For reliable reception of transmitted data in digital communication systems using quadrature amplitude modulation (QAM) schemes, one of the critical signal processing blocks is an accurate carrier frequency offset estimator and tracker. Failure to either accurately provide a good initial estimate of carrier phase and/or frequency offset or to track slowly varying phase and/or frequency offset results in poor fidelity of receiving the transmitted bits. While many phase-locked loop (PLL) based and Kalman filter-based phase/frequency offset estimation and tracking algorithms have been proposed, they are typically highly sensitive to random noise and other sources of distortion.
Embodiments are described in which carrier phase offset and frequency offset estimation and tracking (e.g., Kalman filter-based) with initial state search and hard-decision trust algorithms for M-QAM modulated signals are employed.
In one aspect the present invention provides a method for determining coarse carrier phase and frequency offsets of an initial block of received symbols modulated using an M quadrature amplitude modulation (M-QAM) symbol map. The method includes creating a grid of discrete candidate phase offset values. The method also includes, for each candidate phase offset value in the grid: applying the candidate phase offset value to each of the M-QAM symbols, applying a respective hard decision from the symbol map to each of the candidate phase offset value-applied M-QAM symbols, and computing a figure of merit for the candidate phase offset value based on the candidate phase offset value-applied M-QAM symbols and their respective hard decisions. The method also includes selecting the discrete candidate phase offset value from the grid having the best figure of merit as an initial phase offset estimate. The method also includes computing an initial frequency offset estimate using the M-QAM symbols updated with the initial phase offset estimate, their respective hard decisions, and an approximation of the complex exponential function.
In another aspect the present invention provides a method for tracking carrier phase and frequency offsets associated with a series of blocks of received symbols modulated using an M quadrature amplitude modulation (M-QAM) symbol map. The method includes, for each M-QAM symbol of a current block, setting a binary trust weight for the M-QAM symbol based on comparison of a computed parameter with a threshold. The method also includes using the binary trust weights to compute a phase offset error and a frequency offset error associated with the current block. Each of the M-QAM symbols having a zero-valued binary trust weight is non-contributory in the computation of the phase and frequency offset errors.
In yet another aspect the present invention provides a non-transitory computer-readable medium having instructions stored thereon that when executed by a processor, cause the processor to perform operations of a method. The method is for determining coarse carrier phase and frequency offsets of an initial block of received symbols modulated using an M quadrature amplitude modulation (M-QAM) symbol map. The operations include creating a grid of discrete candidate phase offset values. The operations also include, for each candidate phase offset value in the grid: applying the candidate phase offset value to each of the M-QAM symbols, applying a respective hard decision from the symbol map to each of the candidate phase offset value-applied M-QAM symbols, and computing a figure of merit for the candidate phase offset value based on the candidate phase offset value-applied M-QAM symbols and their respective hard decisions. The operations also include selecting the discrete candidate phase offset value from the grid having the best figure of merit as an initial phase offset estimate. The operations also include computing an initial frequency offset estimate using the M-QAM symbols updated with the initial phase offset estimate, their respective hard decisions, and an approximation of the complex exponential function. Or, the method is for tracking carrier phase and frequency offsets associated with a series of blocks of received symbols modulated using an M quadrature amplitude modulation (M-QAM) symbol map. The operations include for each M-QAM symbol of a current block, setting a binary trust weight for the M-QAM symbol based on comparison of a computed parameter with a threshold. The operations also include using the binary trust weights to compute a phase offset error and a frequency offset error associated with the current block. Each of the M-QAM symbols having a zero-valued binary trust weight is non-contributory in the computation of the phase and frequency offset errors.
Generally, the success of a phase/frequency offset tracker is contingent upon having a good initial estimate of the carrier phase offset and frequency offset of the received signal. A poor initial estimate, as seen in many conventional solutions, typically results in complete failure of the phase/frequency offset tracking algorithms. Embodiments are described of an initial phase/frequency offset estimation method that ensures the tracking algorithms following the initial state estimator work reliably even under noisy conditions. This is achieved by performing a suitable grid search for initial phase estimation that ensures that most hard decisions on the symbols in the first block are correct.
Additionally, the carrier phase/frequency offset tracking algorithm of the described embodiments is much more robust against random noise in the received QAM modulated signal. The phase tracker achieves the robustness by incorporating a binary trust-factor of either 0 or 1 for each received symbol in the block of QAM symbols being used for estimating the updated value of the carrier phase offset and frequency offset.
For efficient computation of closed form solutions of the phase/frequency offset parameters, a complex exponential function is approximated as a cubic polynomial for a domain that is restricted to a set of pre-determined values determined by the M-QAM scheme which depends on the maximum supported carrier frequency offset for the M-QAM symbol map. The maximum supported carrier frequency offset for the M-QAM symbol map is dependent upon the number of symbols in a block. Embodiments use a cubic approximation of the complex exponential by restricting the domain of the complex exponential to pre-determined values based on the maximum supported carrier frequency offset for a given modulation scheme. The approach to cubic approximation by exploiting the knowledge of the supported domain of the approximated function results in a much lower approximation error than, for example, using a third order Taylor series approximation (see
To aid the in the description of the embodiments described herein, a signal model will now be developed that is similar to the formulation by Takashi Inoue and Shu Namiki, “Carrier recovery for M-QAM signals based on a block estimation process with Kalman filter.” Optics express 22.13 (2014): 15376-15387.
Let the received symbols be modeled per equation (1) (ignoring noise and other distortion).
rk,n=sk,n exp jϕk,n (1)
In equation (1), rk,n is the nth received symbol in the kth block, where n∈[−L, L], and the number of symbols in a block is 2 L+1; sk,n is the corresponding transmitted clean symbol; and ϕk,n is the phase rotation in the received symbol, relative to the ideal position of the symbol. In the present disclosure, generally when expressing ranges of numbers, the notation [A, B] denotes the set of all numbers between A and B, including A and B as well; (A, B] denotes the set of all numbers between A and B, excluding A but including B; [A, B) denotes the set of all numbers between A and B, including A but excluding B; and (A, B) denotes the set of all numbers between A and B, excluding A and B.] Equation (2) expresses the phase rotation in the received symbol. In the present disclosure, the units of phase, θ, and frequency, ω, terms, when used as a part of the argument of a complex exponential have units of radian and radians per symbol, respectively, unless otherwise specified.
ϕk,n=θk+nωk;n∈[−L,L] (2)
K+1 is the total number of blocks.
Assume that from one block to the next, the phase offset θk and frequency offset ωk associated with the block k evolves according to equation (3A):
where nf is random phase noise and models slow variation of frequency offset over received blocks of symbols.
Equivalently, in terms of matrix variables, the state evolves according to equation (3B):
The state evolution model in equation (3) assumes that blocks are non-overlapping. For tracking fast changes in the phase-frequency offset state, the evolution model can be changed as according to equation (4A):
Equivalently, in terms of matrix variables, the state evolves according to equation (4B):
xk+1=Fxk+nS (4B)
where, the matrix F is re-defined according to equation (4C):
The value assigned to V controls the number of overlapping symbols in consecutive blocks. For example, setting V=L+1 means consecutive blocks have no overlap. On the other extreme, V=1 means that L symbols are common between two consecutive blocks.
Let the noisy observations of the state vector xk be yk such that:
Referring now to
Referring now to
The first stage 402 processes the 0th block of QAM symbols to obtain a good estimate of the carrier phase and frequency offsets for the block. The first stage 402 is described in pseudocode below as Algorithm 1. The embodiments use a cubic approximation of a complex exponential, as expressed in equation (6),
e−jωn≈α0+α1ωn+α2ω2n2+α3ω3n3 (6)
by restricting the domain of the complex exponential to pre-determined values based on the maximum supported carrier frequency offset for a given modulation scheme, such as shown in Table 1 above. Table 2 below lists, as an example, values of the polynomial coefficients of the cubic approximation of the complex exponential of equation (6) for different M-QAM constellations with block size of 17, i.e., L=8.
Algorithm 1 performs estimation of phase and frequency offsets of the zeroth block of received symbols and may also be referred to as “Initial Phase and Frequency Offsets Estimation,” where the term “Initial” indicates that the estimated quantities are for the zeroth block of received QAM symbols. Advantageously, Algorithm 1 discretizes candidate phase offset values, as described in more detail below, e.g., with respect to Derivation 1, which may reduce the computational intensity associated with estimating the initial phase and frequency offsets, i.e., the phase and frequency offset estimates for the zeroth block of M-QAM symbols. In the present disclosure, a quantity with a superscript “*” denotes the complex conjugate of a complex number. For example, (2+j1)*=2−j1, where j is equal to √{square root over (−1)}.
The second stage 404, forward phase and frequency offset tracking, processes all the blocks of received QAM symbols in chronological order, performing Kalman filter-based carrier phase and frequency offset tracking in the forward direction. The second stage 404 is described in pseudocode below as Algorithm 2. The phase tracker in the second stage 404 incorporates a binary trust-factor of either 0 or 1, for each received symbol in the block of QAM symbols being used for estimating the updated values of the carrier phase and frequency offsets.
The third stage 406, pre-processing for reverse phase and frequency offset tracking, processes the symbols for reverse tracking. The third stage 406 is described in pseudocode below as Algorithm 3.
The last stage 408, reverse phase and frequency offset tracking, processes the symbols in the reverse direction. The fourth stage 408 is described in pseudocode below as Algorithm 4.
Derivations of certain aspects of the algorithms described above are now presented.
Derivation 1—Phase and Frequency Offsets Estimation and Correction on the Zeroth Block of Received QAM Symbols
Derivation Outline:
The details of steps 1, 2 and 3 are now described.
Step 1: Coarse Phase Offset Estimation and Correction
Since the transmit QAM constellation has a square geometry, the range of coarse phase offset to be estimated reduces from [−180°, +180°) to [−45°, +45°).
The true phase offset can be any real number in the continuum between −45° to +45°. However, at this step the goal is not to estimate the true phase offset. A coarse phase estimate that results in error free hard decision suffices. This motivates conversion of the continuum [−45°, +45°) into a set of discrete values, with a common step size, Δθ, as shown in
θ(i)−45°+Δθ×i:i∈{0,1,2, . . . ,iLAST},θ(i)∈[−45°,+45°)
Referring now to
The maximum supported frequency offset
implies that with a maximum residual phase offset error of Δθ/2, the phase of the last symbol in the block is
For error free hard decisions, the phase needs to be less than the maximum supported phase offset, θMAX as listed in Table 1, as expressed here:
This leads to the condition imposed on the selected phase grid step size, as expressed here:
Δθ<2(1−f)θMAX
The coarse phase offset estimate is then obtained as described in Steps 3 and 4 of Algorithm 1, which are reproduced here.
Compute a figure of merit, g(i), for all candidate phase values θ(i) as follows:
FOR All i∈{0, 1, 2, . . . , iLAST}
Apply phase −θ(i) to all symbols in the block: r0,n←r0,ne−jθ
Apply hard decision: H(r0,n)=hn(i): n∈[−L, +L]
Compute RMS error metric:
END FOR
Initial phase estimate is the candidate phase with minimum RMS Error: θ0←θ(l): g(l)=min {g(i)}i=0i
Step 2: Correct the received symbols for the estimated phase offset estimate θ0 obtained in Step 1 above to get:
r0,n←r0,ne−jθ
Apply hard decision on the resultant symbols and obtained resultant clean symbols as follows:
H(r0,n)=hn:n∈[−L,+L]
The phase corrected received symbols r0,n and corresponding hard decision symbols hn are used to compute the frequency offset and residual phase offset as described in Algorithm 1 (Steps 6 and 7) reproduced below. The derivation of these estimates is a special case of Derivation 2 with Hard Decision Trust Weights ρ0,n set to 1 for all symbols in the block.
Frequency offset estimate for the zeroth block of symbols is:
where
Residual phase offset estimate for the zeroth block is: arg(Σn=−L+L r0,nhn*e−jω
Step 3: Return the corrected phase offset estimate for the zeroth block as the sum of the coarse phase offset estimate from Step 1 and the residual phase offset estimate from Step 2 above.
Derivation 2—Cubic Polynomial Approximation of the Complex Exponential
This section derives the expressions for the coefficients of the cubic polynomial approximation of the complex exponential e−jω in the domain ω∈[−ω0, +ω0] for a given parameter ω0>0. Note that the letters in boldface denote vectors and upper-case letters in boldface denote matrices.
Start with the approximation equation:
exp−jω=α0+α1ω+α2ω2+α3ω3+h(ω)=wTα+h(ω)
where ω∈[−ω0, +ω0] is the domain of approximation, h(ω) is the approximation error, αm∈[0,3] are the unknown coefficients of the cubic polynomial, α[α0 α1α2 α3]T and w [1 ω ω2 ω3]T.
Optimal coefficients minimize the energy of the approximation error h(co) integrated over the domain [−ω0, +ω0].
To find the α that minimizes the cost function Λ(α), find the gradient vector of Λ(α) with respect to α and set that equal to the zero vector.
Integrate and expand the matrices and vectors to get:
which yields the desired optimal coefficients of the cubic polynomial that approximates the complex exponential:
Table 2 above lists, as an example, values of the coefficients produced by the described cubic polynomial approximation embodiment for different M-QAM constellations with block size of 17, i.e., L=8.
Referring now to
In contrast, the cubic polynomial approximation according to embodiments described herein is expressed as e−jω≈α0+α1ω+α2ω2+α3ω3, and expressions for the coefficients are described above with respect to Algorithm 1 and Derivation 2. With respect to
where e−jω is the complex exponential to be approximated, ƒ(ω) is the chosen polynomial approximation function, and [−ω0, +ω0] is the domain of approximation, where ω0(%)ω0 (degrees)×100/360.
Derivation 3—Phase and Frequency Offset Error Estimates for Blocks of Symbols Past the Zeroth Block
This section derives the expressions for the phase and frequency offset errors used in Algorithm 2. The expressions are reproduced below:
Frequency Offset Error:
Phase Offset Error:
Refer to Algorithm 2 for the definitions of the terms appearing in these expressions.
Start by defining the cost function that must be minimized by selecting appropriate values for the phase offset error and frequency offset error.
where ρ is the “hard decision trust” vector, [ρ−k,L . . . ρk,+L]T:ρk,n ∈{0, 1}
Minimizing Λmin (δωk, δθk; ρ) by choosing appropriate δωk, δθk is equivalent to minimizing
because |rk,n−|2+|hk,n−|2 are independent of δωk, δθk.
Therefore, the task is the same as that of maximizing
by choosing appropriate δωk, δθk.
Setting the partial derivative of Λmax(δωk, δθk; ρ) with respect to δωk yields:
The second expression uses the cubic polynomial approximation of the complex exponential. Simplify this further by ignoring all the terms that involve the product δθkδωk or other higher order products of the phase offset error and frequency offset error.
Then
This results in a quadratic equation in δωk which can be solved to yield two solutions. The solution for which the second derivative of Λmax(δωk, δθk; ρ) is negative is the estimate for the frequency offset error, and is given by:
Once the frequency offset error estimate is obtained, phase offset estimate error is computed as:
The described carrier phase and frequency tracking system and method embodiments may enjoy the following advantages relative to conventional systems. First, the carrier phase and frequency tracking system may be more robust against random noise in received QAM modulated symbols. Second, the carrier phase and frequency tracking system may have a higher range of maximum frequency offset that can be tracked. Third, the carrier phase and frequency tracking system may acquire and maintain stable constellation lock, i.e., obtain a good initial coarse estimate of the phase and frequency offsets, in a fewer number of received QAM symbols. More specifically, embodiments described here may provide significant performance improvements over conventional systems, such as PLL-based carrier phase and frequency trackers, in terms of the ability to recover QAM symbols without error, the carrier frequency offset estimation bias in no-noise conditions and under noisy conditions with various signal to noise ratios (SNR), the standard deviation and the RMS error in the estimated carrier frequency offset under various SNR conditions for QAM signals with different constellation sizes, e.g., from M=4 to 4096.
Referring now to
It is noted that different and/or additional components from those depicted in
It is still further noted that the functional blocks, components, systems, devices, and/or circuitry described herein can be implemented using hardware, software, or a combination of hardware and software. For example, the disclosed embodiments can be implemented using one or more programmable integrated circuits that are programmed to perform the functions, tasks, methods, actions, and/or other operational features described herein for the disclosed embodiments. The one or more programmable integrated circuits can include, for example, one or more processors and/or PLDs (programmable logic devices). The one or more processors can be, for example, one or more central processing units (CPUs), controllers, microcontrollers, microprocessors, hardware accelerators, ASICs (application specific integrated circuit), and/or other integrated processing devices. The one or more PLDs can be, for example, one or more CPLDs (complex programmable logic devices), FPGAs (field programmable gate arrays), PLAs (programmable logic array), reconfigurable logic circuits, and/or other integrated logic devices. Further, the programmable integrated circuits, including the one or more processors, can be configured to execute software, firmware, code, and/or other program instructions that are embodied in one or more non-transitory tangible computer-readable mediums to perform the functions, tasks, methods, actions, and/or other operational features described herein for the disclosed embodiments. The programmable integrated circuits, including the one or more PLDs, can also be programmed using logic code, logic definitions, hardware description languages, configuration files, and/or other logic instructions that are embodied in one or more non-transitory tangible computer-readable mediums to perform the functions, tasks, methods, actions, and/or other operational features described herein for the disclosed embodiments. In addition, the one or more non-transitory tangible computer-readable mediums can include, for example, one or more data storage devices, memory devices, flash memories, random access memories, read only memories, programmable memory devices, reprogrammable storage devices, hard drives, floppy disks, DVDs, CD-ROMs, and/or any other non-transitory tangible computer-readable mediums. Other variations can also be implemented while still taking advantage of the new frame structures described herein.
Referring now to
At block 1002, a receiver receives the zeroth/initial block of noisy M-QAM symbols modulated by a transmitter using an M-QAM symbol map (e.g., 4-QAM, 16-QAM, 256-QAM, etc.). Operation continues to block 1004.
At block 1004, a grid of discrete candidate phase offset values is created. The grid may be created by generating a sequence of values in increments of a step size within a search range, e.g., −45 degrees to +45 degrees. The step size may be computed based on a selectable fraction of a maximum supported frequency offset. The maximum supported frequency offset may be determined by a maximum supported phase offset for the M-QAM symbol map, e.g., as determinable from Table 1, and by the block length of the zeroth/initial block of M-QAM symbols. Operation continues to block 1006.
At block 1006, for each candidate phase offset value in the grid created at block 1004, the following is performed. First, the receiver applies the candidate phase offset value to each M-QAM symbol in the zeroth/initial block. Next, the receiver applies a respective hard decision from the symbol map to each of the M-QAM symbols to which the candidate phase offset value has been applied. Then, the receiver computes a figure of merit for the candidate phase offset value based on the M-QAM symbols to which the candidate phase offset value has been applied and their respective hard decisions. In one embodiment, the figure of merit is a root mean square (RMS) error across the M-QAM symbols to which the candidate phase offset value has been applied relative to their respective hard decisions. Operation continues to block 1008.
At block 1008, the receiver selects the discrete candidate phase offset value from the grid having the best figure of merit (e.g., minimum RMS error) as an initial phase offset estimate. Operation continues to block 1012.
At block 1012, the receiver computes an initial frequency offset estimate using the M-QAM symbols updated with the initial phase offset estimate, their respective hard decisions, and an approximation of the complex exponential function. In one embodiment, the approximation of the complex exponential function is a cubic polynomial whose domain of approximation is restricted to a set of predetermined values that depend upon a maximum supported carrier frequency offset for the M-QAM symbol map that depends upon the value of M and in which the coefficients of the cubic polynomial are chosen to minimize a mean squared error approximation error averaged over the restricted domain of approximation. Operation continues to block 1014.
At block 1014, the receiver refines the initial phase offset using the initial frequency offset estimate, the updated M-QAM symbols, and their respective hard decisions.
Referring now to
At block 1102, a receiver receives a series of blocks of noisy M-QAM symbols modulated by a transmitter using an M-QAM symbol map (e.g., 4-QAM, 16-QAM, 256-QAM, etc.). The blocks are segregated by the receiver as overlapping blocks of a particular size and having a number of symbols of overlap. The block size and number of symbols of overlap is selectable. Operation continues to block 1104.
At block 1104, for each M-QAM symbol of a current block, the following operations are performed by the receiver. First, the receiver determines a first hard decision symbol of the symbol map closest to the M-QAM symbol. Next, the receiver determines a second hard decision symbol of the symbol map second closest to the M-QAM symbol. Then, the receiver computes a parameter based on a ratio of first and second distances. The first distance is between the M-QAM symbol and the first hard decision symbol, and the second distance is between the M-QAM symbol and the second hard decision symbol. Finally, the receiver sets a binary trust weight for the M-QAM symbol based on a comparison of the computed parameter with a threshold. Operation continues to block 1106.
At block 1106, the receiver uses the binary trust weights to compute a phase offset error and a frequency offset error associated with the current block. The phase offset error may be computed using an approximation of the complex exponential function as described above, e.g., at block 1012. Each M-QAM symbol having a zero-valued binary trust weight does not contribute to the computation of the phase and frequency offset errors, e.g., because the binary trust weight is a multiplicative factor of a product, and the product is an addend of a summation. The computation of the frequency offset error also uses the received M-QAM symbols of the previous block corrected by predicted phase and frequency offsets of the previous block, hard decisions on the corrected received M-QAM symbols of the previous block, the binary trust weights for the corrected received M-QAM symbols of the previous block, and the block size. The computation of the phase offset error also uses the approximation of the complex exponential. Operation continues to block 1108.
At block 1108, the receiver computes a noise covariance matrix of the previous block using the received M-QAM symbols of the previous block corrected by predicted phase and frequency offsets of the previous block, hard decisions on the corrected received M-QAM symbols of the previous block, the binary trust weights for the corrected received M-QAM symbols of the previous block, and the block size. The receiver than predicts a noise covariance matrix of the current block using the computed noise covariance matrix of the previous block. Operation continues to block 1112.
At block 1112, the receiver computes the Kalman gain matrix for the current block using the predicted noise covariance matrix of the current block and a covariance matrix, e.g., covariance matrix R of Algorithm 2. Operation continues to block 1114.
At block 1114, the receiver estimates the phase offset and frequency offset of the current block using the predicted phase and frequency offsets of the current block, the computed phase and frequency offset errors of the current block, and the computed Kalman gain matrix of the current block. Operation continues to block 1116.
At block 1116, the receiver performs the operations of blocks 1104 to 1114 on the symbols of multiple blocks of the series in a forward direction, and then performs the operations of blocks 1104 to 1114 on the symbols of multiple blocks of the series in a reverse direction to produce transmitted M-QAM symbols.
Further modifications and alternative embodiments of this invention will be apparent to those skilled in the art in view of this description. It will be recognized, therefore, that the present invention is not limited by these example arrangements. Accordingly, this description is to be construed as illustrative only and is for teaching those skilled in the art the manner of carrying out the invention. It is to be understood that the forms of the invention herein shown and described are to be taken as present embodiments. Various changes may be made in the implementations and architectures. For example, equivalent elements may be substituted for those illustrated and described herein, and certain features of the invention may be utilized independently of the use of other features, all as would be apparent to one skilled in the art after having the benefit of this description of the invention.
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Inoue, Takashi et al. “Carrier Recovery for M-QAM Signals Based on a Block Estimation Process with Kalman Filter.” National Institute of Advanced Industrial Science and Technology (AIST) Ibaraki, Japan. Published Jun. 17, 2014, vol. 22, No. 13. Optics Express. pp. 15376-15387. |
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20210176109 A1 | Jun 2021 | US |