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This section presents scatter plots that illustrate how blocks in the RF Impairments library distort a signal modulated by 16-ary quadrature amplitude modulation (QAM). The usual 16-ary QAM constellation without distortion is shown in the following figure.

As the scatter plots show, the first two blocks distort both the magnitude and angle of points in the constellation, while the last two alter just the angle.

You can create these scatter plots with models similar to the following, which produces the scatter plot for the Memoryless Nonlinearity block:

The model uses the Rectangular QAM Modulator Baseband block,
from AM in the Digital Baseband Modulation sublibrary of the Modulation
library. You control the power of the block's output signal with the **Normalization
method** parameter. To ```
open
this model
```

, enter `doc_16qam_plot`

at
the MATLAB^{®} command line.

You can generate the next scatter plot by replacing the Memoryless
Nonlinearity block in the 16-ary QAM Model with the I/Q Imbalance
block. Set the block's **I/Q amplitude imbalance (dB) **parameter
to `10`

and the **I/Q phase imbalance (deg)** parameter
to `30`

.

For more examples of scatter plots produced using this block, see the I/Q Imbalance block reference page.

You can generate the next scatter plot by replacing the Memoryless
Nonlinearity block in the 16-ary QAM Model with the Phase/Frequency
Offset block. Set the block's **Frequency offset (Hz) **parameter
to `0`

and the **Phase offset (deg)** parameter
to `70`

.

The **Frequency offset (Hz) **parameter adds
a constant to the phase of the signal. The scatter plot corresponds
to the standard constellation rotated by a fixed angle of 70 degrees.

The **Frequency offset (Hz) **parameter determines the rate of change of the
signal's phase. In this example, **Frequency offset (Hz)** is set to
`0`

, so the scatter plot always falls on the grid shown in the
preceding figure. If you set **Frequency offset (Hz)** to a positive
number, the points on the scatter plot fall on a rotating grid, corresponding to the
standard constellation, which revolves at a constant rate in the counterclockwise
direction. For an example, see the Phase/Frequency Offset block reference
page.

You can generate the next scatter plot by replacing the Memoryless
Nonlinearity block in the 16-ary QAM Model with the Phase Noise block.
Set the **Phase noise level (dBc/Hz) **parameter
to `-60`

and the **Frequency offset (Hz)** parameter
to `100`

.

The phase noise adds a random error to the signal's phase, so that the points in the scatter plot are spread in a radial pattern around the constellation points.

The RF Impairments library contains two blocks that simulate phase/frequency offsets and phase noise:

The Phase/Frequency Offset block applies phase and frequency offsets to a signal.

The Phase Noise block applies phase noise to a signal.

The Phase/Frequency Offset block and the Phase Noise block alter only the phase and frequency of the signal.

The RF Impairments Library contains two blocks that simulate signal impairments due to thermal noise and signal attenuation due to the distance from the transmitter to the receiver:

The Receiver Thermal Noise block simulates the effects of thermal noise on a complex baseband signal.

The Free Space Path Loss block simulates the loss of signal power due to the distance from the transmitter and signal frequency.

The following two blocks model signal impairments due to nonlinear devices or imbalances between the in-phase and quadrature components of a modulated signal:

The Memoryless Nonlinearity block models the AM-to-AM and AM-to-PM distortion in nonlinear amplifiers.

The I/Q Imbalance block models imbalances between the in-phase and quadrature components of a signal caused by differences in the physical channels carrying the separate components.

These blocks distort both the phase and amplitude of the signal.

The Memoryless Nonlinearity block applies a nonlinear distortion
to the input signal. This distortion models the AM-to-AM and AM-to-PM
conversions in nonlinear amplifiers. The block provides several methods,
which you specify by the **Method** parameter, for
modeling the nonlinear characteristics of amplifiers:

Cubic polynomial

Hyperbolic tangent

Saleh model

Ghorbani model

Rapp model

In the model shown in the preceding figure, the **Method** parameter
is set to `Ghorbani model`

. The following
figure shows the scatter plot the model generates.

For another example of a scatter plot produced using this block, see the Memoryless Nonlinearity block reference page.

This example shows the effects that spectral and phase noise have on a 100 kHz sine wave.

**Open Example Model and Explore Its Contents**

Open the example model `slex_phasenoise`

.

A Sine Wave block generates a 100 kHz tone. A Phase Noise block adds phase noise of:

`-85`

dBc/Hz at a frequency offset of`1e3`

Hz`-118`

dBc/Hz at a frequency offset of`9.5e3`

Hz`-125`

dBc/Hz at a frequency offset of`19.5e3`

Hz`-145`

dBc/Hz at a frequency offset of`195e3`

Hz

To analyze the spectrum and phase noise, the model includes three Spectrum Analyzer blocks. The Spectrum Analyzer blocks use the default `Hann`

windowing setting, the units are set to `dBW/Hz`

, and the number of spectral averages is set to `10`

.

Additionally, the model includes blocks that calculate and display the RMS phase noise. The subsystem that calculates the RMS phase noise finds the phase error between the pure and noisy sine waves, then calculates the RMS phase noise in degrees. In general, to accurately determine the phase error, the pure signal must be time aligned with the noisy signal. However, the periodicity of the sine wave in this model makes this step unnecessary.

**Run the Model to Generate Results**

In the Simulink Editor, click **Run** to simulate the model.

When the resolution bandwidth is 1 Hz, the `dBW/Hz`

view for the spectrum analyzer shows the tone at 0 dBW/Hz. The Spectrum Analyzer block corrects for the power spreading effect of the Hann windowing.

The visual average of the phase noise achieves the spectrum defined by the Phase Noise block.

When the resolution bandwidth is 10 Hz, the `dBW/Hz`

view for the spectrum analyzer shows the tone at -10 dBW/Hz. That same tone energy is now spread across 10 Hz instead of 1 Hz, so the sine wave PSD level reduces by 10 dB. With the resolution bandwidth at 10 Hz, the visual average of the phase noise still achieves the phase noise defined by the Phase Noise block.

The Spectrum Analyzer block still corrects for the power spreading effect of the Hann window, and it achieves better spectral averaging with the wider resolution bandwidth. For more information, see Why Use Windows? (Signal Processing Toolbox).

**Further Exploration**

In the Phase Noise block, change the **Phase noise level (dBc/Hz)** parameter, rerun the model, and notice how the spectrum shape changes. With more noise, the side lobes increase in amplitude. As more phase noise is added, the 100 Hz signal becomes less distinct and the measured RMS phase noise increases.

This model applies RF impairments to a signal modulated by differential quadrature phase shift keying (DQPSK). To demonstrate and visualize the RF impairments, levels applied in this model are exaggerated and not representative of typical levels for modern radios.

A random signal is DQPSK modulated and various RF impairments are applied to the signal. The model uses impairment blocks from the RF Impairments library. 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. Since 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. To analyze the constellation, the model includes Constellation Diagram blocks after modulation, before correction, and after correction.

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.

**Open Example Model and Explore Its Contents**

Open the example model `slex_rcvrimpairments_dqpsk`

.

The `After Modulation`

diagram, shows the clean reference DQPSK modulated constellation.

The transmitted signal is distorted by various RF impairments. The `Before Correction`

diagram shows the attenuated and distorted constellation.

The signal on the correction path is adjusted by the DC Blocker, AGC Block, and I/Q Imbalance Compensator blocks. The `After Correction`

diagram shows the constellation has been amplified and improved after the correction blocks.

**Display the BER for the signal with and without correction.**

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

**Further Exploration**

To run the model yourself, open the example using the button provided or by entering `open slex_rcvrimpairments_dqpsk`

at the MATLAB® command prompt. Adjust the RF impairments, rerun the model, and notice the changes to the constellation diagrams and error rates. Consider 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.

[1] Simon, M. K., and Alouini, M. S., *Digital
Communication over Fading Channels – A Unified Approach to Performance
Analysis*, 1st Ed., Wiley, 2000.

[2] 3rd Generation Partnership Project, Technical Specification Group Radio Access Network, Evolved Universal Terrestrial Radio Access (E-UTRA), Base Station (BS) radio transmission and reception, Release 10, 3GPP TS 36.104, v10.0.0, 2010-09.

[3] 3rd Generation Partnership Project, Technical Specification Group Radio Access Network, Evolved Universal Terrestrial Radio Access (E-UTRA), User Equipment (UE) radio transmission and reception, Release 10, 3GPP TS 36.101, v10.0.0, 2010-10.