Determining crossmodulation interference is an important factor in the design of a code division multiple access (CDMA) mobile transceiver. The crossmodulation interference level directly affects the performance of the receiver's ability to overcome the spurious interferences, such as what the IS98^{1} calls single tone desensitization.
A few papers have been published, characterizing the crossmodulation caused by the low noise amplifier (LNA) in a receiver front end, from empirical equations to rigorous mathematical derivations.^{25} The crossmodulation levels characterized by these papers differ by as much as 10 dB. Through a straightforward mathematical concept and a simple mathematical model, this article addresses these differences and presents a result that is in close agreement with the empirical expressions. The expressions derived in this article have been verified through many commercial CDMA handset designs.
Crossmodulation Spectrum
In a mobile phone that utilizes CDMA technology, the transmitter and receiver operate in a fullduplex mode. The transmit (Tx) power will be present at the LNA input subject to the attenuation of the duplexer in front of the LNA. The interference signal in the receive (Rx) band will also be present at the LNA input. The Tx signal crossmodulating with the interference signal produces an interference that falls in the desired Rx signal band.
Since the Tx signal is a spread spectrum, with its signal energy distributed across its signal bandwidth (1.25 MHz), the power spectrum density (PSD) of the Tx power can be assumed to be approximately expressed as a rectangular function, as shown in Figure 1 , where B is the frequency bandwidth of the transmit spectrum.
It is known from the Fourier transform that, for a signal to have such a spectrum, the signal in the time domain must be a Sinc function,6
where = 2 f_{TX} and f_{TX} is the frequency of the Tx carrier signal. When the Tx signal is present at the LNA input along with an interference signal (onetone interference signal), the Tx signal is squared by the nonlinearity of the LNA, producing a low frequency (baseband) product that has twice the bandwidth (B) of the original Tx signal. This baseband product has a PSD that is a triangular function because it is a result of the convolution of the rectangular PSD by itself.^{7} Crossmodulation is produced when the baseband Tx product is modulated onto the onetone interference signal, resulting in a spectrum, as shown in Figure 2 , where f_{c} is the center frequency of the desired CDMA signal, f_{I} is the frequency of the interference signal and f = f_{I}  f_{c}.
Note that the crossmodulation product of a CDMA signal will not fall completely in the entire band of the desired signal because of the frequency offset ( f), and the crossmodulation product has a bandwidth that is twice the bandwidth of the desired CDMA signal. The PSD of the crossmodulation product may be expressed mathematically as
where
A = a function of the interferer power level, P_{I}
IIP3 = input thirdorder intercept point of the LNA
The function A describes the nonlinearity of the LNA.
The crossmodulation power that falls in the band of the desired CDMA signal may be calculated from
Let the CDMA crossmodulation factor, C, be the ratio between the crossmodulation power that falls in the desired band and the total crossmodulation power, or
where
assuming unity impedance at the LNA input. In Equation 5, it is also assumed that the LNA distributes the harmonic products of the Tx power with onehalf of its harmonic power at the baseband, crossmodulated with the onetone interference, while the other half power is at its secondharmonic frequency.
Note that the crossmodulation factor, C, is a function of the frequency offset, f, and the channel bandwidth, B. It is not related to the nonlinear characteristics of the device. The frequency offset, f, must be less than 1.5B for the crossmodulation product to fall in the Rx band. C should always be less than unity.
Crossmodulation Power Level
The intermodulation product of singletone signals may be calculated as^{8}
P_{(2AB)} = 2P_{A} + P_{B}  2IIP_{3} (6)
where
P_{A} = power level (dBm) of the signal that is squared
P_{B} = singletone interference (dBm)
IIP_{3} = input thirdorder intercept point of the nonlinear device (dBm)
All power levels are referred to the input of the device. Note that for a singletone intermodulation interference, all the interference power falls in the band of the desired signal.
It is easy to demonstrate that intermodulation and crossmodulation are generated by the same nonlinearity, A(P_{I},IIP_{3}), of the LNA. Therefore, Equation 6 also applies for crossmodulation when the crossmodulation product falls entirely on the Rx band.
To apply Equation 4 in the CDMA crossmodulation calculation, let
P_{A} = P_{Tx} (7)
P_{B} = P_{I} (8)
P_{(2AB) }= P_{XMod}(total) = P_{XMod}(in Rx band)  C(dB) (9)
where
Substituting Equations 7 through 9 into Equation 6 gives
P_{XMod}(in Rx band) = 2P_{Tx} + P_{I}  2IIP_{3} + C(dB) (11)
As per IS98^{1} and JSTD18,^{9} the singletone desensitization is measured with the onetone interference at 900 kHz and 1.25 MHz offset from the channel frequency for cellular and PCS bands, respectively, or
For cellular band, f = 900 kHz, B = 1.23 MHz, C(dB) = 8.32 dB, or
P_{XMod}(in Rx band) = 2P_{Tx} + P_{I}  2IIP_{3}  8.31 (for cellular band) (12)
For PCS band, f = 1.25 MHz, B = 1.23 MHz, C(dB) = 12.33 dB, or
P_{XMod}(in Rx band) = 2P_{Tx} + P_{I}  2IIP_{3}  12.33 (for PCS band) (13)
Examining Equation 11, note that the crossmodulation interference is reduced by C (dB) because the interference does not occupy the entire Rx band as a result of spectrum spreading that distributes the energy over a frequency band.
Equations 12 and 13 are in close agreement with the empirical equations derived by Draxler^{2}
P_{XMod}(in Rx band) = 1.906P_{Tx} + 0.949P_{I}  1.852IIP_{3}  8.4 (for cellular band) (14)
P_{XMod}(in Rx band) = 1.913P_{Tx} + 0.949P_{I}  1.864IIP_{3}  12.7 (for PCS band) (15)
The empirical equations were derived through a series of discrete timedomain simulations using Omnisys (Agilent Technologies) and curvefittings with a linear regression. Other references, such as AliAhmad^{10}, would yield an expression similar to Equation 11, but with C = 3 dB for the cellular band, in which it is assumed that the crossmodulation product is onehalf in band and onehalf out of band, and hence the 3 dB factor. Such a treatment over estimates the interference contribution due to crossmodulation.
Table 1  
Tx = 33.0 dBm, Jammer = 32.0 dBm, IIP3 = 9.0 dBm  

Cellular band 
PCS band 
Draxler's empirical Equations 
118.3 
123.0 
This article (Equations 12 & 13) 
124.3 
128.3 
Difference (%) 
4.8 
4.2 
Comparison
As an example, let the Tx power level presented at the input of the LNA be 33.0 dBm, the onetone interferer be 32.0 dBm and the LNA input IP3 be +9.0 dBm. Table 1 illustrates the crossmodulation levels at the LNA input as calculated from Equations 12 through 15. The difference between the results of this article and the empirical equations are 4.8 and 4.2 percent for the cellular and PCS frequency bands, respectively.
The analysis presented in this article assumes that the crossmodulation power is evenly distributed at the baseband and the second harmonic of the Tx frequency. Also, the nonlinearity higher than the second order has been ignored. Such treatments may underpredict the crossmodulation power level, which accounts for the difference indicated in the data.
Equations 14 and 15 were used to design over 10 different models of CDMA handsets for both cellular and PCS frequency bands. Laboratory tests show that Equations 14 and 15 over estimate the crossmodulation contribution, resulting in a larger design margin than budgeted. Using Equations 12 and 13 will lead to a more accurate system link budget and relax the specifications for the duplexer and LNA, reducing the cost of the CDMA handsets.
References
1. Recommended Minimum Performance Standards for Dualmode Spread Spectrum Cellular Mobile Stations, TIA/EIA/IS98B Standard, Section 9.4.2., August 1998.
2. P. Draxler, private email communication, Qualcomm Inc., July 23, 1997.
3. R. Mohindra, "Part 1: Crossmodulation and Linearization in CDMA Mobile Phone Transceivers," Wireless Symposium/ Portable by Design Conference Digest , San Jose, CA, 1999.
4. V. Aparin, B. Butler and P. Draxler, "Crossmodulation Distortion in CDMA Receivers," IEEE Transactions on Microwave Theory and Techniques , Vol. 48, No. 12, December 2000, pp. 100109.
5. "The Effect of CDMA Receiver Minimum Standards on RF Systems," Application Notes, Maxim Integrated Products Inc., September 2000.
6. J.D. Gaskill, Linear Systems, Fourier Transforms and Optics , John Wiley & Sons Inc., New York, NY 1978.
7. E.O. Brigham, The Fast Fourier Transform and Its Applications , PrenticeHall Inc., Englewood Cliffs, NJ 1988, pp. 6065.
8. H. Kinley, "More About Intermodulation," Mobile Radio Technology , April 1995, p. 8.
9. Recommended Minimum Performance Requirements for 1.8 to 2.0 GHz Code Division Multiple Access Personal Stations, TIA/EIA/JSTD018 Interim Standard, Section 3.4.2, January 1996.
10. W.Y. AliAhmad, "RF System Issues Related to CDMA Receiver Specifications," RF Design , September 1999.
Yang Zhang received his BSEE degree (with high honors) from China University of Electronics Science & Technology, Chengdu, China, in 1982, and his MS and PhD degrees in electrical engineering from Duke University, Durham, NC, in 1984 and 1990, respectively. Since 1990, he has worked with Duke University as a research associate, conducting research in microwaveinduced hyperthermia for cancer treatment; Otsuka Electronics, Fort Collins, CO, as a principal engineer, developing transceivers for MRI systems; and Ericsson Inc., Lynchburg, VA, as a receiver special team leader, leading a group of engineers to design land mobile radios and base stations. Zhang joined Qualcomm Inc. in 1996, and is currently a principal engineer, leading a team of engineers designing various 2G and 3G CDMA handsets of multiple modes and frequency bands. He has designed over 10 different models of CDMA cell phones marketed in both North America and China. Zhang has authored more than 30 technical papers and holds two US patents. He can be reached at (858) 6518741 or via email at yzhang@qualcomm.com.