Consider a nonlinear element with the following characteristic of the voltage amplitude V [4]:

If amplitude V is small and coefficient µ is finite, this characteristic is linear. At high values of V ƒ(V) is limited by margins ±A0. At intermediate values of V it resembles the soft limiter characteristic. By that measure, “softness” is determined by the dispersion σ2. Parameter σ is the root-mean-square (rms) deviation of a signal. The coefficient μ defines the linear part of Eq. 1., for which the output signal phase completely resembles the input signal phase. If an input signal applied to the nonlinear part of Eq. 1 is the arbitrary narrowband signal V(t) = V(t)cos0(t) + φ(t)] with a spectrum Sin(ω), the output signal spectrum in the vicinity of the center frequency ω0 can be represented in the form[4]:

The functions Θ(ω) = S(ω) Ⓧ S(-ω), Θn (ω) = Θn-1 (ω) Ⓧ Θ(ω) are convolutions of the input signal spectrum components and the function Θ0(ω) = δ(ω) is the delta-function:

n! = 1*2*3* …*n ; n!! = 1*3*5*7* …*n.

Equation 2 shows that the output spectrum of an input signal with Sin(ω) passed through the nonlinear element of Eq. 1 comprises the linearly scaled input signal (first term in Eq. 2) and distorted term of the spectrum noted as ΔS(ω). The term ΔS(ω) represents the multiple self-convolution of an input signal. If the dispersion of an input signal is defined as and an output signal is defined as then the output signal can also be expressed as[5]:

The functions K(x) and D(x) are the full elliptic integrals[5], and they can be represented in series form:

Dispersion associated with the undistorted first term in Eq. 2 is given by[4]:

The difference between (5) and (3) is then:

So, the output power (dispersion) associated with the distorted part of a signal in Eq. 2 can be simply found out by Eq. 6. It comprises both so-called in-channel and out-of-channel distorted components in the vicinity of ω0 (harmonics are not taken into account). The maximum level of the distorted power, η1 = 1 π / 4, is reached when μ = 0 (for hard limiter), so lim x → 1 [K(x) - D(x)] = 1. Thus, 21.5% of the total output power of a signal passed through a hard limiter is spread over the limits of the useful signal, decreasing the in-channel signal-to-noise ratio and elevating spectrum side lobes. This deteriorates the so-called spectral purity for GSM systems or the adjacent-channel power ratio (ACPR) for other digital communication standards. Extending the linear part of Eq. 1 (elevating parameter), the distorted part of the power defined by Eq. 6 decreases and in limit becomes 0 at μ → ∞, so lim x → 0 [K(x) - D(x)] = π/4. To evaluate exact values of spectral components at specified frequencies, Eq. 2 must be resolved knowing the input signal spectrum for each particular case. Note that equations 2 through 6 are valid for any kind of narrowband signals passed through the nonlinear part of Eq. 1. For instance, the dispersion associated with the distorted part of a spectrum for a five-channel forward link CDMA signal passed through the hard limiter (with linear phase characteristic) has been measured as 21%. By that, the output signal envelope becomes practically constant. Only 0.5% of the distorted power is assumed to reduce the in-channel signal-to-noise ratio by 0.022 dB. In general, the CDMA-like random signals are more resistant to the in-channel noise growth imposed by nonlinearity, although, for out-of-channel spectral re-growth, an opposite affirmation is valid[6-8].

Now consider a signal with a Gaussian spectrum applied to the nonlinear element in Eq. 1:

In this case[4]:

Again, the first term in Eq. 8 is an appropriately linearly scaled input signal, and the sum represents the distorted part of a spectrum. In this case, the maximum value of the distorted component imposed by the nonlinearity at any frequency does not exceed[4]:

The function ζ(p) is the Riman zetta-function[4]. Equation 9 shows that, even for an ideal hard limiter, the ideal Gaussian spectrum signals passed have values of η2 < 1.29 ≈ 1.1dB for any component considered, and an out-of-channel spectral re-growth is not the main concern for these signals passed through the ideal nonlinearity in Eq. 1. The measure of an in-channel noise growth influence onto a whole GSM system performance is the bit error rate (BER) and frame error rate (FER)[3]. However, the GSM transmitter performance influenced by this noise is the phase error. Considering the constellation diagram in Figure 1, the maximum phase error is seen to be:

The term a is the undistorted component and b is the additional noise component. By this approach the maximum phase error can reach αmax = 2sin -1(0.29/2) = 16.7°. However, simply choosing μ > 0.815 for the linear part of Eq. 1 avoids compliance violations with the maximum average phase error of 5° and the peak phase error of 20° specified for GSM. So, when μ = 0.815, it defines a margin for the nonlinear characteristic of Eq. 1, below which phase error may cause trouble. For a PA near the saturation region, this condition results in back-off operation from the maximal achievable power with some efficiency reduction. To reach the phase error of 5°, the next correlation to be carried out should be — b / a = -10.6 dB (a is the total channel power at limits ± 100 kHz and b is the total additional noise at the same limits for the worst case of the phase correlation). While the worst case of the noise components phase correlation will rarely be achieved, predicting this correlation in a particular case is difficult. Note that the spectral re-growth defined by Eq. 9 for each frequency component and the phase error determined in Eq. 10 are independent on the input signal dispersion σ2.

### Statistical consideration of distortion

So far, the ideal limiter characteristic described in Eq. 1 with a linear phase characteristic and the “pure” Gaussian spectrum signal defined by Eq. 7 have been considered. A real PA transfer characteristic should take into account amplitude modulation (AM) and phase modulation (PM), considering not only AM-to-AM conversion but also AM-to-PM, PM-to-PM, PM-to-AM, memory effects, frequency dependence of matching and biasing networks and non-symmetry of IMD products[9]. The effects noted are especially important at high power levels with high efficiency where the gain compression and power saturation become perceptible and the phase characteristic of the PA varies sharply. These effects result in deterioration of an out-of-channel spectrum purity and in-channel noise level growth, even for the constant-envelope Gaussian signals.

As an example, Figure 2 represents the spectrum measured for a 4 W GSM MMIC PA (f = 880 MHz, filter rolloff α = 0.3, full data rate 270.833 kbps, PSAT = 36.5 dBm and spectral components were measured at 30 kHz resolution bandwidth, pulsed-modulation disabled). In the main spectral lobe, the difference between the input and output spectral components does not exceed 1.1 dB. However, starting from 350 kHz, this difference becomes obvious (in this particular case, the maximum difference is 8 dB at 450 kHz).

When representing the linearity of a PA for Gaussian-like signal applications, manufacturers usually include different orders of two-tone IMD products at some frequency spacing and harmonics level in the datasheet. However, the system performance requires other measures of quality, such as the out-of-channel spectral purity and phase error for GSM standard, and this raises several questions. For example, what is the relation between these different concepts and how to evaluate them? How can a circuit be tuned to satisfy all requirements? What part of the transmitter determines spectral re-growth at particular frequencies?

Consider a statistical signal with the power spectral density PSD(ƒ). If the envelope and average power P0 of a signal are constants, one can represent PSD(ƒ) as the probability that the spectral component with power P0 appears at frequency ƒ:

In this case, the average power is:

So, g(ƒ) is the density of a power probability distribution function (distribution throughout the frequency). The power spectrum is solid owing to non-periodic driving of the PA. The probability of an appearance of the spectral component with power P0 at the frequency limits 1 is g(ƒ1)dƒ1 and, at limits 2, it is equal to g(ƒ2)dƒ2. These simultaneous power components passing through a nonlinear PA create different order instantaneous IMD products with the probability of g(ƒ1)g(ƒ2)dƒ12. The frequency spacing 1 - ƒ2) defines the frequency offsets ƒ0 at which different order IMD products will appear. The nth-order IMD product is denoted as by the driving of the PA input by two signals with each carrier level equal to P0. For the fixed offset, ƒ0 and fixed order of IMD, the resulting effect is the sum of the input spectral component scaled by the gain value, and IMD defined at different spacing as an integral:

The term G(ƒ0) is the gain of the PA at a certain input signal level. The term n is the order of IMD products (for the frequencies of interest n = 2m - 1 and m is an integer >2). Eq. 13 represents the same physical entity as Eq. 2, but with only one convolution procedure. Repeating the procedure defined by Eq. 13 for many times, the results of Eqs. 13 and 2 will be entirely equal. The function Fn(ƒ0) plays the role of a weighting coefficient at a particular frequency offset ƒ0 for each order of IMD product specified at P0 input drive level (analogous to results in[6-8]).

Consider the ideal Gaussian spectrum signal defined in Eq. 7 (solid line in Figure 3). The frequency offset ƒ0 is normalized to the rms deviation σ of an input signal. The value σ can be defined knowing α = BbT of a Gaussian filter used in a system (Bb is the filter bandwidth and T is the period of the baseband signal). The Gaussian filter complex transfer characteristic is represented in the form:

Here, ƒ00/2π is the center frequency of the filter; Δ is the filter-equivalent bandwidth and τd is time delay of the filter. Assuming ω >> Δ, ω0 >> Δ and, for simplicity τd=0, the following is derived:

At ω - ω0 = Δ: K(ω) = e-π/2= 0.208045

Normalizing this point to Gaussian distribution (0.399043*0.208045 = 0.083019), one can get from the table[5]:

In this case, the result of the appearance of several weighting coefficients Fn0) in Eq. 13 is shown in Figure 5; Fn0) is equal to the square restricted by the appropriate curve and the x-axis. The term Δƒ is the frequency spacing at which IMD products are defined. Note that the Fn(Δƒ) distribution at a fixed ƒ0 is a Gaussian one with clearly observed maximums and different dispersions, while also noting the power distribution is not Gaussian. The resulting weighting functions for three IMD product orders n are presented in Figure 3. The frequency spacing at which the IMD contribution is maximum is presented in Figure 4 by solid lines for the same IMD. Accordingly, IMD products must be measured (with equal two-tone drive levels) at 4 dB back-off from the real output-power level P0, recalling that the main lobe spectrum shape is almost a Gaussian one (Eq. 8), and the maximum spectral density is 4 dB less than the total power (Eq. 9). In general, the simple rule to calculate the slope of the frequency spacing Δƒ vs. the frequency offset ƒ0 is:

The frequency margins at which IMD products must be defined can be represented in the form. These functions are presented in Figure 4 by dashed lines. The power restricted by these margins comprises 95.5% of the total power determined by the second term in Eq. 13 if the variations of IMD through Δƒ are absent. The simple rule to define σn is:

It is important to note that the two-tone frequency spacing for the IMD contribution to the total distortion does not depend on the frequency offset for the fixed order of IMD (in opposite, for flat spectrum shape signals[6, 7], the frequency spacing decays with an increase in offset). Results in Figure 3 and Figure 4 have been calculated by use of a single convolution procedure defined in Eq. 13. However, this procedure precisely accounts for the main contribution of IMD toward spectral re-growth.

In Figure.3, the maximum values of IMD relate to the carrier power maximum spectral density by the rule of Eq. 18. The shape of IMD curves vs. ƒ0 is also a Gaussian one with different dispersions calculated as:

For a properly designed PA, IMD products do not usually vary within frequency spacing margins. On the contrary, if these variations are observed, this is a sign of improperly matched circuits at baseband frequencies, most likely the biasing circuits. It is known that two-tone IMD products may elevate sharply for very small frequency spacing[9]. Considering Figure 3 and Figure 4 (and analogous results in[6, 7]) confirms that the Gaussian average power spectrum signals are more robust to the out-of-channel spectral re-growth than flat average-power spectrum signals.

Figure 3 presents the spectral mask at frequencies specified for GSM (DCS, PCS) mobile and small base station PAs. The difference between the IMD curves and the spectral mask in Figure 3 shows the margin between IMD products (in decibel values) at specified output power levels and the level beyond which the system spec becomes invalid (if the influence of only one specified order of IMD is considered). These margins for GSM (DCS, PCS) mobile and small base station PAs are presented in Figure 6. In Figure 3, it can be seen that the spectral mask resembles the properly weighted IM3 characteristic. Usually, this is likely for Class A solid-state PAs operating very close to saturation. For other classes of operation, the shape of the side-lobe spectrum may differ from IM3 presented in Figure 3 due to an increased influence of higher-order IMD products. Spectral re-growth problems can be considered through simulation or measurement of the different orders of IMD products in accordance with Figure 3 for frequency offsets specified in Figure 4 and Equations 18 and 19. Then, the entire circuit can be tuned to eliminate the source of distortion.

Figure 6 shows that, for GSM (DCS, PCS) mobile and small base station PAs, the frequency offsets contributing the most to the deterioration of a spectrum are placed around 400 kHz, which is a well-known “weak” point confirmed by the real PA testing presented in Figure 2. In this case, the frequency spacing margins Δƒ at which IMD products should be considered are presented in Figure 7. Therefore, when designing GSM PAs, special care should be taken to match networks at baseband frequencies specified by Figure 7. At these frequencies, the IM5 and higher-order IMD products play a decisive role in spectral re-growth. In most practical cases, the margins shown in Figure 6 rarely can be achieved for classes A, AB and B PAs, even very close to saturation. However, for high efficiency switching-type PAs, such as modes E, F and other rectangular-shaped voltage or current RF output signals[10], it is problematic to achieve the maximum efficiency for a GSM modulated signal due to an increased level of intermodulation products (except for the nonlinear transmitter approach used in[11] or other linearization techniques[9]), because the spectrum mask requirements will be violated.

Figure 3 shows that up to the saturation region, the additional IMD-imposed noise components are far from the relation of b/a = -10.6 dB at ±100 kHz channel bandwidth, as indicated previously, and that IMD products have a small influence on phase error in GSM systems. The main source of a phase error is the phase-transient response, not the steady state AM-PM conversion, and the amplitude transient response of the PA, which is statistically the amount of time an output signal exists outside the input signal trajectory when passing from one phase state to another. During the operation of a real GSM transmitter, the average phase error contribution imposed by the PA's nonlinear characteristic rarely exceeds 2°. The peak phase error usually is imposed by pulse shaping during the burst operation. However, discussion of this topic is beyond the scope of this article.

Figure 3 also presents the channel offset for several different existing and old digital communications standards with Gaussian-like signals (DECT, CDPD, CT2 and Mobitex) for nominal modulation data rates and standard filters. The paging system (Mobitex) has the lowest sensitivity to IMD products, and its PA can be driven closer to saturation.

Finally, to properly account for noise in the PA, the noise power spectral density term must be added to Eq. 13. To evaluate in-channel noise growth, the previous approach can be extended up to the center of the frequency spectrum[8].

The burst mode of a PA results in the additional spectral components within the spectrum. Though not discussed here, these components would depend on the burst repetition rate and the particular pulse-shaping characteristics of the PA.

### Transmitter intermodulation

One of the important parameters of a GSM transmitter is the transmitter intermodulation (IM) characteristic when a low-level RF signal from an antenna connector reaches the output of an RF power amplifier. This small signal mixes with the large signal of the PA, resulting in unwanted signals on the transmitter output. The GSM standard places severe restrictions on the radiation of these unwanted signals[3]. Usually, the check of transmitter intermodulation characteristics is carried out after completing design of a whole transmitter. However, the power amplifier specification often fails to include this important parameter. Except for a small contribution by passive components like output filters, duplexers and power combiners, the main source of intermodulation distortion is the nonlinear output transconductance of the final stage of the transmitter amplifier. The challenge then becomes selecting components for this stage to avoid this difficulty.

An output transistor's transconductance can be represented as:

Driving this transconductance with a two-tone voltage signal is represented as follows:

The current through this transconductance is calculated as:

Assuming ω1 is the large-signal carrier frequency and ω2 is the small-signal interference signal, and restricting analysis to only third-order IMD, which usually gives the highest level of transmitter intermodulation) results in two IMD current levels at the transistor output reference plane:

The passive linear part of a transmitter between the final stage of the PA and the antenna connector cannot change nonlinearity. So, it is convenient to do analysis at the output of the PA, where 50 Ω matching has usually been established. Applying the GSM standard transmitter intermodulation margin[3] defined in Eq. 23, one can define the nonlinearity requirements for the PA's final transistor stage for different power levels and frequency offsets. The calculated results of these nonlinearity requirements for mobile transmitters and base station transmitters are presented in Figure 8 and Figure 9. The IMD caused by frequency offsets between 1.2 MHz and 6 MHz should be verified first.

For DCS mobiles, the minimum nonlinearity requirement is G3 = 0.009335V-2 for the maximum rated power of 33 dBm and the transmitter IM is not a concern when designing the PA. Therefore, different kinds of transistors can be used in that situation. For GSM mobiles and mid- and low-power base stations, requirements for nonlinearity are restrictive. However, up to 43 dBm rated power, regular MESFET transistors with flat epi-doping profiles can still be used[12]. For higher power levels, only increased linearity transistors like doping-improved MESFET, HEMT or LDMOS should be used to pass an intermodulation spec. At power levels higher than 49 dBm, the best choice seems to be LDMOS transistors.

Certainly, transmitter output loss and filtering characteristics are not taken into account by this consideration. However, Figures 8 and 9 give clear insights into the design of the transmitter output chain when the nonlinear characteristics of the transistor chosen for the PA output stage are known.

### References

1. F. Amoroso, R.A. Monzingo. “Digital Data Signal Spectral Side Lobe Regrowth in Soft Saturating Amplifiers.” Microwave Journal, February 1998, No. 2, p. 27-32.

2. J. Duclercq. “GSM Base Station Power Amplifier Module Linearity.” Microwave Journal, April 1999, No. 4, p.116-127.

3. Digital Cellular Telecommunication Systems (Phase 2+), Radio Transmission and Reception (GSM05.05 version 8.5.0 release 1999), Draft ETSI EN300910.

4. Y.A. Evsikov, V.V. Chapursky. “Random Process Transformation in Radio-technical Systems,” Moscow, Vyschaya Shkola, 1977.

5. G.A. Korn, T.M. Korn. “Mathematical Handbook for Scientists and Engineers,” McGraw-Hill Book Company, 1968.

6. O. Gorbachov. “IMD Products and Spectral Regrowth in CDMA Power Amplifiers.” Microwave Journal, March 2000, No. 3, p. 96-108.

7. O. Gorbachov, Y. Cheng, J.S.W. Chen. “Noise and ACPR Correlation in CDMA Power Amplifiers.” RF Design, May 2001, p. 48-56.

8. O. Gorbachov, J.S.W. Chen. “Evaluate Noise in GSM PAs” Microwaves & RF, February 2001, p. 69-74.

9. A. Katz. “SSPA Linearization.” Microwave Journal, April 1999, No. 4, p. 22-44.

10. F.J. Ortega-Gonzalez, J.L. Jimenez-Martin, A.Asenisio-Lopez, G. Torregrosa-Penalva. “High-Efficiency Load-Pull Harmonic Controlled Class-E Power Amplifier.” IEEE Microwave and Guided Wave Letters, Vol. 8, No. 10, October 1998, p. 348-350.

11. M. Heimbach. “Digital Multimode Technology Redefines the Nature of RF Transmission.” Applied Microwave & Wireless, August 2001.

12. J.A. Higgins, R.L. Kuvas. “Analysis and Improvement of Intermodulation Distortion in GaAs Power FET.” IEEE Trans. on MTT, vol. MTT-28, No. 1, January 1980, p. 9-17.