ofdmmod

Modulate using OFDM method

Syntax

Y = ofdmmod(X,nfft,cplen)

Y = ofdmmod(X,nfft,cplen,nullidx)

Y = ofdmmod(X,nfft,cplen,nullidx,pilotidx,pilots)

Y = ofdmmod(X,nfft,cplen,___,OversamplingFactor=Value)

Description

Y = ofdmmod(X,nfft,cplen) modulates the frequency-domain input data subcarriers in X using the orthogonal frequency division multiplexing (OFDM) method with an FFT size specified by nfft and cyclic prefix length specified by cplen. For information, see OFDM Modulation.

Y = ofdmmod(X,nfft,cplen,nullidx) inserts null subcarriers into the frequency domain input data signal prior to performing OFDM modulation. The null subcarriers are inserted at index locations from 1 to nfft, as specified by nullidx. For this syntax, the number of rows in the input X is the number of used data subcarriers and must equal nfft – length(nullidx). Use null carriers to account for guard bands and DC subcarriers. For information, see Subcarrier Allocation, Guard Bands and Guard Intervals.

Y = ofdmmod(X,nfft,cplen,nullidx,pilotidx,pilots) inserts null and pilot subcarriers into the frequency domain input data symbols prior to performing OFDM modulation. The null subcarriers are inserted at the index locations specified by nullidx. The pilot subcarriers, pilots, are inserted at the index locations specified by pilotidx. For this syntax, the number of rows in the input X must equal nfft – length(nullidx) – length(pilotidx). The function assumes pilot subcarrier locations are the same across each OFDM symbol and transmit stream.

Y = ofdmmod(X,nfft,cplen,___,OversamplingFactor=Value) specifies the optional oversampling factor name-value argument in addition to input arguments in previous syntaxes. The oversampling factor for an upsampled output signal must be specified as a positive scalar, and the products (OversamplingFactor×nfft) and (OversamplingFactor×cplen) must both result in integers. For example, ofdmmod(X,nfft,cplen,OversamplingFactor=2) upsamples the output signal by a factor of two. The default value for OversamplingFactor is 1.

Tip

If you set the oversampling factor to a noninteger rational number, specify a fractional value rather than a decimal value. For example, with an FFT length of 12 and an oversampling factor of 4/3, their product is the integer 16. However, rounding 4/3 to 1.333 when setting the oversampling factor results in a noninteger product of 15.9960, which results in a code error.

Examples

OFDM Modulation over Two Streams

OFDM-modulate a fully packed input over two transmit streams.

Initialize input parameters and generate a random data input signal assigning data subcarriers to all FFT bins. Perform OFDM modulation.

M = 16;       % Modulation order for 16QAM
nfft = 128;   % FFT length
cplen = 16;   % Cyclic prefix length
nSym = 5;     % Number of symbols per RE
nStreams = 2; % Number of transmit streams
dataIn = randi([0 M-1],nfft,nSym,nStreams);
qamSig = qammod(dataIn,M,'UnitAveragePower',true);
y1 = ofdmmod(qamSig,nfft,cplen);

Apply OFDM Assigning Null Subcarriers

Apply OFDM modulation assigning null subcarriers.

Initialize input parameters and generate random data.

M = 16;     % Modulation order for 16QAM
nfft  = 64; % FFT length
cplen = 16; % Cyclic prefix length
nSym  = 10; % Number of symbols per RE

nullIdx  = [1:6 33 64-4:64]';
numDataCarrs = nfft-length(nullIdx);
inSym = randi([0 M-1],numDataCarrs,nSym);

QAM modulate data. Perform OFDM modulation.

qamSig = qammod(inSym,M,'UnitAveragePower',true);
outSig = ofdmmod(qamSig,nfft,cplen,nullIdx);

Perform OFDM Modulation Varying Cyclic Prefix per Symbol

Perform OFDM modulation to input frequency domain data signal varying cyclic prefix length applied to each symbol.

Initialize input parameters and generate random data.

M = 16; % Modulation order for 16QAM
nfft  = 64;
cplen = [4 8 10 7 2 2 4 11 16 3];
nSym  = 10;
nullIdx  = [1:6 33 64-4:64]';
numDataCarrs = nfft-length(nullIdx);
inSym = randi([0 M-1],numDataCarrs,nSym);

QAM modulate the data symbols and perform OFDM modulation to the QAM signal.

qamSig = qammod(inSym,M,UnitAveragePower=true);
outSig = ofdmmod(qamSig,nfft,cplen,nullIdx);

Apply OFDM to QPSK Signal Spatially Multiplexed over Two Streams

Apply OFDM modulation to a QPSK signal that is spatially multiplexed over two transmit streams.

Initialize input parameters and generate random data for each stream.

M = 4; % Modulation order for QPSK
nfft = 64;
cplen = 16;
nSym = 5;
nStreams = 2;
nullIdx = [1:6 33 64-4:64]';
pilotIdx = [12 26 40 54]';
numDataCarrs = nfft-length(nullIdx)-length(pilotIdx);
pilots = repmat(pskmod((0:M-1).',M),1,nSym,nStreams);

stream1 = randi([0 M-1],numDataCarrs,nSym);
stream2 = randi([0 M-1],numDataCarrs,nSym);

QPSK modulate data individually for each stream. Perform OFDM modulation.

qpskSig(:,:,1) = pskmod(stream1,M);
qpskSig(:,:,2) = pskmod(stream2,M);
y1 = ofdmmod(qpskSig,nfft,cplen,nullIdx,pilotIdx,pilots);

OFDM Modulation with Null and Pilot Packing

OFDM-modulate data input, specifying null and pilot packing.

Initialize input parameters, defining locations for null and pilot subcarriers. Generate random data, apply 16-QAM to data, QPSK to pilots, and perform OFDM modulation.

M = 16;     % Modulation order
nfft = 64;  % FFT length
cplen = 16; % Cyclic prefix length
nSym  = 10; % Number of symbols per RE

nullIdx  = [1:6 33 64-4:64]';
pilotIdx = [12 26 40 54]';

numDataCarrs = nfft-length(nullIdx)-length(pilotIdx);
dataSym = randi([0 M-1],numDataCarrs,nSym);
qamSig = qammod(dataSym,M,UnitAveragePower=true);
pilots = repmat(pskmod((0:3).',4),1,nSym);

y2 = ofdmmod(qamSig,nfft,cplen,nullIdx,pilotIdx,pilots);

OFDM Modulate with Upsampling and Nulls

Apply OFDM modulation to symbols. Insert nulls in the OFDM grid and oversample the output signal.

Initialize variables for the modulation order, oversampling factor, FFT size, cyclic prefix length, and null indices.

M = 64;      % Modulation order
osf = 3;     % Oversampling factor
nfft = 256;  % FFT length
cplen = 16;  % Cyclic prefix length

nullidx  = [1:6 nfft/2+1 nfft-5:nfft]';
numDataCarrs = nfft-length(nullidx);

Generate data symbols, apply QAM, and OFDM-modulate the data.

x = randi([0 M-1],numDataCarrs,1);
qamSig = qammod(x,M,UnitAveragePower=true);
y = ofdmmod(qamSig,nfft,cplen,nullidx,OversamplingFactor=osf); 

Input Arguments

X — Input baseband signal

array | dlarray object

Input baseband signal, specified as an N_DataSC-by-N_Sym-by-N_Streams array, or a dlarray (Deep Learning Toolbox) object. For more information, see Array Support.

• N_DataSC is the number of data subcarriers. N_DataSC = nfft – length(nullidx) – length(pilotidx).

• N_Sym is the number of OFDM symbols per transmit stream.

• N_Streams is the number of transmit streams and must be less than or equal to the number of transmit antennas.

• Symbols input data to an OFDM modulator are typically created with a baseband digital modulator, such as qammod.

Data Types: double | single
Complex Number Support: Yes

nfft — FFT length

integer greater than or equal to 8

FFT length, specified as an integer greater than or equal to 8. nfft is the upper bound for the number of subcarriers (N_DataSC) used in the modulation process. N_DataSC = nfft – length(nullidx) – length(pilotidx)

Data Types: double

cplen — Cyclic prefix length

scalar | row vector of length N_Sym

Cyclic prefix length, specified as a scalar or row vector of length N_Sym. Cyclic prefix length must be nonnegative.

• When you specify cplen as a scalar, the value must be in the range [0, nfft]. In this case, all symbols through all streams have the same cyclic prefix length.

• When you specify cplen as a row vector of length N_Sym, the cyclic prefix length can vary across symbols but remains the same length through all streams.

N_Sym is the number of symbols per stream.

For more information, see Subcarrier Allocation, Guard Bands and Guard Intervals.

Data Types: double

nullidx — Indices of null subcarrier locations

column vector

Indices of null subcarrier locations, specified as a column vector with element values in the range [1, nfft].

Data Types: double

pilotidx — Indices of pilot subcarrier locations

column vector

Indices of pilot subcarrier locations, specified as a column vector with element values in the range [1, nfft].

Data Types: double

pilots — Pilot subcarriers

3-D array | dlarray object

Pilot subcarriers, specified as an N_Pilot-by-N_Sym-by-N_Streams array of symbols, or a dlarray (Deep Learning Toolbox) object. For more information, see Array Support.

• N_Pilot must equal the length of pilotidx.

• N_Sym is the number of OFDM symbols per transmit stream.

• N_Streams is the number of transmit streams and must be less than or equal to the number of transmit antennas.

Tip

The function assumes pilot subcarrier locations are the same across each OFDM symbol and transmit stream. Use the comm.OFDMModulator System object™ to vary pilot subcarrier locations across OFDM symbols or streams.

Data Types: double | single

Output Arguments

Y — OFDM-modulated baseband signal

matrix of complex symbols | dlarray object

OFDM-modulated baseband signal, returned as an (osf × N_Out)-by-N_Streams matrix of complex symbols or a dlarray (Deep Learning Toolbox) object. The output, Y, is a dlarray if either X or pilots is a dlarray. For more information, see Array Support.

• osf is the oversampling factor, as determined by OversamplingFactor.

• When cplen is a scalar, N_CP = cplen and N_Out = N_Sym × (nfft + N_CP).

• When cplen is a row vector, N_CPTotal = sum(cplen) and N_Out = (N_Sym × nfft) + N_CPTotal.

• N_Sym is the number of symbols per transmit stream.

• N_Streams is the number of transmit streams and is less than or equal to the number of transmit antennas.

Data Types: double | single
Complex Number Support: Yes

More About

Array Support

The ofdmmod function supports input signals represented in an 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.

• X — The input data subcarriers can be an array of up to four dimensions, specified as an nfft-by-N_Sym-by-N_Streams-by-N_B array.

• pilots — The pilot subcarriers can be an array of up to four dimensions, specified as an N_Pilot-by-N_Sym-by-N_Streams-by-N_B array.

N_Pilot is the number of pilots, specifically the length of pilotidx. N_Sym is the number of OFDM symbols per data stream. N_Streams is the number of transmit streams and must be less than or equal to the number of transmit antennas.

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

Algorithms

OFDM Modulation

OFDM belongs to the class of multicarrier modulation schemes. Because the operation can transmit multiple carriers simultaneously, noise does not influence OFDM to the same degree as single-carrier modulation.

OFDM operation divides a high-rate data stream into low-rate data substreams by decomposing the transmission frequency band into a number of contiguous individually modulated subcarriers. This set of parallel and orthogonal subcarriers carry the data stream occupying almost the same bandwidth as a wideband channel. By using narrow orthogonal subcarriers, the OFDM signal gains robustness over a frequency-selective fading channel and eliminates adjacent subcarrier interference. Intersymbol interference (ISI) is reduced because the lower data rate substreams have symbol durations larger than the channel delay spread.

This image shows a frequency domain representation of orthogonal subcarriers in an OFDM waveform.

The transmitter applies inverse fast Fourier transform (IFFT) to N symbols at a time. Typically, the output of the IFFT is the sum of the N orthogonal sinusoids:

$$x(t)={\displaystyle \sum _{k=0}^{N-1}{X}_k{e}^{j2\pi k\Delta ft}},0\le t\le T,$$

where {X_k} are data symbols, and T is the OFDM symbol time. The data symbols X_k are typically complex and can be from any digital modulation alphabet (for example, QPSK, 16-QAM, 64-QAM, etc.).

Note

The MATLAB^® implementation of the discrete Fourier transform normalizes the output of the IFFT by 1/N. For more information, see Discrete Fourier Transform of Vector on the ifft reference page.

The subcarrier spacing is Δf = 1/T, ensuring that the subcarriers are orthogonal over each symbol period:

$$\frac{1}{T}{\displaystyle {\int }_0^T{\left(e^{j2\pi m\Delta ft}\right)}^{*}\left(e^{j2\pi n\Delta ft}\right)}dt=\frac{1}{T}{\displaystyle {\int }_0^T{e}^{j2\pi (m-n)\Delta ft}}dt=0\text{for}m\ne n.$$

An OFDM modulator consists of a serial-to-parallel conversion followed by a bank of N complex modulators, individually corresponding to each OFDM subcarrier.

Subcarrier Allocation, Guard Bands and Guard Intervals

Individual OFDM subcarriers are allocated as data, pilot, or null subcarriers.

As shown here, subcarriers are designated as data, DC, pilot, or guard-band subcarriers.

• Data subcarriers transmit user data.

• Pilot subcarriers are for channel estimation.

• Null subcarriers transmit no data. Subcarriers with no data provide a DC null and serve as buffers between OFDM resource blocks.

  • The null DC subcarrier is the center of the frequency band with an index value of (nfft/2 + 1) if nfft is even, or ((nfft + 1) / 2) if nfft is odd.

  • The guard bands provide buffers between adjacent signals in neighboring bands to reduce interference caused by spectral leakage.

Null subcarriers enable you to model guard bands and DC subcarrier locations for specific standards, such as the various 802.11 formats, LTE, WiMAX, or for custom allocations. You can allocate the location of nulls by assigning a vector of null subcarrier indices.

Similar to guard bands, guard intervals protect the integrity of transmitted signals in OFDM by reducing intersymbol interference.

Assignment of guard intervals is analogous to the assignment of guard bands. You can model guard intervals to provide temporal separation between OFDM symbols. The guard intervals help preserve intersymbol orthogonality after the signal passes through time-dispersive channels. You create guard intervals by using cyclic prefixes. Cyclic prefix insertion copies the last part of an OFDM symbol as the first part of the OFDM symbol.

OFDM benefits from the use of cyclic prefix insertion as long as the span of the time dispersion does not exceed the duration of the cyclic prefix.

Inserting a cyclic prefix results in a fractional reduction of user data throughput because the cyclic prefix occupies bandwidth that could be used for data transmission.

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™.

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

Version History

Introduced in R2018a

R2025a: C code generation support with deep learning array inputs

The ofdmmod function adds support for C code generation with input signals stored as dlarray (Deep Learning Toolbox) objects.

R2024a: Allow cyclic prefix length greater than FFT length

When the ofdmmod function specifies the cyclic prefix length as a vector, you can now assign OFDM symbol cyclic prefix lengths greater than the FFT length.

R2023a: Add deep learning array support

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

R2023a: Add GPU array support

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