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## CPM Phase Tree

This model shows how to use the Discrete-Time Eye Diagram Scope block to view the phase trajectory, phase tree, and instantaneous frequency of a CPM modulated signal.

Structure of the Example

This example uses various Communications System Toolbox™, DSP System Toolbox™ and Simulink blocks to model a baseband CPM signal.

In particular, the example model includes these blocks:

• The Random Integer block, which is a source of uniformly distributed random integers between 0 and M-1, where M is the constellation size of the CPM signal

• The Integer to Bit Converter block

• The CPM Modulator Baseband block

• The Complex to Magnitude-Angle Converter block

• The Phase Unwrap block

• The Zero-Order Hold block

• The Discrete Transfer Fcn block

• The Gain block

• Multiple copies of the Discrete-Time Eye Diagram Scope block

Results and Displays

When you run the example, several Discrete-Time Eye Diagram Scope blocks show how the CPM signal changes over time:

• The Modulated Signal block displays the in-phase and quadrature signals. Double-click the block to open the scope. The modulated signal is easy to see in the eye diagram only when the Modulation index parameter in the CPM Modulator Baseband block is set to 0.5. If you set the Modulation index to another value, for example 2/3, the features of the modulated signal are difficult to decipher for this more complex modulation. Unwrapping the phase and plotting it is another way to illustrate these more complex CPM modulated signals.

• The Phase Trajectory block displays the CPM phase. Double-click the block to open the scope. The Phase Trajectory block reveals that the signal phase is also difficult to view because it drifts with the data input to the modulator.

• The Phase Tree block displays the phase tree of the signal. The CPM phase is processed by a few simple blocks to make the CPM pulse shaping easier to view. This processing holds the phase at the beginning of the symbol interval and subtracts it from the signal. This resets the phase to zero every three symbols. The resulting plot shows the many phase trajectories that can be taken by the signal from any given symbol epoch.

• The Instantaneous Frequency block displays the instantaneous frequency of the signal. The CPM phase is differentiated to produce the frequency deviation of the signal. Viewing the CPM frequency signal enables you to observe the frequency deviation qualitatively, as well as make quantitative observations, such as measuring peak frequency deviation.

Exploring the Example

To learn more about the example, try changing the following parameters in the CPM Modulator Baseband block:

• Change Pulse length to a value between 1 and 6.

• Change Frequency pulse shape to one of the other settings, such as Raised Cosine or Gaussian.

You can observe the effect of changing these parameters on the phase tree and instantaneous frequency of the modulated signal.