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?

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 waveform shown in Figure 1A represents 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 logarithmic 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. A curves of both a 16QAM and affects its power characteristics. QPSK signal transfers only two QPSK signal obtained by the HP Using CCDF curves, we can fully bits per symbol, while a 16QAM E4406A VSA. We can quickly verify characterize the power statistics signal transfers four bits per that the 16QAM signal has a more of different modulation formats, symbol. Therefore, the bit rate stressful CCDF curve than that of and compare the results of of a 16QAM signal is twice that the QPSK signal. While 16QAM is choosing one modulation format of a QPSK signal for a given capable of transmitting more bits over another. symbol rate. The 16QAM signal per state than QPSK for a given appears to have higher peak-to- symbol rate, it also produces In Figure 3, the vector plot of a average power ratios, but it is greater peak-to-average ratios Quadrature Phase Shift Keying difficult to make quantitative than does QPSK. (QPSK) signal is compared to that observations from this plot.

6 High-alpha root- raised-cosine filter (a) 100%

10% QPSK signal 1%

OCQPSK signal 0.1%

0.01%

0.001% 0 5 10 0.0001% 0.00 10.00 dB Res BW 5.00000 MHz Low-alpha root- Figure 5: CCDF curves of a QPSK signal and a OCQPSK signal. raised-cosine filter (b)

This implies that the amplifiers and components used for a 16QAM signal will have different design needs than those for QPSK signals because of the higher peak-to-average statistics.

Similarly, we can use CCDF curves to compare between QPSK and Orthogonal Complex QPSK (OCQPSK). The CCDF curves in 0 5 10 Figure 5 immediately tell us that OCQPSK is less stressful than Figure 6: Impluse response of a high-alpha QPSK. This is one of the reasons and a low-alpha filter. why 3G cellular systems will use OCQPSK in the mobile stations.

7 (a) TRACE A: Ch 1 Main Time A Mkr 5.91518 us 24.403 mV -19.552 deg 40 mV Signal A I-Q

8 mV Signal A: Signal B: /div Filter alpha at 0.22 Filter alpha at 0.75 Root-raised-cosine filter Root-raised-cosine filter

-40 mV (c) -52.287583351 mV 52.2875833511 mV 100%

10% (b) A TRACE B: Ch 1 Main Time 1% B Mkr 5.91518 us 20.055 mV 79.470 deg 40 mV Signal B 0.1% I-Q 0.01% 8 mV /div B 0.001%

0.0001%

-40 mV -52.287583351 mV 52.2875833511 mV

Figure 7: CCDF curve and vector plots of high-alpha filter and low-alpha filter.

Modulation filtering time-trace vector plot squared The filtering parameters for a would represent the power of particular modulation format can the signals. Signal A clearly significantly affect the CCDF looks more spread out in the curves of the signal. Although vector plot due to the lower low-alpha (low-a) filters require filter alpha. The CCDF plot less bandwidth than high-alpha confirms that the low-a filter filters, they have a longer produces higher peak-power response time and more severe events than the high-a filter. ringing (Figure 6). This time- However, as mentioned above, domain ringing results in the high-a filters also require a addition of more symbols, which larger bandwidth. causes higher peak-power events. The CCDF curve can be used For example, Figure 7c shows to troubleshoot signals. Higher the CCDF plots of QPSK signals or lower CCDF curves give an with different filtering a factors, indication of the filtering pres- captured on an E4406A VSA. ent on the signal. An incorrect Also shown (Figures 7a and 7b) curve could identify incorrectly are the vector diagrams of the coded baseband signals. two signals. The magnitude of the

8 The same effect applies to multiple Set A: Set B: Set C: 4-tone signals 64-tone signals 1000-tone signals digitally modulated signals at different frequencies sent Tones have random phase relationships through a single amplifier or 100% antenna. Many of today’s schemes to achieve multi-carrier 10% signals must consider such

1% effects. For instance, an amplifier designed for a GSM signal only 0.1% transmits a constant envelope AWGN signal during a time slot. However, A 0.01% an amplifier designed for multiple B GSM signals at different frequen- 0.001% C cies will need to handle signals with significantly more stressful 0.0001% 0.00 10.00 dB CCDF curve power statistics. Res BW 5.00000 MHz Looking at Figure 8 again, notice Figure 8: CCDF curves of multiple frequency signals. that as the number of tones increases, the CCDF curves begin Multiple-frequency signals waveform with fluctuating ampli- to resemble the Additive White Combining multiple signals of tude, similar to an AM signal. The Gaussian Noise (AWGN) curve. different frequencies increases the CCDF curve of a two-CW signal This becomes an important appli- power in the resulting multi-tone differs significantly from that of cation in Noise Power Ratio signal, creating problems for the the single CW signal. (NPR) measurements. overall system. Figure 8 shows the sets of NPR is a common characterization A single Continuous Wave (CW) CCDF curves of 4-CW, 64-CW, and for nonlinear performance in signal has a substantially constant 1000-CW signals displayed on communications circuits. The envelope power, which means the E4406A VSA. These tones all NPR test requires a Gaussian that its peak envelope power have random phase relationships, noise stimulus with a notch located nearly equals its average power. which is why the number of tones in the center of the band. The Theoretically, the CCDF curve is somewhat ambiguous when stimulus is generated using large of this signal should be a point specifying the power characteris- filters and components. An alter- at 0 dB and 100%. On the other tics of multi-tone signals (see native NPR test stimulus, which hand, two CW signals at different phases affect multi-tone signals, can reduce costs and maximize frequencies create a time-domain below). However, as the number workspace, is an arbitrary wave- of tones increases, we expect the form signal generator capable peak-power events to also increase. of generating a multi-tone CW signal. To simulate a notch in the multi-tone signal, tones are omitted at the frequencies where the notch is desired. As we saw in Figure 10, one of the 1000-tone signals (the one in the middle) closely approximates the AWGN up to about 9.5 dB. The CCDF curve allows us to easily determine how closely the multi-tone signal matches a true AWGN signal.

9 Phases affect multi-tone signals it is possible to generate a signal As mentioned in the previous with fewer tones that is as stress- section, the phase relationships ful or sometimes more stressful of multi-tone signals can either than a signal with more tones. For significantly reduce or increase the instance, Figure 9 shows that a power statistics. We saw above 4-tone signal with all phases that a signal with more tones is aligned is slightly more stressful likely to have more stressful than one particular 64-tone signal power characteristics. However, with random phase relationships.

A multi-tone signal in which the 100% phases of all tones are aligned 4 tones yields the maximum CCDF curve 10% possible. Figure 10 shows two

1% spectrally identical 8-tone signals. Signal A has initial phases aligned 64 tones 0.1% at 0 degrees, while Signal B has AWGN phases offset by 45 degrees (that 0.01% is, 0, 45, 90, … , 315 degrees). The aligned phase signal (Signal A) 0.001% has a significantly more stressful 0.0001% CCDF curve. 0.00 15.00 dB Res BW 5.00000 MHz

Figure 9: CCDF curves for a phase-aligned 4-tone signal and a 64-tone signal with random phases.

Signal A: 8 tones have phases all alighed

Signal B: 8-tones have phases offset by 45 degrees

100% A 10%

0.1%

0.01%

B 0.001%

0.0001% 0.00 15.00 dB Res BW 5.00000 MHz

Figure 10: CCDF curves of 8-tone signals

10 A. QPSK Summed QPSK 1.5 1.5 1.5 Vectors 1 1 1 0.5 0.5 0.5

-1.5 -1 -0.5 0.5 1 1.5 -1.5 -1 -0.5 0.5 1 1.5 -1.5 -1 -0.5 0.5 1 1.5 -0.5 -0.5 -0.5 -1 -1 -1 -1.5 -1.5 -1.5

B. QPSK Summed QPSK 1.5 1.5 1.5 Vectors 1 1 1 0.5 0.5 0.5

-1.5 -1 -0.5 0.5 1 1.5 -1.5 -1 -0.5 0.5 1 1.5 -1.5 -1 -0.5 0.5 1 1.5 -0.5 -0.5 -0.5 -1 -1 -1 -1.5 -1.5 -1.5

Figure 11: (A) Addition and (B) cancellation of vectors in spread-spectrum systems.

Spread-spectrum systems systems send each bit to both I We can view spread-spectrum and Q, while 3G systems send signals as multiple QPSK signals every other bit to I or Q.) The summed together at baseband. vectors associated with all the Each QPSK signal represents a active codes then superimpose unique user channel or informa- upon one another to form a tion stream. Since these user spread-spectrum signal. Finally, channels are all transmitted at the receiver end, most systems together in the same frequency apply a known scrambling code band, the system relies on orthog- in order to distinguish and to onal coding in order to separate retrieve the desired user the information for different users. information.

Orthogonal codes allow spreading Adding up the QPSK vectors of of the information bits from each multiple codes defines the peak user. IS-95 cdmaOne format uses power for a symbol at any given Walsh codes. A "1" information bit time. As displayed in Figure 11, sends all 64 bits of the appropri- the vectors sometimes add up ate Walsh code. A "0" information (when in the same direction), and bit sends the inverse 64 bits of sometimes cancel each other out the appropriate Walsh code. Next, (when in opposing directions). the Walsh-coded information With the presence of more codes, translates into a QPSK vector. the potential peak power of the (cdmaOne spread-spectrum signal rises. The CCDF curve of a signal reflects this fact.

11 (a) TRACE A: D 1 Main Time

50 mV Signal A I-Q

10 mV /div Signal A: cdmaOne signal–Pilot

Signal B: cdmaOne signal–Pilot plus eight other channels -50 mV -65.359479189 mV 65.3594791889 mV (c) 100%

(b) 10% TRACE B: D 1 Main Time A 50 mV 1% Signal B B I-Q 0.1%

10 mV 0.01% /div 0.001%

0.0001% 0.00 15.00 dB -50 mV Res BW 5.00000 MHz -65.359479189 mV 65.3594791889 mV

Figure 12: CCDF curves and vector plots of a single-channel cdmaOne signal and a nine-channel cdmaOne signal.

Number of active codes In Figure 12c, we compare the in spread spectrum CCDF curves of two cdmaOne The number of active codes in a signals. Signal A (pilot channel) spread-spectrum system affects has a significantly lower statistical the power statistics of the signal. power ratio than does Signal B Increasing the number of channels (pilot plus 8 other randomly in a spread-spectrum system chosen channels) as indicated by increases the power peaks, thus the CCDF curves and the vector incurring higher stress on system plots. As expected, the signal components. with higher channel usage has a more stressful power CCDF curve.

12 Signal A (9 cdmaOne codes): Signal B (9 worst spacing codes): Pilot: code 0 Pilot: code 0 Sync: code 32 Sync: code 32 Paging: code 1 Paging: code 7 Traffic: codes 8, 9, 10, 11, 12, 13 Traffic: codes 15, 22, 39, 47, 55, 63

100%

10%

B 1%

A 0.1%

0.01% ~2.3 dB difference

0.001%

0.0001% 0.00 15.00 dB Res BW 5.00000 MHz

Figure 13: Effect of code selection on CCDF curves.

Orthogonal coding effects In Figure 13, we compare the Given a fixed number of active CCDF curves of two cdmaOne CDMA code channels, the peak- signals. Signal A and Signal B to-average power ratios of a signal each have nine active codes. depend on the specific selection However, Signal A consists of of the channels used. This is a randomly chosen codes, while consequence of orthogonal coding Signal B consists of one of the effects. The deterministic con- worst (most stressful) 9-code struction process for Walsh codes combinations possible. There is a gives the codes some symmetric significant difference between the properties. The symmetric prop- CCDF curves of the two signals. erties are the main reason that certain code combinations have more stressful CCDF curves than others. These code combinations are also deterministic.

13 Signal A (9 cdmaOne codes): Output signal of amplifier passes ACPR test. Suppose an amplifier designer ACPR IS-95A Averages: 19 PASS was asked to design an amplifier with certain ACPR characteristics Ref 3.62 dBm Spectrum (Total Pwr Ref) for a 9-channel CDMA signal. If 10.00 the CCDF curve of the signal was dB/ not defined, the amplifier designer would encounter ambiguity in the MaxP 3.7 design specifications. The designer could design the amplifier to work ExtAt with Signal A, but the customer 0.0 might test the amplifier with Center 1.00000 GHz Span 10.00 MHz Signal B and fail the amplifier Total Pwr Ref: 3.62 dBm/ 10.00 MHz (Figure 14). From the CCDF ACPR curve, the designer would have to Lower Upper Offset Freq Integ Bw dBc dBm dBc dBm design the amplifier with about 750.00 kHz 30.00 kHz -45.73 -42.11 -45.86 -42.24 2.3 dB more range for Signal B. 1.98 Mhz 30.00 kHz -72.13 -68.52 -71.29 -67.67 CCDF curves can help amplifier designers avoid ambiguity in design specifications. Signal B (9 worst spacing codes): Output signal of amplifier fails ACPR test

ACPR IS-95A Averages: 19 FAIL

Ref 3.65 dBm Spectrum (Total Pwr Ref) 10.00 dB/

MaxP 2.7

ExtAt 0.0 Center 1.00000 GHz Span 10.00 MHz

Total Pwr Ref: 3.65 dBm/ 10.00 MHz ACPR Lower Upper Offset Freq Integ Bw dBc dBm dBc dBm 750.00 kHz 30.00 kHz -44.40 F -40.76 -44.83 F -41.18 1.98 Mhz 30.00 kHz -70.08 -66.43 -69.74 -66.10

Figure 14: ACPR measurements of amplified signals.

14 Signal A: One 9-active-code cdmaOne signal

Signal B: Three 9-active-code cdmaOne signals

100%

10%

0.1% B 0.01%

A 0.001%

0.0001% 0.00 15.00 dB Res BW 5.00000 MHz

Figure 15: CCDF curves for a multi-carrier cdmaOne signal and a single-carrier cdmaOne signal

Multi-carrier signals Figure 15 compares the CCDF As discussed earlier, multi-tone curve of a 9-code cdmaOne signal signals cause greater stress on to that of three 9-code cdmaOne components such as amplifiers signals combined (in adjacent and mixers than single-tone signals. frequency channels), measured This effect also applies to multi- with the E4406A VSA. Clearly, carrier signals in spread-spectrum the multi-carrier signal B has systems, since multi-carrier signals the more stressful CCDF curve. use distinct frequency bands. Many Using CCDF curves, power companies are now creating amplifier designers know exactly amplifier designs that can handle how stressful a signal the amplifier combined multi-carrier cdmaOne will need to handle. Systems signals. developers also need to consider this power increase to ensure proper operation of the system.

15 CCDF in component design

1: Off 1: Power Log Mag 2.0 dB/Ref 4.40 dB m 2: S21 Fwd Trans Log Mag 0.2 dB/Ref 24.40 dB 2: Off dB dBm 25.2 12 Meas1: Mkr1 -27.98 dBm Meas2: Mkr2 -13.52 dBm -3.187 dBm 25 23.934dB 10 1 2 24.8 1 8

24.6 6

2: 1:

24.2 2

24 2 0

23.8 2 -2 1 23.6 -4

Start -28.00 dBm CW 1 000.000 MHz Stop -13.00 dBm Start -28.00 dBm CW 1 000.000 MHz Stop -13.00 dBm

(a) Gain versus input power (b) Output power versus input power

Figure 16: Characterization of power amplifier

Compression of signal by Figure 16 shows two characteri- nonlinear components zations of the same amplifier. The In nonlinear components, com- left display shows the amplifier pression of signals may occur. For gain versus input power. The instance, an amplifier compresses right display shows the same a signal when the signal exceeds information portrayed as a plot the amplifier power limitations. To of output power versus input avoid compression, an engineer power. Consider a noise-like signal needs to know the [optimum] passing through this amplifier. If input power level for the amplifier. both the input and output signals We will see how CCDF curves can remain within the power con- be used to determine input power straints of the amplifier, then levels and serve as a guide for RF the output would be a linear power amplifier designers and amplification of the input signal. systems engineers. However, if either the input or output signal exceeds the power limitations of the amplifier, the signal undergoes compression.

16 (a) Amplifier input signal Unfortunately, it is difficult to see Ref–15.00 dBm RF Envelope compression effects in the time 4.00 dB/ domain (Figure 17), because we cannot see appreciable amounts MaxP of clipping. Even if we were able -9.0 to perceive some clipping, we have no convenient way of describing ExtAt the compression quantitatively in 0.0 the time domain. However, we can easily detect compression of a signal by comparing the power CCDF curves of the input signal and the amplified output signal. Trig Free 0.00 ms 1.00 ms Res BW 100.000 KHz Gaussian Samples 1365 points @733.33ns

(b) Output power versus input power

Ref 10.00 dBm RF Envelope 4.00 dB/

MaxP 4.0

ExtAt 0.0

Trig Free 0.00 ms 1.00 ms Res BW 100.000 KHz Gaussian Samples 1365 points @733.33ns

Figure 17: Power-versus-time plots for input and output signals of an amplifier.

17 100%

10% Amplifier Gain vs Input Power (with 1GHz CW signal)

1% 25.5 Input signal 25 0.1% 24.5 24 0.01% 23.5 Amplified signal 23 22.5 0.001% 22 21.5 0.0001% 21 0.00 15.00 dB -30 -28 -26 -24 -22 -20 -18 -16 -14 -12 -10 Res BW 5.00000 MHz Input Power (dBM) Figure 18: Compression effects displayed by CCDF curves.

100%

As seen in Figure 18, CCDF 10%

curves are clearly an excellent Input 1%

tool for displaying compression signal 0.1% CCDF effects. By definition, CCDF 0.01% curves measure how far and 0.001% how often a signal exceeds the Correlating the amplifier 0.0001% average power. If a signal passing curves with the CCDF plot 0.00 15.00dB Res BW 5.00000 MHz through an amplifier were per- Figure 19 correlates the input fectly (linearly) amplified, the signal CCDF curve to the amplifier gain versus input power plot. The Figure 19: Correlation of amplifier gain output waveform of the signal in and input signal CCDF curve. the time domain would perfectly power axes of the CCDF curve resemble the input waveform, and the gain curve have the same will spend at least 3% of the time with a gain in power. Both the scale, allowing us to line up the compressed by 0.6 dB or more. By average and envelope power two curves. We have shifted the viewing these curves, an amplifier of the amplified signal would graph to line up the 0 dB point of designer can easily check the increase by a common factor. the CCDF curve with the average amplifier power performances Therefore, the peak-to-average power of the input signal on the against the desired specifications. power ratio would not change, gain curve. Now, we can correlate and the two CCDF curves would the two graphs. As seen from the CCDF curve appear identical. However, when comparisons, the output signal the output signal exceeds the For example, the input signal has significantly distorted peak- power limitations of the amplifier, spends 3% of its time at or above power events. This distortion will clipping occurs; the output wave- 6 dB relative to the average affect the Bit Error Rate (BER) or form no longer resembles an power (from the CCDF curve). Frame Error Rate (FER) of the amplified version of the input Since the average power of the system. Therefore, the CCDF waveform, and the peak-to-aver- signal equals –21 dBm, peaks will curve can be a good tool for age ratios change. The CCDF occur above –15 dBm 3% of the determining the optimum signal curve of the clipped signal time. At –15 dBm on the gain level for a particular amplifier. decreases and no longer matches curve, the signal is compressed the original input signal. This about 0.6 dB. Therefore, the signal effect makes the CCDF a good indicator of compression.

18 (a) Amplifier input signal

ACPR IS-95A Averages: 20 PASS Compression affects other key measurements Ref–21.19 dBm Bar Graph (Total Pwr Ref) The result of this compression 10.00 dB/ will also affect key amplifier and MaxP transmitter system specifications -12.3 such as Adjacent Channel Power ExtAt Ratio (ACPR) and code-domain 0.0 power (CDP). Figure 20 shows Center 1.00000 GHz two ACPR measurements com- Total Pwr Ref: -21.19 dBm/ 1.23 MHz ACPR paring the input and output sig- Lower Upper nals of an amplifier. The com- Offset Freq Integ Bw dBc dBm dBc dBm 750.00 kHz 30.00 kHz -71.00 -92.20 -71.38 -92.57 pression has caused approximate- 1.98 MHz 30.00 kHz -74.05 -95.24 -74.22 -95.41 ly 15 dB degradation in the ACPR of the signal.

(b) Amplifier output signal Figure 21 (page 20) shows the CDP plots of the input and output ACPR IS-95A Averages: 20 PASS signal of the amplifier. The ampli- Ref 3.47 dBm Bar Graph (Total Pwr Ref) fied signal (with compression) 10.00 has significantly reduced the dB/ Signal-to-Noise Ratio (SNR) of MaxP 3.7 the active channels versus the ExtAt inactive channels. 0.0 Center 1.00000 GHz Measurements such as ACPR and Total Pwr Ref: 3.47 dBm/ 1.23 MHz CDP are well complemented by ACPR Lower Upper CCDF curves. While ACPR and Offset Freq Integ Bw dBc dBm dBc dBm CDP give us information regard- 750.00 kHz 30.00 kHz -45.42 -41.95 -45.28 -41.82 1.98 MHz 30.00 kHz -71.27 -67.80 -70.96 -67.50 ing overall average power and individual channel power, CCDF tells us the more detailed infor- Figure 20: ACPR increases as the signal passes through the amplifier. mation about the peaks of a signal.

19 (a) Amplifier input signal Applications Code Domain IS-95A Averages: 10 CCDF curves can help to confirm Ref 0.00 dB Bar Graph (Total Pwr Ref) whether or not a component 5.00 dB/ design performs as specified. For example, a power amplifier Sync designer can compare the CCDF Esec curve of a signal at the input and output of an RF amplifier. If the 0 32 63 Act Set Th -20.00 dB design is correct, the curves will Time Ofs: -11755.48 us Pilot: -7.01 dB Avg AT: -10.58 dB coincide. However, if the amplifier Freq Error: 7.24 Hz Paging: -7.27 dB Max IT: -47.28 dB compresses the signal, the peak- 1.98 MHz -30.13 dB Sync: -13.25 dB Avg IT: -49.18 dB to-average ratio of the signal is lower at the output of the amplifier, which means the CCDF curve (b) Amplifier output signal decreases. The amplifier designer Code Domain IS-95A Averages: 10 would need to increase the range of the amplifier to account for Ref 0.00 dB Bar Graph (Total Pwr Ref) the power peaks, or the system 5.00 dB/ designers could lower the average power of the signal to meet the Sync Esec power limitations of the amplifier.

Similarly, CCDF curves can also 0 32 63 Act Set Th -20.00 dB be used to troubleshoot a system Time Ofs: 9917.39 us Pilot: -7.01 dB Avg AT: -10.59 dB or subsystem design. Making Freq Error: 8.21 Hz Paging: -7.25 dB Max IT: -37.12 dB 1.98 MHz -27.53 dB Sync: -13.33 dB Avg IT: -43.70 dB CCDF measurements at several points in the system allows system engineers to verify performance Figure 21: CDP increases as the signal passes through the amplifier. of individual subsystems and components; also, circuit components or subsystems can be tested for linearity non-intrusively.

20 Summary CCDF measurements are becoming In particular, CCDF curves are a valuable tool in the communica- a valuable measurement tool tions industry. The need for CCDF for spread-spectrum systems. arises primarily when dealing For instance, the number of with digitally modulated signals in active codes in a CDMA signal spread-spectrum systems such as significantly affects the power cdmaOne, cdma2000, and W-CDMA. statistics. Furthermore, different Because these types of signals are combinations of active codes yield noise-like, CCDF curves provide a different power CCDF curves, due useful characterization of the signal to orthogonal coding effects. power peaks. CCDF is a statistical Multi-carrier signals also cause method that shows the amount of a significant change in CCDF time the signal spends above any curves, similar to the effect of given power level. The mathemat- multi-tone signals. CCDF is ical origins of CCDF curves are becoming a necessary design and the familiar PDF and CDF curves testing tool in 3G communications that most engineering students systems. have seen in an introductory probability and statistics course. Perhaps the most important contribution of CCDF curves is in The CCDF measurement is an setting the signal power specifica- excellent way to fully characterize tions for mixers, filters, amplifiers, the power statistics of a digitally and other components. The CCDF modulated signal. Modulation for- measurement can help determine mats can be compared via CCDF the optimum operation point for in terms of how stressful a signal components. As this new tool is on a component such as an becomes more prevalent, we can amplifier. The CCDF curve can also expect to see correlation studies be used to determine the impact between CCDF curve degradation of filtering on a signal. From the and digital radio system parameters CCDF curves of multi-tone signals, such as BER, FER, CDP, and ACPR. amplifier designers know exactly how much headroom to allow for the peak power excursions to avoid compression. Offsetting the phases of a multi-tone signal can minimize unwanted power peaks. CCDF curves are a powerful way to view and to characterize how various factors affect the peak amplitude excursions of a digitally modulated signal.

21 References 1. Nick Kuhn, Bob Matreci, and Peter Thysell, Proper Stimulus Ensures Accurate Tests of CDMA Power, (article reprint) Microwaves & RF, January 1998, literature number 5966-4786E.

2. Digital Modulation in Communications Systems – An Introduction, Application Note 1298, literature number 5965-7160E.

3. Testing and Troubleshooting RF Communications Transmitter Designs, Application Note 1313, literature number 5968-3578E.

4. Performing cdma2000 Measurements Today, Application Note 1325, literature number 5968-5858E.

5. Understanding CDMA measurements for Base Stations and Their Components, Application Note 1311, literature number 5968-0953E.

22 Symbols 3G—Third Generation 8VSB—8-level Vestigial Side Band and Acronyms ACPR—Adjacent Channel Power Ratio AM—Amplitude Modulation AMPS—Advanced Mobile Phone System AWGN—Additive White Gaussian Noise BER—Bit Error Rate CCDF—Complementary Cumulative Distribution Function CDF—Cumulative Distribution Function CDMA (cdmaOne, cdma2000)—Code Division Multiple Access CDP—Code Domain Power COFDM—Coded Orthogonal Frequency Division Multiplexing CW—Continuous Wave dB—Decibel dBm—Decibels relative to 1 milliwatt. (10log(power/1mW)) DSB-LC—Double-SideBand Large-Carrier DSP—Digital Signal Processing EDGE—Enhanced Data for GSM Evolution FER—Frame Error Rate FM—Frequency Modulation GSM—Global System for Mobile communications HDTV—High Definition TeleVision HPSK—Hybrid Phase Shift Keying I and Q—In-phase and Quadrature NADC—North American Digital Cellular NPR—Noise Power Ratio OCQPSK—Orthogonal Complex Quadrature Phase Shift Keying PDF—Probability Density Function QAM—Quadrature Amplitude Modulation QPSK—Quadrature Phase Shift Keying RF—Radio Frequency SNR—Signal to Noise Ratio TACS—Total Access Communications System VSA—Vector Signal Analyzer W-CDMA—Wideband-Code Division Multiple Access

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