhari (Post author)

      Yes Graham, carrier phase synchronization is what I am going to talk about in the near future.

      Reply

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