OFDM is based upon 52 independently modulated subcarriers, four of which are BPSK pilots. The remaining 48 subcarriers are modulated as 64 quadrature amplitude modulation, 16QAM, QPSK or BPSK, depending on the data rate. The subcarriers are spaced 312.5 kHz apart and each symbol lasts 4 microseconds (all carriers are synchronized in phase and symbol period). Similar to the single-carrier modes, the physical-layer data rate is set by both M-ary size and level of coding, while the symbol rate remains constant. The resultant spectrum appears nearly rectangular and is often referred to as a "Bart's head" for this reason. Spectral regrowth causes diagonal side lobes to develop, but error vector magnitude (EVM) limits will occur at the higher data rates long before spectral mask is exceeded (see Figure 1b).

The spectral mask for 802.11b is defined as -30 dBr beyond plus/minus 11 MHz and -50 dBr beyond 22 MHz. The spectral mask for OFDM is 0 dBr to 9 MHz, -20 dBr at 11 MHz offset, -28 dBr at 20 MHz offset and -40 dBr beyond 30 MHz. The EVM is essentially equivalent to signal-to-noise ratio (SNR). The 802.11b standard specified its single-carrier modes in terms of percentage (voltage scale) while 802.11a/g specifies its OFDM modes as decibel relationships. There is no fundamental difference and the two are mutually relatable, since EVM in decibels is 20Log10 the voltage-scale EVM percentage. Transmit EVM is of particular concern because it is a measure of total transmitter performance.

The 802.11b specification requires the peak EVM to be less than 35 percent. Depending on the distribution and correlation of the error voltage, this is usually on the order of 15 to 20 percent rms EVM. However, most 802.11b transmitters produce better than 10 percent rms EVM. The faster 802.11g PBCC modes (PBCC-22 and PBCC-33, both 8-PSK) require an even more accurate modulation performance of about 7 percent rms.

The 802.11a/g standard's OFDM EVM is measured in fundamentally the same way; it is a composite of all of the subcarriers, mutually derotated. However, because most OFDM modes contain less energy per bit of information than most constant-carrier modes at the same power level, proper operation requires a much higher SNR.

Linearity approximations
The restricted linearity of semiconductor devices is a very important characteristic that ultimately limits the capabilities of a system in many regards. Models are used to characterize the nonlinear behavior of a system or device in order to predict the resultant signal properties. While such models often become extremely complicated when great accuracy is required, very simple polynomial approximations are useful for determining roughly how the system will behave.

The polynomial approximation is a nonlinear transfer function that disregards time-varying and memory effects. It is based upon the Taylor Series Expansion and assumes one output voltage for each input voltage. Typically, the first-order (gain), second-order (squaring) and third-order (cubing) terms are considered, though higher-order terms become important as a system is driven harder. Its coefficients are either determined or implied based upon the measured power levels that the resultant-order term would produce.

The first-order term produces a scaled version of the input spectra while the second-, third- and higher-order terms generate extraneous products. The second-order products consist of the frequency-domain sum and difference of each spectral line. In the two-tone case, f1 and f2, the result is f1 + f2 and f1 - f2, which fall far outside of the desired passband. The resultant second-order products are referred to as IM2 (or second-order intermodulation-distortion product) for the second order, IM3 for the third order and so forth.

The close-in third-order products, for the same two input tones, are 2x(f2-f1) and 2x(f1-f2). These third-order products, as well as all odd-order products, are likely to produce energy that falls near or in the desired band and are very important considerations in WLAN performance specifications. Similarly, fifth-order products also produce two close-in products: 3x(f1-2f2) and 3x(f2-2f1).

Ideally (this is a simplified model), IM2 tones grow at a 2:1 slope, IM3 at a 3:1 slope (see Figure 2) and IMx at an X:1 slope in decibels as device input power increases. As input power increases, however, the device will become increasingly saturated, thereby compressing the output waveform. This waveform compression is also compression in gain-the point at which the device gain is 1 dB less than its low-power gain is called P1dB (1-dB compression point).

Although gain compression makes it impossible to keep increasing the input power until the IMx tones are the same level as the two desired tones, it is useful to geometrically extrapolate those points in order to imply the value of their respective polynomial coefficients. Those extrapolated points are called IP2, IP3 and so on. Equations 1 and 2 express this relationship between the third- and fifth-order intercept points, the resultant intermodulation power of each tone and the single-tone power of the two-tone inputs.

There exists a fairly uniform relationship between the point of 1-dB compression and the various intercept points for a specific device. Typically, IP3 is about 10 to 15 dB above P1dB, while IP5 is usually similar or higher.

Modulation and linearity
Although P1dB indicates the point where the gain of a system is compressed by 1 dB, this characteristic is mostly a voltage-domain function and the voltage-limiting characteristic may be gradual or sharp. This is important to consider because real communication systems generally do not use simple waveforms and those used to measure IP3- and P1dB-modulated signals can contain a great deal of time-varying voltage fluctuation.

Signaling for 802.11a/b/g is highly dynamic and its peak voltage is drastically different from its average voltage, which is used to measure average power. An OFDM signal contains voltage excursions that are greater than five times its average value (a 14-dB power ratio), while single-carrier modes exhibit a ratio of about 1.4 (3 dB, but the exact ratio depends upon pulse shaping). It is for this reason that system components need to be "backed off"; the average output power of a system must be lower than its linear continuous-wave power-handling capabilities so that these voltage excursions are not excessively distorted.

The distortion of voltage excursions is the reason for spectral regrowth; compression of modulation causes an expansion in the modulated-signal bandwidth. Exactly what happens to the two adjacent tones (for a two-tone IP3 measurement) is what happens to the modulated carrier, just in convoluted fashion: It is valid to model a modulated signal as many discrete tones next to one another that mix together. Just as the two measurement tones mix, and produce higher-order products, the many discrete slices of the modulated-signal spectra produce new discrete tones whose distance from the two original tones bears the same relationship (see Figure 3).

For single carrier, the first side lobe is primarily a function of IP3 (it is really IM3 + IM5 + IM_N, but the IM3 product dominates at reasonable signal levels), while the second side lobe is primarily IP5. Therefore, the relative height of the second side lobe can be roughly estimated according to the same rules used to measure the two-tone IP3; the second and third side lobes rise at a rate of 3 and 5 dB, respectively, for every decibel increase in main-lobe output power. To fit Equations 1 and 2 (online), which are based on the individual powers of each of two input tones, 3 dB must be added to the IMx products in order to compensate for the total modulated power, which is one spread tone.

Hence, if the total output power is X dB less than IP3, the first side lobe will be about 2X - 3 dB below the main lobe. This approximation assumes flat spectra in the main lobe and is somewhat dependent upon channel filtering, but provides a reasonable estimate so long as the basic IP3 model is valid for the system.

Following this approximation, the -30-dBr spectral mask for the first side lobe is met roughly when output power is at least 16.5 dB below the system's output IP3. IP5 can be derived in the same way: For a second side lobe at -50 dBr, IP5 should be greater than (4X-3)/4 above the system output power, which is a 13.5-dB IP5 backoff. Expressed as functions, this approximation is stated as shown in Equations 3 and 4.

Nonlinearity is not usually an overwhelming factor in EVM for single-carrier systems—the spectral mask is exceeded long before EVM is usually degraded beyond allowable limits for Barker, CCK or PBCC modulation. However, spectral regrowth is highly deleterious to OFDM EVM. This is because OFDM utilizes many independent carriers whose intermodulation and regrowth cause mutual interference to one another.

Each individual subcarrier will be subject to the IM3 products from all of the various combinations (there are half as many IM3 combinations as OFDM carriers) that will result when this many signals are lined up next to one another—the total IM3 SNR degradation is the Log10(26) greater than Pout-IM3. Again, since the IP3 model is based on the individual power of two input tones and it is more appropriate to work with the total OFDM power, the IM3 product must be scaled an additional 3 dB. Based upon this model, it is possible to estimate the relationship between IP3 and IM3-induced (regrowth-induced) SNR.

Again, recalling the IP3/IM3 relationship where IM3 tones are twice the distance below the input level as the IP3 is above, this input-to-IM3 level is IM3-induced SNR. When there are 52 random superimposed IM3 N (noise) products falling atop of each subcarrier, the total IM3 N is 10Log₁₀(26) plus the 3-dB single-tone scaling factor, then the individual IM3 tone is scaled by 17 dB. Therefore, IM3-induced EVM is estimated as 17 dB greater than half the difference between IP3 and total output power. Equation 5 (online) expresses this relationship.

For the minimum 54-Mbit/s OFDM EVM of -25 dB, IP3 should be at least half of 25 + 17. This is 21 dB above the desired output power, which corresponds to a 6- to 11-dB backoff from P1dB.

EVM, of course, is a composite of many different sources of error; it is not just a function of in-band spectral regrowth. However, in an efficient system, regrowth-induced EVM dominates, thereby making any other contribution negligible. Each individual contributor to the composite voltage EVM combines in an rms fashion (Figure 4).

When one observes a signal at a level far within the linearity limits of the transmitter system, this baseline (residual) EVM will be observed. As power increases and the regrowth-induced EVM equals the baseline EVM, the total EVM will degrade by 3 dB. As power rises further, regrowth EVM will dominate. For example, when the regrowth-only EVM is 10 dB above baseline, only a 0.4-dB error results. Beyond that, the excess EVM due to spectral regrowth is nearly the same as the composite EVM—baseline EVM becomes insignificant. If a designer wishes, regrowth EVM may be de-embedded from the baseline EVM by taking the square root of the square of the composite minus the square of the baseline, but simple subtraction is not valid.

The EVM/IP3 relationship further denotes a 2:1 relationship between output power and EVM: If output power rises 1 dB and the 26 IM3 products on falling on each tone rise 3 dB, then a net 2 dB has been lost in terms of SNR. Fig. 4 is an actual measurement of EVM vs. output power of a power amplifier vs. the predicted 2:1 slope using the IP3 EVM model. Regrowth EVM dominated the composite EVM once the baseline EVM was significantly exceeded.

About the Author
Jeffrey Feigin (jeffrey.feigin@skyworksinc.com) is principal applications engineer at Skyworks Solutions Inc. He has an MSEE from Worcester Polytechnic Institute (Worcester, Mass.).