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The issue of noise is one of the most important parameters when considering overall communication system quality. For wireless systems, major noise sources are the transmitter, the air interface and receiver noise.

In code-division multiple access (CDMA) systems, the signal-to-noise ratio determines the bit error rate (BER) or frame error rate (FER) numbers that represent a digital information loss¹ . In transmitters, one of the critical noise sources is a power amplifier (PA). Usually, the PA's non-linear characteristics associate with spectrum re-growth. An adjacent-channel power ratio (ACPR) imposed by the non-linear effect of the PA has been investigated by many authors² . Some use the clipping noise power approach³ , while others use a hypothetical third-order intercept point approach⁴ . Both methods are used to evaluate the system degradation characteristics due to an increased noise level for digitally modulated signals. Direct in-channel noise measurements are impossible because of the presence of a useful signal. Therefore, a statistical approach is used based upon CDMA systems. This approach allows a simple evaluation of increased noise on a basis of an ACPR measurement (or simulation) due to the digital signals passed through a non-linear PA.

Consider a PA driven by a statistical signal with the idealized, solid rectangular flat shape of an average power spectrum at a frequency band Δf (see Figure 1). This signal can be represented statistically in the frequency domain with a flat probability distribution function in margins^{5, 6} . It implies that each average-power spectral component inside the flat shape is created from the instantaneous input signal on the same frequency with the additional probability of an appearance of a signal on that frequency. The probability of an appearance of the spectral component inside the limits df₁ is equal to the formula in equations (1) on page 44.

In (1), g(P_in ) is the density of a probability distribution function for a PA input power. At the same time, the spectral component inside the limits df₂ exists with the similar probability. It can be assumed that the spectral components at limits df₁ and df₂ are not correlated. These simultaneous spectral components cause instantaneous inter-modulation distortions (IMD) of a PA output signal at frequency offsets determined by a difference between f₁ and f₂ (the particular case in Figure 1 presents the IM₃ definition). For the out-of-channel spectral re-growth, the IMD contribution has been considered in [5] and [6]. For in-channel noise power elevation due to IMD products, the situation differs from the previous one. Again, referring to Figure 1, the IMD appearance at a frequency offset, f₀ is possible from two parts of a spectrum: left, δf₁ and right, δf₂ (below, the left side is determined at δf₁ > Δf/2, and the right side at δf₂ < Δf/2). The probability of the appearance of IMD products at different frequency offsets may be represented as the equation in formula (2).

In (2), g(IMD) is the density of a probability distribution function for IMD imposed by an instantaneous input signal. Assume that the main contribution to IMD is from frequency components with equal power levels (it has been verified experimentally for metal semiconductor field-effect transistor [MESFET] PAs). Then the effects occur at a frequency band near the saturation region for the average value of IMD at different frequency offsets (see equation (3)).

The first term in (3) represents the left side spectrum contribution to the total IMD value, and the second term represents the right side (see Figure 1). Remember that the total probability of an IMD appearance equals the sum of particular probabilities. The integral limits in (3) determining the two-tone frequency spacing are defined by the equations of the formula. The frequency-spacing margins, where the IMD products are placed, are at a frequency offset f₀ (see Figure 2 — top). These margins have been previously defined in [5] and [6] for out-of-channel spectral re-growth (at f₀ /Δf > 0.5 ). For in-channel (in margins of Δf) IMD products, the solid lines belong to the left side of a spectrum and dashed lines to the right side.

It can be observed that, at a specified frequency offset f₀ , the total IMD product (3) combines components with different frequency spacing starting from near-to-zero and up to margins determined by (4) and Figure 2 — top. For the lower order of IMD, the frequency spacing range is wider than it is for the higher ones. CDMA signals are narrow-band and the difference in IMD products at a frequency offset is small through the range of the frequency spacing mentioned above. In this case, consider the mean values and equation (3) transforms into the formula in equation (5). The function F(f₀ ) plays the role of a weighting coefficient for the IMD average value at different frequency offsets and its shape is presented in Figure 2 — bottom (at f₀ /Δf > 0.5, it has been defined in [5] and [6]). So, IMD products from (5), together with the proper noise of a PA (without input signal) and with a spectral density of N₀ , create the total noise spectral component at a frequency f₀ , as determined by the formula in equation (6). The total in-channel noise is given by the formula in equation (7).

The function (5) has been determined in [5] and [6] for CDMA signals by the formula in equation (8).

In (5), IMD[mW] and P_in [mW] are the IMD power at a specified input power in milliwatts. g(P_in [dBm] is the density of a power probability distribution function for an input signal (usually, it has a logarithmic-normal distribution [7]). Ta = g(P_in [dBm] is the auxiliary transfer function for the IMD [5]. The integral from the function Ta is proportional to the IMD at a given frequency offset, f₀ . The function T_a versus P_in is convenient for a graphical representation of equation (8). It directly represents a contribution from the input signal statistics and IMD characteristics of a PA at each instantaneous drive power levels to the in-channel noise spectral power at f₀ . The square restricted by the curve T_a and P_in axis is equal to the additive noise with the accuracy of a coefficient. Considering curves T_a versus P_in , The circuit can be tuned to decrease the additive noise imposed by the PA's non-linear characteristic at input power levels with high noise contribution.

Observing the weighting functions F(f₀ ), and assuming the constancy of IMD products through the frequency spacing defined above, it can be concluded that an in-channel noise power distribution is not additive white Gaussian noise (AWGN)-type. The noise-power spectral density elevates by 3 dB at spectrum margins over a center frequency. The noise influence on a system's performance is defined by the integral (7). In this case, one can represent an in-channel noise as AWGN with mid-values of a noise spectral density. For each IMD product contribution to a total noise, the mean weighting functions F(f₀ ) are presented in Figure 2 — bottom by horizontal dashed lines (-8.0 dB for IM₃ ; -11.2 dB for IM₅ , and -13.6 dB for IM₇ ).

Then, comparing the in-channel noise power elevation defined by (8) and the out-of-channel spectral re-growth [5], [6] imposed by IMD products yields the following: It can be concluded that the mechanism of the appearance of these two figures of merit is virtually the same except for the different weighting coefficients. Therefore, ACPR measurements can estimate for in-channel noise power (manifold equipment exists for that⁸ ).

For the small output power of a PA, the IM₃ virtually determines the total noise power. In this case, it is possible to measure an ACPR at “X” frequency offset, and use the IM₃ weighting function in figure 2b to define the reference between the noise and ACPR. However, it is necessary to check the smoothness of the ACPR shape. It should resemble the shape presented in Figure 2 — bottom. Furthermore, base-band matching networks (usually biasing ones) can sharply elevate IMD products and ACPR at certain frequencies [9]. To note, at small signals, the mean in-channel noise spectral component equals the ACPR spectral component at a frequency offset of f₀ /Δf = 0.72, or f₀ = 885 kHz for CDMA-one systems (exactly the spectral mask frequency specified for PCS base station PA). However, to use this test point, it is important to to check the input signal spectrum quality. Even 10 dB difference between ACPR measurements for input and output signals, due to an improper filtering of an input signal, results in 0.4 dB “alleged growth” of an in-channel noise power. The best frequency offset can be chosen at f₀ /Δf = 0.8 to 0.9. The reference position for the noise correlation will be as 1.0 to 2.4 dB.

There are some issues with ACPR measurement and simulation for their correlation with different orders' IMD products. For class AB MESFET PA modules driven by multi-carrier CDMA-one and W-CDMA (4.096 Mch/s) signals up to the systems' specified margins (defined by ACPR) at frequencies of 0.8, 1.8, 2.4, and 3.5 GHz the IM₃ related noise power exceeds the IM₅ -related noise power by more than 6 to dB. This means that, if choosing the ACPR measurement frequency offset at f₀ /Δf = 0.8, the mean in-channel noise spectral density will be equal to ACPR +1 dB, with an accuracy of 0.4 to 0.6 dB for maximal output power levels (ACPR +0.4 to 0.6 dB). Higher values of a frequency offset are undesirable because of an increased uncertainty of measurements for noise and ACPR correlation. In fact, the weighting functions in figure 2 — bottom represent the clipping noise power imposed by a PA non-linear characteristic, and this noise is distributed through frequencies and divided on different terms. Usually, each term varies insignificantly throughout the frequency, and the total additive noise power spectral density is not “white.” The high portion of a clipping power is placed out of the channel margins and defined as Δf. For example, the related IM₃ noise power inside Δf comprises only -3.1 dB of a total noise power produced by IM₃ . For IM₅ , this figure is -4.8 dB and for IM₇ , -6.3 dB. Exact values of each noise power component may be evaluated by the formula in equation (8) by means of the IMD products' measurement. The validity of a proposed correlation method between ACPR and IMD has been verified experimentally for W-CDMA PA⁵ .

Certainly, the method proposed is valid for multi-bearer signals when the signal statistics result in a shorter, proper time for a random process¹⁰ . In this case, for a linear PA, the amplitude and phase modulation (AM/PM) characteristic may be excluded from consideration owing to a fairly high-output power back off³ . But, even for a single-user, CDMA reverse-link channel, one approach proposed allows the tuning of a PA circuit for the best BER (FER) performance.

The method is also valid for personal digital cellular/communications (PDC) and personal handiphone systems (PHS) communication systems with a flat shape of the signal power spectrum⁶ . But, if a PA circuit is tuned for the best ACPR at a frequency offset of f₀ /Δf = 0.8 to 0.9, it automatically results in the lowest added in-channel noise power level for CDMA PA (including W-CDMA systems). This should not be considered a correlation between the in-channel noise power and ACPR for the center of an adjacent channel. It may result in an erroneous conclusion. However, the PA circuit's tuning needs a trade-off between ACPR and noise for PDC and PHS systems. For the best noise, the minimizing of IM₃ is important and, for the best ACPR, the higher orders of IMD products play a decisive role⁶ .

For the multilevel quadrature amplitude modulation (QAM) and hybrid phase-shift keying (PSK)/QAM, the results obtained may be applied directly.

For manifold orthogonal frequency division multiplexing (OFDM) modulation formats, an approach proposed is applicable. However, it requires some modifications due to the frequency hopping time and frequency guard intervals introduced, which result in a different statistical correlation between the individual spectral components.

The method also works with non-flat power spectrum signals such as Gaussian signals¹¹ , however, the power spectrum statistics differ.

For a strict result, the multiple self-convolutions of a signal should be considered¹¹ . However, this simplified approach only takes into account the first convolution procedure. The formula in equation (8) represents the main contribution of IMD to a noise growth.

The in-channel noise growth evaluation imposed by a PA in CDMA systems is based on the two-tone IMD product approach and signal statistics analysis in a frequency domain that allows a noise evaluation by ACPR measurements. The ACPR test frequency offset is defined as f₀ /Δf = 0.8 to 0.9 for the best correlation accuracy. The method proposed is also valid for other digitally modulated communication systems with a flat power spectrum.

The analysis addresses and clarifies PA tuning procedures to achieve the best noise characteristics in the whole system. Finally, to avoid confusion, the base-band matching networks should be considered as a first order.

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

3. A. Chini, Y. Wu, M. El-Tananu, S. Mahmoud. “Hardware Nonlinearities in Digital TV Broadcasting Using OFDM Modulation.” IEEE Trans. on Broadcasting, vol.44, March 1998, p.12-21.

4. S. Merchan, A.G. Armada, J.L. Garcia. “OFDM Performance in Amplifier Nonlinearity.” IEEE Trans. on Broadcasting, vol.44, no.1, March 1998, p.106-113.

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

6. O. Gorbachov. “Determine IMD Products for Digital Communication Standards.” Microwaves & RF, Aug. 2000.

7. J.F. Sevic. “Statistical Characterization of RF Power Amplifier Efficiency for Wireless Communication Systems.” WFB Workshop Proceedings of the 1997 MTT-S Symposium.

8. HP Application Note 1311.“Understanding CDMA Measurements for Base Stations and Their Components.”

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

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

11. O. Gorbachov. “Analyzing Linearity in Communication Amplifiers for Constant Envelope Gaussian Signals.” Submitted to Microwave Journal.

Oleksandr Gorbachov has a Ph.D. and a MS in electrical engineering from Kiev Polytechnic Institute, Ukraine. He is senior engineering director for the GaAs Division at Gatax Technology, Taipei, Taiwan, R.O.C.

Jason Shen-Whan Chen received a Ph.D. from the University of Illinois at Chicago in 1992. He received master and bachelor degrees from Marquette and National Tsing-Hwa University. He is currently vice president of Gatax Technology.

Yu Cheng received a BS degree from the department of physics, Fu-Jen Catholic University, Taipei, Taiwan, and a MS degree in electronics engineering from National Tsing-Hwa University, Hsinchu, Taiwan. Currently, she is working toward Ph.D. degree at Tsing-Hwa University.

Address: Gatax Technology Co. 6F-1, No.160, Sec.6, Min-Chuan E. Road, Taipei, Taiwan, R.O.C. - 114.
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