# An Essential Step to Generate a Modulation Signal – CCDF

If you are an audiophile, you get high-Res audio music to test the performance of your sound system. Of course, you need to have sharp ears first. Before doing that, you need to know the characteristics of the music. Otherwise, loud sound can cause the system and your ears damage.

Have you taken the same step when you are using a signal generator?

Many of the digitally modulated signals appear noise-like in the time and frequency domains. The signals are typically difficult to quantify because of their inherent randomness and inconsistency. It is critical to completely characterize and understand the power of the digital modulation in your signals. This can be easily done by using the Complementary Cumulative Distribution Function (CCDF), a statistic of digitally modulated signals.

In this blog, you'll learn what CCDF is and what its impacts are to your RF measurements.

## What Are CCDF Curves?

In modern wireless communications, modulation schemes are becoming more. Complex modulated signals result in a higher peak-to-average power ratio and higher nonlinear distortion. Therefore, to extract useful power-related information from the complex signals, we need a statistical analysis of the power levels.

Power CCDF curves characterize the higher-level power of a signal and provide critical information such as the peak-to-average ratio (PAR). PAR is often used for evaluating nonlinearities in power amplifiers and transmitters. Let's construct a CCDF curve step by step to fully understand how it can help for your tests.

### Construction of a CCDF Curve

Let’s start with a signal waveform. Figure 1 below illustrates how the power CCDF curve is derived from a baseband I/Q waveform. We are interested in knowing the probability that the signal level exceeds a specific level above the average power. Let’s construct a CCDF curve step by step.

Figure 1: Construction of CCDF curve

Step 1: A modulated waveform varies voltage with time. The waveform shows the voltage of either I or Q in the time domain.

Step 2: Convert the I or Q waveform into a probability density function (PDF) curve. The probability of a noise-like modulation signal for I or Q is a Gaussian-like probability distribution.

Step 3: Overlay the I and Q waveform probability curve in x- and y-axis.

Step 4: The power of the waveform is proportional to the sum of the square of I and square of Q. The summation of the square of two normal distributions is a Chi-squared distribution with two degrees of freedom. The curve is called power PDF.

Step 5: To integrate the PDF curve, the resulting function indicates the probability of the power is below the upper integration limit. This integrated function is called the Cumulative Distribution Function (CDF).

Step 6: Subtract the CDF from 100% probability and obtain the “Complementary” CDF (CCDF = 1 - CDF), which indicates how likely the power is to be at or exceed a given level.

Step 7: Normalize the function to the average power.

Step 8: Display the probability axis in log scale in order to get more resolution on the high-power signal levels that occur at very low probability.

The right-most point of the CCDF curve is the peak power of the waveform and you can read out the ratio which is called peak-to-average ratio (PAR). This value is important for R&D engineers before inputting the signal into a device under test.

## CCDF Plot of a Signal Generator

When you simulate a digital modulation signal with a signal generator, you need to make sure the output signal won’t be saturated by the signal generator. You can use the CCDF plot capability of a signal generator to identify the power distribution curve of a signal waveform, as shown in Figure 2. The signal waveform shown here is a 64 QAM with symbol rates at 1 Msps and RRC (root-raised-cosine) baseband filter waveform. The PAR is 5.95 dB as shown at left-down. If the output amplitude set to 0 dBm (average power), the peak envelope power (PEP) is equal to output amplitude plus PAR, i.e. +5.95 dBm.

PEP = Pout + PAR

where Pout is the amplitude setting on a signal generator (average output power of the signal).

Figure 2: CCDF plot from waveform utility of Keysight’s N5182B signal generator

### Gain Compression of a Signal Generator

If the output power of the signal generator is saturated, it impacts not only the output power level accuracy but also the modulation quality due to AM-to-AM compression. For a high PAR signal, the amplitude level setting on a signal generator can't be greater than the maximum output power (i.e. PEP) of the signal generator minus the peak-to-average ratio. For example, the maximum output power of the signal generator is +18 dBm. The PAR of the signal is 10 dB. The maximum amplitude setting (average power) you can go with the signal generator is +8 dBm (18 - 10 = 8). This prevents the signal generator’s power amplifier from being saturated.

### When is CCDF Used?

CCDF curves are an excellent tool to characterize the power distributions of digital modulation signals. CCDF curves apply to many design applications as below.

• Visualizing the effects of modulation formats.
• Combining multiple signals via system components.
• Evaluating spread-spectrum systems (e.g. CDMA).
• Designing and testing RF components.

To better understand the performance of your device under test, you can follow below three steps to characterize and troubleshoot your designs.

First, you need to know the power characteristic of the simulated waveform so that you can apply the right amplitude level to your signal generator.

Second, you can measure and compare CCDF curves of input and output signals to see if the DUT’s output is compressed. This can be done by using a power meter or a signal analyzer easily.

Third, you can analyze the output signal distortion performance and modulation quality. Using a signal analyzer helps to troubleshoot your design. Performing these three steps will ensure you make better design choice and create a great product.