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. |
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The subject of this article is a technique

for characterizing the transient behavior of pulse power

amplifiers (PPAs). The technique is based on signal processing

applied to time-domain waveforms sampled directly at RF using a

fast digital storage oscilloscope (DSO). Spectral properties of

the output signal can be estimated locally on the waveforms,

and can indicate the correct strategy to fix the distortion.

The paper defines and discusses the local Fourier transform

(LFT) and wavelet transform (WT), and analyzes an L-band 150 W

PPA working at 1025 MHz. Signal processing information was used

to find and fix the distortion generation mechanism of a

spurious 700 MHz signal. LFT proves to be more effective than

WT and provides an effective insight into the spurious

generation mechanism.

The RF designer developing power amplifiers often faces the

occurrence of spurious amplifier responses”the amplifier output

signal is not an amplified replica of the input signal but

exhibits substantial differences (distortion). These

differences may be associated with a number of causes,

overlapping in several cases. Among the possible causes of

distortion are: amplifier non-linearity, oscillations due to

the coupling of the RF transistor with input/output matching

networks, and oscillations in the power supply lines.

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.

Consider the case of an amplitude modulated (AM) signal

s_{in} (t) at the input of a power amplifier, where:

(1)

a_{in} (t) is the time-dependent signal amplitude,

w _{c} = 2p f_{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 a

_{1}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**. This distortion results mainly in spectrum enlargement

2

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

As explained in the introduction, there are DSOs available

with sampling rates of several GS/s. This kind of instrument is

well suited for sampling signals in the RF/MW range, so that

both amplitude and phase can be directly measured on the RF

waveform. This analysis is not possible with PPWMs and SAs.

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

**Transforms: Principles and Practical
Applications**

Time-domain data measured by the DSO were sequences of 8-bit

samples with spacing T

_{s}= 1/f

_{s}= 0.25 ns.

Transforms applied to the sampled signal highlight features

that are necessary to understand the physical phenomena.

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

_{s}

^{2}

multiplications, in principle. If N

_{s}is a power of

2, the FFT algorithm can be applied and only N

_{s}

log

_{2}N

_{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_{t} <>_{s} .

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

2^{16} 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 2^{16} points

signal, N_{w} = 2^{9} , 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^{jw t} ); 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.

This article has illustrated a technique for characterizing

the transient behavior of PPAs. The technique is based on

numerical processing applied to time-domain waveforms sampled

at RF using a fast DSO.

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.

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