Orthogonal Frequency Division Multiplexing

class ElementType(value)

Bases: SerializableEnum

Type of resource element.

REFERENCE = 0

Reference element within the time-frequency OFDM grid

DATA = 1

Data element within the time-frequency OFDM grid

NULL = 2

Empty element within the time-frequency OFDM grid

class PrefixType(value)

Bases: SerializableEnum

Type of prefix applied to the OFDM resource

CYCLIC = 0

Cyclic prefix repeating the resource waveform in time-domain

ZEROPAD = 1

Prefix zero-padding the prefix in time-domain

NONE = 2

No prefix applied

class FrameElement(type, repetitions=1)

Bases: Serializable

yaml_tag: Optional[str] = 'FrameElement'

YAML serialization tag.

serialized_attributes: Set[str] = {'repetitions', 'type'}

Set of object attributes to be serialized.

type: ElementType

repetitions: int = 1

class FrameResource(repetitions=1, prefix_type=PrefixType.CYCLIC, prefix_ratio=0.0, elements=None)

Bases: Serializable

Configures one sub-section of an OFDM symbol section in time AND frequency.

yaml_tag: Optional[str] = 'OFDM-Resource'

YAML serialization tag.

serialized_attributes: Set[str] = {'elements', 'prefix_type'}

Set of object attributes to be serialized.

prefix_type: PrefixType

Prefix type of the frame resource

elements: List[FrameElement]

Individual resource elements

property repetitions: int

Number of block repetitions along the frequency axis.

Returns:

Number of repetitions.

Return type:

int

property prefix_ratio: float

Ratio between full block length and prefix length.

Returns:

The ratio between zero and one.

Return type:

float

Raises:

ValueError – If ratio is less than zero or larger than one.

property num_subcarriers: int

Number of occupied subcarriers.

Returns:

Number of occupied subcarriers.

Return type:

int

property num_symbols: int

Number of data symbols this resource can modulate.

Returns:

Number of modulated symbols.

property num_references: int

Number of references symbols this resource can modulate.

Returns:

Number of modulated symbols.

property mask: ndarray

Boolean mask selecting a specific type of element from the OFDM grid.

Returns:

Mask of dimension num_element_types`x`num_subcarriers.

Return type:

np.ndarray

class FrameSection(num_repetitions=1, frame=None)

Bases: object

OFDM Frame configuration time axis.

property frame: OFDMWaveform | None

OFDM frame this section belongs to.

Returns:

Handle to the OFDM frame. None if this section is considered floating.

property num_repetitions: int

Number of section repetitions in the time-domain of an OFDM grid.

Returns:

The number of repetitions.

Return type:

int

property num_symbols: int

Number of data symbols this section can modulate.

Returns:

The number of symbols

Return type:

int

property num_references: int

Number of data symbols this section can modulate.

Returns:

The number of symbols

Return type:

int

property num_words: int

Number of OFDM symbols, i.e. words of subcarrier symbols this section can modulate.

Returns:

The number of words.

Return type:

int

property num_subcarriers: int

Number of subcarriers this section requires.

Returns:

The number of subcarriers.

Return type:

int

property resource_mask: ndarray

abstract property num_samples: int

Number of samples within this OFDM time-section.

Returns:

Number of samples

Return type:

int

place_symbols(data_symbols, reference_symbols)

Return type:

ndarray

pick_symbols(grid)

Return type:

Tuple[ndarray, ndarray]

abstract modulate(symbols)

Modulate this section into a complex base-band signal.

Parameters:

symbols (np.ndarray) – The palced complex symbols encoded in this OFDM section. This includes both reference and data symbols to be transmitted.

Returns:

The modulated signal vector.

Return type:

np.ndarray

abstract demodulate(signal)

Demodulate a time section of a complex OFDM base-band signal into data symbols.

Parameters:

signal (np.ndarray) – Vector of complex-valued base-band samples.

Return type:

ndarray

Returns: Sequence of demodulated data and reference symbols.

class FrameSymbolSection(num_repetitions=1, pattern=None, frame=None)

Bases: FrameSection, Serializable

yaml_tag: str = 'Symbol'

YAML serialization tag.

serialized_attributes: Set[str] = {'pattern'}

Set of object attributes to be serialized.

pattern: List[int]

property num_symbols: int

Number of data symbols this section can modulate.

Returns:

The number of symbols

Return type:

int

property num_references: int

Number of data symbols this section can modulate.

Returns:

The number of symbols

Return type:

int

property num_words: int

Number of OFDM symbols, i.e. words of subcarrier symbols this section can modulate.

Returns:

The number of words.

Return type:

int

property num_subcarriers: int

Number of subcarriers this section requires.

Returns:

The number of subcarriers.

Return type:

int

modulate(symbols)

Modulate this section into a complex base-band signal.

Parameters:

symbols (np.ndarray) – The palced complex symbols encoded in this OFDM section. This includes both reference and data symbols to be transmitted.

Returns:

The modulated signal vector.

Return type:

np.ndarray

demodulate(signal)

Demodulate a time section of a complex OFDM base-band signal into data symbols.

Parameters:

signal (np.ndarray) – Vector of complex-valued base-band samples.

Return type:

ndarray

Returns: Sequence of demodulated data and reference symbols.

property resource_mask: ndarray

property num_samples: int

Number of samples within this OFDM time-section.

Returns:

Number of samples

Return type:

int

class FrameGuardSection(duration, num_repetitions=1, frame=None)

Bases: FrameSection, Serializable

yaml_tag: Optional[str] = 'Guard'

YAML serialization tag.

property duration: float

Guard section duration in seconds.

Returns:

Duration in seconds.

Return type:

float

property num_samples: int

Number of samples within this OFDM time-section.

Returns:

Number of samples

Return type:

int

modulate(symbols)

Modulate this section into a complex base-band signal.

Parameters:

symbols (np.ndarray) – The palced complex symbols encoded in this OFDM section. This includes both reference and data symbols to be transmitted.

Returns:

The modulated signal vector.

Return type:

np.ndarray

demodulate(baseband_signal)

Demodulate a time section of a complex OFDM base-band signal into data symbols.

Parameters:

signal (np.ndarray) – Vector of complex-valued base-band samples.

Return type:

ndarray

Returns: Sequence of demodulated data and reference symbols.

class OFDMWaveform(subcarrier_spacing=1000.0, num_subcarriers=1024, dc_suppression=True, resources=None, structure=None, **kwargs)

Bases: ConfigurablePilotWaveform, Serializable

Generic Orthogonal-Frequency-Division-Multiplexing with a flexible frame configuration.

The internally applied FFT size is OFDMWaveform.num_subcarriers() times WaveformGenerator.oversampling_factor().

The following features are supported:

• The modem can transmit or receive custom-defined frames. Frames may contain UL/DL data symbols, null carriers, pilot subcarriers, reference signals and guard intervals.

• SC-FDMA can also be implemented with a precoder.

• Subcarriers can be modulated with BPSK/QPSK/16-/64-/256-QAM.

• Cyclic prefixes for interference-free channel estimation and equalization are supported.

This implementation has currently the following limitations:

• All subcarriers use the same modulation scheme

Parameters:

• subcarrier_spacing (float, optional) – Spacing between individual subcarriers in Hz. \(1~\mathrm{kHz}\) by default.

• num_subcarriers (int, optional) – Maximum number of assignable subcarriers. Unassigned subcarriers will be assumed to be zero. \(1024\) by default.

• dc_suppression (bool, optional) – Suppress the direct current component during waveform generation. Enabled by default.

• resources (List[FrameResource], optional) – Frequency-domain resource section configurations.

• structure (List[FrameSection], optional) – Time-domain frame configuration.

• **kwargs (Any) – Waveform generator base class initialization parameters. Refer to WaveformGenerator for details.

yaml_tag: str = 'OFDM'

YAML serialization tag.

dc_suppression: bool

property resources: List[FrameResource]

OFDM grid resources.

Returns: List of resources.

property structure: List[FrameSection]

OFDM frame configuration in time domain.

Returns: List of frame elements.

property modulation_order: int

Access the modulation order.

Returns:

The modulation order.

Return type:

int

add_resource(resource)

Add a OFDM frequency resource to the waveform.

Parameters:

resource (FrameResource) – The resource description to be added.

Return type:

None

add_section(section)

Add a frame section to the OFDM structure.

Parameters:

section (FrameSection) – The section to be added.

Return type:

None

property pilot_section: PilotSection | None

Static pilot section transmitted at the beginning of each OFDM frame.

Required for time-domain synchronization and equalization of carrier frequency offsets.

Returns:

FrameSection of the pilot symbols, None if no pilot is configured.

property pilot_signal: Signal

Model of the pilot sequence within this communication waveform.

Returns:

The pilot sequence.

Return type:

Signal

property subcarrier_spacing: float

Subcarrier spacing between frames.

Returns:

Spacing in Hz.

Return type:

float

property symbols_per_frame: int

Number of symbols per OFDM frame.

property words_per_frame: int

Number of words per OFDM frame.

property references_per_frame: int

Number of reference symbols per OFDM frame.

property frame_duration: float

Length of one data frame in seconds.

Returns:

Frame length in seconds.

Return type:

float

property samples_in_frame: int

The number of discrete samples per generated frame.

Returns:

The number of samples.

Return type:

int

map(data_bits)

Map a stream of bits to data symbols.

Parameters:

data_bits (np.ndarray) – Vector containing a sequence of L hard data bits to be mapped onto data symbols.

Returns:

Mapped data symbols.

Return type:

Symbols

unmap(symbols)

Map a stream of data symbols to data bits.

Parameters:

symbols (Symbols) – Sequence of K data symbols to be mapped onto bit sequences.

Returns:

Vector containing the resulting sequence of L data bits In general, L is greater or equal to K.

Return type:

np.ndarray

modulate(symbols)

Modulate a stream of data symbols to a base-band signal containing a single data frame.

Parameters:

data_symbols (Symbols) – Singular stream of data symbols to be modulated by this waveform.

Returns:

Signal model of a single modulate data frame.

Return type:

Signal

demodulate(signal)

Demodulate a base-band signal stream to data symbols.

Parameters:

signal (np.ndarray) – Vector of complex-valued base-band samples of a single communication frame.

Return type:

Symbols

Returns:

The demodulated communication symbols

property bandwidth: float

Bandwidth of the frame generated by this generator.

Used to estimate the minimal sampling frequency required to adequately simulate the scenario.

Returns:

Bandwidth in Hz.

Return type:

float

property bits_per_frame: int

Number of bits required to generate a single data frame.

Returns:

Number of bits

Return type:

int

property bit_energy: float

Returns the theoretical average (discrete-time) bit energy of the modulated baseband_signal.

Energy of baseband_signal \(x[k]\) is defined as \(\sum{|x[k]}^2\) Only data bits are considered, i.e., reference, guard intervals are ignored.

property symbol_energy: float

The theoretical average symbol (discrete-time) energy of the modulated baseband_signal.

Energy of baseband_signal \(x[k]\) is defined as \(\sum{|x[k]}^2\) Only data bits are considered, i.e., reference, guard intervals are ignored.

Returns:

The average symbol energy in UNIT.

property power: float

Returns the theoretical average symbol (unitless) power,

Power of baseband_signal \(x[k]\) is defined as \(\sum_{k=1}^N{|x[k]|}^2 / N\) Power is the average power of the data part of the transmitted frame, i.e., bit energy x raw bit rate

property num_subcarriers: int

Maximum number of subcarriers.

Sometimes also referred to as FFT-size.

Returns:

Number of subcarriers.

Return type:

int

property sampling_rate: float

Rate at which the waveform generator signal is internally sampled.

Returns:

Sampling rate in Hz.

Return type:

float

class PilotSection(pilot_elements=None, frame=None)

Bases: FrameSection, Serializable

Pilot symbol section within an OFDM frame.

Parameters:

• pilot_elements (Optional[Symbols], optional) – Symbols with which the subcarriers within the pilot will be modulated. By default, a pseudo-random sequence from the frame mapping will be generated.

• frame (Optional[WaveformGeneratorOfdm], optional) – The frame configuration this pilot section belongs to.

yaml_tag: Optional[str] = 'OFDM-Pilot'

YAML serialization tag

property num_samples: int

Number of samples within this OFDM time-section.

Returns:

Number of samples

Return type:

int

property pilot_elements: Symbols | None

Symbols with which the subcarriers within the pilot will be modulated.

Returns:

A stream of symbols. None, if no subsymbols where specified.

Raises:

ValueError – If the configured symbols contains multiple streams.

modulate(_=None)

Modulate this section into a complex base-band signal.

Parameters:

symbols (np.ndarray) – The palced complex symbols encoded in this OFDM section. This includes both reference and data symbols to be transmitted.

Returns:

The modulated signal vector.

Return type:

np.ndarray

demodulate(_)

Demodulate a time section of a complex OFDM base-band signal into data symbols.

Parameters:

signal (np.ndarray) – Vector of complex-valued base-band samples.

Return type:

ndarray

Returns: Sequence of demodulated data and reference symbols.

classmethod to_yaml(representer, node)

Serialize a serializable object to YAML.

Parameters:

• representer (SafeRepresenter) – A handle to a representer used to generate valid YAML code. The representer gets passed down the serialization tree to each node.

• node (PilotSection) – The channel instance to be serialized.

Return type:

MappingNode

Returns: The serialized YAML node.

classmethod from_yaml(constructor, node)

Recall a new serializable class instance from YAML.

Parameters:

• constructor (SafeConstructor) – A handle to the constructor extracting the YAML information.

• node (Node) – YAML node representing the PilotSection serialization.

Return type:

PilotSection

Returns: The de-serialized object.

class SchmidlCoxPilotSection(pilot_elements=None, frame=None)

Bases: PilotSection

Pilot Symbol Section of the Schmidl Cox Algorithm.

Refer to Schmidl and Cox[1] for a detailed description.

Parameters:

• pilot_elements (Optional[Symbols], optional) – Symbols with which the subcarriers within the pilot will be modulated. By default, a pseudo-random sequence from the frame mapping will be generated.

• frame (Optional[WaveformGeneratorOfdm], optional) – The frame configuration this pilot section belongs to.

yaml_tag: Optional[str] = 'SchmidlCoxPilot'

YAML serialization tag

property num_samples: int

Number of samples within this OFDM time-section.

Returns:

Number of samples

Return type:

int

demodulate(_)

Demodulate a time section of a complex OFDM base-band signal into data symbols.

Parameters:

signal (np.ndarray) – Vector of complex-valued base-band samples.

Return type:

ndarray

Returns: Sequence of demodulated data and reference symbols.

class OFDMSynchronization(waveform_generator=None)

Bases: Synchronization[OFDMWaveform]

Synchronization Routine for OFDM Waveforms.

Parameters:

waveform_generator (WaveformGenerator, optional) – The waveform generator this synchronization routine is attached to.

class OFDMCorrelationSynchronization(threshold=0.9, guard_ratio=0.8, *args, **kwargs)

Bases: CorrelationSynchronization[OFDMWaveform]

Correlation-Based Pilot Detection and Synchronization for OFDM Waveforms.

Parameters:

• threshold (float, optional) – Correlation threshold at which a pilot signal is detected.

• guard_ratio (float, optional) – Guard ratio of frame duration.

• *args (Any) – Synchronization base class initialization parameters.

yaml_tag: Optional[str] = 'OFDM-PilotCorrelation'

YAML serialization tag.

class SchmidlCoxSynchronization(waveform_generator=None)

Bases: OFDMSynchronization

Schmidl-Cox Algorithm for OFDM Waveform Time Synchronization and Carrier Frequency Offset Equzalization.

Applying the synchronization routine requires the respective waveform to have a SchmidlCoxPilotSection pilot symbol section configured.

Refer to Schmidl and Cox[1] for a detailed description.

Parameters:

waveform_generator (WaveformGenerator, optional) – The waveform generator this synchronization routine is attached to.

yaml_tag: Optional[str] = 'SchmidlCox'

YAML serialization tag

synchronize(signal)

Simulates time-synchronization at the receiver-side.

Sorts base-band signal-sections into frames in time-domain.

Parameters:

signal (np.ndarray) – Vector of complex base-band samples of with num_streams`x`num_samples entries.

Return type:

List[int]

Returns:

List of time indices indicating the first samples of frames detected in signal.

Raises:

RuntimeError – If the synchronization routine is floating

class ReferencePosition(value)

Bases: SerializableEnum

Applied channel estimation algorithm after reception.

IDEAL = 0

IDEAL_PREAMBLE = 1

IDEAL_MIDAMBLE = 2

IDEAL_POSTAMBLE = 3

class OFDMIdealChannelEstimation(reference_position=ReferencePosition.IDEAL, *args, **kwargs)

Bases: IdealChannelEstimation[OFDMWaveform], Serializable

Ideal channel state estimation for OFDM waveforms.

Parameters:

reference_position (ReferencPosition, optional) – Assumed location of the reference symbols within the ofdm frame.

yaml_tag: Optional[str] = 'OFDM-Ideal'

YAML serialization tag.

serialized_attributes: Set[str] = {'reference_position'}

Set of object attributes to be serialized.

reference_position: ReferencePosition

Assumed position of the reference symbol within the frame.

estimate_channel(symbols)

Estimate the wireless channel of a received communication frame.

Parameters:

symbols (Symbols) – Demodulated communication symbols.

Return type:

Tuple[StatedSymbols, ChannelStateInformation]

Returns: The symbols and their respective channel states.

class OFDMLeastSquaresChannelEstimation(waveform_generator=None)

Bases: ChannelEstimation[OFDMWaveform], Serializable

Least-Squares channel estimation for OFDM waveforms.

Parameters:

waveform_generator (WaveformGenerator, optional) – The waveform generator this estimation routine is attached to.

yaml_tag: Optional[str] = 'OFDM-LS'

YAML serializtion tag

estimate_channel(symbols)

Estimate the wireless channel of a received communication frame.

Parameters:

symbols (Symbols) – Demodulated communication symbols.

Return type:

Tuple[StatedSymbols, ChannelStateInformation]

Returns: The symbols and their respective channel states.

class OFDMChannelEqualization(waveform_generator=None)

Bases: ChannelEqualization[OFDMWaveform], ABC

Channel estimation for OFDM waveforms.

Parameters:

waveform_generator (WaveformGenerator, optional) – The waveform generator this equalization routine is attached to.

yaml_tag: Optional[str] = 'OFDM-NoEqualization'

YAML serialization tag.

class OFDMZeroForcingChannelEqualization(waveform_generator=None)

Bases: ZeroForcingChannelEqualization[OFDMWaveform]

Zero-Forcing channel equalization for OFDM waveforms.

Parameters:

waveform_generator (WaveformGenerator, optional) – The waveform generator this equalization routine is attached to.

yaml_tag: Optional[str] = 'OFDM-ZF'

YAML serialization tag

[1] (1,2)

T.M. Schmidl and D.C. Cox. Robust frequency and timing synchronization for ofdm. IEEE Transactions on Communications, 45(12):1613–1621, 1997. doi:10.1109/26.650240.