1
`
`Asad Abidi
`
`I am currently a Distinguished Chancellor’s Professor of Electrical Engineering at the Uni-
`versity of California, Los Angeles (“UCLA”). I have been a professor at UCLA since 1985. Prior
`to UCLA, I was at Bell Laboratories as a member of their technical staff in the Advanced LSI De-
`velopment Laboratory. I was also a visiting faculty researcher at Hewlett Packard Laboratories
`in 1989.
`
`My research interests at present are focused on circuit methods to overcome fundamental
`limitations on the performance of the radio portions of single-chip wireless receivers and
`transmitters. I also maintain active research on circuits and architectures for analog-to-digital
`converters, which includes sample-and-hold circuits.
`
`In 2007, I was elected to the National Academy of Engineering for my contributions
`to the development of single-chip radios that have enabled the handheld wireless devices
`of today. This is the highest peer recognition that an engineer receives in the US. A copy
`of my curriculum vitae, which describes in further detail my qualifications, responsibilities,
`employment history, honors, awards, professional associations, invited presentations, and
`publications is Ex. 1005.
`
`I have reviewed United States Patent Nos. 6,061,551 (“the ‘551 patent”) to Sorrells et
`al., Exhibits 1001; 6,266,518 (“the ‘518 patent”) to Sorrells et al., Exhibits 1002; and 6,370,371
`(“the ‘571 patent”) to Sorrells et al., Exhibits 1003. I have also reviewed the patents and printed
`publications cited in the endnotes of this declaration.
`
`

`

`I have been informed by counsel for the petitioner that the level of ordinary skill in the art
`is evidenced by the references. I have further been informed that the parties in the Qualcomm
`litigation appear to have generally agreed that one of ordinary skill in the art (sometimes
`referred to herein as “one skilled in the art”) would have “a Bachelor’s of Science degree in
`Electrical Engineering and four years of experience in the wireless communications industry”.
`This is consistent with the level of skill evidenced by the references cited herein.
`
`2
`
`Shannon’s sampling theorem is a cornerstone of modern electrical engineering. It
`underlies all digital communications and digital signal processing. In his groundbreaking
`paper of 1949 [1], Shannon states the theorem thus:
`
`THEOREM 1: If a function f (t) contains no frequencies higher than W cps, it is
`completely determined by giving its ordinates at a series of points spaced 1/2W
`seconds apart.
`
`but adds that
`
`“A similar result is true if the band W does not start at zero frequency but at some
`higher value, and [this result] can be proved by a linear translation (corresponding
`physically to single-sideband modulation) of the zero-frequency case.”
`
`This lays the foundation for what has come to be known as sub-sampling or undersampling.
`However, the concept would be developed in expositions of the sampling theorem by Shannon’s
`co-workers, such as by Black in 1953. He points out [2, p. 56] that
`
`THEOREM X: When sampling a band of frequencies displaced from zero, the
`minimum sampling rate fr , for a band of width B and highest in-band frequency
`f2, is 2f2/m where m is the largest integer not exceeding f2/B.
`
`Black goes on to say that
`
`“... the minimum sampling rate is not in general twice the highest frequency in the
`band but is
`
`fr = 2Bµ1+
`
`where k = f2/B ° m.”
`This is illustrated in Fig. 3.1.
`
`(3.1)
`
`∂
`
`k m
`
`

`

`3
`
`B >
`
`2B
`
`0
`
`fS MfS
`
`(M+1)fS
`
`Figure 3.1: Black’s criterion on selection of subsampling frequency, fS, prevents aliasing distortion from
`spectral overlap after downconversion to baseband.
`
`Why is this practically important? Black describes the process of receiving and trans-
`mitting an information signal that occupies a bandwidth B around some high frequency
`carrier.
`
`“Often a signal band does not include zero frequency. By standard modulation
`techniques a band extending from f1 to f1+ B can be translated to the range 0 to
`B (and following signal processing) restored to the original range by an inverse
`translation. These techniques include modulators (mixers), carrier generators (RF
`oscillators), and band separating filters, as well as amplifiers. For simplicity a direct
`sampling process which avoids shifting the band would be preferred.”
`
`Detailed discussions of sampling applied to bandpass signals are also to be found in
`[3]. This concept became well-entrenched in the community of circuit designers working in
`communications applications. For example, a 1988 article [4] on uses for a high-speed A/D
`converter notes that
`
`“A naive reading of Nyquist’s theorem might imply that ±10 kHz information side-
`bands around a 200-MHz carrier can only be recovered by sampling at >400.020
`MHz. However, synchronous undersampling makes it possible to recover these
`sidebands as baseband signals. This undersampling produces aliases and makes it
`possible, with appropriate ratio of sampling rate to carrier, to select any spectral re-
`gion, such as the band-limited sidebands of the carrier. For example, if a 200-MHz
`carrier is sampled at 200 MHz (one-half the Nyquist rate), the first alias will be at
`0 (i.e., dc); any sidebands associated with the carrier are returned to baseband ...
`In contrast to the simplistic view that undersampling creates useless aliases, the
`sideband signal is aliased down to baseband, and this alias is meaningful.”
`
`See also [5, p. 52], [6, p. 239]
`
`In RF applications, f2 is the carrier frequency, say 900 MHz, while B in, say, GSM is
`as low as 200 kHz; that is, B ø f2. In this limit, the minimum sampling frequency fr ! 2B.
`Thus, in this example, the information spectrum at RF can be downconverted by sampling at
`
`

`

`4
`
`a frequency slightly greater than 400 kHz. This works because sampling, ideally using a train
`
`of impulses separated by the period TS and denoted by the symbol X≥ tTS¥, creates a train
`
`
`of impulses in its frequency spectrum X≥ ffS¥ that denote the presence of tones at frequency
`fS = 1/TS and its integer multiples. When two of these tones separated in frequency by 2B
`bracket the frequency interval [f2°B, f2] containing the information of interest, then following
`convolution, the band of interest is translated approximately to [0,B]—that is, to baseband. In
`simple terms, Black’s Theorem X above may be understood as follows: it gives the numerical
`condition on the relevant frequencies to guarantee that the train of sampling tones brackets the
`frequency interval of interest, without causing one of the tones to lie in that interval (Fig. 3.1).
`This principle is well understood by those who design sampling circuits (e.g. [7, Sec. 3.2]).
`
`The sampling theorem, as originally stated, requires a waveform to be multiplied by a
`periodic train of impulses, or delta functions. The delta function is of zero width but infinite
`height, and encloses an area of one. This multiplication leads to a train of impulses of unequal
`area, where now the area is the instantaneous value of the waveform at the uniformly spaced
`time instants where the delta function lie. Although each impulse is infinitely tall, its area
`captures the instantaneous value of the waveform. In engineering and physics, this area is
`sometimes informally called the “energy” in the impulse. Of course it is not energy in the
`proper sense; its dimensions are Volts or Amperes, not Joules. Mathematically, the formal way
`to capture a single point f (tk) on a continuous waveform f (t) is to multiply it by the impulse
`±(t ° tK ) located at t = tK .
`But of course impulses of infinite height do not exist in real-life. This was very clear to
`the Bell Labs engineers who first put to use sampling of speech waveforms in the telephone
`system to multiplex by time-division many voice channels over long-distance cables, and
`later to digitize these waveforms in modern digital telephony. An obvious approximation is
`to substitute a zero width impulse with a pulse of practically achievable height and non-zero
`width ø in time. After multiplication, this leads to a train of pulses of fixed width but variable
`height, called Pulse Amplitude Modulation. This gives a faithful sampling of the waveform—
`except for a filtering effect arising from the finite width of the pulse. This so-called aperture
`effect is understood as follows. If the waveform being sampled changes appreciably over the
`width of one pulse, then clearly the modulated area of the pulse captures some sort of average
`of that varying snippet of the waveform. This can be modelled by passing the original waveform
`through an appropriate filter, and then capturing a sample.
`
`This phenomenon arising from sampling with finite pulses is well known. For example,
`the first published report on the sampling oscilloscope spells out this limitation to bandwidth
`clearly in the text [8, p. 57] and illustrates it in an accompanying figure:
`
`

`

`5
`
`(a)
`
`(b)
`
`Figure 4.1: Plots of the sinc(x) function: (a) Linear scales, (b) Log scales.
`
`“When the oscillation time of the voltage to be examined becomes of the same
`order as the duration ø of the pulses, then the conversion conductance of the
`mixing process and thus the sensitivity of the instrument rapidly diminishes. The
`frequency limit fmax may therefore be said to be of the order of 1/ø.”
`
`Black gives a fundamental theorem on the aperture effect in sampling [2, p. 55]. Also Bennett [9,
`p. 245], who was involved in the early uses of sampling in the telephone system, says that
`
`“The falloff in gain with signal frequency is called aperture effect, after the similar
`effect observed in television from scanning with an aperture of finite size.”
`
`If F (f ) is the frequency spectrum (Fourier Transform) of the waveform f (t), then the pulses
`capture with perfect fidelity samples of a filtered waveform F§(f ), where
`F§(f )= H(f )F (f )= sinc(f ø)F (f )
`For reference, the sinc function is defined as
`sinc(x) , sin(ºx)
`, where sinc(0)= 1
`(ºx)
`and is plotted in Fig. 4.1. H(f ) is a linear filter which passes frequencies up to f = 1/(2ø) with
`an attenuation of less than 4 dB, but beyond it rolls off dramatically until an input frequency of
`f = 1/ø is completely nulled. This acts as a sampler pre-filter, filtering the original waveform
`before it is sampled perfectly by a (fictitious) train of impulses. This sinc pre-filter will appear
`in much of the following discussion.
`
`(4.1)
`
`(4.2)
`
`The concept of Sample-and-Hold (S/H) appears shortly after the idea of sampling
`(see, for example, [9, Fig. 6-1-3]). Instead of modulating the heights of voltage pulses, the
`sampled voltages can be held constant between adjacent samples. Thus a piecewise constant
`
`1
`
`0.8
`
`jsinc(x)j
`
`0.4
`
`0.6
`
`0.2
`
`0
`0
`
`0.5
`
`1
`x
`
`1.5
`
`2
`
`100
`x
`
`101
`
`0
`-5
`-10
`-15
`-20
`-25
`-30
`10-1
`
`jsinc(x)j(dB)
`
`

`

`6
`
`vout[fIF]
`
`+ -
`
`RL
`
`vLO[fLO]
`RON
`
`C
`
`RS+RON=R
`RS
`
`+ -
`
`vout[fIF] vin[fRF]
`
`vLO[fLO]
`
`+ -
`
`RL
`
`C
`
`RS
`
`+ -
`
`vin[fRF]
`
`fLO = fs= 1/Ts
`
`fRF = Mfs+fIF
`
`Figure 4.2: (Left) Schematic of Sample-and-Hold circuit. (Right) Equivalent circuit, with terms defined.
`
`approximation is obtained for the original waveform. This comprises what is formally known
`as a zero-order S/H [10], since the waveform is being approximated by a set of polynomials
`in time raised to the power zero (constant). It is particularly straightforward to realize this in
`a practical circuit, since a switch turned on by the sampling pulse can connect a capacitor
`C, ideally with no leakage, to a voltage source with continuous waveform Vin(t) and source
`resistance R. When the switch turns off, the capacitor holds its voltage steady until the next
`sample, realizing to a very good extent a zero-order S/H (Fig. 4.2).
`In subsequent analysis, we will refer to the sampling duty cycle as the ratio of the switch on
`time ø to the time between successive samples TS,
`D , ø/TS
`
`(4.3)
`
`Obviously the sampling rate fS = 1/TS.
`While this is a remarkably simple circuit—signal source, switch, pulse generator, and
`capacitor—as a survey of the literature shows, it has not been analyzed comprehensively. Partial
`analyses date back to the 1960’s [11, 12] and continue to appear in the literature through the
`years [13, Sec. 2.4] [14, Fig. 3]. We have not found analysis published in the technical literature
`by the patentees or others at ParkerVision.
`A full analysis reveals the detailed action of this linear but time-varying (that is, periodically
`switched) circuit. The circuit carries three frequencies.
`1. There is the input frequency f , which consists, in the RF application, of a modulation
`band B around the carrier frequency fRF .
`
`2. To keep the analysis simple, an offset frequency fI F from the carrier models a single tone
`of interest in the modulation sidebands. In Black’s notation, typically |fI F|∑ ½B.
`3. There is the sampling frequency, which for our purposes is a subharmonic of the carrier
`frequency, i.e. fS = fRF /M. That is, if the carrier were being sampled in the absence of any
`modulation, every sample would lie on the same point on the carrier sinewave; in other
`words, it would be subsampled to zero frequency, or DC. Then the modulation around
`the carrier is subsampled to frequencies close to, but not equal to, zero1. Therefore, after
`1A complex subsampling arrangement can discriminate between positive and negative frequencies.
`
`

`

`7
`
`Vout
`
`+
`
`Figure 4.3: Signal Flow Graph for sampler based on switch and capacitor. Mixing with impulse train is
`not idealization, but models the periodic isolation of the capacitor from the source when
`switch turns off.
`
`subsampling the only frequency of interest that remains at baseband is fI F (as noted, for
`instance by [4] and quoted in Section 3 above).
`
`It should be noted that, although the output waveform is analog, it is strictly speaking a
`discrete-time waveform that updates every TS seconds. We will treat it as an analog waveform
`that is defined continuously on the time axis. These transformations, and the accompanying
`shifts in frequency, are captured by a signal flow graph (Fig. 4.3). As in a radio receiver where the
`signal of interest undergoes frequency shifts, it is not sensible to write down a single transfer
`function that describes the entire chain of signal processing. Rather, it is customary to write a
`transfer function HRF that the channel of interest experiences through the RF section, then
`multiply this by the mixer’s conversion gain, and then further multiply this by the transfer
`function HI F of the signal of interest through the IF section. We will extract these functions
`from the signal flow graph.
`
`(4.4)
`
`1
`
`ø2
`
`sinc(fRF ø)e°j !RF
`
`Vout
`Vin
`
`(fRF , fI F )=
`
`ø
`RC
`
`,
`
`(4.5)
`
`z
`}|
`{
`
`e j !RF°Ts° ø2¢
`fI F
`1+ j
`fc
`|
`{z
`}
`HI F (fI F )
`
`1 D
`
`1°°1° øRC¢e°j !I F Ts
`
`When fRF = M fs + fI F , e j !RF Ts = e j !I F Ts º 1+ j !I F Ts because !I F Ts < 1 This simplifies (4.4)
`as follows:
`
`e j !RF°Ts° ø2¢
`1+ j !I F Ts °°1° øRC¢
`
`
`e j !RF°Ts° ø2¢
`= sinc(fRF ø)
`|
`{z
`}
`1+ j 2ºRC fI F
`HRF (fRF )
`
`Vout
`Vin
`
`(fRF , fI F ) º
`
`ø
`RC
`
`sinc(fRF ø)
`
`= sinc(fRF ø)
`
`Ts
`ø
`
`
`
`where fc = 12ºRC , D = ø
`
`Ts
`
`.
`
`Delay
`
`

`

`8
`
`It is seen that the RF input is pre-filtered by an aperture-like sinc function, then mixed
`with an ideal impulse train in frequency. One of the impulses will lie at the carrier frequency,
`and mixing with it will convert the carrier to DC. Thus, after mixing, the modulation will appear
`at a frequency fI F at baseband. Just as the sinc pre-filter applied to the input frequency f which
`lies in the RF band, the post-sampling filter, or post-filter, applied to the baseband frequencies
`fI F . If this is to transmit a 1 MHz modulation bandwidth (B) superimposed on a 1 GHz carrier
`frequency, the pre-filter must pass the carrier frequency (and its modulation sidebands, which
`will occupy a very small fractional bandwidth) with acceptably small attenuation, and the
`post-filter must transmit the bandwidth B (strictly speaking ½B) also with little attenuation.
`
`D ø 1
`The most long-lasting and widespread use of undersampling is in the sampling oscillo-
`scope. This was first demonstrated at Philips in 1950 [8], and it made it possible for 1960’s-era
`oscilloscopes to display repetitive signals at microwave frequencies [15]. The idea here is to
`sample points on wideband waveforms that repeat periodically on a slow time scale. The sam-
`pling oscilloscope is able to display waveforms with a bandwidth that reaches the microwave
`region, say 10 GHz, but at a sample update rate that is very slow by comparison, say 200 kHz.
`This is fast enough for the intended purpose of displaying points of the waveform on a screen.
`However it corresponds to a very low duty cycle D. This has repercussions of worsened Noise
`Figure, but for the purposes of waveform display on a sampling oscilloscope, noise is usually of
`secondary importance. Indeed users of sampling oscilloscopes know at the time of purchase
`that the instrument trades off poor noise for wide bandwidth.
`
`The sampling oscilloscope is designed to acquire wideband signals with the highest
`possible bandwidth. While the operation of the instrument is best understood, and always
`illustrated, as assembling samples in the time domain [8, 15], other aspects are more readily
`understood in the frequency domain2. In terms of the analysis above, a good sampling os-
`cilloscope requires that the aperture-limited pre-filtering HRF and the bandwidth of the IF
`sections HI F should both be comparable and high: either can become the bottleneck to the
`instrument’s final bandwidth. The narrowest aperture is obtained using the fastest possible
`sampling switch driven by the narrowest pulse (ø as small as possible), while the IF section
`must use the lowest possible hold capacitor C. This balance between the RF and IF bandwidth
`is implicit in an early paper describing the design of a sampler for oscilloscopes [16].
`
`2It is standard teaching and practice in Electrical Engineering to view signals and systems in the time and/or
`frequency domains, using one or both as is appropriate and convenient.
`
`

`

`9
`
`D < 1
`Samplers have been employed for the specific purpose of characterizing narrowband
`modulated waveforms, as would be typical in RF and microwave communications. [11] de-
`scribes a sampler-based phase detector for such an instrument. When the instrument is phase
`locked to a sinewave, the detector produces a DC output. When it locks to a modulated carrier,
`the phase detector passes the downconverted modulation to its output—in this instance, the
`downconversion is by subsampling. Unlike the wideband sampling oscilloscope, this is a
`narrowband use: the modulation bandwidth is usually much lower than the carrier frequency.
`Therefore in this sampler the bandwidth of HRF must be wide enough to pass the carrier
`frequency, while the bandwidth of HI F need only be wide enough to pass the modulation.
`These bandwidths are typically different by several orders of magnitude.
`
`In the case of greatly unequal bandwidths in the RF and IF sections, a new concept
`comes into play: sampling efficiency. To understand this, suppose that for the sake of small
`aperture ø, the switch is turned on for a very short time. But the hold capacitor is large since it
`is designed for a small IF bandwidth, and the source resistance R cannot possibly charge it over
`one aperture to the instantaneous source voltage Vin. Then a sampling efficiency (¥) can be
`defined, for example, as the ratio of the voltage acquired by an initially discharged capacitor in
`one aperture when driven by a DC Vin. It is easy enough to show that under these conditions,
`
`¥=
`It is possible to envisage cases of very high carrier frequency but very narrow modulation
`bandwidth that ¥ could be very small. The question is: will the sampler show a large signal loss
`after it has subsampled RF to IF? This is answered in the next section.
`
`(5.1)
`
`ø
`RC
`
`Downconversion by subsampling works best and is easiest to explain in narrowband
`cases when the carrier frequency downconverts to, or close to, DC, and the modulation appears
`as a complex spectrum at baseband, or a real spectrum at some low IF. This means that the
`sample rate is synchronous with the carrier frequency, that is,
`
`fS = fRF /M, where M is an integer > 1
`We will consider the output in steady-state, that is after a large number of sampling events
`have taken place.
`
`(5.2)
`
`If the input is a pure sinewave of frequency fRF every sample will lie on the same point
`on the sinewave; we will call the voltage at this point V §in. The sampling switch is bi-directional,
`
`

`

`10
`
`w/o RL
`w/ RL
`
`t t t
`
`vn+2
`
`vin
`
`vLO
`
`vout
`
`(cid:87)
`
`Tn
`
`Ts
`
`vn
`
`vn+1
`
`Tn+1
`
`Tn+2
`
`Figure 5.1: Illustrating synchronous sampling of a sinewave, and buildup to equilibrium. Waveforms
`show the effect of adding a leakage resistor RL in parallel with the hold capacitor.
`
`which means that until the held voltage on the capacitor Vout becomes equal to V §in, on every
`closure of the switch current will flow from the source into the capacitor. This process will
`reach equilibrium when Vout = V §in. In other words, when we wait long enough the capacitor
`voltage will acquire the sampled carrier voltage with a gain of 1 (Fig. 5.1).
`
`It follows from this argument that the smaller the ø, the more sampling events it takes to
`reach this final value. Suppose the amplitude of the input sinewave is suddenly doubled. Then,
`after waiting long enough, the capacitor voltage will arrive at the new steady-state value of
`2£V §in. This qualitative discussion tells us that the voltage gain for the simple switch-followed-
`by-capacitor S/H is at the carrier frequency, and it follows that if the amplitude is modulated at
`very low fI F which changes slowly enough that the capacitor can track it, the gain will remain
`close to one. However, for higher fI F the capacitor will be unable to keep pace, and the gain
`will fall off. Qualitatively this defines the bandwidth of the overall transfer function; more
`specifically, this is the bandwidth specified by the function HI F in the expression (4.5). This
`bandwidth applies to either amplitude or phase/frequency modulation.
`This is readily seen in (4.5), by noting that |HRF| ! 1 when fRF ø ø 1, and |HI F| ! 1 when
`fI F = 0. The °3 dB IF bandwidth BI F , defined by |HI F (BI F )| = 1/p2, limits the passband seen
`by the modulation at baseband. It is given by
`
`ø
`BI F = D fC =
`TS £ 2ºRC
`The sampling duty cycle D affects the bandwidth otherwise defined by R and C—it is as if the
`duty cycling of the switch has raised the value of the source resistance from R to R/D.
`
`(5.3)
`
`We may conclude that synchronously subsampling a sinewave with a very small aper-
`ture leads, in steady-state, to a sampler voltage gain of one. Of course this assumes that the
`sampling aperture is aligned with the peak of the sinewave. In reality any downconversion to
`zero IF requires sampling into two channels actuated by quadrature phases of the sampling
`
`

`

`clock, and the envelope of the two downconverted outputs, when viewed as a complex number,
`always shows a gain of one for any location of the sampling apertures relative to the peak.
`Fig. 5.2 shows excellent agreement between the expressions given here and periodic steady-
`state simulations of the actual circuit using the simulator SPECTRE-RF.
`
`11
`
`The discussion of any circuit intended for use in an RF receiver is quite incomplete
`without consideration of its Noise Factor. For any circuit with two ports, including circuits with
`internal frequency translation, Noise Factor (F ) is defined as [17, Sec. 14.10], [18]
`F , S/N(input port)
`S/N(output port)
`
`(5.4)
`
`Noise Figure is the appropriate representation of F in decibels. Since random waveforms are
`specified in the frequency domain in terms of spectral density, Noise Figure is calculated as
`10log(F ). F is defined at a single frequency, and is therefore a function of fI F . F is also defined
`with respect to a reference source resistance, conventionally 50≠.
`
`First let us examine qualitatively how noise is sampled and downconverted. We will
`refer to the signal flow through the sampler as shown in Fig. 4.3. White noise occupies all
`frequencies with a uniform spectral density. We are interested in the noise voltage originating
`in the source resistance R, which is given by the spectral density
`Svn = 4kT R V2/Hz
`The standard method to analyze white noise propagating through linear circuits [17] is to divide
`up the frequency axis into 1 Hz wide intervals, and assume that each interval is populated by a
`sinewave of a known mean-square voltage, as given by (5.5), but uniformly distributed random
`phase. Then, as each frequency traverses the circuit the known signal transfer function scales
`it in amplitude, shifts in phase, and possibly translates it in frequency. Using superposition,
`the noise spectral density at the output is assembled and a noise transfer function may be
`developed.
`
`(5.5)
`
`The sampler processes noise as described by the following series of steps (Fig. 5.3):
`
`1. White noise originating in the resistor R can be modelled as a series of sinewaves in the
`voltage source VS. This enters the sampler through the pre-filter HRF (fRF ) whose noise
`bandwidth (as defined in [18, p. 33]) is (1/ø) Hz.
`
`2. The sinewaves comprising the noise are sampled by the impulse frequency comb X°f /fS¢.
`There are (1+ 1/D) discrete tones that will lie in the noise bandwidth. Each tone will
`downconvert the two noise sinewaves lying at a frequency offset ±fI F from it to the
`
`

`

`12
`
`1pF / 1MΩ
`
`550ps / 18pF / 2kΩ
`
`
`
`AnalysisAnalysis
`
`
`SimSim
`
`ulatioulatio
`
`
`
`nn
`
` 10
` 1
`IF frequency [MHz]
`
` 100
`
`(b)
`
` 0
`
`-5
`
`-10
`
`-15
`
`-20
`
`-25
`
`-30
`
`-35
`
`-40
` 0.1
`
`Gain [dB]
`
`S i m
`S i m
`
`ulatio n
`ulatio n
`
`10ps10ps
`
`1 0 p s
`1 0 p s
`
`4ps4ps
`
`
`
`AA
`
`
`
`nn
`
`aly sis
`aly sis
`
`4 p s
`4 p s
`
`
`
`11
`
`
`
`22
`
`
`
`pp
`
`
`
`pp
`
`
`
`ss
`
`
`
`ss
`
` 10
` 1
`IF frequency [MHz]
`
` 100
`
`(a)
`
` 0
`
`-5
`
`-10
`
`-15
`
`-20
`
`-25
`
`-30
`
`-35
`
`-40
`0.1
`
`Gain [dB]
`
`550ps / 18pF / No leak path
`
`
`
`AnalysisAnalysis
`
`
`SimSim
`
`ulatioulatio
`
`
`
`nn
`
` 10
` 1
`IF frequency [MHz]
`
` 100
`
`(c)
`
` 0
`
`-5
`
`-10
`
`-15
`
`-20
`
`-25
`
`-30
`
`-35
`
`-40
` 0.1
`
`Gain [dB]
`
`Figure 5.2: Comparing expressions for gain with simulated sampler circuit gain. RL, C, and ø swept
`over given values. In all cases, R = 50 ≠, carrier frequency fRF = 900 MHz, sample rate fS =
`100 MHz.
`
`

`

`13
`
`Figure 5.3: Wideband noise passes into the sampler through pre-filter is sampled by multiple frequen-
`cies, and accumulates at IF by spectral overlap.
`
`
`
`
`
`Duty=0.01Duty=0.01Duty=0.01
`
`
`
`
`
`Duty=0.05Duty=0.05Duty=0.05
`
`AA
`
`A
`
`n
`n
`n
`
`aly sis
`aly sis
`aly sis
`
`
`Duty=0.10Duty=0.10
`Duty=0.10
`
`Output Noise Density [nV/sqrtHz]
`
` 8
` 7
` 6
` 5
` 4
` 3
` 2
` 1
` 0
` 1e+06
`
`
`
`
`
`Duty=0.50Duty=0.50Duty=0.50
`
`SimulationSimulationSimulation
`
`
` 1e+07
`IF Frequency [Hz]
`
` 1e+08
`
`Figure 5.4: Comparing output noise voltage spectral density from (5.6) with PNOISE simulations, vary-
`ing sampling duty cycles D.
`
`baseband frequency fI F . Thus noise accumulates at fI F , and since each noise sinewave
`is uncorrelated with the others because of its random phase, they will add in a mean
`square sense.
`
`3. The accumulated noise at fI F will then be subject to the transfer function H(fI F ).
`
`This process results in an output spectral density of
`
`(5.6)
`
`∂
`
`1 D
`
`¢2µ1+
`
`1D
`
`4kT R
`
`1+°2ºRC fI F
`
`∂=
`
`1 D
`
`¥2µ1+
`
`4kT R
`
`1D
`
`fc
`
`1+≥ fI F
`
`Svout,n (fI F )=
`
`The results compare almost exactly with PNOISE simulations of the actual circuit on SPECTRE-
`RF (Fig. 5.4). With a reliable expression for output noise, the Noise Factor may now be calculated
`with respect to a source resistance of 50 ≠,
`
`SNRin
`
`SNRout = °1+ 1D¢
`
`sinc2(fRF ø)µ1+
`
`F =
`
`Ron
`
`
`
`
`
`Rs ∂= °1+ 1D¢sinc2(MD)µ1+
`
`Ron
`
`Rs ∂
`
`(5.7)
`
`f
`
`f
`
`0
`
`0
`
`-1/(2
`
`τ
`
`)
`
`1/(2
`
`τ
`
`)
`
`1/
`
`τ
`
`1/
`
`T
`
`s
`
`2/
`
`T
`
`s
`
`-1/
`
`T
`
`s
`
`-2/
`
`T
`
`s
`
`-1/
`
`τ
`
`...
`
`...
`
`Pre-filter
`
`Sampling
`
`

`

`14
`
`M=fRF/fs (No leak path)
`
`M=10M=10
`
`M=5M=5
`
`M=2M=2
`
`M=1M=1
`
`M=0M=0
`
` 1
`
` 2
` 0.01
`
` 0.1
`Duty Cycle D
`
`(b)
`
` 20
`
` 18
`
` 16
`
` 14
`
` 12
`
` 10
`
` 8
`
` 6
`
` 4
`
`Noise Figure [dB]
`
`fRF=100MHz / fs=100MHz / No leak path
` 20
`
`
`
`SimulationSimulation
`
`Analysis
`Analysis
`
`
`
`7.2dB7.2dB
`
` 10
`
`Noise Figure [dB]
`
`=0.4MHz=0.4MHz
`
`ff
`
`IF
`IF
`
`5
` 0.01
`
` 0.1
`Duty Cycle D
`
`(a)
`
`Figure 5.5: Noise Figure of sampling mixer: (a) Variation with duty cycle, theory matches simulation.
`(b) Worsens with subsampling factor M. The curve for M = 0 is given as a baseline for the
`others.
`
`It is seen that as an aperture ø ! 0, which means that D ! 0, will cause the Noise Factor to
`grow without bound. Circuit simulation using PNOISE verifies the dependence on M. Noise
`figure is plotted versus duty cycle for various M.
`
`We may conclude that, although any aperture with synchronous sampling will give a
`gain of one at zero IF, the smaller the aperture the worse the Noise Factor. Therefore, for use in
`a wireless receiver where low Noise Factor is at a premium, we should use the largest possible
`aperture. From the plot of the sinc function Fig. 4.1, an aperture equal to half the RF carrier’s
`period attenuates the signal through the prefilter HRF by 4 dB, and may be thought of as the
`cutoff frequency of a sinc filter3. As it happens, this aperture was used to illustrate Fig. 5.1.
`
`D ' 0.5
`It is not unreasonable to use an aperture that is half the period of the carrier wave. For
`example, [10, p. 27] says that
`
`“... carrier-modulated systems using phase detectors can be treated as sampled
`systems ... The phase detector operates on the basis that once (or twice) each
`cycle of the carrier a linear detector charges a load condensor [capacitor] to a
`
`3This choice of 4 dB is, of course, somewhat arbitrary, since it is close to the 3 dB loss that usually defines the
`cutoff of most filters. It is used for the sinc filter because at this loss the input frequency lies at half the null
`frequency.
`
`

`

`15
`
`peak voltage proportional to the amplitude of the latest carrier cycle. In this
`manner, a signal datum is produced once (or twice) per cycle of carrier and, in
`effect, a sampling process of the usual form results. The sampling frequency can be
`considered equal to the carrier frequency or twice the carrier frequency, depending
`on whether a half- or full-wave detector is used.”
`
`Although with this aperture the sampler’s pre-filtering HRF will attenuate the signal of interest
`by 4 dB, it is now possible to choose a range of D ª 0.5! 0.25 which may allow the Noise Factor
`can be lowered to the levels needed in a narrowband wireless receiver.
`
`We will list three instances from the literature of a sampling downconverter that use a
`half-cycle aperture in a narrowband wireless receiver:
`
`1. Estabrook [19] uses a diode mixer in a zero IF receiver. The diode, a two-terminal device,
`is driven by the sum of a weak RF input and a strong local oscillator (LO) sinewave tuned
`to the carrier frequency. The LO waveform will dominate the sum, and it will turn on
`the diode for (say) positive half cycles of the LO. This produces a synchronous switching
`action at fLO with a duty cycle of D = 0.5. A capacitor at the diode output stores samples
`of the modulation.
`
`2. Avitabile [20] uses a diode bridge switch as a downconversion mixer. Unlike a diode
`mixer, the diode bridge is a full-fledged three-terminal switch which can be turned on
`and off by an independent switching pulse. Avitabile uses an aperture of 40% of the
`period of the RF carrier to sample a modulated waveform on to a hold capacitor. When
`sampling a 1030 MHz RF input (fI F = 30 MHz), this sampler uses ø= 400 ps and TS = 4 ns,
`thereby obtaining a duty cycle D = 0.1.
`3. Weisskopf [6] uses a subharmonic sampler to downconvert an 18.5 GHz carrier signal to
`baseband. Weisskopf samples at the 10th subharmonic and uses a sample aperture of
`50% of the period of the RF carrier: ø= 27 ps and TS = 540 ps, thereby obtaining a duty
`cycle D = 0.05.
`These publications demonstrate that downconversion is indeed possible with wide apertures.
`
`So far we have assumed a hold capacitor C with no load resistance across it. What, then,
`is the impact of a load resistance RL that might leak away charge from C between samples?
`The analysis that led to (4.5) can be repeated by including RL, and it results in the following
`modified transfer function:
`
`(5.8)
`
`ø2
`
`1
`1+∏
`
`e°j !RF
`1+ j fI F
`
`1D
`
`fc
`
`Vout
`Vin
`
`(fRF , fI F )' sinc(fRF ø)£
`
`1
`1+ ∏
`
`

`

`16
`
`D
`
`where ∏ = D0
`Rs
`and D0 = 1° D, D0 TsRLC ø 1. Thus, the gain of the IF post-filter falls at all
`
`RL
`frequencies by the factor (1+ ∏) attributable to leakage, while the IF bandwidth rises by the
`same amount because, effectively, a smaller resistance is driving the hold capacitor. The drop
`in gain may be seen qualitatively in Fig. 5.1, where RL causes the held voltage to droop between
`samples thereby lowering the average value of the sampled IF output voltage.
`To the first order a small leakage (∏< 1