Coarse Frequency Compensator

Compensate for carrier frequency offset in PAM, PSK, or QAM

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Libraries:
Communications Toolbox / RF Impairments Correction
Communications Toolbox / Synchronization

Description

The Coarse Frequency Compensator block compensates for carrier frequency offset in QAM, 8-PSK, BPSK, OQPSK, PAM, and QPSK modulation schemes.

Examples

Compensate Signal Frequency and Phase Offset in Simulink

Compensate for a frequency offset imposed on a noisy 8-PSK channel by using the Coarse Frequency Compensator block.

The doc_coarsefreqcomp model compares a frequency offset impaired signal to one that has coarse frequency compensation applied.

A spectrum analyzer plot shows the transmitted signal before and after applying coarse frequency compensation.

Estimated frequency offset is 2000.1220704.2 Hz.

Ports

Input

In — Input signal

column vector

Input signal, specified as a column vector.

Data Types: single | double
Complex Number Support: Yes

Output

Out — Compensated output signal

complex column vector

Compensated output signal, returned as a complex column vector with the same dimensions and data type as the input In.

Data Types: single | double

FreqEst — Estimate of frequency offset

scalar

Estimate of the frequency offset, returned as a scalar.

If you set the Modulation type of input signal parameter to OQPSK or QAM, or set the Estimation algorithm parameter to FFT-based, the returned frequency offset estimate is an integer multiple of the Frequency resolution (Hz) parameter.

Dependencies

To add this port, select Estimated frequency offset output port.

Data Types: single | double

Parameters

To edit block parameters interactively, use the Property Inspector. From the Simulink^® Toolstrip, on the Simulation tab, in the Prepare gallery, select Property Inspector.

Modulation type of input signal — Modulation type

QAM (default) | 8PSK | BPSK | OQPSK | PAM | QPSK

Modulation type, specified as QAM, 8PSK, BPSK, OQPSK, PAM, or QPSK.

Estimation algorithm — Algorithm used to estimate frequency offset

FFT-based (default) | Correlation-based

Specify the frequency offset estimation algorithm as FFT-based or Correlation-based. For more information, see Algorithms.

Note

When you set the Modulation type of input signal parameter to 'OQPSK' or 'QAM', the block uses the FFT-based estimation algorithm.

Dependencies

This parameter appears when you set Modulation type of input signal to 8PSK, BPSK, PAM, or QPSK.

Frequency resolution (Hz) — Frequency resolution

100 (default) | positive scalar

Frequency resolution in Hz, specified as a positive real scalar.

Dependencies

This parameter applies when you set Modulation type of input signal to OQPSK or QAM, or you set Estimation algorithm to FFT-based.

Samples per symbol — Number of samples per symbol

4 (default) | even integer greater than or equal to 4

Number of samples per symbol, specified as an even integer scalar greater than or equal to 4.

Dependencies

This parameter appears when you set Modulation type of input signal to OQPSK.

Maximum frequency offset (Hz) — Maximum measurable frequency offset

5e3 (default) | positive scalar

Maximum measurable frequency offset in Hz, specified as a positive scalar.

Dependencies

This parameter appears when you set Estimation algorithm to Correlation-based.

Estimated frequency offset output port — Option to add estimated frequency offset output port

on (default) | off

Select this check box to add the FreqEst output port. The FreqEst port outputs the estimated frequency offset in Hz.

Simulate using — Type of simulation to run

Code generation (default) | Interpreted execution

Type of simulation to run, specified as Code generation or Interpreted execution.

• Code generation — Simulate the model by using generated C code. The first time you run a simulation, Simulink generates C code for the block. The model reuses the C code for subsequent simulations unless the model changes. This option requires additional startup time, but the speed of the subsequent simulations is faster than with the Interpreted execution option.

• Interpreted execution — Simulate the model by using the MATLAB^® interpreter. This option shortens startup time, but the speed of subsequent simulations is slower than with the Code generation option. In this mode, you can debug the source code of the block.

For more information, see Interpreted Execution vs. Code Generation (Simulink).

Block Characteristics

Data Types	double | single
Multidimensional Signals	no
Variable-Size Signals	no

Algorithms

Correlation-Based Estimation

Reference [ 1 ] describes the correlation-based estimation algorithm used to estimate the frequency offset for PSK and PAM signals. To determine the frequency offset, Δf, the algorithm performs a maximum likelihood (ML) estimation of the complex-valued oscillation exp(j2πΔft). The observed signal, r_k, is represented as

$$r_k=e^{j\left(2\pi \Delta fk{\text{T}}_s+\theta \right)},1\le k\le N$$

T_s is the sampling interval, θ is an unknown random phase, and N is the number of samples. The ML estimation of the frequency offset is equivalent to seeking the maximum of the likelihood function,

$$\Lambda (\Delta f)\approx {\left|{\displaystyle \sum _{i=1}^N{r}_i{e}^{-j2\pi \Delta fi{T}_s}}\right|}^2={\displaystyle \sum _{k=1}^N{\displaystyle \sum _{m=1}^N{r}_k{r}_m^{*}{e}^{-j2\pi \Delta f{T}_s(k-m)}}}$$

After simplifying, the problem is expressed as a discrete Fourier transform, weighted by a parabolic windowing function. It is expressed as

$$\mathrm{Im}\left\{{\displaystyle \sum _{k=1}^{N-1}k(N-k)R(k)e^{j2\pi \Delta \widehat{f}{T}_s}}\right\}=0$$

R(k) denotes the estimated autocorrelation of the sequence r_k and is represented as

$$R(k)\triangleq \frac{1}{N-k}{\displaystyle \sum _{i=k+1}^N{r}_i{r}_{i-k}^{*}},0\le k\le N-1$$

The term k(N–k) is the parabolic windowing function. In [ 1 ], it is shown that R(k) is a poor estimate of the autocorrelation of r_k when k = 0 or when k is close to N. Consequently, the windowing function can be expressed as a rectangular sequence of 1s for k = 1, 2, ..., L, where L ≤ N – 1. The result is a modified ML estimation strategy in which

$$\mathrm{Im}\left\{{\displaystyle \sum _{k=1}^{L}R(k)e^{-j2\pi \Delta \widehat{f}k{T}_s}}\right\}=0$$

This equation results in an estimate of $$\Delta \widehat{f}$$ in which

$$\Delta \widehat{f}\cong \frac{f_{samp}}{\pi (L+1)}\arg \left\{{\displaystyle \sum _{k=1}^{L}R(k)}\right\}$$

The sampling frequency, f_samp, is the reciprocal of T_s. The number of elements used to compute the autocorrelation sequence, L, are determined as

$$L=\mathrm{round}\left(\frac{f_{samp}}{f_{max}}\right)-1$$

f_max is the maximum expected frequency offset and round is the nearest integer function. The frequency offset estimate improves when L ≥ 7 and leads to the recommendation that f_max ≤ f_samp / (4M).

FFT-Based Estimation

FFT-based estimation algorithms can be used to estimate the frequency offset, as an integer multiple of the frequency resolution, for various modulation types. The coarse frequency compensator implementation supports these modulation methods by using the algorithm noted.

• For BPSK, QPSK, 8PSK, PAM, or QAM modulation, the coarse frequency compensator uses the FFT-based algorithm described in [ 2 ]. The algorithm estimates $$\Delta \widehat{f}$$ by using a periodogram of the mth power of the received signal and is given as

  $$\Delta \widehat{f}=\frac{f_{samp}}{N\cdot m}\arg \underset{f}{\max }\left|{\displaystyle \sum _{k=0}^{N-1}{r}^m(k)e^{-j2\pi kt/N}}\right|,\left(-\frac{R_{sym}}{2}\le f\le \frac{R_{sym}}{2}\right)$$

  where m is the modulation order, r(k) is the received sequence, R_sym is the symbol rate, and N is the number of samples. The algorithm searches for a frequency that maximizes the time average of the mth power of the received signal multiplied by various frequencies in the range of [–R_sym / 2, R_sym / 2]. Because the form of the algorithm is the definition of the discrete Fourier transform of r^m(t), searching for a frequency that maximizes the time average is equivalent to searching for a peak line in the spectrum of r^m(t). The number of points required by the FFT is

  $$N=2^{\lceil {\log }_2\left(\frac{f_{samp}}{f_r}\right)\rceil }$$

  where f_r is the desired frequency resolution.

• For OQPSK modulation, the coarse frequency compensator uses the FFT-based algorithm described in [ 4 ]. The algorithm searches for spectral peaks at ±200 kHz around the symbol rate. This technique locates desired peaks in the presence of interference from spectral content around baseband frequencies due to filtering.

References

[1] Luise, M., and R. Reggiannini. “Carrier Frequency Recovery in All-Digital Modems for Burst-Mode Transmissions.” IEEE^® Transactions on Communications 43, no. 2/3/4 (Feb. 1995): 1169–78.

[2] Wang, Y., et al. “Non-Data-Aided Feedforward Carrier Frequency Offset Estimators for QAM Constellations: A Nonlinear Least-Squares Approach.” EURASIP Journal on Advances in Signal Processing 2004, no. 13 (Dec. 2004): 856139. https://doi.org/10.1155/S1110865704403175.

[3] Nakagawa, Tadao, et al. “Non-Data-Aided Wide-Range Frequency Offset Estimator for QAM Optical Coherent Receivers.” Optical Fiber Communication Conference/National Fiber Optic Engineers Conference 2011, OMJ1. OSA, 2011. https://doi.org/10.1364/OFC.2011.OMJ1.

[4] Olds, Jonathan. Designing an OQPSK demodulator.

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using Simulink® Coder™.

Version History

Introduced in R2015b

R2025a: Estimated frequency offset returned as integer multiple of frequency resolution

When configured to use the FFT-based algorithm, the Coarse Frequency Compensator block now returns the estimated frequency offset, FreqEst, as an integer multiple of the Frequency resolution (Hz) parameter value.

R2023b: Default settings change

The Coarse Frequency Compensator block parameters have new default values.

Parameter	New Default	Previous Default
Frequency resolution (Hz)	100	0.001
Maximum frequency offset (Hz)	5e3	0.05

When executing code created in a previous release, confirm the coarse frequency compensator settings. The new default settings change your results. To produce results using the previous default settings, you can manually update the settings of your block.