Discrete Spectrum Techniques for Spectral Regrowth Analysis
Introduction
CDMA signals are increasing in wireless communication to
facilitate high data rates. CDMA is presently used in the IS95
system, and it is included in the European IMT2000
proposal. Spectral regrowth, a key issue for these
systems, is a phenomena that occurs if a bandlimited
timevarying envelope signal is passed through an oddorder
nonlinear circuit. In this situation, the output spectrum is
wider than the input spectrum"spectral regrowth"and cochannel
and adjacentchannel distortion results. Discretespectrum
approaches are popular techniques for analyzing spectral
regrowth phenomena. However, these approaches may produce errors
compared to continuousspectrum signals with respect to
spectral bandwidth and spectral regrowth power level. This
paper investigates some of the implications of this error
introduction.
FrequencyRelated Aspects
In general, the signals involved in spectralregrowth
analysis are, more or less, random and continuous, and thus
have a continuous frequencydomain representation. This is not
the case when you use a Fourier analysis that produces
approximate techniques. This type of signal has a discrete
spectrum description. Reference uses a Fourierseries approach to represent
the CDMA signal in IS95. Using a Fourier series means using a
number of discrete frequency points to approximate the signal,
which is generally continuous in a certain bandwidth 2B.
Considering a deterministic and discrete spectrum approach, such as the one in Maas, gives the result in Figure 1a. In this figure, the input is consists of the sum of three cosine signals, which results in a discrete inputfrequency spectrum with components at the input frequencies. The distortion components are also illustrated in Figure 1a for a thirdorder nonlinear device. Observe that the input consists of the three tones, but there are only two tones in the upper and lower spectral regrowth bands at and below f_{0}  2Df, and at and above f_{0} + 2Df.
Now assume that a continuous frequencydomain signal occupies the frequency band f_{0}  B < f < f_{0} + B, where f_{0} >> B. Applying this signal as input to a nonlinear device means that oddorder nonlinearities generally spreads the spectrum at the output. An mth (odd) order nonlinearity results in an output spectrum that occupies the frequency band f_{0}  m B < f < f_{0} + m B no matter how small B is"this is due to spectrum folding. Next we have the Fourier series representation used for stochastic signal description. Assume for now that B is extremely small, for example, B = 0.5 Hz. Intuitively, it should be sufficient to represent this very narrowband signal by a single frequency component at f_{0}, which represents the signal in the frequency band f_{0}  B < f < f_{0} + B. Since this signal is represented by a single frequency component, the output from an mth (odd) order nonlinearity also occupies the frequency band f_{0}  B < f < f_{0} + B. Compared to the continuous frequencydomain signal, this is obviously incorrect. The conclusion of this analysis is that no matter how small the input bandwidth, a "number" of discrete frequency points must always be used to obtain a fair representation of the output signal from a nonlinear device. Figure 1b illustrates the bandwidth problem for a stochastic and discrete spectrum approach with three frequency points representing the input signal. The lower and upper spectral regrowth sidebands each have a bandwidth of 2Df, whereas the correct bandwidths are 3Df.
Figure 1: Output spectrum of a thirdorder nonlinearity. (a) An input consisting of three cosine tones with frequencies (f_{1}, f_{2}, f_{3}) along with their intermodulation components. (b) The power spectral density representing an arbitrary signal, where each frequency component represents the signal in a Df bandwidth.
Assume the input signal is represented at #_{Q} frequency points such that 2B = x #_{Q}, where x is the frequency resolution in the discrete spectrum approximation. This means that the output signal from an mth (odd) order nonlinearity occupies the frequency band . Thus, the discrete spectrum bandwidth is 100 (m1)/(m#_{Q}) percent too low. As shown in Figure 2, the percentage error in bandwidth is quite large"even for high Q's. Also notice from Figure 2 that the order dependence only has minor impact on bandwidth error.
Figure 2: Percentage bandwidth error between the approximate discrete spectrum approach and the true continuous spectrum approach.
PowerRelated Aspects
Using a discretebandwidth analysis introduces a frequency
bandwidth error. There is also some error in the power
estimation, since a discrete spectrum is approximating a
continuous spectrum"Figure 3 illustrates this concept.
The figure is calculated using the techniques described in . In Figure 3, we use only three
frequency points to describe a CDMA signal with a bandwidth of
1.25 MHz. The technique by Wu et al. is the reference for this operation, since it
includes the correct spectral folding of the continuous
spectrum signals. As shown in Figure 3, the discrete
spectrum approach does not predict the correct bandwidth, as
discussed in the previous section of this article.
Figure 3: Power spectral density for the output of a nonlinear power amplifier for IS95 applications.
However, there is also an error in amplitude"in particular at the lower and upper edges of the spectral regrowth. For the discrete spectrum analysis in , the predicted total 3rdorder spectral regrowth power is shown in Figure 4, along with the reference set by the Wu technique. The percentage error in total power of the lower/upper regrowth sidebands is shown in Figure 5. Figures 4 and 5 show that the power predicted by the discrete spectrum approach rapidly converges towards the reference for an increasing number of frequency points in the discretespectrum representation. Figure 5 shows that the error is less than 1% for Q's above 5.
Figure 4: The total spectral regrowth powers in both lower and upper adjacent sidebands.
Figure 5: Percentage error in total upper/lower sideband power between the discrete spectrum approach and the continuous spectrum approach.
Conclusions
This article has shown that care must be taken when using
discretespectrum techniques to analyze spectral regrowth in
nonlinear devices. Using too few frequency points to represent
an otherwise continuous spectrum signal gives an
underestimation in output bandwidth and, more importantly, an
error in power level. We show an example to illustrate these
concepts for spectral regrowth in an IS95 CDMA power
amplifier. The example shows that using 11 frequency points in
the discrete spectrum approach, gives a bandwidth error of
approximately 6% whereas the spectral regrowth power error is
well below 1%. Since power estimation is the most important
parameter, this is considered to be acceptable. The example can
be considered typical in the sense that thirdorder nonlinear
effects are dominated by the firstorder (linear) effect. When
choosing the number of frequency points, you need to consider
two issues:
