Decision Feedback Equalizer
Equalize modulated signals using decision feedback filtering
Libraries:
Communications Toolbox /
Equalizers
Description
The Decision Feedback Equalizer block uses a decision feedback filter tap delay line with a weighted sum to equalize modulated signals transmitted through a dispersive channel. Using an estimate of the channel modeled as a finite input response (FIR) filter, the block processes input frames and outputs the estimated signal.
This icon shows the block with all ports enabled for configurations that use the LMS or RLS adaptive algorithm.
This icon shows the block with all ports enabled for configurations that use the CMA adaptive algorithm.
Examples
DF Equalize QPSKModulated Signal in Simulink
Apply decision feedback equalization using the least mean squares (LMS) algorithm to recover QPSK symbols passed through an AWGN channel.
The slex_dfeq_qpsk_signal
model generates an M=4 sequence using the Random Integer Generator
block. The sequence is modulated with the MPSK Modulator Baseband
block, filtered with the Discrete FIR Filter
block, and then impaired with the AWGN
block. The Decision Feedback Equalizer
block equalizes the data sequence, the data is demodulated with the MPSK Demodulator Baseband
block, and the bit error rate is computed. The signal path out of the modulator is split to a Selector
block, which provides the first 1000 symbols of the modulated signal to the equalizer as an initial training sequence.
No delay is introduced between the transmitted and received signal because the maximum tap value is the first tap of the discrete FIR filter and the equalizer reference tap is 1
. The equalizer converges after around 1000 symbols so this value is used for the computation delay of the Error Rate Calculation
block.
The computed error rate is displayed and plots show the equalized constellation, equalized tap weights, and signal error magnitude.
Computed error rate = 0.00077778
Extended Examples
Adaptive Equalization with Filtering and Fading Channel
The behavior of the selected adaptive equalizer in a communication link that has a fading channel. The transmitter and receiver have root raised cosine pulse shaped filtering. A subsystem block enables you to select between linear or decision feedback equalizers that use the least mean square (LMS) or recursive least square (RLS) adaptive algorithm.
Ports
Input
in — Input signal
column vector
Input signal, specified as a column vector. The vector length of in must be equal to an integer multiple of the Number of input samples per symbol parameter. For more information, see Symbol Tap Spacing.
Data Types: double
Complex Number Support: Yes
Desired — Training symbols
column vector
Training symbols, specified as a column vector. The vector length of
Desired must be less than or equal to the length of input
in. The Desired input port is ignored when
the Train input port is 0
.
Dependencies
To enable this port, set the Adaptive algorithm parameter
to LMS
or RLS
.
Data Types: double
Complex Number Support: Yes
Train — Train equalizer flag
boolean scalar
Train equalizer flag, specified as 1
or 0
.
The block starts training when this value changes from 0
to
1
(at the rising edge). The block trains until all symbols in the
Desired input port are processed.
Dependencies
To enable this port, set the Adaptive algorithm parameter
to LMS
or RLS
and select the Enable
training control input parameter.
Data Types: Boolean
Update — Update tap weights flag
1
 0
Update tap weights flag, specified as 1
or
0
. The tap weights are updated when this value is
1
.
Dependencies
To enable this port, set the Adaptive algorithm parameter
to CMA
and the Source of adapt weights flag
parameter to Input port
.
Data Types: Boolean
Reset — Reset equalizer flag
1
 0
Reset equalizer flag, specified as 1
or 0
.
If Reset is set to 1
, the block resets the tap weights
before processing the incoming signal. The block performs initial training until all
symbols in the Desired input port are processed.
Dependencies
To enable this port, select the Enable reset input parameter.
Data Types: Boolean
Output
Out — Equalized symbols
column vector
Equalized symbols, returned as a column vector that has the same length as input signal in.
This port is unnamed until you select the Output error signal or Output taps weights parameter.
Err — Error signal
column vector
Error signal, returned as a column vector that has the same length as input signal in.
w — Tap weights
column vector
Tap weights, returned as an N_{Taps}by1 vector, where N_{Taps} is equal to the sum of the Number of forward taps and Number of feedback taps parameter values. w contains the tap weights from the last tap weight update.
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.
Structure parametersNumber of forward taps — Number of forward equalizer taps
5
(default)  positive integer
Number of forward equalizer taps, specified as a positive integer. The number of forward equalizer taps must be greater than or equal to the value of the Number of input samples per symbol parameter.
Number of feedback taps — Number of feedback equalizer taps
3
(default)  positive integer
Number of feedback equalizer taps, specified as a positive integer.
Signal constellation — Signal constellation
pskmod(0:3,4,pi/4)
(default)  vector
Signal constellation, specified as a vector. The default value is a QPSK
constellation generated using this code: pskmod(0:3,4,pi/4)
.
Number of input samples per symbol — Number of input samples per symbol
1
(default)  positive integer
Number of input samples per symbol, specified as a positive integer. Setting this
parameter to any number greater than 1
effectively creates a
fractionally spaced equalizer. For more information, see Symbol Tap Spacing.
Adaptive algorithm — Adaptive algorithm
LMS
(default)  RLS
 CMA
Adaptive algorithm used for equalization, specified as one of these values:
LMS
— Update the equalizer tap weights using the Least Mean Square (LMS) Algorithm.RLS
— Update the equalizer tap weights using the Recursive Least Square (RLS) Algorithm.CMA
— Update the equalizer tap weights using the Constant Modulus Algorithm (CMA).
Step size — Step size
0.01
(default)  positive scalar
Step size used by the adaptive algorithm, specified as a positive scalar. Increasing the step size reduces the equalizer convergence time but causes the equalizer output estimates to be less stable.
Tunable: Yes
Dependencies
To enable this parameter, set Adaptive algorithm to
LMS
or CMA
.
Forgetting factor — Forgetting factor
0.99
(default)  scalar in the range (0, 1]
Forgetting factor used by the adaptive algorithm, specified as a scalar in the range (0, 1]. Decreasing the forgetting factor reduces the equalizer convergence time but causes the equalizer output estimates to be less stable.
Tunable: Yes
Dependencies
To enable this parameter, set Adaptive algorithm to
RLS
.
Initial inverse correlation matrix — Initial inverse correlation matrix
0.1
(default)  scalar  matrix
Initial inverse correlation matrix, specified as a scalar or an
N_{Taps}byN_{Taps}
matrix. N_{Taps} is equal to the sum of the
Number of forward taps and Number of feedback
taps parameter values. If you specify this value as a scalar,
a, the equalizer sets the initial inverse correlation matrix to
a times the identity matrix:
a(eye
(N_{Taps})).
Dependencies
To enable this parameter, set Adaptive algorithm to
RLS
.
Reference tap — Reference tap
3
(default)  positive integer
Reference tap, specified as a positive integer less than or equal to the Number of forward taps parameter value. The equalizer uses the reference tap location to track the main energy of the channel.
Input signal delay (samples) — Input signal delay
0
(default)  nonnegative integer
Input signal delay in samples relative to the reset time of the equalizer, specified
as a nonnegative integer. If the input signal is a vector of length greater than 1, then
the input delay is relative to the start of the input vector. If the input signal is a
scalar, then the input delay is relative to the first call of the block and to the first
call of the block after the Reset input port toggles to
1
.
Dependencies
To enable this parameter, set Adaptive algorithm to
LMS
or RLS
.
Source of adapt weights flag — Source of adapt tap weights request
Property
(default)  Input port
Source of the adapt tap weights request, specified as one of these values:
Property
— Specify this value to use the Adaptive algorithm parameter to control when the block adapts tap weights.Input port
— Specify this value to use the Update input port to control when the block adapts tap weights.
Dependencies
To enable this parameter, set Adaptive algorithm to
CMA
.
Adapt tap weights — Adapt tap weights
on
(default)  off
Select this parameter to adaptively update the equalizer tap weights. If this parameter is cleared, the block keeps the equalizer tap weights unchanged.
Dependencies
To enable this parameter, set Adaptive algorithm to
CMA
and Source of adapt weights flag to
Property
.
Initial tap weights source — Source for initial tap weights
Auto
(default)  Property
Source for initial tap weights, specified as one of these values:
Auto
— Initialize the tap weights to the algorithmspecific default values, as described in the Initial weights parameter.Property
— Initialize the tap weights using the Initial weights parameter value.
Initial weights — Initial tap weights
0
or [0;0;1;0;0]
(default)  scalar  column vector
Initial tap weights used by the adaptive algorithm, specified as a scalar or an
N_{Taps}by1 vector.
N_{Taps} is equal to the sum of the
Number of forward taps and Number of feedback
taps parameter values. The default is 0
when the
Adaptive algorithm parameter is set to LMS
or
RLS
. The default is [0;0;1;0;0]
when the
Adaptive algorithm parameter is set to
CMA
.
If you specify Initial weights as a vector, the vector length must be N_{Taps}. If you specify Initial weights as a scalar, the equalizer uses scalar expansion to create a vector of length N_{Taps} with all values set to Initial weights.
Dependencies
To enable this parameter, set Initial tap weights source to
Property
.
Tap weight update period (symbols) — Tap weight update period
1
(default)  positive integer
Tap weight update period in symbols, specified as a positive integer. The equalizer updates the tap weights after processing this number of symbols.
Enable training control input — Enable training control input
off
(default)  on
Select this parameter to enable input port Train. If this parameter is cleared, the block does not reenter training mode after the initial tap training.
Dependencies
To enable this parameter, set Adaptive algorithm to
LMS
or RLS
.
Update tap weights when not training — Update tap weights when not training
on
(default)  off
Select this parameter to use decision directed mode to update equalizer tap weights. If this parameter is cleared, the block keeps the equalizer tap weights unchanged after training.
Dependencies
To enable this parameter, set Adaptive algorithm to
LMS
or RLS
.
Enable reset input — Enable reset input
off
(default)  on
Select this parameter to enable input port Reset. If this parameter is cleared, the block does not reset the tap weights before processing the incoming signal.
Output error signal — Enable error signal output
off
(default)  on
Select this parameter to enable output port Err containing the equalizer error signal.
Output taps weights — Enable tap weights output
off
(default)  on
Select this parameter to enable output port w containing tap weights from the last tap weight update.
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 theInterpreted 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 theCode 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 

Multidimensional Signals 

VariableSize Signals 

More About
Symbol Tap Spacing
You can configure the equalizer to operate as a symbolspaced equalizer or as a fractional symbolspaced equalizer.
To operate the equalizer at a symbolspaced rate, specify the number of samples per symbol as
1
. Symbolrate equalizers have taps spaced at the symbol duration. Symbolrate equalizers are sensitive to timing phase.To operate the equalizer at a fractional symbolspaced rate, specify the number of input samples per symbol as an integer greater than
1
and provide an input signal oversampled at that sampling rate. Fractional symbolspaced equalizers have taps spaced at an integer fraction of the input symbol duration. Fractional symbolspaced equalizers are not sensitive to timing phase.
Algorithms
Decision Feedback Equalizers
A decision feedback equalizer (DFE) is a nonlinear equalizer that reduces intersymbol interference (ISI) in frequencyselective channels. If a null exists in the frequency response of a channel, DFEs do not enhance the noise. A DFE consists of a tapped delay line that stores samples from the input signal and contains a forward filter and a feedback filter. The forward filter is similar to a linear equalizer. The feedback filter contains a tapped delay line whose inputs are the decisions made on the equalized signal. Once per symbol period, the equalizer outputs a weighted sum of the values in the delay line and updates the weights to prepare for the next symbol period.
DFEs can be symbolspaced or fractional symbolspaced.
For a symbolspaced equalizer, the number of samples per symbol, K, is 1. The output sample rate equals the input sample rate.
For a fractional symbolspaced equalizer, the number of samples per symbol, K, is an integer greater than 1. Typically, K is 4 for fractional symbolspaced equalizers. The output sample rate is 1/T and the input sample rate is K/T. Tap weight updating occurs at the output rate.
This schematic shows a fractional symbolspaced DFE with a total of N weights, a symbol period of T, and K samples per symbol. The filter has L forward weights and NL feedback weights. The forward filter is at the top, and the feedback filter is at the bottom. If K is 1, the result is a symbolspaced DFE instead of a fractional symbolspaced DFE.
In each symbol period, the equalizer receives K input samples at the forward filter and one decision or training sample at the feedback filter. The equalizer then outputs a weighted sum of the values in the forward and feedback delay lines and updates the weights to prepare for the next symbol period.
Note
The algorithm for the Adaptive Algorithm block in the schematic jointly optimizes the forward and feedback weights. Joint optimization is especially important for convergence in the recursive least square (RLS) algorithm.
For more information, see Equalization.
Least Mean Square (LMS) Algorithm
For the LMS algorithm, in the previous schematic, w is a vector of all weights w_{i}, and u is a vector of all inputs u_{i}. Based on the current set of weights, the LMS algorithm creates the new set of weights as
w_{new} = w_{current} + (StepSize) ue*.
The step size used by the adaptive algorithm is specified as a positive scalar. Increasing the
step size reduces the equalizer convergence time but causes the equalized output signal to
be less stable. To determine the maximum step size allowed when using the LMS adaptive
algorithm, use the maxstep
object
function. The * operator denotes the complex conjugate and the error calculation e = d 
y.
Recursive Least Square (RLS) Algorithm
For the RLS algorithm, in the previous schematic, w is the vector of all weights w_{i}, and u is the vector of all inputs u_{i}. Based on the current set of inputs, u, and the inverse correlation matrix, P, the RLS algorithm first computes the Kalman gain vector, K, as
$$K=\frac{Pu}{(ForgettingFactor)+{u}^{H}Pu}.$$
The forgetting factor used by the adaptive algorithm is specified as a scalar in the range (0, 1]. Decreasing the forgetting factor reduces the equalizer convergence time but causes the equalized output signal to be less stable. H denotes the Hermitian transpose. Based on the current inverse correlation matrix, the new inverse correlation matrix is
$${P}_{\text{new}}=\frac{(1K{u}^{H}){P}_{\text{current}}}{ForgettingFactor}.$$
Based on the current set of weights, the RLS algorithm creates the new set of weights as
w_{new} = w_{current}+K*e.
The * operator denotes the complex conjugate and the error calculation e = d  y.
Constant Modulus Algorithm (CMA)
For the CMA adaptive algorithm, in the previous schematic, w is the vector of all weights w_{i}, and u is the vector of all inputs u_{i}. Based on the current set of weights, the CMA adaptive algorithm creates the new set of weights as
w_{new} = w_{current} + (StepSize) u*e.
The step size used by the adaptive algorithm is specified as a positive scalar. Increasing the
step size reduces the equalizer convergence time but causes the equalized output signal to
be less stable. To determine the maximum step size allowed by the CMA adaptive algorithm,
use the maxstep
object
function. The * operator denotes the complex conjugate and the error calculation
e = y(R 
y^{2}), where R is a constant related to the signal
constellation.
Extended Capabilities
C/C++ Code Generation
Generate C and C++ code using Simulink® Coder™.
Version History
Introduced in R2019a
See Also
Blocks
Objects
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