ofdmEqualize

Equalize OFDM signals

Since R2022b

Syntax

[eqsym,csi]= ofdmEqualize(rxsym,heff,nvar)

[eqsym,csi]= ofdmEqualize(rxsym,heff)

[eqsym,csi]= ofdmEqualize(___,Name=Value)

Description

[eqsym,csi]= ofdmEqualize(rxsym,heff,nvar) returns equalized symbols eqsym and soft channel state information csi after performing minimum mean squared error (MMSE) equalization on input OFDM symbols rxsym. heff specifies the effective estimated channel information. nvar specifies the estimated noise variance.

[eqsym,csi]= ofdmEqualize(rxsym,heff) performs MMSE equalization with the estimated noise variance equal to 0.

[eqsym,csi]= ofdmEqualize(___,Name=Value) specifies options using one or more name-value arguments in addition to the input arguments in the previous syntaxes. For example, ofdmEqualize(rxsym,heff,Algorithm="zf") equalizes the input OFDM symbols using the zero-forcing algorithm.

Examples

OFDM Equalization for MIMO Channel with Beamforming

Obtain the channel response for a beamformed OFDM signal with ${\mathit{N}}_{\mathrm{S}}$ parallel data streams filtered through an ${\mathit{N}}_{\mathrm{T}}$-by-${\mathit{N}}_{\mathrm{R}}$ MIMO channel. Use the channel response to apply OFDM equalization to the received OFDM-demodulated signal.

Define simulation variables.

rng(1);
numStreams = 2;  % Number of parallel data streams (Ns)
numTx = 4;       % Number of transmit antennas (Nt)
numRx = 3;       % Number of receive antennas (Nr)
bps = 6;         % Bits per QAM symbol (and OFDM data subcarrier)
nfft = 256;      % FFT length
cpLen = 16;      % Cyclic prefix length
numOFDMSym = 10; % Number of OFDM symbols
SNRdB = 40;      % Signal-to-noise ratio

Check that the number of data streams is no greater than either the number of transmit antennas or the number of receive antennas.

if numStreams > min(numTx,numRx)
    error('numStreams must be equal to or less than numTx and numRx.');
end

Configure OFDM subcarriers.

ofdmNullIdx = ...   % Guard bands and DC subcarrier
    [1:9 (nfft/2+1) (nfft-8+1:nfft)]'; 
numDataSC = ...     % Number of data subcarriers
    nfft-length(ofdmNullIdx);

Generate an array of data symbols consisting of ${\mathit{N}}_{\mathrm{S}}$ parallel data streams, QAM-modulate the symbols, and then OFDM-modulate the QAM-modulated symbols.

dataBits = randi([0,1],[numDataSC*bps numOFDMSym numStreams]);
M = 2^bps; % Modulation order
qamTx = qammod(dataBits,M, ...
    InputType="bit", ...
    UnitAveragePower=true);
ofdmOut = ofdmmod(qamTx,nfft,cpLen,ofdmNullIdx);

Beamforming expands the dimensionality of the transmit signal to improve link performance over multipath channels. The data streams are fed through a beamformer that focuses the transmit energy over an ${\mathit{N}}_{\mathrm{T}}$-by-1 transmit antenna array where the number of antennas ${\mathit{N}}_{\mathrm{T}}\ge {\mathit{N}}_{\mathrm{S}}$.

Form a beamformer matrix from steering vectors acting on each stream.

% Beamform the transmitted signal
fc = 1e9;
lambda = physconst('LightSpeed')/fc;
beamAngles = 15;
antIdx = (0:numTx-1);
antDelay = 2*pi*sin(2*pi*beamAngles*(0:numStreams-1).'/360)/lambda;
B = exp(1i*antIdx.*antDelay);  % Ns x Nt beamformer matrix
txOut = ofdmOut * B;

Filter the OFDM-modulated signal through a MIMO channel to get the channel coefficients.

mimoChannel = comm.MIMOChannel( ...
    SampleRate=1e6, ...
    PathDelays=[0 3e-6 5e-6], ...
    AveragePathGains=[0 0.5 0.2], ...
    MaximumDopplerShift=0, ...
    SpatialCorrelationSpecification="None", ...
    NumTransmitAntennas=numTx, ...
    NumReceiveAntennas=numRx, ...
    PathGainsOutputPort=true);
[channelOut,pathGains] = mimoChannel(txOut);

In a practical system, the channel must be sounded to estimate the channel response. Instead of sounding, the ofdmChannelResponse function computes the exact channel response using the path gains and path filters that are available after you pass data through the MIMO channel System object. Use channel path gains returned by the MIMO channel object, and the path filters and timing offset returned by the info object function, to obtain the OFDM channel response. If ${\mathit{N}}_{\mathrm{T}}<{\mathit{N}}_{\mathrm{R}}$, the channel forms an over-determined system (there are more receive antennas than necessary to adequately decode the transmitted signals). Call ofdmChannelResponse with the pathGains from the MIMO channel function to get the channel response.

mimoChannelInfo = info(mimoChannel);
pathFilters = mimoChannelInfo.ChannelFilterCoefficients;
toffset = mimoChannelInfo.ChannelFilterDelay;
h = ofdmChannelResponse(pathGains,pathFilters,nfft,cpLen, ...
    setdiff(1:nfft,ofdmNullIdx),toffset); % Nsc x Nsym x Nt x Nr

[rxIn,nVar] = awgn(channelOut,SNRdB,"measured");

Before demodulating the OFDM signal, account for the timing offset and symbol offset, remove the initial samples, and then pad with zeros to keep the signal length unchanged.

zeropadding = zeros(toffset,numRx);
ofdmDemodIn = [rxIn(toffset+1:end,:); zeropadding];
symOffset = cpLen/2;

OFDM-demodulate and equalize the received signal.

rxSym = ofdmdemod(ofdmDemodIn,nfft,cpLen,symOffset,ofdmNullIdx);

The effects of beamforming and the MIMO channel affect the received data streams. The effective channel (${\mathit{H}}_{\mathrm{eff}}$) is an ${\mathit{N}}_{\mathrm{S}}$-by-${\mathit{N}}_{\mathrm{R}}$ matrix defined as the product of the transmit beamformer and the MIMO channel coefficients.

To use the OFDM channel response when equalizing the OFDM-demodulated signal, you must reshape the ${\mathit{N}}_{\mathit{SC}}$-by-${\mathit{N}}_{\mathrm{OFDMsym}}$-by-${\mathit{N}}_{\mathrm{T}}$-by-${\mathit{N}}_{\mathrm{R}}$ array to an ${\mathit{N}}_{\mathrm{QAMsym}}$-by-${\mathit{N}}_{\mathrm{T}}$-by-${\mathit{N}}_{\mathrm{R}}$ array. Form the effective channel using the beamformer matrix B and the reshaped channel coefficients h. Equalize the received OFDM signal using the calculated effective channel, the noise variance, and the MMSE algorithm.

hReshaped = reshape(h,[],numTx,numRx);

hEff = zeros(numDataSC*numOFDMSym,numStreams,numRx);
for k = 1:numOFDMSym*numDataSC
    hEff(k,:,:) = B * squeeze(hReshaped(k,:,:));
end

eqSym = ofdmEqualize(rxSym,hEff,nVar);

Show the received OFDM-demodulated symbols (rxSym) and the equalized OFDM-demodulated symbols (eqSym). The constellation of the OFDM-demodulated symbols before equalization does not resemble the QAM constellation. After equalization, the constellation points lay near the reference constellation.

figure(1);
scatterplot(rxSym(:));

figure(2);
scatterplot(eqSym(:));

Show Equivalence of 2-D and 3-D Data Format for OFDM Equalization

Initialize variables for simulation of a MIMO system and a 120-resource-element subset of the OFDM subcarrier-symbol grid.

Nre = 120;  % Number of resource elements
Ns = 4;     % Number of data streams
Nr = 8;     % Number of receive antennas
nvar = 0.1; % Noise variance

Create random signals for a 2-D symbol array and a channel estimate.

rxsym2d = complex(randn(Nre,Nr),randn(Nre,Nr));
Hest = complex(randn(Nre,Ns,Nr),randn(Nre,Ns,Nr));

Apply OFDM equalization to the 2-D signal contained in an 120-by-8 symbol array.

[eqsym2d,csi2d] = ofdmEqualize(rxsym2d,Hest,nvar,DataFormat="2-D");

Reshape the 2-D signal to a 30-by-4-by-8 symbol array. Apply OFDM equalization to the 3-D signal. Compare the results of OFDM equalization for the 30-by-4-by-8 symbol array with OFDM equalization for the 120-by-8 symbol array. As the isequal function result shows, the equalized symbols and soft channel state information returned for the 30-by-4-by-8 and 120-by-8 symbol arrays are equal.

rxsym3d = reshape(rxsym2d,30,4,Nr);
[eqsym3d,csi3d] = ofdmEqualize(rxsym3d,Hest,nvar,DataFormat="3-D");
isequal(eqsym3d,reshape(eqsym2d,30,4,Ns))

ans = logical
   1

isequal(csi3d,csi2d)

ans = logical
   1

Reshape the 2-D signal to a 60-by-2-by-8 symbol array. Apply OFDM equalization to the 3-D signal. Compare the results of OFDM equalization for the 60-by-2-by-8 symbol array with OFDM equalization for the 120-by-8 symbol array. The isequal function result confirms the equalized symbols and soft channel state information returned for the 60-by-2-by-8 and 120-by-8 symbol arrays are equal.

rxsym3d2 = reshape(rxsym2d,60,2,Nr);
[eqsym3d2,csi3d2] = ofdmEqualize(rxsym3d2,Hest,nvar,DataFormat="3-D");
isequal(eqsym3d2,reshape(eqsym2d,60,2,Ns))

ans = logical
   1

isequal(csi3d2,csi2d)

ans = logical
   1

Input Arguments

rxsym — Received symbols

numeric array | dlarray object

Received symbols, specified as a numeric array or a dlarray (Deep Learning Toolbox) object. For more information, see Array Support.

• If DataFormat is set to "3-D", the function expects rxsym to be specified as an N_SC-by-N_Symbols-by-N_R array. N_SC represents the number of OFDM subcarriers, N_Symbols represents the number of OFDM symbols, and N_R represents the number of receive antennas.

• If DataFormat is set to "2-D", the function expects rxsym to be specified as an N_RE-by-N_R array. N_RE represents the number of resource elements in an irregular subset of the OFDM subcarrier symbol grid. A resource element comprises one subcarrier in the frequency domain and one OFDM symbol in the time domain.

Data Types: double | single
Complex Number Support: Yes

heff — Effective channel estimate

numeric array | dlarray object

Effective channel estimate, specified as a 3-D numeric array, or a dlarray (Deep Learning Toolbox) object. heff can be a dlarray object only when the number of data streams N_S == 1. For more information, see Effective Channel and Array Support.

• If DataFormat is set to "3-D", the function expects heff to be an N_SC-by-N_S-by-N_R or an (N_SC×N_Symbols)-by-N_S-by-N_R array.

  • If heff is an N_SC-by-N_S-by-N_R array, all OFDM symbols in rxsym are equalized by the same channel estimate. N_SC represents the number of OFDM subcarriers, N_S represents the number of data streams, and N_R represents the number of receive antennas.

  • If heff is an (N_SC×N_Symbols)-by-N_S-by-N_R array, each OFDM symbol in rxsym is equalized by the corresponding entry in heff. N_Symbols represents the number of OFDM symbols.

• If DataFormat is set to "2-D", the function expects heff to be an N_RE-by-N_S-by-N_R array. Each OFDM symbol in rxsym is equalized by the corresponding entry in heff. N_RE represents the number of resource elements in an irregular subset of the OFDM subcarrier symbol grid.

Data Types: double | single
Complex Number Support: Yes

nvar — Noise variance

0 (default) | nonnegative scalar

Noise variance estimate for MMSE equalization, specified as a nonnegative scalar.

Dependencies

The noise variance setting is used only when you set Algorithm to "mmse".

Data Types: double | single

Name-Value Arguments

Specify optional pairs of arguments as Name1=Value1,...,NameN=ValueN, where Name is the argument name and Value is the corresponding value. Name-value arguments must appear after other arguments, but the order of the pairs does not matter.

Before R2021a, use commas to separate each name and value, and enclose Name in quotes.

Example: ofdmEqualize(rxsym,heff,DataFormat="2-D") equalizes an N_RE-by-N_R input OFDM symbol array using the MMSE algorithm.

Algorithm — Equalization algorithm

"mmse" (default) | "zf"

Equalization algorithm, specified as "mmse" or "zf".

• When this argument is set to "mmse", the function equalizes using the MMSE algorithm.

• When this argument is set to "zf", the function equalizes using the zero-forcing algorithm. When using the zero-forcing algorithm, the nvar argument value is ignored.

DataFormat — Format of signals

"3-D" (default) | "2-D"

Format of the signals, specified as "3-D" or "2-D".

When this argument is set to "3-D", OFDM subcarriers and OFDM symbols use two separate dimensions in the representation of rxsym and eqsym.

• The rxsym input must be an N_SC-by-N_Symbols-by-N_R array.

• The eqsym output is returned as an N_SC-by-N_Symbols-by-N_S array.

When this argument is set to "2-D", OFDM subcarriers and OFDM symbols use one combined dimension in the representation of rxsym and eqsym.

• The rxsym input must be an N_RE-by-N_R array.

• The eqsym output is returned as an N_RE-by-N_S array.

N_SC represents the number of OFDM subcarriers. N_Symbols represents the number of symbols. N_RE represents the number of resource elements in an irregular subset of the OFDM subcarrier symbol grid. N_S represents the number of data streams. N_R represents the number of receive antennas.

Output Arguments

eqsym — Equalized symbols

3-D array | 2-D array

Equalized symbols, returned as a 3-D or 2-D numeric array, or a dlarray (Deep Learning Toolbox). For more information, see Array Support.

• If DataFormat is set to "3-D", the function returns an N_SC-by-N_Symbols-by-N_S array. N_SC represents the number of OFDM subcarriers, N_Symbols represents the number of OFDM symbols, and N_S represents the number of data streams.

• If DataFormat is set to "2-D", the function returns an N_RE-by-N_S array. N_RE represents the number of resource elements in an irregular subset of the OFDM subcarrier symbol grid.

csi — Soft channel state information

matrix

Soft channel state information, returned as a matrix with size(csi,1) = size(heff,1) and size(csi,2) = NS = size(heff,2). N_S represents the number of data streams. This output is a log-likelihood ratio (LLR) scaling factor that accounts for the SNR in each resource element. A resource element comprises one subcarrier in the frequency domain and one OFDM symbol in the time domain.

The returned matrix can be a numeric array, or a dlarray (Deep Learning Toolbox). For more information, see Array Support.

More About

Array Support

The ofdmEqualize function supports input signals represented in a numeric array, dlarray (Deep Learning Toolbox), or gpuArray (Parallel Computing Toolbox). If inputs are specified as a combination of dlarray and gpuArray, the returned matrix is a dlarray object on the GPU.

The number of batch observations (N_B) is an optional dimension that can be added to these inputs for all supported data types.

• rxsym — The received symbols can be an:

  • N_SC-by-N_Symbols-by-N_R-by-N_B array, if DataFormat is specified as "3-D".

  • N_RE-by-N_R-by-N_B array, if DataFormat is specified as "2-D".

• heff — The channel estimate can be an:

  • N_SC-by-N_S-by-N_R-by-N_B or (N_SC×N_Symbols)-by-N_S-by-N_R-by-N_B array, if DataFormat is specified as "3-D".

  • N_RE-by-N_S-by-N_R-by-N_B array, if DataFormat is specified as "2-D".

• nvar — The noise variance can be a real vector with number of elements equal to number of batch observations, N_B.

N_SC is the number of OFDM subcarriers, N_Symbols is the number of symbols, N_RE is the number of resource elements in an irregular subset of the OFDM subcarrier symbol grid, N_S is the number of data streams, and N_R is the number of receive antennas.

For a list of Communications Toolbox™ features that support dlarray objects, see AI for Wireless.

Algorithms

Represent a MIMO, SISO, SIMO, or MISO communications system as Y = XH_eff + N, where:

• X is an N_SC-by-N_S transmitted signal.

• H_eff is an N_S-by-N_R MIMO channel matrix.

• N is an N_R-by-1 AWGN signal at receiver.

• Y is an N_SC-by-N_R received signal.

• N_SC is the number of OFDM subcarriers.

• N_S is the number of data streams.

• N_R is the number of receive antennas.

OFDM Equalization

In an OFDM system, Y = XH_eff + N applies to each frequency subcarrier. Equalization estimates the transmitted OFDM signal X and the soft channel state information by using a zero-forcing (ZF) equalizer or a minimum mean square estimate (MMSE) equalizer.

ZF Equalizer

With a ZF equalizer, the estimate of X is

$$\widehat{X}=Y{\widehat{H}}_{\text{eff}}^{†},$$

where $${\widehat{H}}_{\text{eff}}^{†}$$ is the pseudo inverse of Ĥ_eff.

When N_S < N_R, the soft channel state information is

$$CSI=\mathrm{diag}({\widehat{H}}_{\text{eff}}{\widehat{H}}_{\text{eff}}^{\text{'}})$$

and when N_S ≥ N_R

$$CSI=1/(\mathrm{diag}{({\widehat{H}}_{\text{eff}}{\widehat{H}}_{\text{eff}}^{\text{'}})}^{-1}).$$

MMSE Equalizer

With an MMSE equalizer, the estimate of X is

$$\widehat{X}=Y{\widehat{H}}_{\text{eff}}^{\text{'}}{({\widehat{H}}_{\text{eff}}{\widehat{H}}_{\text{eff}}^{\text{'}}+N_{0}I)}^{-1}.$$

The soft channel state information is

$$CSI=1/(\mathrm{diag}{({\widehat{H}}_{\text{eff}}{\widehat{H}}_{\text{eff}}^{\text{'}}+N_{0}I)}^{-1}).$$

Effective Channel

The effective channel, H_eff = PBH, measures the channel from the transmitted data streams to the receive antennas and results in an N_S-by-N_R matrix.

• P is an N_S-by-N_S precoding matrix. When there is no precoding, the precoding matrix is an N_S-by-N_S identity matrix.

• B is an N_S-by-N_T beamforming matrix. When there is no beamforming, N_S=N_T, and the beamforming matrix is an N_S-by-N_S identity matrix.

• H is an N_T-by-N_R matrix containing the estimated channel information.

Extended Capabilities

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

GPU Arrays

Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.

Usage notes and limitations:

• For the dimensions of input signals, if N_S ≤ N_R, then N_R must not exceed 32. N_S represents the number of data streams. N_R represents the number of receive antennas.

This function supports GPU array inputs. For more information, see Run MATLAB Functions on a GPU (Parallel Computing Toolbox).

Version History

Introduced in R2022b

R2023b: Add deep learning array support

The ofdmEqualize function adds support for dlarray (Deep Learning Toolbox) object processing for deep learning applications.

R2023b: Add GPU array support

The ofdmEqualize function adds support for gpuArray (Parallel Computing Toolbox) object processing to run code on a graphics processing unit (GPU).