Characterizing Digitally Modulated Signals with CCDF Curves Application Note Table of Contents 1. Introduction When are CCDF curves used? 2. What are CCDF curves? Statistical origin of CCDF curves Why not crest factor? 3. CCDF in communications Modulation formats Modulation filtering Multiple-frequency signals Phases affect multi-tone signals Spread-spectrum systems Number of active codes in spread spectrum Orthogonal coding effects Multi-carrier signals 4. CCDF in component design Compression of signal by nonlinear components Correlating the amplifier curves with the CCDF plot Compression affects other key measurements Applications 5. Summary 6. References 7. Symbols and acronyms 2 Introduction The move to 3G systems will yield This application note examines signals with higher peak-to-average the main factors that affect power ratios (crest factors) than power CCDF curves, and previously encountered. Analog describes how CCDF curves Digital modulation has created a FM systems operate at a fixed are used to help design systems revolution in RF and microwave power level: analog double-side- and components. communications. Cellular systems band large-carrier (DSB-LC) AM have moved from the old analog systems cannot modulate over When are CCDF curves used? AMPS and TACS systems to second 100% (a peak value 6 dB above Perhaps the most important appli- generation NADC, CDMA, and GSM the unmodulated carrier) without cation of power CCDF curves is standards. Today, the cellular distortion. Third generation systems, to specify completely and without industry talks of moving to third on the other hand, combine ambiguity the power character- generation (3G) digital systems multiple channels, resulting in a istics of the signals that will be such as W-CDMA and cdma2000. peak-to-average ratio that is mixed, amplified, and decoded Outside of cellular communications, dependent upon not only the in communication systems. For the broadcast industry is also number of channels being com- example, baseband DSP signal moving into the era of digital bined, but also which specific designers can completely specify modulation. High Definition channels are used. This signal the power characteristics of Television (HDTV) systems will characteristic can lead to higher signals to the RF designers by use 8-level Vestigial Side Band distortion unless the peak power using CCDF curves. This helps (8VSB) and Coded Orthogonal levels are accounted for in the avoid costly errors at system Frequency Division Multiplexing design of system components, integration time. Similarly, system (COFDM) digital modulation for- such as amplifiers and mixers. manufacturers can avoid mats in various parts of the world. ambiguity by completely specify- Power Complementary Cumulative ing the test signal parameters to Distribution Function (CCDF) their amplifier suppliers. curves provide critical information about the signals encountered in CCDF curves apply to many 3G systems. These curves also design applications. Some of provide the peak-to-average power these applications are: data needed by component • Visualizing the effects of designers. modulation formats. • Combining multiple signals via systems components (for example, amplifiers). • Evaluating spread-spectrum systems. • Designing and testing RF components. 3 What are CCDF curves? A. Ref–10.00 dBm RF Envelope A CCDF curve shows how much 3.00 time the signal spends at or above dB/ 3 a given power level. The power 2 level is expressed in dB relative to 1 the average power. For example, Avg each of the lines across the wave- form shown in Figure 1A repre- sents a specific power level above the average. The percentage of time the signal spends at 0.00 ms 1.00 ms or above each line defines the probability for that particular Figure 1A: Construction of a CCDF curve power level. A CCDF curve is a plot of relative power levels ver- The signal power exceeds the average by at least sus probability. B. 7.5dB for 1% of the time. Y-axis Figure 1B displays the CCDF is the 100% percent curve of the same 9-channel of time 10% 1 cdmaOne signal captured on the the signal power E4406A VSA. Here, the x-axis is 1% 2 is at scaled to dB above the average OR ABOVE 3 the power 0.1% signal power, which means we are specified actually measuring the peak-to- by the 0.010% x-axis. average ratios as opposed to 0.0010% absolute power levels. The y-axis 0.00010% is the percent of time the signal 0.00 15.00 dB Res BW 5.00000 MHz spends at or above the power level specified by the x-axis. For Band-limited Gaussian noise X-axis is dB above CCDF reference line. average power example, at t = 1% on the y-axis, the corresponding peak-to-average Figure 1B: CCDF curve of a typical cdmaOne signal ratio is 7.5 dB on the x-axis. This means the signal power exceeds Figure 1A shows the power versus Unfortunately, the signal in the the average by at least 7.5 dB for time plot of a 9-channel cdmaOne form shown in Figure 1A is 1 percent of the time. The posi- signal. This plot represents the difficult to quantify because of tion of the CCDF curve indicates instantaneous envelope power its inherent randomness and the degree of peak-to-average defined by the equation: inconsistencies. In order to deviation, with more stressful extract useful information from Power = I2 + Q2 signals further to the right. this noise-like signal, we need where I and Q are the in-phase a statistical description of the and quadrature components of power levels in this signal, and the waveform. a CCDF curve gives just that. 4 Statistical Rotate PDF Ref–10.00dBm RF Envelope origin of 3.00 dB/ 50 MaxP %Probability 40 CCDF -9.0 30 Avg ExtHt 20 0.0 50 10 dB above average 40 curves %Probability30 dB above average 0 20 10 -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 Trig 0 -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 dB above average Free 0.00 ms 1.00 ms 0 Res BW 100.000 kHz Guassian Samples 1365points @733.33 ns 50 40 30 20 10 %Probability CDF Now that we’ve seen a practical PDF 120 100 CCDF curve (Figure 1A), let’s 50 %Probability 40 80 briefly investigate the mathematics 30 60 20 40 10 Probability 20 of CCDF curves. Let’s start with a = Cumulative 0 dP 0 -3 -2 -1 0 1 2 3 set of data that has the Probability dB above average -3 -2 -1 0 1 2 3 Density Function (PDF) shown in dB above average Figure 2. To obtain the Cumulative Distribution Function (CDF), we CCDF compute the integral of the PDF. CDF 120 Finally, inverting the CDF results 120 100 100 80 in the CCDF. That is, the CCDF 80 60 60 40 40 is the complement of the CDF Probability 20 Cumulative Probability 20 Cumulative – 0 = -3 -2 -1 0 1 2 3 0 (CCDF = 1 – CDF). To generate dB above average -3 -2 -1 0 1 2 3 1 1 - the CCDF curve in the form dB above average shown in Figure 1A, convert the Figure 2: Mathematical origin of CCDF y-axis to logarithmic form and begin the x-axis at 0 dB. A logarith- mic y-axis provides better resolu- This measurement is of question- In short, crest factor places too tion for low-probability events. able value for several reasons: much emphasis on a signal’s The reason we plot CCDF instead instantaneous peak value. To of CDF is that CCDF emphasizes • Error-correction coding will mitigate this, the peak value is peak amplitude excursions, while often remove the effect of a sometimes considered to be the CDF emphasizes minimum values. single error-causing peak in level where 99.8 percent of the the system. waveform is below the peak value, Why not crest factor? • A single peak that is much and 0.2 percent of the waveform As the need for characterizing higher than the prevailing is above it. While this lessens the noise-like signals becomes signal level sometimes is emphasis on a single point of a greater, we search for the most treated as an anomalous waveform, the use of a single convenient, useful, and reliable condition and ignored. level (such as 0.2 percent) is still measurement to meet this need. • A peak that occurs infre rather arbitrary. CCDF gives us Traditionally, a common measure quently causes little spectral more complete information about of stress for a stimulating signal regrowth, which makes the high signal levels than does has been the crest factor. Crest repeatable measurements crest factor [1]. factor is the ratio of the peak difficult. voltage to its root-mean-square • The highest peak value that value. occurs for true noise depends on the length of the measurement time. 5 CCDF in communications A. QPSK signal TRACE A: Ch1 Main Time A Mkr 0.0000 s 28.144 mV -145.59 deg 40 mV I - Q 8 mV/div -40 mV -52.287583351 mV 52.2875833511 mV 100% B. 16QAM signal TRACE B: Ch1 Main Time 10% QPSK signal 40 mV 1% I - Q 0.1% 16QAM signal 8 mV/div 0.01% 0.001% 0.0001% 0.00 10.00 dB -40 mV Res BW 5.00000 MHz -52.287583351 mV 52.2875833511 mV Figure 3: Vector plots of (A.) QPSK signal Figure 4: CCDF curves of a QPSK signal and a 16QAM signal and (B.) 16QAM signal Modulation formats of a 16-Quadrature Amplitude Figure 4 shows the power CCDF The modulation format of a signal Modulation (16QAM) signal.