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# Memoryless Nonlinearity

Apply memoryless nonlinearity to complex baseband signal

RF Impairments

## Description

The Memoryless Nonlinearity block applies a memoryless nonlinearity to a complex, baseband signal. You can use the block to model radio frequency (RF) impairments to a signal at the receiver.

This block accepts a column vector input signal.

 Note:   All values of power assume a nominal impedance of 1 ohm.

The Memoryless Nonlinearity block provides five different methods for modeling the nonlinearity, which you specify by the Method parameter. The options for the Method parameter are

• `Cubic polynomial`

• `Hyperbolic tangent`

• `Saleh model`

• `Ghorbani model`

• `Rapp model`

The block implements these five methods using subsystems underneath the block mask. For each of the first four methods, the nonlinearity subsystem has the same basic structure, as shown in the following figure.

Nonlinearity Subsystem

For the first four methods, each subsystem applies a nonlinearity to the input signal as follows:

1. Multiply the signal by a gain factor.

2. Split the complex signal into its magnitude and angle components.

3. Apply an AM/AM conversion to the magnitude of the signal, according to the selected Method, to produce the magnitude of the output signal.

4. Apply an AM/PM conversion to the phase of the signal, according to the selected Method, and adds the result to the angle of the signal to produce the angle of the output signal.

5. Combine the new magnitude and angle components into a complex signal and multiply the result by a gain factor, which is controlled by the Linear gain parameter.

Each subsystem implements the AM/AM and AM/PM conversions differently, according to the Method you specify. The Rapp model does not apply a phase change to the input signal. The nonlinearity subsystem for Rapp model has following structure:

Nonlinearity Subsystem for Rapp Model

The Rapp Subsystem applies nonlinearity as follows:

1. Multiply the signal by a gain factor.

2. Split the complex signal into its magnitude and angle components.

3. Apply an AM/AM conversion to the magnitude of the signal, according to the selected Method, to produce the magnitude of the output signal.

4. Combine the new magnitude and angle components into a complex signal and multiply the result by a gain factor, which is controlled by the Linear gain parameter.

If you want to see exactly how the Memoryless Nonlinearity block implements the conversions for a specific method, you can view the AM/AM and AM/PM subsystems that implement these conversions as follows:

1. Right-click on the Memoryless Nonlinearity block and select Mask > Look under mask. This displays the block's configuration underneath the mask. The block contains five subsystems corresponding to the five nonlinearity methods.

2. Double-click the subsystem for the method you are interested in. This displays the subsystem shown in the preceding figure, Nonlinearity Subsystem.

3. Double-click on one of the subsystems labeled AM/AM or AM/PM to view how the block implements the conversions.

### AM/PM Characteristics of the Cubic Polynomial and Hyperbolic Tangent Methods

The following illustration shows the AM/PM behavior for the ```Cubic polynomial``` and `Hyperbolic tangent` methods:

The AM/PM conversion scales linearly with input power value between the lower and upper limits of the input power level (specified by Lower input power limit for AM/PM conversion (dBm) and Upper input power limit for AM/PM conversion (dBm)). Beyond these values, AM/PM conversion is constant at the values corresponding to the lower and upper input power limits, which are zero and , respectively.

### AM/AM and AM/PM Characteristics of the Saleh Method

The following figure shows, for the Saleh method, plots of

• Output voltage against input voltage for the AM/AM conversion

• Output phase against input voltage for the AM/PM conversion

### Example with 16-ary QAM

You can see the effect of the Memoryless Nonlinearity block on a signal modulated by 16-ary quadrature amplitude modulation (QAM) in a scatter plot. The constellation for 16-ary QAM without the effect of the Memoryless Nonlinearity block is shown in the following figure:

You can generate a scatter plot of the same signal after it passes through the Memoryless Nonlinearity block, with the Method parameter set to `Saleh Mode`l, as shown in the following figure.

This plot is generated by the model described in Illustrate RF Impairments That Distort a Signal with the following parameter settings for the Rectangular QAM Modulator Baseband block:

• Normalization method set to ```Average Power```

• Average power (watts) set to `1e-2`

The following sections discuss parameters specific to the Saleh, Ghorbani, and Rapp models.

### Parameters for the Saleh Model

The Input scaling (dB) parameter scales the input signal before the nonlinearity is applied. The block multiplies the input signal by the parameter value, converted from decibels to linear units. If you set the parameter to be the inverse of the input signal amplitude, the scaled signal has amplitude normalized to 1.

The AM/AM parameters, alpha and beta, are used to compute the amplitude gain for an input signal using the following function:

`${F}_{AM/AM}\left(u\right)=\frac{\text{alpha}*u}{1+\text{beta}*{u}^{2}}$`

where u is the magnitude of the scaled signal.

The AM/PM parameters, alpha and beta, are used to compute the phase change for an input signal using the following function:

`${F}_{AM/PM}\left(u\right)=\frac{\text{alpha}*{u}^{2}}{1+\text{beta}*{u}^{2}}$`

where u is the magnitude of the scaled signal. Note that the AM/AM and AM/PM parameters, although similarly named alpha and beta, are distinct.

The Output scaling (dB) parameter scales the output signal similarly.

### Parameters for the Ghorbani Model

The Input scaling (dB) parameter scales the input signal before the nonlinearity is applied. The block multiplies the input signal by the parameter value, converted from decibels to linear units. If you set the parameter to be the inverse of the input signal amplitude, the scaled signal has amplitude normalized to 1.

The AM/AM parameters, [x1 x2 x3 x4], are used to compute the amplitude gain for an input signal using the following function:

`${F}_{AM/AM}\left(u\right)=\frac{{x}_{1}{u}^{{x}_{2}}}{1+{x}_{3}{u}^{{x}_{2}}}+{x}_{4}u$`

where u is the magnitude of the scaled signal.

The AM/PM parameters, [y1 y2 y3 y4], are used to compute the phase change for an input signal using the following function:

`${F}_{AM/PM}\left(u\right)=\frac{{y}_{1}{u}^{{y}_{2}}}{1+{y}_{3}{u}^{{y}_{2}}}+{y}_{4}u$`

where u is the magnitude of the scaled signal.

The Output scaling (dB) parameter scales the output signal similarly.

### Parameters for the Rapp Model

The Linear gain (dB) parameter scales the input signal before the nonlinearity is applied. The block multiplies the input signal by the parameter value, converted from decibels to linear units. If you set the parameter to be the inverse of the input signal amplitude, the scaled signal has amplitude normalized to 1.

The Smoothness factor and Output saturation level parameters are used to compute the amplitude gain for the input signal:

`${F}_{AM/AM}\left(u\right)=\frac{u}{{\left(1+{\left(\frac{u}{{O}_{sat}}\right)}^{2S}\right)}^{1/2S}}$`

where u is the magnitude of the scaled signal, S is the Smoothness factor, and Osat is the Output saturation level.

The Rapp model does not apply a phase change to the input signal.

The Output saturation level parameter limits the output signal level.

## Parameters

Method

The nonlinearity method.

The following describes specific parameters for each method.

Linear gain (db)

Scalar specifying the linear gain for the output function.

IIP3 (dBm)

Scalar specifying the third order intercept.

AM/PM conversion (degrees per dB)

Scaler specifying the AM/PM conversion in degrees per decibel.

Lower input power limit (dBm)

Scalar specifying the minimum input power for which AM/PM conversion scales linearly with input power value. Below this value, the phase shift resulting from AM/PM conversion is zero.

Upper input power limit (dBm)

Scalar specifying the maximum input power for which AM/PM conversion scales linearly with input power value. Above this value, the phase shift resulting from AM/PM conversion is constant. The value of this maximum shift is given by:

Linear gain (db)

Scalar specifying the linear gain for the output function.

IIP3 (dBm)

Scalar specifying the third order intercept.

AM/PM conversion (degrees per dB)

Scalar specifying the AM/PM conversion in degrees per decibel.

Lower input power limit (dBm)

Scalar specifying the minimum input power for which AM/PM conversion scales linearly with input power value. Below this value, the phase shift resulting from AM/PM conversion is zero.

Upper input power limit (dBm)

Scalar specifying the maximum input power for which AM/PM conversion scales linearly with input power value. Above this value, the phase shift resulting from AM/PM conversion is constant. The value of this maximum shift is given by:

Input scaling (dB)

Number that scales the input signal level.

AM/AM parameters [alpha beta]

Vector specifying the AM/AM parameters.

AM/PM parameters [alpha beta]

Vector specifying the AM/PM parameters.

Output scaling (dB)

Number that scales the output signal level.

Input scaling (dB)

Number that scales the input signal level.

AM/AM parameters [x1 x2 x3 x4]

Vector specifying the AM/AM parameters.

AM/PM parameters [y1 y2 y3 y4]

Vector specifying the AM/PM parameters.

Output scaling (dB)

Number that scales the output signal level.

Linear gain (db)

Scalar specifying the linear gain for the output function.

Smoothness factor

Scalar specifying the smoothness factor

Output saturation level

Scalar specifying the the output saturation level.

I/Q Imbalance

## Reference

[1] Saleh, A.A.M., "Frequency-independent and frequency-dependent nonlinear models of TWT amplifiers," IEEE Trans. Communications, vol. COM-29, pp.1715-1720, November 1981.

[2] A. Ghorbani, and M. Sheikhan, "The effect of Solid State Power Amplifiers (SSPAs) Nonlinearities on MPSK and M-QAM Signal Transmission", Sixth Int'l Conference on Digital Processing of Signals in Comm., 1991, pp. 193-197.

[3] C. Rapp, "Effects of HPA-Nonlinearity on a 4-DPSK/OFDM-Signal for a Digitial Sound Broadcasting System", in Proceedings of the Second European Conference on Satellite Communications, Liege, Belgium, Oct. 22-24, 1991, pp. 179-184.