The Microwave Transition Analyzer:
`A New Instrument Architecture for
`Component and Signal Analysis
`
`The microwave transition analyzer brings time-domain analysis to RF and
`microwave component engineers. A very wide-bandwidth, dual-channel
`front end, a precisely uniform sampling interval, and powerful digital
`signal processing provide unprecedented measurement flexibility,
`including the ability to measure magnitude and phase transitions as fast
`as 25 picoseconds.
`
`by David J. Ballo and John A. Wendler
`
`As signal.processing capabilities advance, modern micro(cid:173)
`wave and radio frequency (RF) systems are becoming more
`and more sophisticated. Pulsed-RF signals, once used only
`for radar applications, are increasingly being used in com(cid:173)
`munication systems as well. These signals routinely have
`complex modulation within the pulse, especially frequency
`and phase variations (see Fig. 1). Operating frequencies an~
`bandwidths continue to increase, placing additional demands
`on the components of the systems.
`
`Engineers responsible for the design and testing of such
`components and systems often need to measure them under
`the same dynamic conditions as those in which they are
`used. For example, it may be necessary to measure a de(cid:173)
`vice's response to phase coding or linear frequency chirp
`inside an RF pulse.
`
`Measurements with traditional frequency-domain instrumen(cid:173)
`tation are often insufficient to characterize and understand
`fully the operation of components in dynamic signal environ(cid:173)
`ments. Before the microwave transition analyzer introduced
`in this article, no single instrument could handle the diverse
`range of measurements required for dynamic tes~g at micro(cid:173)
`wave frequencies. In addition to the new measurements it
`makes, this analyzer can perform many of the measurements
`previously requiring the use of network, spectrum, dynamic
`signal, and modulation analyzers, as well as oscilloscopes,
`counters, and power meters.
`
`Importance of the 1ime Domain
`A key benefit of the microwave transition analyzer is that it
`brings time-domain analysis to RF and microwave compo(cid:173)
`nent engineers. In addition to its use in pulsed-RF testing,
`the time domain is essential to characterizing and under(cid:173)
`standing nonlinear devices because one can clearly and intu(cid:173)
`itively see the relationship between the input and output
`signals. As an example, both signals in Fig. 2 would appear
`identical if displayed on a spectrum analyzer. Even if the
`phase 9f the hannonics were known, the differences be(cid:173)
`tween the signals would not be immediately obvious. When
`viewed in the time domain, however, it is clear that signal I
`
`is clipped (the output of a limiter, say), while signal 2 has
`crossover distortion (what might be seen at the output of a
`Class-B amplifier,' for example). Without the time domain,
`engineers have had to guess at the underlying causes of ob(cid:173)
`served frequency-domain behavior. The ability to view micro(cid:173)
`wave signals in the time domain has also proved to be ex(cid:173)
`tremely valuable to designers that are using· CAE microwave
`design simulators, such as HP's MDS. Now simulations
`based on circuit models can be easily compared to actual
`measurements in both the time domain and the frequency
`domain.
`
`Historically, most measurements on high-frequency non(cid:173)
`linear devices have been performed in the frequency domain.
`Often, this has been because of inadequacies in time-domain
`instrumentation. When frequency-domain information is of
`prime concern, spectrum analyzers are superb in their abil(cid:173)
`ity to dispfay hannonic, modulation, and spurious signals ·
`with a large dynamic range. However, without the phase of
`the frequency components, the time-domain signal cannot
`be reconstructed. Network analyzers are excellent for per(cid:173)
`forming linear, small-signal, frequency-domain testing, but
`they are limited in their ability to characterize nonlinear
`devices. The addition of hannonic and offset sweep capabil(cid:173)
`ity in network analyzers has helped, but the time-domain
`perspective is still missing.
`
`For envelope analysis of pulsed-RF signals, spectrum ana(cid:173)
`lyzers off er some limited time-domain capability. Recently,
`network analyzers have been adapted for pulsed-RF time(cid:173)
`domain testing as well. Because of the architecture of these
`instruments, the intermediate frequency (IF) bandwidth
`imposes an upper limit on the measurement bap.dwidth. The
`result is rurumum measurable edge tiines of greater than
`100 ns. The microwave transition analyzer's architecture
`does not have this restriction. Edge speed is limited only by
`the RF bandwidth. Consequently, magnitude and phase mea(cid:173)
`surements on pulses with rise times as fast as 25 ps are pos(cid:173)
`sible. Fig. 3 shows an example of a microwave transition.
`analyzer measurement.
`
`48 October 1992 Hewlett-Packard Journal
`
`RPX-Farmwald Ex. 1043, p 1
`
`

`
`may be specified indirectly as magnitude and phase flatne~
`versus frequency. By transforming the input and output
`pulses to the frequency domain with the built-in fast Fourier
`transform (FFT) and computing their ratio, the transfer
`function is obtained. From this, familiar results of magni(cid:173)
`tude and group delay versus frequency can be displayed.
`Network analyzers are only able to measure the phase and
`group delay of frequency translation components relative to
`a reference or "golden" device.
`
`It is much easier to measure nonlinear devices at low fre(cid:173)
`quencies than at RF and microwave frequencies. At low fre(cid:173)
`quencies, general-purpose oscilloscopes readily show time(cid:173)
`domain behavior, and dynamic signal analyzers provide both
`magnitude and phase in the frequency domain. The only tool
`available for high-speed time-domain measurements before
`the microwave transition analyzer has been the high(cid:173)
`frequency sampling oscilloscope. Initially, sampling _oscillo(cid:173)
`scopes were purely analog instruments, and in the past few
`years have incorporated digital storage and other enhance(cid:173)
`ments such as markers. However, these instruments have
`not enjoyed widespread acceptance from RF and microwave
`engineers for several reasons. The first is the difficulties
`involved in achieving reliable external triggering at high fre(cid:173)
`quencies and small signal levels. High-speed sampling oscil(cid:173)
`loscopes have enjoyed the most success for use with digital
`signals where voltage levels are generally large and triggers
`are not difficult to obtain. Secondly, traditional sampling
`oscilloscopes are not very sensitive, especially compared to
`network and spectrum analyzers. The microwave transition
`analyzer incorporates selectable filters to decrease noise
`without limiting the signal bandwidth·. The resulting increase
`in sensitivity combined with internal triggering across the
`full RF bandwidth greatly aids in the measurement of small
`signals.
`
`Excellent sensitivity also helps overcome a limitation of
`sampling oscilloscopes for high-input-impedance measure(cid:173)
`ments (»50 ohms). Until recently, it has been very difficult
`to obtain probes with low enough parasitic capacitance to
`be useful at microwave frequencies. Companies now offer
`solutions for high-frequency passive probing, but signal at(cid:173)
`tenuation is significant. This signal attenuation is not a prob(cid:173)
`lem for the microwave transition analyzer because of its
`high sensitivity. This has been especially beneficial for prob(cid:173)
`ing monolithic microwave integrated circuits (MMICs) at the
`wafer level.
`
`Finally, the operation of high-speed oscilloscopes has not
`been optimized for RF and microwave applications, where
`terminology is often different from that used in digital design.
`The user interface of the microwave transition analyzer uses
`units and formats that are familiar to RF and microwave
`engineers. For example, log-magnitude displays of pulsed(cid:173)
`RF signals are readily available, and marker annotation can
`be in dBm or dBc as well as volts.
`
`Microwave Transition Analyzer
`The HP 71500A microwave transition analyzer (Fig. 4) is a
`two-channel, sampler-based instrument with an RF band(cid:173)
`width covering from de to 40 GHz. The instrument is called
`a transition analyzer because of its ability to measure very
`fast magnitude and phase transitions under pulsed-RF con(cid:173)
`ditions. However, this name does not encompass the full
`
`October 1992 Hewlett-Packard Journal
`
`49
`
`Tr2=Ch1
`30 mV/div
`0 V ref
`
`525 ns
`Tr3=Ch1
`90 deg/div
`-27 deg ref
`
`125 nsldiv
`
`Tr1=Ch1
`30 mVldiv
`0 V ref
`(a)
`
`Tr1=Ch1
`Y0 mV/div
`0 U ref
`
`Tr2=Ch1
`Y0 mVldiv
`0 U ref
`
`23 . 92 us
`Tr3=FM(Ch1)
`650 kHz/div
`3 GHz ref
`
`5 us/div
`
`(b)
`
`Fig. 1. Examples of complex modulation. (a) A phase coded RF
`pulse. The waveform and magnitude demodulation are shown in
`the upper half. The carrier's phase with respect to a CW reference
`is shown in the lower half. (b) Frequency modulation inside an RF
`pulse. The waveform and magnitude demodulation are shown at
`the top, the frequency demodulation is shown in the middle, and
`the magnitude spectrum of the pulse is shown at the bottom.
`
`The ability to meas"Qre narrow pulses in the time domain can
`also be used to determine the impulse response (and there(cid:173)
`fore magnitude, relative phase, and group delay) of frequency
`translation components such as mixers and receivers. By
`stimulating these devices with a narrow pulse of RF energy,
`time-domain distortion can be directly observed. Often, it is
`the time-domain distortion that is of interest, even though it
`
`RPX-Farmwald Ex. 1043, p 2
`
`

`
`Signal 1
`
`Signal2
`
`Sum=
`Fundamental
`+
`3rd Harmonic
`
`Time
`
`Time
`
`Fig. 2. The importance of phase information in nonlinear design. Signals 1 and 2 wduld appear identical on a spectrum analyzer display.
`
`range of its measurement capability. The microwave transi(cid:173)
`tion. analyzer can best be described as a cross between a
`high-frequency sampling oscilloscope, a dynamic signal
`analyzer, and a network analyzer.
`
`Like a digital sampling oscilloscope, the microwave transi(cid:173)
`tion analyzer acquires a waveform by repetitively sampling
`the input, that is, one or more cycles of the periodic input
`signal occur between consecutive sample points. However,
`unlike an oscilloscope, the sampling instant is not determined
`by an external high-frequency trigger circuit. Instead, the
`sampling frequency is synthesized, based on the frequency
`of the input signal and the desired time scale. A synth~sized
`sampling rate is an attribute that the microwave transition
`analyzer shares with dynamic signal analyzers. Also in com(cid:173)
`mon is an abundance of digital signal processing capability.
`The FFI', for example, allows simultaneous viewing of the
`time waveform and its frequency spectrum. However, unlike
`a dynamic signal analyzer, the microwave transition analyzer
`
`M1(*)
`M2(t)
`
`31.8529 ns
`31.7032 ns
`
`326 mU
`92.Y9 mU
`
`rise= 1Y9.63
`
`s
`EXT
`
`Tr1=Ch1
`125 mV/div
`0 U ref
`
`33.01 ns
`
`Tr3=Ch1
`125 mV/div
`0 V ref
`
`500 psldiv
`
`Fig. 3. The microwave transition analyzer can meac;ure edge
`speeds on modulated waveforms as fast as 25 ps.
`
`50 October 1902 Hf'wlett-Packard Journal
`
`does not have an anti-aliasing filter at its input. The sampling
`frequency is automatically adjusted to achieve a controlled
`aliasing of the ftequency components of the input signal.
`Finally, like a network analyzer, the microwave transition
`analyzer can be configured to control a synthesized signal
`source for the characterization of devices over frequency or
`power· ranges. It can also receive a frequency that is offset
`from or a harmonic of the source frequency, and it can pro(cid:173)
`vide frequency and power sweeps at a particular point within
`a pulse of RF, on pulses as narrow as 1 ns.
`
`Architecture
`Fig. 5 shows a simplified block diagram of the microwave
`transition analyzer. The analyzer has two identical signal
`processil).g channels. Each channel samples and digitizes
`signals over an input bandwidth of de to 40 GHz. The chan(cid:173)
`nels are sampled simultaneously (within 10 ps), permitting
`accurate ratioed amplitude and phase measurements. A
`single synthesized low-noise oscillator drives a step recov(cid:173)
`ery diode, the output of which is split into two pulse trains
`that drive the microwave samplers. The microwave sam(cid:173)
`plers and the analog-to-digital converters (ADCs) are run at
`the same frequency. The maximum sampling frequency is
`20 MSa/s (20 million samples per second).
`
`Tf1-e signal at the output of the samplers is processed by a
`10-MHz-bandwidth low-pass IF strip. The IF (intermediate
`frequency) circuitry includes a programmable shaping am(cid:173)
`plifier to compensate for the sampler's IF response roll-off,
`60 dB of step gain to optimize the signal level into the ADC,
`and variable low-pass filtering to remove noise and sampler
`feedthrough. The trigger circuitry is at the end of the analog
`path. Triggering on IF signals (instead of RF input signals)
`allows the microwave transition analyzer to be internally
`triggered to 40 GHz. Enhancements to the hardware trigger
`are available through the use of digital signal processing.
`
`Periodic Sampling
`The mathematical analysis of periodic functions was begun
`in the early 19th century by Jean-Baptiste-Joseph Fourier.
`Fourier's theorem introduced the techniques for decomp.osing
`
`RPX-Farmwald Ex. 1043, p 3
`
`

`
`Filtering, a convolution oppration in the time domain, is
`morP Pasily intPrpreterl as frequency-domain multiplication.
`AlternatiVPly, a mixer multiplies two signals in the time do(cid:173)
`main, but the result is exprpssed as frequency-domain
`translation, a convolution operation. Why convolution is the
`analytical mechanism for realizing frequency translation is
`explainPd in "Frequency Translation as Convolution" on
`page 61.
`
`An ideal sampler driven by a periodic sampling pulse can be
`considPred a switch that briefly connects the input port to
`the output port at a periodic rate. When the switch is closed
`the output signal is the input signal multiplied by unity.
`'
`When the switch is open, the output signal is grounded, that
`is, the input signal is multiplied by zero. Thus, the signal at
`the sampler's output is formed as the product of the input
`signal and the periodic pulse defining the switch state as a
`function of time. As in the mixer example on page 61, time(cid:173)
`domain multiplication results in frequency-domain convolu(cid:173)
`tion. The frequency spectrum of the sampler's input signal is
`convolved with the spectrum of the periodic pulse to produce
`the spectrum of the sampler's output (IF) signal.
`
`The frequency spectrum of a periodic pulse is composed of
`delta functions at the fundamental repetition frequency and
`all multiples (harmonics) of this frequency. This infinite set
`of impulses in the frequency domain, sometimes called a fre(cid:173)
`quency comb, inherits a magnitude and phase profile accord(cid:173)
`ing to the time-domain pulse shape. A narrow, rectangular
`pulse imparts a sin(f)/f roll-off characteristic to tl#equency
`comb. The first null of the response occurs at a frequency
`equal to the reciprocal of the pulse width and the 3-dB atten(cid:173)
`uation frequency occurs at 0.443 times this value. Funda(cid:173)
`mental to wide-bandwidth sampling is achieving a very nar(cid:173)
`row sampling pulse or aperture. The sampling aperture in
`the microwave transition analyzer is less than 20 ps.
`
`The sampling front end of the microwave transition analyzer
`converts the high-frequency input signal to a low-frequency
`IF signal suitable for digitization and subsequent numerical
`processing. Depending on the application, three different in(cid:173)
`terpretations of the sampling process are possible: frequency
`translation, frequency compression, and a combination of
`translation and compression.
`
`Fig. 4. Named for its ability to measure vny fast magnitmle and
`phase transitions under pulsed-RF ronditions, the HP 71500A
`micro\\'ave transition analyzer (top instrument) is part high(cid:173)
`frequen<'y sampling oscillos<'ope, part dynamic signal analyzer,
`and part network analyzer. The HP 7lfiOOA consists of the HP
`78020A microwave transition analyzer module and the HP 70004A
`mainframe. The bottom instmrnent shown herP is the HP 8:3640A
`synthesized sweeper.
`
`any periodic waveform into a sum of harmonically related
`sinusoids. The Fourier series is a frequency-domain repre(cid:173)
`sentation of the original time function and is used to sim(cid:173)
`plify the description and provide insight into thP function's
`underlying characteristics.
`·
`
`The sampler in the microwave transition analyzer is driven
`by a constant-frequency sampling signal. BPcause the sam(cid:173)
`pler drive is periodic, Fourier analysis can be usPd to under(cid:173)
`stand the sampler's operation. Often, periodic signals or
`systems responding to periodic signals are described and
`analyzed in the frequency domain. Transformations be(cid:173)
`tween the time and frequency domains replace convolution
`operations in one domain with multiplication in thP other.
`
`Sample Rate
`Synthesizer
`
`Microwave
`Samplers
`
`· Switchable
`Low-Pass
`Filters
`
`IF
`Step Gain
`Amplifiers
`
`Sample-
`and-Hold
`Circuits
`
`Fig. 5. Simplified block diagram of the HP 71 GOOA 1nicrowm·e trm s1tion analyzer.
`
`0 C"tober 1092 Hcwlett-Pal'kard .Journal
`·t
`
`51
`
`RPX-Farmwald Ex. 1043, p 4
`
`

`
`_...__.__(] -
`
`---+! t-(cid:173)
`BW
`
`· f
`
`____..tl__.___--+-1 _
`
`(a)
`
`(b)
`
`0
`
`0
`
`Sampler Output
`
`... tJ /1 fJ (lf\ ;iltJ r1 tJ r1 tJ r1 .. : f
`
`0
`
`(c)
`
`ADC Input
`
`(d)
`
`0
`
`• f
`
`Fig. 6. Sampling used to translate a frequency band. (a) Input
`spectrum. (b) Sampling comb. (c) The sampler output spectrum
`is the convolution of the waveforms in (a) and (b). (d) Filtered
`output.
`
`Frequency Translation
`N onrepetitive or single-shot events can be captured by sam(cid:173)
`pling the input signal at a rate greater than twice the input
`bandwidth. This is known as the Nyquist criterion. However,
`maintaining this criterion does not imply that the sampling
`rate must be g:r:eater than twice the input signal's highest
`frequency. If the RF bandwidth of the sampler is adequate,
`narrowband information on a high-frequency carrier can be
`captured by low-frequency sampling, as long as a sampling
`rate of approximately twice the modulation bandwidth is
`maintained. Sampling the high-frequency signal translates
`the signal to baseband.
`
`Samplers are often used in place of mixers for frequency
`conversion-for example, in the front ends of many general(cid:173)
`purpose network analyzers. In the case of translation only, a
`given narrow frequency band is converted to baseband by
`an appropriate choice of sampling frequency. Fig. 6 diagrams
`the conversion process. The spectrum of the input signal is
`shown in Fig. 6a and the frequency comb of the sampling
`pulse is shown in Fig. 6b. The sampling frequency, that is,
`the spacing between the teeth of the frequency comb, is
`chosen such that the input spectrum lies appropriately posi(cid:173)
`tioned between adjacent comb teeth. The convolution result
`is shown in Fig. 6c.
`
`Two important considerations in the choice of sampling fre(cid:173)
`quency can be seen from these diagrams. First, the input
`signal bandwidth (Fig. 6a) must be less than one half the
`sample rate. Second, the sample rate must be chosen so the
`input spectrum is entirely contained in a frequency range
`bounded by the nearest sampling harmonic and the frequency
`halfway to the ne~ higher or lower harmonic. If these crite(cid:173)
`ria are not met, the sampler will translate or alias more than
`one component of the input spectrum to the same output
`frequency, causing uncorrectable errors. The maximum sam(cid:173)
`pling rate of the microwave transition analyzer is 20 MSa/s.
`The rate is continuously adjustable (in 1-mHz steps) down
`
`52 October 1992 Hewlett-Packard Journal
`
`to a minimum rate of 1 Sais and can be phase-locked to an
`external 10-MHz reference.
`
`The signal at the output of the sampler is amplified and low(cid:173)
`pass filtered before analog-to-digital conversion. This filter(cid:173)
`ing virtually restores the original input spectrum, but it is now
`centered in the much lower IF range (Fig. 6d). Because the
`filter transition from passband to stopband is not immedi(cid:173)
`ate, some undesired high-frequency energy may be included
`in the signal presented to the ADC. In this case, the band(cid:173)
`width of the signal at the ADC exceeds half the sample rate.
`Aliasing occurs as the highest-frequency components are
`folded back on top of the original translated spectrum by
`the sample-and-hold circuit of the ADC. However, unlike
`the aliasing problems mentioned in the previous paragraph,
`the effects of this aliasing can be predicted and corrected
`in software because the aliased components represent
`redundant information.
`
`In summary, using a sampler with a bandwidth many times
`the sample rate allows the capture of single-shot events in
`the modulation on a high-frequency carrier (see Fig. 7). The
`analysis bandwidth is limited to half the sample rate,.
`
`Frequency Compression
`A second, fundamentally different perspective of the sam(cid:173)
`pling process is useful in the measurement of periodic high(cid:173)
`frequency signals. Traditionally, these measurements have
`required trigger-based repetitive sampling techniques. In the
`microwave transition analyzer, precision RF trigger circuitry
`is not used. Periodic sampling' alone is used to convert a
`strictly periodic input with harmonic components spread
`across a very wide bandwidth to a low-frequency signal with
`harmonic components spread over the narrow IF range. This
`is accomplished by choosing a sampling frequency that con(cid:173)
`verts each component of the input signal into the IF such that
`the harmonic ordering, magnitude, and phase relationships
`of the original input are preserved in the IF signal. The sam(cid:173)
`pling process effectively compresses the Wide-bandwidth
`input signal into a low-frequency signal at the IF.
`
`Tr1=Ch1
`15 mUldiv
`0 U ref
`
`Tr3=Ch1
`15 mU/div
`0 U ref
`
`0 5.
`
`50 us/div
`
`Fig. 7. Turn-on characteristic of a synthesizer's output amplifier.
`This single-shot measurement was internally triggered on the
`signal that originated from the enabling of the RF output of the
`synthesizer. The carrier frequency is 5 GHz.
`
`RPX-Farmwald Ex. 1043, p 5
`
`

`
`Compression Factor. The signal at the IF is a replica of the
`input signal, but at a much lower fundamental frequency.
`When this signal is digitized and displayed, the waveshape
`matches that of the input. The time range indicated on the
`display is calculated by dividing the real time (sample period
`times trace points) by the compression factor (input frequency
`x l/x, where x corresponds to the fundamental frequency at
`the IF-see Fig. 8):
`
`S
`.
`rme pan=
`T
`
`(Sample Period) (Number of Trace Points)
`(Input Frequency)/x
`·
`
`When the microwave transition analyzer is used for repeti(cid:173)
`tive sampling, the input signal must be strictly periodic, and
`the period must be lmown to high accuracy. If the frequency
`that the analyzer assumes for the input signal is near but not
`exactly equal to the frequency of the signal being measured,
`the IF will be shifted in frequency by an amount equal to the
`difference. The resulting measurement will show an erro(cid:173)
`neous time· scale, the error equal in percentage to the fre(cid:173)
`quency error of the IF signal. Thus, a small RF inaccuracy
`can result in a very large time-scale error. The ability to fre(cid:173)
`quency-lock the microwave transition analyzer's sampling
`rate to the signal being measured (by sharing a common
`reference frequency with the stimulus), removes this source
`of error. The resulting time scale accuracy is specified to 1
`ps-better than any current trigger-based oscilloscope.
`
`Triggering. To keep the display "triggered," low-frequency
`trigger circuitry is connected to the IF signal and used to
`initiate the storage of a data record relative to a rising or
`falling edge. Data samples in the buffer before the trigger
`occurrence are displayed as negative time (pretrigger view).
`Through the combination of periodic sampling and a low(cid:173)
`frequency trigger circuit, the microwave transition analyzer
`is able to trigger internally on periodic signals across the full
`40-GHz input bandwidth and offer negative-time capability
`without delay lines.
`
`IF Filtering for Noise Reduction. As mentioned earlier, the sig(cid:173)
`nal at the output of the sampler is low-pass filtered before
`analog-to-digital conversion. In Fig. 8d the bandwidth cho(cid:173)
`sen for this filtering is less than half the sampling rate. Any
`IF components above the band edge of the filter correspond
`to input harmonic components beyond the specified input
`bandwidth of 40 GHz and may be filtered off. Filtering the IF
`signal to a bandwidth narrower than half the sampling rate
`means that not all of the noise across the 40-GHz input band(cid:173)
`width is converted to noise on the IF signal. Thus, noise is
`removed from the displayed signal without affecting the
`
`Fig. 9. Periodic sampling in the time domain.
`
`October 1992 Hewlett-Packard Journal
`
`53
`
`2nf5
`
`3nf5
`
`Sampler Output
`
`nf5
`
`(c)
`
`... j ll
`
`(d)
`
`Fig. 8. Sampling used to frequency compress a periodic input
`signal. (a) Input signal spectrum. (b) Sampling comb. (c) Expand(cid:173)
`ed frequency scale showing the relationship between the input and
`the sampling signal components. ( d) The sampler output signal is
`the convolution of the waveforms in (a) and (b).
`
`Fig. 8 illustrates the concept in the frequency domain. The
`input spectrum and frequency comb of the sampling pulse
`(including the RF response roll-off) are shown in Figs. Ba
`and 8b. Fig. 8c provides a close-up view of the relative posi(cid:173)
`tioning of the comb lines with respect to the input signal.
`The sampling rate is chosen such that a given harmonic (the
`nth) is positioned x Hz below the input's fundamental fre(cid:173)
`quency. Then, the (2n)th sampling harmonic will be posi(cid:173)
`tioned 2x below the input's second harmonic, the (3n)th
`sampling harmonic will be 3x below the input's third har(cid:173)
`monic, and so on. Fig. 8d shows ~e result of the convolu(cid:173)
`tion. Each harmonic of the input is converted to a corre(cid:173)
`sponding harmonic of the low-frequency signal at the IF.
`
`The sampler does not have infinite bandwidth, and the
`sin(t)/f roll-off of the sampling comb attenuates the IF re(cid:173)
`sponses that correspond to input components at the higher
`frequencies. Small amounts of attenuation may be compen(cid:173)
`sated for in software, however, after the signal is digitized.
`The combination of a very narrow sampling aperture and
`software corrections allow the microwave transition analyzer
`to specify a flat response to 40 GHz.
`
`Viewing this process in the time domain, the sample interval
`is set to be a multiple of the input period plus a small amount
`equal to the effective time between points (Fig. 9). Since the
`sampling interval is not an exact multiple of the input period,
`the sampling instant moves with respect to the input at a
`prescribed increment as the samples are acquired. The effec(cid:173)
`tive time between points is determined by how close the sam(cid:173)
`pling frequency is to a subharmonic of the input frequency.
`
`RPX-Farmwald Ex. 1043, p 6
`
`

`
`For a given pulsed-RF input signal with an arbitrary carrier
`frequency, the values of x and y cannot be independently
`controlled by adjustments in the sampling rate alone. If the
`sampling rate is set to achieve the desired compression fac(cid:173)
`tor (PRF/x), there is no remaining degree of freedom for
`adjusting the spectral offset (y) to avoid overlap. One solu(cid:173)
`tion is to provide a mechanism for automatically adjusting
`the carrier frequency under control of the microwave transi(cid:173)
`tion analyzer. In many cases, the microwave transition ana(cid:173)
`lyzer is used in a stimulus-response configuration similar to
`that of a network analyzer. If the carrier source is under
`control, the carrier frequency control can be used to adjust
`the spectral offset independent of the sampling rate.
`
`Often, however, the microwave transition analyzer does not
`control the carrier source, or it is desired that the carrier
`frequency not be modified. In these cases, the simultaneous
`requirements on the sample rate are achieved by slight mod(cid:173)
`ifications to either the requested time span or the number of
`trace points. The parameter to be modified is determined by
`the user. Remembering that the displayed time span is equal
`
`a)
`
`b)
`
`Sampling
`Comb -
`
`c)
`
`d)
`
`e)
`
`0
`
`40GHZ
`
`1---- PRF
`i - - PRF-x = fs
`2
`-t 14- -t 14- - t 14-
`y- x
`y+x
`Y
`
`Sampler Output
`
`f--x
`
`.. f
`
`Fig. 11. Sampling used to analyze periodic wideband modulation.
`(a) Input signal spectrum. (b) Sampling comb. (c) Expanded fre(cid:173)
`quency scale showing the relationship between input and sampling
`signal components. (d) The sampler output signal is the convolution
`of the waveforms in (a) and (b). (e) The IF spectrum on an expand(cid:173)
`ed frequency scale, showing the spacing of the signal components.
`
`Tr1=Ch1
`2.03 mU/div
`-486 uU ref
`
`Tr2=Mem1
`2.03 mU!div
`-648 uU ref
`
`200 psldiv
`
`Fig. 10. Filtering the IF signal removes noise but retains the
`underlying wave shape.
`
`waveshape. The result is cleaner displays and improved
`sensitivity (by more than 20 dB) compared to conventional
`trigger-based sampling oscilloscopes (see Fig. 10).
`
`Translation and Compression
`The perspectives of translation and compression are com(cid:173)
`bined to analyze the third use of the microwave transition
`analyzer's sampling front end. The application is measuring
`signals composed of broadband, periodic modulation on a
`high-frequency carrier. Examples include pulsed-RF signals
`with narrow pulse widths or fast edge speeds. Proceeding as
`before, the spectrum of the input signal and the frequency
`comb of the sampling pulse are shown in Figs. 1 la and 1 lb,
`respectively. Fig. Ile has an expanded frequency scale show(cid:173)
`ing the relative positioning of the input's spectral lines and
`those of the sampling pulse. Two variables, x and y, are intro(cid:173)
`duced in this figure, and are related to the concepts of com(cid:173)
`pression and translation, respectively. The sampling frequen(cid:173)
`cy is chosen such that the signal's pulse repetition frequency
`(PRF) is slightly greater (x Hz) than a multiple of the sam(cid:173)
`pling rate. In other words, the t1me between sampling
`instants is slightly greater than an integral number of input
`pulse repetition periods. As can be seen from the diagram,
`the frequency separation between a given signal component
`and the nearest sampling harmonic increments by x Hz
`when considering the next-higher signal component. Conse(cid:173)
`quently, the spacing of the corresponding components in the
`sampler's output signal is x Hz, resulting in a compression
`factor of PRF/x.
`
`In Fig. 1 k, the spectral center of the input signal is shown
`to be offset by y Hz from the nearest sampling harmonic.
`Therefore, the signal at the output of the sampler is centered
`at y Hz, as shown in Fig. 1 ld. If the offset y is allowed to
`decrease by a change in the input carrier frequency, the sam(cid:173)
`pler output components are translated toward one another
`as indicated by the dashed arrows. If y becomes too small,
`the components will partially overlap and distort the spec(cid:173)
`trum. Likewise, if y is increased, the sampler's output com(cid:173)
`ponents move opposite to the dirpctions indicated and will
`overlap as y approaches half the sampling rate.
`
`54 Ortober lfl92 Hewlett -Packard Journal
`
`RPX-Farmwald Ex. 1043, p 7
`
`

`
`se c / d i v:
`
`EXT
`
`/'
`
`!'
`
`.
`
`Tr1 = Ch1
`25 rn\J/div
`r e f
`(a) 121
`\J
`
`37.Lf ns
`
`5121 121 ps / div
`
`::35.4600 ns
`
`:37 . !3600 ns
`
`40 .4 600 ns
`
`.
`
`;
`
`:•
`
`~. · .:.._ __ _ __: - -- -~-- -
`
`---::----t-~ ... -':~..__:rr-.;;......:-~-'.'T_,,_~~-~+-:--"--~-t-_,_~ ........ ~t-~--''----t-c-~ ·~:----
`,__ ....... ......,,==;;;;:j::::~~~~·-~~.-~,~~ .. ~· ~.~~/·~"
`
`1 .. : .... -
`
`....
`
`.
`
`·\,:.
`
`--·~~----!-----'----~'----·-· --,--
`
`-
`
`----+:-- - -· .. ~· -~~---~--:___
`
`(b) ~~,;,e~ase
`
`·2s . oo mv o i t s /d iv
`SOC p s /d