Characterizing an L-Band Pulse Amplifier with Local FFT and Wavelet Transforms

	ABOUT THE AUTHOR Lorenzo Carbonini has 12 years of experience in RF/MW design and testing. He did achieve the Physics degree cum laude at the University of Turin in 1989, then the Mathematics degree cum laude at the same university in 1992. From 1989 to 1996 he has been RF/EMC design engineer at Alenia Avionic Equipment Division. From 1996 to 1998 he has been Product Development Manager for EMC testing products at Thermo Voltek Europe. In 1998 he joined Marconi Mobile covering different positions, he is presently the Material Engineering Manager. He is the inventor of three patents in the RF Design field.

Signal distortion must be below some specified level. Since this distortion often cannot be efficiently analyzed by simulation, you must identify and eliminate it during the experimental phase of the design.

Instruments typically available to the RF designer are a power meter (PWM) and a spectrum analyzer (SA). The spectrum analyzer is particularly efficient in determining the occurrence of distortions since, in several cases, these distortions lead to spectral deformation of the output signal or to spurious spectral lines.

The output spectrum of a power amplifier is subjected to various constraints, including those associated with the maximum level of harmonics or of undesired spectral lines, or a spectral mask limiting the spectrum degradation close to the carrier frequency f_c.

When dealing with pulse power amplifiers (PPAs), the situation becomes more complex due to the abrupt variations in the input and output signal. For these cases, the information provided by PWMs and SAs is not sufficient to investigate, in depth, the phenomena causing distortion. The peak power meter (PPWM) is a useful instrument in this case, measuring the instantaneous signal power at a (repetitive) sampling rate of several Megahertz. You can use the PPWM to study the signal amplitude distortion with a dynamic range of about 30 dB.

Recently, equipment vendors have introduced real-time SAs with the capability of performing instantaneous spectrum measurements over a bandwidth of several MHz (for example, the Tektronix 3000 series). This kind of instrument exhibits a wide dynamic range, but is not sufficient to analyze, in detail, narrow pulses (for example, shorter than 10 ms).

Digital sampling oscilloscopes (DSOs) with a 8 to 16 GS/s non-repetitive sampling rate and input analog bandwidth of 1.5 to 2 GHz have been introduced. Examples of these instruments include the Agilent Infiniium 54845A and Lecroy WavePro 960. This kind of instrument has the advantage of capturing the signal amplitude and phase variations directly at RF, even for very narrow pulses. The main limitation of this solution is the achievable dynamic range (normally about 40-50 dB) which is determined by the DSO's ADC. The measurement technique proposed in this article is based on a numerical analysis of waveforms sampled by a DSO"a technique particularly well suited to investigate the physical mechanisms underlying signal distortion. As a consequence, an adequate analysis of the sampled waveform can be valuable to the RF designer in determining the reason for a distortion and in implementing adequate countermeasures.

Throughout this article, an example using an L-band PPA illustrates the DSO waveform-sampling technique.

(1)

a_in(t) is the time-dependent signal amplitude, w_c = 2pf_c is the angular frequency, and f_c is the carrier frequency.

In the general case, the signal at the power amplifier output is:

(2)

In Equation 2, both a_out(t) and f(t) depend on the present and all the preceding values of a_in(t) (dependence on previous values indicates a "memory effect" of the system).

A physical constraint on the form of s_out(t) is that a_out(t) = 0 if t<t_g, t_g>0, which is the group delay. In other words, the amplifier output is zero before the input signal is applied and propagated through the amplifier. Another condition is that f(t) = w _c t_g + t w ₁(t), where w ₁(t) is an instantaneous frequency variation.

The mechanisms causing distortion may be very complex and a general treatment is very difficult. However, two main phenomena can occur:

• Linear Distortion
  In this case S_out(w) = F(w) S_in(w), where F(w) is a transfer function. This distortion occurs mainly across the signal frequency band, but also spectral lines out of band may occur if the transfer function exhibits poles at values of w very close to the real axis. The effects of linear distortion are shown in Figure 1, and result in the occurrence of resonance frequencies (w_spur) and time-domain waveform distortion.
  
  
• Non-Linear Distortion
  In the particular case of a memory-less system with polynomial transfer function of degree M, the output signal is . This implies that the output spectrum is:
  
  (3)
  
  The coefficient a₁ is equivalent to the small signal gain; implies the occurrence of the second harmonic; implies the occurrence of the third harmonic together with third order inter-modulation distortion; and so on. The effect of non-linear distortion is summarized in Figure 2. This distortion results mainly in spectrum enlargement close to the carrier and occurrence of harmonics in the frequency domain, together with signal clipping with sharper rise and fall times in the time domain.

Figure 1: Input and output signal for linear distortion

Figure 2: Input and output signal for non-linear distortion

In practice, what frequently happens is that the amplifier outputs spurious frequencies due to linear distortion and inter-modulation products between the input spectrum. You can model these phenomena as the effect of a linear distortion passing through a memory-less non-linear distortion. The effect is shown in Figure 3, where the time-domain output signal exhibits linear distortion and signal clipping.

However, the simplified models this article has outlined have limited validity. Another effect is phase distortion due to the amplifier's nonlinear behavior. This distortion results in amplitude modulation (AM) to phase modulation (PM) conversion, and a spurious phase modulation is visible at the amplifier output. This effect is particularly visible on the rising and falling edges of the waveform.

Another effect is power-supply-line oscillation during the rise and fall of the waveform. This effect is particularly prevalent in Class C amplifiers since the amplifier current increases abruptly when the RF input signal is present.

Up to this point, we note the following:

• Non-linear amplitude distortion results in spectral broadening, inter-modulation, and the presence of output harmonics.
• Linear amplitude distortion may result in some in-band distortion and is the cause of spurious spectral lines. Linear phase distortion results in, essentially, in-band effects.
• Non-linear phase distortion causes a variation of the instantaneous carrier frequency, hence contributing to peak broadening. This type of distortion occurs evenly during rise and fall times of the input waveform.
• Power-supply-line oscillation occurs during the waveform's rise and fall times and may be different during the rise and fall times.

Figure 3: Input and output signal for complex distortion

The main drawback of DSOs is the low sensitivity normally available, due to the range of the instruments' input ADCs.

The main parameters to be considered when evaluating DSOs for measuring RF signals are:

• The oscilloscope sampling rate f_s
• Number of bits N_b of the ADC.

In order to correctly sample the input signal one should have, according to the sampling theorem, the relation f_s > 2f_MAX must hold, assuming that the signal spectrum is confined below f_MAX.

If a characterization of some harmonics of the input signal is needed, the required sampling rate grows rapidly.

If the input signal contains high-frequency components which cannot be properly sampled by the DSO, you should connect a low-pass filter (LPF) with cut-off at f_s/2 to the oscilloscope input to avoid aliasing (under-sampling high-frequency components that may generate in-band noise).

The ADC number of bits N_b affects the DSO instantaneous dynamic range. As a rule of thumb, the theoretical dynamic range of a DSO is given as:

Normally the dynamic range is further limited by other factors. A tradeoff is possible between the system's dynamic range and the signal bandwidth.

Indeed, if the sampling rate is 2^2K times f_MAX, then by numerical filtering the number of bits of the signal can be increased by a factor K (which means that the dynamic range is increased by K 6.02 dB). This function is often implemented in the DSO firmware.

The DSO used to perform the tests is a LeCroy LC684DM with 8 GS/s maximum sampling rate, 1.5 GHz analog input bandwidth, and 8-bit input ADC (leading to a theoretical 48 dB spurious-free dynamic range).

The analyzed pulse was essentially a rectangular RF pulse with rise time of about 200 ns, fall time of about 300 ns, carrier frequency f_c of 1025 MHz, and width of about 8 ms. It was the output signal of a RF amplifier with 150 W peak output power; this amplifier was the driver stage of a 1 kW pulse amplifier.

Figure 4 shows a schematic of a typical bipolar class C common base stage of a power amplifier. M_in and M_out are, respectively, the input and output matching networks, L_in and L_out are inductances (normally suitable microstrip lines) necessary to guarantee the correct transistor polarization.

Figure 4: Schematic of a bipolar Class C amplifier stage

The output signal was sampled at 4 GS/s in order to capture the whole pulse width with a reasonably small output file. With this sampling rate, a maximum signal frequency of 2 GHz would be correctly sampled, exceeding the DSO input passband; therefore, in this case no LPF was required.

The test results, which will be analyzed throughout this article, are shown in Figures 5 and 6. These are output signals sampled at the driver output. The first signal was measured at the beginning of the amplifier optimization, and includes a spurious resonance at 700 MHz; in the second signal, this resonance was suppressed. The difference between the two signals is negligible.

Figure 5: Driver output signal at 1025 MHz before spurious elimination

Figure 6: Driver output signal at 1025 MHz after spurious elimination

The most important and well-known transform is the Fourier transform (FT), in its computationally efficient form, the fast Fourier transform (FFT). FTs are important because they present a spectrum of the signal that would be obtained by an ideal SA.

More specifically, if s_k = s(k T_s) (k = 0,..,N_s-1) is the sampled signal and w_k = 2p k f_s/N_s, then:

(4)

Equation 4 shows the discrete FT (DFT) of the signal s(t).

To compute the DFT of the signal according to Equation 4, you would need N_s² multiplications, in principle. If N_s is a power of 2, the FFT algorithm can be applied and only N_s log₂N_s multiplications are necessary.

Two important aspects need to be pointed out. The first is that Equation 4 should be applied to periodic signals, with period N_s T_s (in other words, s(t) = s(t + N_s T_s)). If this condition is not met, you get in-signal aliasing, which means that the spectrum of the sampled signal is not a true sampling of the spectrum of the original signal. In the case of an isolated pulse, this condition is easily met provided that the pulse is well centered in the time window; in other word, there is some integer k_s for which when k<k_s or k>N_s-k_s. Furthermore, when dealing with measured data some aliasing is present in any case because of the quantization noise of the signal close to the zero value.

The residual aliasing may be avoided by a proper windowing, by defining a signal where if k_s<k<N_s-k_s, w_k=0 if k<k_t or k>N_s-k_t, with k_ts.

The choice of the window w_k must be such that it minimally influences the signal spectrum. A good review of classical windows used for FT is reported in.

The FFT technique has been applied to the measured signals shown in Figures 5 and 6. The signals were truncated to 2¹⁶ samples without any window, due to limited pulse length, and then a FFT was performed. Figure 7 shows the spectrum of the measured PA driver output signal before optimization; a strong spurious resonance is present at 700 MHz with level -25 dBc. Referring to the Distortion in Power Amplifiers discussion, due to the nonlinear behavior of the class C amplifier an inter-modulation product is visible at 1350 MHz with level -35 dBc. The main problem with this distortion was that the final stages of the power amplifier were particularly sensitive to the resonance, which even resulted in a reduction of output power.

Figure 7: Spectrum of the driver output signal at 1025 MHz before spurious elimination

Using the technique described in the introduction, we found the reasons for the distortion mechanism and were able to reduce the spurious signal.

Figure 8 shows the spectrum after optimization; the spurious level at 700 MHz is now about -50 dBc, while the inter-modulation at 1350 MHz is reduced below -76 dBc.

Figure 8: Spectrum of the driver output signal at 1025 MHz after spurious elimination

Note that the spectra obtained by a simple FFT are similar to what can be obtained by SA tests: as such, the information is global and is not related to the details of the time-domain waveform. Hence no indication about the distortion mechanism is provided by the measurement and the only possible analysis is purely experimental.

Localized Fourier Transform
Detailed knowledge of the signal spectral properties along different portions of the waveform, such as rise or fall edges, can be valuable in understanding the physical mechanism of distortion in pulse power amplifiers.

This can be accomplished quite easily by applying a localized FT (LFT) to the signal. You can define the LFT as follows:

• Define a time internal of N_w samples over which the localized spectrum is computed
• Choose a proper window w_k (k = 0, .., N_w-1) to smooth the data
• Choose a time step of N_st T_s for the spectral analysis.

Defining t_l = l t_s N_st, the result of a localized LFT is the following:

(5)

S_LFT may be defined as a spectrogram (time-dependent spectrum).

N_w determines the detail to which the spectral analysis is performed, in other words, the interval over which the FFT is performed. A large value of N_w implies a low resolution in the time domain and high resolution in the frequency domain, and vice versa. N_st determines the granularity in the time domain of the spectral estimate.

Figure 9: Spectrogram of the driver output signal at 1025 MHz before spurious elimination

The figure was obtained based on a 2¹⁶ points signal, N_w = 2⁹, and N_st = 394. The spectrum granularity is 7.8125 MHz, the time window is 128 ns wide, and the window is a squared raised cosine with expression:

(6)

It is evident from Figure 9 that the resonance at 700 MHz is not a transient close to the transistor rise and fall edges. Due to this conclusion, resonant features of the supply lines can be excluded along with S-parameter phase variations due to the varying input power.

The attention was then focused on the input and output matching networks of the transistor. Figure 10 shows that a resistor placed in parallel with the input inductance Lin was sufficient to suppress the resonance. The inter-modulation at 1350 MHz, due to the amplifier non-linearity, was also reduced.

Figure 10: Spectrogram of the driver output signal at 1025 MHz after the spurious elimination

The spectrogram analysis shows also that the major part of other spurious responses occur solely at the pulse's falling edge, a memory effect. These responses are probably due to the transient response of the power supply lines when the amplifier's current goes from a peak level to zero.

The spectrograms show also a slight increase of the spurious levels along the pulse around about 750-800 MHz. This effect is probably due to a temperature rise in the transistor die and is a further memory effect in the system.

Wavelet Analysis
As shown in the preceding section, the use of transforms with local features can be quite useful for the analysis of transient signals. A relatively recent technique is represented by the wavelet transform. The following discussion only covers basic wavelet theory. Interested readers can consult the references for a deeper introduction to the subject (for a practical introduction and a wider list of references, see ).

The wavelet transform (WT) technique and synthesis is well suited for a variety of problems in which standard Fourier analysis is inadequate or impractical. Applications include image processing, signal compression, noise reduction, and digital transmission techniques.

Wavelet theory is a functional analysis technique in which a signal is projected on a basis of functions with compact or almost compact support (in other words, functions that are essentially localized). In the case of Fourier analysis, the basis function upon which the signal is projected is not localized (e^jwt); hence, wavelet techniques are particularly well suited to analyze the transient features of signals. Moreover, signals with abrupt variations and/or discontinuities in the derivatives exhibit a very broad Fourier spectrum. You can construct wavelets, which exhibit discontinuities in the derivatives, and describe in a more efficient way these kinds of signals.

Wavelet analysis is based on a wavelet function denoted as y(t), satisfying some regularity conditions. Different wavelets have been found in the literature. The wavelet we will use in the analysis is the complex Morlet wavelet y_M(t):

(7)

Looking at the wavelet shapes, it is evident from Equation 7 that the Morlet wavelet is very similar to a windowed Fourier basis function.

Given a wavelet y(t), the corresponding wavelet transform of the signal s(t) is:

(8)

The parameter a, which must be strictly positive, is the scale of the signal, while the parameter b is the signal's position. a is somehow related with the frequency of the signal (although only Fourier transforms provide rigorous frequency-content information) and defines, at the same time, the width of the region over which the signal analysis is performed. b identifies the position at which the analysis is performed. A discrete version of Equation 8 may also be formulated. The frequency can be associated to the parameter a by the following formula:

(9)

In Equation 9, the frequency f_y is defined as the frequency of the wavelet, in other words the frequency at which the wavelet FT reaches a maximum. For the Morlet wavelet, .

We are now ready to interpret the results of wavelet transforms of the signals of Figures 11 and 12.

Figure 11: Morlet wavelet transform of the driver output signal before spurious elimination

Figure 12: Morlet wavelet transform of the driver output signal after spurious elimination

As shown in Figures 11 and 12, the differences in the Morlet wavelet transforms before and after spurious elimination are negligible; in particular, the resonance at 700 MHz (scale of about 2.86) is not very evident.

The advantage of this technique is that spectral properties can be estimated locally on the waveforms, providing considerable insight into the physical mechanism at the basis of signal distortions. This is a considerable advantage over standard SA measurements and reduces the need for long experimental optimization time (normally performed by trial and error).

Some of the possible causes of signal distortion, including linear and nonlinear distortion and supply line oscillation have been described in this article.

The article also defined and discussed two different transform techniques, LFT and WT; in the WT case, we used the Morlet and order 2 Daubechies wavelets.

An L-band 150 W PPA working at 1025 MHz was analyzed. The distortion generation mechanism of a spurious signal at 700 MHz was found, and the information available through numerical signal processing suggested a strategy for suppressing the resonance (working on the input/output matching networks).

LFT proves to be more effective than WT in analyzing the RF signals and provides an effective insight into the spurious generation mechanism.