Memoryless Nonlinearity

Apply amplifier models to complex baseband signal

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Libraries:
Communications Toolbox / RF Impairments and Components

Description

The Memoryless Nonlinearity block applies amplifier models to a complex baseband signal. Use this block to model memoryless nonlinear impairments caused by signal amplification in the radio frequency (RF) transmitter or receiver. For more information, see Memoryless Nonlinear Impairment Models.

Examples

Add Saleh Model of Power Amplifier to 16-QAM Signal in Simulink

This example applies the Saleh method of memoryless nonlinearity to a 16-QAM modulated signal. To show the Saleh model of a power amplifier, the example applies exaggerated levels that are not typical for modern radios.

The cm_mnl_saleh_16qam model 16-QAM modulates a signal containing a complete set of constellation points and passes them to Memoryless Nonlinearity blocks configured to apply the Saleh method with the default AM/AM and AM/PM distortion settings, AM/AM distortion alone, and AM/PM distortion alone. The model includes Constellation Diagram blocks after each Memoryless Nonlinearity block so you can analyze the impact of each impairment on the constellation.

In the model, you can click the Plot Transfer Curve block to run the testPA.m helper function. The testPA helper function plots this amplifier transfer curve for the Saleh method in its default configuration to show the nonlinear input-to-output signal. After amplification, plotted constellation points get displaced according to the input-to-output characteristics of the amplifier model. The voltage for each constellation point determines the direction and magnitude of distortion for each point in the constellation.

This set of constellation diagrams shows the constellation distortion for the amplifier configured to distort the amplitude and phase individually and together.

• The AMAM diagram shows the signal with amplitude-to-amplitude distortion but no amplitude-to-phase distortion. AM/AM distortion displaces constellation points radially away from or toward the origin. For the Saleh method and specified operating characteristics, the inner corner constellation points move away from the origin and the outer corner constellation points move toward the origin. The noncorner constellation points move a negligible amount.

• The AMPM diagram shows the signal with amplitude-to-phase distortion but no amplitude-to-amplitude distortion. AM/PM distortion causes rotation of constellation points. Constellation points farther from the origin are more displaced than points closer to the origin.

• The AMAM and AMPM diagram shows the signal with amplitude-to-amplitude distortion and amplitude-to-phase distortion. In this diagram, the constellation points move away from or toward the origin and rotate counter-clockwise.

This scatter plot of the constellation points shows points displaced by AM/AM distortion only. The scatterplot function used here plots constellation point locations labeled to highlight the relative distance each point has moved from the ideal constellation point location. To analyze the magnitude and direction each constellation point moved, you would need to replot the input and output characteristic curve for the current operating characteristics.

To explore the model try adjusting settings for the Memoryless Nonlinearity blocks to:

• Apply different impairment model methods.

• Apply different levels of impairments.

• Update the testPA.m helper function to plot and analyze input and output characteristics at nondefault Memoryless Nonlinearity block settings.

Add RF Impairments to DQPSK Signal

This example applies RF impairments to a signal modulated by the differential quadrature phase shift keying (DQPSK) method. To show the RF impairments, the example applies exaggerated levels that are not typical levels for modern radios.

In this example, the slex_rcvrimpairments_dqpsk model DQPSK-modulates a random signal and applies various RF impairments to the signal. The model uses impairment blocks from the RF Impairments library. The InitFun callback function initializes simulation variables. For more information, see Model Callbacks (Simulink).

After the impairment blocks, the signal forks into two paths. One path applies DC blocking, automatic gain control (AGC), and I/Q imbalance compensation to the signal before demodulation. The signal on the correction path is adjusted by the DC Blocker, AGC, and I/Q Imbalance Compensator blocks. Because the signal is DQPSK modulated, no carrier synchronization is required. The second path goes directly to demodulation. After demodulation, an error rate calculation is performed on both signals. The model includes Constellation Diagram blocks after modulation, before correction, and after correction so that you can analyze the constellation.

When the model runs, constellation diagrams plot the signal at these stages in the simulation:

• The After Modulation constellation diagram shows the reference DQPSK-modulated signal constellation.

• The Before Correction constellation diagram shows the attenuated and distorted signal constellation.

• The After Correction constellation diagram shows the signal has been amplified and improved after the correction blocks.

The error rate for the demodulated signal without AGC is primarily caused by free space path loss and I/Q imbalance. The QPSK modulation minimizes the effects of the other impairments.

Error rate for corrected signal:   0.000
Error rate for uncorrected signal: 0.042

To explore the model try:

• Adjusting RF impairment settings, rerun the model, and notice the changes to the constellation diagrams and error rates.

• Modifying the model to add an equalizer stage before the demodulation. Equalization has inherent ability to reduce some of the distortion caused by impairments. For more information, see Equalization.

Ports

Input

In — Input RF baseband signal

scalar | column vector | matrix

Input RF baseband signal, specified as a scalar, an N_Samp-by-1 vector, or an N_Samp-by-N_Chan matrix. N_Samp is the number of input samples per channel and N_Chan is the number of channels. Each channel must have the same number of samples.

When you input a variable-size signal, the frame size (number of rows) of the signal can change during simulation but the number of channels (columns) must remain constant. For more information, see Variable-Size Signal Support.

This port is unnamed on the block mask.

Data Types: double | single
Complex Number Support: Yes

Output

Out — Output RF baseband signal

scalar | column vector | matrix

Output RF baseband signal, returned as a scalar, column vector, or matrix. The output is of the same data type and dimension as the input signal, In. This port is unnamed on the block mask.

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.

Main Tab

Model — Amplifier nonlinearity model

cubic (default) | ampm | modified-rapp | saleh

Specify the amplifier nonlinearity model as one of the following methods:

• cubic

• ampm

• modified-rapp

• saleh

For more information, see Memoryless Nonlinear Impairment Models.

Linear power gain (dB) — Linear gain of amplifier

0 (default) | real scalar

Linear gain, specified as a scalar in dB.

Dependencies

To enable this parameter, set Model to cubic or modified-rapp.

Type of Non-linearity — Type of nonlinearity

OIP3 (default) | IIP3 | IP1dB | OP1dB | IPsat | OPsat

Third-order nonlinearity type in dBm, specified as one of these options:

• IIP3 — Input third-order intercept point

• OIP3 — Output third-order intercept point

• IP1dB — Input 1 dB compression point

• OP1dB — Output 1 dB compression point

• IPsat — Input saturation point

• OPsat — Output saturation point

Dependencies

To enable this parameter, set Model to cubic.

IIP3 (dBm) — Input third-order intercept point

Inf (default) | real positive number

Input third-order intercept point, specified as a real positive number in dBm.

Dependencies

To enable this parameter, set Model to cubic and Type of Non-linearity to IIP3.

OIP3 (dBm) — Output third-order intercept point

Inf (default) | real positive number

Output third-order intercept point, specified as a real positive number in dBm.

Dependencies

To enable this parameter, set Model to cubic and Type of Non-linearity to OIP3.

IP1dB (dBm) — Input 1 dB compression point

Inf (default) | real positive number

Input 1 dB compression point, specified as a real positive number in dBm.

Dependencies

To enable this parameter, set Model to cubic and Type of Non-linearity to IP1dB.

OP1dB (dBm) — Output 1 dB compression point

Inf (default) | real positive number

Output 1 dB compression point, specified as a real positive number in dBm.

Dependencies

To enable this parameter, set Model to cubic and Type of Non-linearity to OP1dB.

IPsat (dBm) — Input saturation point

Inf (default) | real positive number

Input saturation point, specified as a real positive number in dBm.

Dependencies

To enable this parameter, set Model to cubic and Type of Non-linearity to IPsat.

OPsat (dBm) — Output saturation point

Inf (default) | real positive number

Output saturation point, specified as a positive real number in dBm.

Dependencies

To enable this parameter, set Model to cubic and Type of Non-linearity to OPsat.

Lookup table (Pin(dBm), Pout(dBm), deg) — Lookup table

[ -25, 5, -1; -10, 20, -2; 0, 27, 5; 5, 28, 12 ] (default) | M-by-3 real matrix

Table lookup entries specified as a real M-by-3 matrix. The table expresses the model output power dBm level in matrix column 2 and the model phase change in degrees in matrix column 3 as related to the absolute value of the input signal power of matrix column 1 for the AM/AM-AM/PM model method. The column 1 input power must increase monotonically.

Dependencies

To enable this parameter, set Model to ampm.

Output saturation level (V) — Output saturation level

1 (default) | real positive number

Voltage output saturation level, specified as a real positive number in volts.

Dependencies

To enable this parameter, set Model to modified-rapp.

Magnitude smoothness factor — Magnitude smoothness factor

2 (default) | real positive number

Magnitude smoothness factor for the AM/AM calculations of the modified Rapp model method, specified as a positive real number.

Dependencies

To enable this parameter, set Model to modified-rapp.

Phase gain (rad) — Phase gain

-0.45 (default) | real scalar

Phase gain for the AM/PM calculations of the modified Rapp model method, specified as a real scalar in radians.

Dependencies

To enable this parameter, set Model to modified-rapp.

Phase saturation — Phase saturation

0.88 (default) | real positive number

Phase saturation for the AM/PM calculations of the modified Rapp model method, specified as a positive real number.

Dependencies

To enable this parameter, set Model to modified-rapp.

Phase smoothness factor — Phase smoothness factor

3.43 (default) | positive scalar | positive two-element vector

Phase smoothness factor for the AM/PM calculations of the modified Rapp model method, specified as a positive scalar or a positive two-element vector.

Dependencies

To enable this parameter, set Model to modified-rapp.

Input scaling (dB) — Scaling factor for input signal level

0 (default) | nonnegative real number

Scaling factor for input signal level for the Saleh model method, specified as a nonnegative real number in dB.

Dependencies

To enable this parameter, set Model to Saleh.

AM / AM parameters [alpha beta] — AM/AM conversion parameters

[ 2.1587, 1.1517 ] (default) | two-element vector

AM/AM two-tuple conversion parameters for the Saleh model method, specified as a two-element vector of nonnegative real numbers.

Dependencies

To enable this parameter, set Model to saleh.

AM / PM parameters [alpha beta] — AM/PM conversion parameters

[ 4.0033, 9.1040 ] (default) | two-element vector

AM/PM two-tuple conversion parameters for the Saleh model method, specified as a two-element vector of real numbers.

Dependencies

To enable this parameter, set Model to saleh.

Output scaling (dB) — Scaling factor for output signal level

0 (default) | nonnegative real number

Scaling factor for output signal level for the Saleh model method, specified as nonnegative real number in dB.

Dependencies

To enable this parameter, set Model to saleh.

Reference load (ohm) — Reference load

1 (default) | positive scalar

Reference load value in ohms, specified as a positive scalar. Use this value to convert between the voltage values and the power values.

Tunable: Yes

Plot power characteristics — Plot power characteristics

button

This button plots the power characteristics based on the parameters specified on the Model tab.

For more information, see Sample Power Characteristic Plots.

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	yes

More About

Memoryless Nonlinear Impairment Models

Memoryless nonlinear impairments distort the amplitude and phase of the input signal. The amplitude distortion is amplitude-to-amplitude modulation (AM/AM) and the phase distortion is amplitude-to-phase modulation (AM/PM). This block implements these model methods to simulate the memoryless nonlinear impairment models.

Model Method	Parameter Value	Memoryless Nonlinear Impairment
Cubic polynomial	cubic	Applies AM/AM and AM/PM
Saleh model	saleh
Modified Rapp model	modified-rapp
Lookup table	ampm	Applies impairment according to [Pin, Pout, ΔΦ] amplifier characteristics specified by the Lookup table (Pin(dBm), Pout(dBm), deg) parameter

The modeled impairments apply the AM/AM and AM/PM distortions differently according to the model method you specify. The models apply the memoryless nonlinear impairment to the input signal by following these steps.

1. Multiply the signal by an input gain factor.

   Note

   You can normalize the signal to 1 by setting the input scaling gain to the inverse of the input signal amplitude.

2. Split the complex signal into its magnitude and angle components. For real-valued input signals, the imaginary component is set to zero.

3. Apply an AM/AM distortion to the magnitude of the signal, according to the selected model method, to produce the magnitude of the output signal.

4. Apply an AM/PM distortion to the phase of the signal, according to the selected model method, to produce the angle of the output signal.

5. Combine the new magnitude and angle components into a complex signal. Then, multiply the result by an output gain factor.

The model methods apply AM/AM and AM/PM impairments as shown in this figure.

Cubic Polynomial Model Method

To apply the cubic polynomial model method, set the Model parameter to cubic. This figure shows the AM/PM conversion behavior for the cubic polynomial and hyperbolic tangent model methods.

The AM/PM conversion scales linearly with an input power value between the lower and upper limits of the input power level. Outside this range, the AM/PM conversion is constant at the values corresponding to the lower and upper input power limits, which are zero and (AM/PM conversion) × (upper input power limit – lower input power limit), respectively.

The cubic polynomial method uses linear power gain to determine the linear coefficient of a third-order polynomial and either IP3, P1dB, or Psat to determine the third-order coefficient of the polynomial. The general form of cubic nonlinearity models the AM/AM characteristics as:

$${\text{F}}_{\text{AM/AM}}{\text{(|u|)=c}}_{\text{1}}\text{×|u|+}\frac{\text{3}}{\text{4}}{\text{c}}_{\text{3}}{\text{×|u|}}^{\text{3}}$$

where FAM/AM(|u|) is the magnitude of the output signal, |u| is the magnitude of the input signal, c₁ is the coefficient of the linear gain term, and c₃ is the coefficient of the cubic gain term. The method takes the results for IIP3, OIP3, IP1dB, OP1dB, IPsat, and OPsat from [1]. It computes the c₃ coefficient for these nonlinearity types by using these parameters and equations:

Parameter	Description	Equation
IIP3 (dBm)	Input third-order intercept point — Input power level at which the power from linear gain is equal to the power from a third-order nonlinearity	$$c_3=-\frac{4c_1}{3\times {10}^{[(IIP3-30)/10]}}$$ IIP3 is given in dBm.
OIP3 (dBm)	Output third-order intercept point — Output power level at which the power from linear gain is equal to the power from a third-order nonlinearity	$$c_3=-\frac{4c_1^3}{3\times {10}^{[(OIP3-30)/10]}}$$ OIP3 is given in dBm.
IP1dB (dBm)	Input 1 dB gain compression power — Input power level at which the output power is one dB less than the power from linear gain	$$c_3=-\frac{2c_1({10}^{\raisebox{1ex}{$19$}\!\left/ \!\raisebox{-1ex}{$20$}\right.}-10)}{15\times {10}^{[(IP1dB-30)/10]}}$$ IP1dB is given in dBm.
OP1dB (dBm)	Output 1 dB gain compression power — Output power level one dB less than the power from linear gain	$$c_3=-\frac{2c_1^3({10}^{\raisebox{1ex}{$19$}\!\left/ \!\raisebox{-1ex}{$20$}\right.}-10)}{15\times {10}^{[(OP1dB-30-LGdB+1)/10]}}$$ OP1dB is given in dBm, and LGdB is the linear gain in dB.
IPsat (dBm)	Input saturation power — Input power at which the output power saturates	$$c_3=-\frac{4c_1}{9\times {10}^{[(IPsat-30)/10]}}$$ IPsat is given in dBm.
OPsat (dBm)	Output saturation power	$$c_3=-\frac{16c_1^3}{81\times {10}^{[(OPsat-30)/10]}}$$ OPsat is given in dBm.

Saleh Model Method

To apply the Saleh model method, set the Model parameter to saleh. As described in [2], the Saleh method is based on a normalized transfer function and uses input and output scaling parameters to adjust the signal levels from their normalized values. With the Saleh method,

• AM / AM parameters [alpha beta] specifies the alpha_AM/AM and beta_AM/AM variables that compute the amplitude gain for an input signal using this equation:

  $$F_{AM/AM}(u)=\frac{alph{a}_{AM/AM}\times \left|u\right|}{1+bet{a}_{AM/AM}\times |u{|}^2}$$

  where |u| is the magnitude of the scaled signal and u is calculated as:

  $$u=InputScale\times u_{\text{in}}$$

• AM / PM parameters [alpha beta] specifies the alpha_AM/PM and beta_AM/PM variables that compute the phase change for an input signal using this equation:

  $$F_{AM/PM}(u)=\frac{alph{a}_{AM/PM}\times |u{|}^2}{1+bet{a}_{AM/PM}\times |u{|}^2}+angle(u)$$

  where |u| is the magnitude of the scaled signal and angle is a function that returns the phase angle of u.

• The scaled output signal, u_out is calculated as:

  $$u_{\text{out}}=F_{AM/AM}\times e^{(F_{AM/PM}\times i)}\times OutputScale$$

Modified Rapp Model Method

To apply the modified Rapp model method, set the Model parameter to modified-rapp. As described in [3], the modified Rapp method applies AM/AM and AM/PM distortion based on normalized transfer functions and uses input and output scaling parameters to adjust the signal levels from their normalized values. The amplitude and phase distortion of the input signal are given by:

$$F_{\text{AMAM}}(u)=\frac{g_{\text{lin}}\times u}{{\left(1+{\left(\frac{g_{\text{lin}}\times u}{O_{\text{sat}}}\right)}^{\text{2}S}\right)}^{\text{1/2}S}}\text{,}$$

$$F_{AM/PM}(u)=\frac{{\rm A}|u{|}^{q_1}}{1+{\left(\frac{\left|u\right|}{B}\right)}^{q_2}}$$

The output signal u_out is calculated as:

$$u_{\text{out}}=F_{AM/AM}(u)\times e^{(F_{{}_{AM/PM}}(u)\times i)}$$

• g_lin is 10^{(Linear power gain (dB)/20))} and is the amplitude gain of the amplifier.

• u is the magnitude of the input signal.

• S is the smoothness factor, specified by the Magnitude smoothness factor parameter.

• O_sat is the output saturation level, specified by the Output saturation level (V) parameter.

• A is the phase gain in radians, specified by the Phase gain (rad) parameter.

• B is the phase saturation, specified by the Phase saturation parameter.

• q is the phase smoothness, specified by the Phase smoothness factor parameter.

  • When the Phase smoothness factor is a positive scalar value, q1 = q2 = q = Phase smoothness factor.

  • When the Phase smoothness factor is a positive two-element vector value, q1 = Phase smoothness factor (1); q2 = Phase smoothness factor (2).

Lookup Table Model Method

To apply the lookup table method, set the Model parameter to ampm. The lookup table method uses a lookup table to specify the amplifier power characteristics. The table returns interpolated or extrapolated values using linear interpolation. Each row in the table expresses the relationship between output power or phase change as a function of input power.

$$u_{\text{out}}=Tabl{e}_{AM/AM}(u)\times e^{((Tabl{e}_{AM/PM}(u)+\angle u)\times i)}$$

where u_out is the output signal and u is the magnitude of the input signal.

Sample Power Characteristic Plots

• To visualize the power characteristics of your model, set the parameters listed in the table and click the Plot power characteristics button. This table shows power characteristic plot samples for the parameter settings listed in the Parameters and Example Values column.

  Model	Parameters and Example Values	Power Characteristics Plot
  Cubic Polynomial	Main tab: Linear power gain (dB): 7 Type of Non-linearity: IIP3 IIP3 (dBm): 33 Simulate using: Code generation
  Saleh	Main tab: Input scaling (dB): 0 AM/AM parameters [alpha beta]: [ 2.1587, 1.1517 ] AM/PM parameters [alpha beta]: [ 4.0033, 9.1040 ] Output scaling (dB): 0 Simulate using: Interpreted execution
  Modified Rapp	Main tab: Linear power gain (dB): 7 Output saturation level (V): 1 Magnitude smoothness factor: 2 Phase gain (rad): -.45 Phase saturation: 0.88 Phase smoothness factor: 3.43 Simulate using: Code generation
  Lookup Table	Main tab: Lookup table (Pin(dBm), Pout(dBm), deg): [-25, 5, -1; -10, 20, -2; 0, 27, 5; 5, 28, 12] Simulate using: Code generation	Note To determine appropriate Pout (dBm) and deg values for any Pin (dBm) values below the value specified in the first row of the Lookup table (Pin(dBm), Pout(dBm), deg) parameter, the block applies the same power gain and deg as the first row, to maintain linearity and phase continuity. For any Pin (dBm) higher than the value in the last row of the Lookup table (Pin(dBm), Pout(dBm), deg) parameter, the block uses linear extrapolation from last two rows of the Lookup table (Pin(dBm), Pout(dBm), deg).

References

[1] Kundert, Ken."Accurate and Rapid Measurement of IP₂ and IP₃," The Designer Guide Community, May 22, 2002.

[2] Saleh, A.A.M. “Frequency-Independent and Frequency-Dependent Nonlinear Models of TWT Amplifiers.” IEEE^® Transactions on Communications 29, no. 11 (November 1981): 1715–20. https://doi.org/10.1109/TCOM.1981.1094911.

[3] Rapp, Ch. "Effects of HPA-Nonlinearity on a 4-DPSK/OFDM-Signal for a Digital Sound Broadcasting System." In Proceedings Second European Conf. on Sat. Comm. (ESA SP-332), 179–84. Liege, Belgium, 1991. https://elib.dlr.de/33776/.

Extended Capabilities

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

Version History

Introduced before R2006a

R2026a: Multichannel Support Added

The block can now process an N_Samp-by-N_Chan matrix signal. N_Samp is the number of input samples per channel and N_Chan is the number of channels. Each channel must have the same number of samples.

R2024b: Memory Less Nonlinearity Block Updates

The block now permits a 2-element phase smoothness factor with independent q₁ and q₂ values for the modified Rapp model method, as expressed in this equation.

$$F_{AM/PM}(u)=\frac{{\rm A}|u{|}^{q_1}}{1+{\left(\frac{\left|u\right|}{B}\right)}^{q_2}}$$

The block now permits negative values in the 2-element AM / PM conversion parameter for the Saleh model method. This update enables you to express negative phase rotation in the AM / PM impairment.

The values in the Model parameter for Memoryless Nonlinearity block have been renamed:

Parameter	Values before R2024b	Values since R2024b
Model	Cubic polynomial	cubic
	AM/AM - AM/PM	ampm
	Modified Rapp	modified-rapp
	Saleh	saleh

The default value for the Type of Non-linearity parameter have been changed:

Parameter	Default Values Before R2024b	Default Values Since R2024b
Type of Non-linearity	IIP3	OIP3

R2023a: Modernize memoryless nonlinearity for amplifier modeling

The Memoryless Nonlinearity block updates the methods to apply memoryless nonlinearity impairments when modeling an amplifier.

The block adds an AM/AM-AM/PM model that uses a lookup table to define the impairment settings, modifies the parameters to apply the cubic polynomial model, and replaces the Rapp model with the modified Rapp model. For amplifier model method details, see Memoryless Nonlinear Impairment Models.

These modeling methods are removed:

• Hyperbolic tangent model method

• Ghorbani model method

• AM/PM conversion settings of the cubic polynomial model method