Download this article in PDF format. (508K) Noise Figure and Logarithmic Amplifiers [Ed.
Monolithic, fully calibrated
logarithmic amplifiers ( Log amps come in three
basic forms. However, in their capacity as - Those employing
multistage
amplification and progressive limiting
^{1}generate a close approximation to the logarithm in a piecewise fashion. Some of these parts also make available the output of the final limiting-amplifier stage, for extracting time-encoded information (PM or FM, baseband bit streams). These include the AD608, AD640/AD641 and the extensive AD8306, AD8307, AD8309, AD8310, AD8311, AD8312, AD8313, AD8314, AD8315, AD8316, AD8317, AD8318, and AD8319 family, as well as closely matched dual log amps, such as the AD8302 (which also measures*phase*), and the ADL5519, having an unprecedented 1 kHz to 10 GHz measurement range.^{†}
These*progressive-compression log amps*include a rectifier (detector) with each of five to 10 low-gain (8 dB to 12 dB) stages, whose outputs are summed to produce a filtered voltage that is a decibel-scaled measure of average power. Where the final hard-limited signal is also provided (as in the 100-dB range products AD8306/AD8309), the logarithmic measure is often viewed as ancillary, and referred to as the*received-signal-strength indicator*(RSSI). - Those using an
*exponential-gain amplifier*(X-AMP^{®}architecture)^{2}with a 60-dB typical gain span, followed by a*single*detector whose filtered output is compared to a reference level; the integrated error generates a voltage, which adjusts the amplifier gain to null the error (see text associated with Figure 6). That voltage is a representation of the decibel value of the applied signal, due to the accurate exponential (sometimes called “linear-in-dB”) gain function. By giving the detector a square-law response, it is the power-equivalent (rms) value of the applied signal that is measured.
This will be recognized as the general form of an*automatic-gain-control*(AGC) amplifier. Accordingly, we can call them*AGC-style log amps*. The AD8362, AD8363, and AD8364 are of this type, the latter two providing the simultaneous measurement and differencing of two input signals. In this type, there is usually no provision to access the amplified signal. An exception is the AD607 (actually a single-chip superheterodyne receiver), whose decibel-scaled RSSI output spans 100 dB, and whose signal outputs are the I/Q components of the demodulated IF. - Those based
on the uncannily reliable translinear properties of a bipolar junction transistor
(BJT)—the precise logarithmic relationship between its base-emitter voltage
(V
_{BE}) and its collector current (*I*) over a range of up to 10_{C}*decades*of current (200 dB!). An early exploitation of this property in conjunction with an op amp was due to Paterson.^{3 }Modern products, now known as*translinear log amps*, are similar, deviating only in the details of implementation. This separate class of log amps, used in fiber-optic communication systems to measure optical power and control the gain of optical-mode amplifiers, only measures essentially*static currents*, from as low as 1 picoamp up to several milliamps. Alternatively, using external input resistors, voltages with a large range of amplitudes can also be measured. Examples include the AD8304, AD8305, ADL5306, and ADL5310.
kT, thus its absolute
operating temperature, T (where k is Boltzmann’s constant). In
one case of general interest, the root source is an antenna, whose noise
results from the electromagnetic coupling into the free-space resistance from
which it receives its signal, and which has the basic value of 377 ohms.
The signal and noise are equally coupled into the system via a first impedance
transformation created by the design of the antenna, and thence conveyed by
a cable of the same impedance. They operate at their greatest power efficiency
in driving, say, a 300-ohm balanced (“twin” or
“ribbon”) feeder, or alternatively a 50-ohm (or
occasionally a As an aside, the minimum
loss in a coaxial cable occurs when its characteristic impedance is 71 ohms.
Above this, the resistance of the thinning As
a power source (really, a Z) + _{A}jIm(Z). Nevertheless,
it acts purely resistive over a generally narrow range of frequencies. Clearly,
the _{A}power it can deliver to an open-circuit—such as an ideal voltage-responding
element—is zero, since none of the available current is extracted from
the source. Similarly, the power into a short circuit—such as an ideal current-responding
element—is zero, since none of the voltage swing is used. The power-transfer
theorem shows that the maximum power that can be delivered to a load connected
to this source occurs when the resistive part of the load impedance is made
equal to R = Re(_{A}Z),_{A}
Log amps intended for
RF power measurement (often referred to simply as Figure 2 shows how this lower bound on the dynamic range can be expressed as power for various choices of impedance. Note that the response, illustrated for a typical scaling of 20 mV/dB (400 mV/decade), is specifically for a sine wave input; a 0-dBV input signifies a sine input whose rms amplitude is 1 V. Below each axis marker is the corresponding power level when the voltage is applied to a termination resistor of 50 ohms or 316 ohms.
In
an earlier monograph, LEIF 2131:080488*, I discuss
how the basic RF log-amp types compare in responding to various other
waveforms. For many years the effect of the signal’s waveform on
the logarithmic intercept (often misleadingly called “offset”)
went largely unnoticed because early log amps were fairly crude and needed
to be manually adjusted *[Ed.
amplifier absorbs the maximum available power, while adding no
noise of its own. But—apart from naturally occurring noise sources in the surroundings—the
antenna will have its own noise, typically referred to the 50-ohm impedance level, just as any resistor generates noise. Note that this is not
the consequence of some particular manufacturing technology, although there
are additional noise mechanisms at work, to varying degrees, in most practical
resistors.Resistor noise was first
noted by Johnson Now consider a real resistor,
R, at an absolute temperature, T, connected to an ideal noise-free resistance,
R ^{2}/(4R)
= kTB.Thus, E = √4_{N}kTRB V rms.Noise-figure specifications assume (somewhat arbitrarily) that an antenna “operates at” a temperature of 290 K (16.85°C). What is really being referred to here is not the actual temperature of the metal elements comprising the antenna, nor the air temperature that surrounds it; even less the temperature of the directionally narrow source of the signal. Rather, it is the average temperature of all the material objects within the full scope of the antenna’s “view,” modified by its polar diagram (sensitivity vs. direction). The background temperature (thus kT), near Stockholm during winter, as perceived by an antenna seeking a source beyond the warm buildings, may actually be much higher than in pointing the antenna to the Nevada sky (though, in fact, air temperature will have a small effect on the antenna’s intrinsic noise figure). At 290 K the open-circuit
VNSD of a 50-ohm antenna, like that of any other resistor, is 894.85
pV/√Hz. Applied
to a noise-free load of 50 ohms, the noise voltage at the load is halved, to 447.43 pV/√Hz, so the noise power is this voltage squared
divided by 50 ohms, or 4 Notice that the impedance
level is arbitrary; the noise floor would still be –174 dBm/Hz if
the antenna were matched into a 75-ohm load. This is evident when we
note that, in the above calculation, the quantity √4kTR was first halved to get [Ed.
As we’ve seen, the open-circuited
signal voltage, V Of course, this is only
possible using a noise-free load. Such ideality is conceivable when the load
is created from reactances. For example, √L/C has the dimensions of resistance, while an L/C network, in principle,
has no loss. Even real L/C networks, have very low loss: they are essentially
nondissipative. (By contrast, resistors convert power into heat, which is then
lost to the universe.) But even when assisted by the magic of L’s and C’s, the
elements essential to providing power gain, the
shot-noise phenomena, emerging from
a different sort of stochastic mechanism, namely, the granularity of an electric
current crossing a potential barrier. This was first observed by Schottky^{8} in the electrons emitted from the cathode of a vacuum diode. Being released
randomly, they form a Poisson sequence of events—each electron, like a honeybee,
faithfully carrying its precise little packet of charge, q = 1.602 × 10^{–19} coulombs.A similar process arises
in injecting carriers from the emitter into the base of a BJT. Fluctuations
in emission/injection are due to the continual tiny changes in the carrier energy
against the work-function of a cathode, or the band-gap energy of a semiconductor
junction. In the latter case (unlike a vacuum diode), some of the injected carriers
recombine in the base You should note that Johnson
noise is due to the random It is readily shown that the magnitude of the spectral
density of the shot noise current, in A/√Hz, is √2qI, where q is the electron charge and I is the mean bias current,
taken as I Now, such an impedance
(not the “collector output resistance”) exists within a transistor. It is the
“incremental emitter resistance,” r At _{e} is not an ohmic resistance, but simply
the partial derivative, ∂V_{BE}/∂I_{C}, and thus it is noise-free (which
is why it is shown using a distinctive symbol). However, it is interesting to
note that the said product of the shot noise current and this resistance is
identical to the noise voltage generated by a real resistance of half
its value. Here, for example, r_{e} is 25.86 ohms, and the noise
of a real 12.93-ohm resistor is also 463 pV/√Hz. This is simply because the “shot-noise-times-r_{e}”
can be written as kT)^{2}/qI = √2kTr_{e}_{e}/2). This quantity equates to √4kTR, the Johnson
noise of a resistor, R, only when R = r_{e}/2. This must clearly “work
out right.” It does leave some perplexing questions, though. Why is there such
an amusing convergence of these two apparently very disparate fundamental noise
processes? That’s a topic for another (long) memo!
_{BB'} and R_{EE'}, are
included in the recipe.Figure 4 shows what at
first appears to be a highly rudimentary and incomplete circuit, little more
than a diode-connected transistor with a resistor, R
The approach may be called
the *[Ed. Now, the strange thing
about this little circuit is that _{F} to track r_{e} in the manner shown,
which means giving it the algorithmic value qI_{C}R_{A}2/kT.
It follows, as is so often the case, that I_{C} must be proportional
to absolute temperature (PTAT) to maintain this match—and a temperature-stable
gain having the signed value 1 – qI_{C}R_{A}/kT.This
can be seen by setting I qR, that is, 517.2
μA = 25.86 mV/50 ohms, when R_{A}_{A} = 50 ohms, the gain becomes
zero (i.e., –∞ dB), after
which it rises, crossing –1 (back to 0 dB again!) at an I_{C} of precisely
1.034 mA (for T = 300 K).From that value onward,
the gain increases. All the while, the input impedance remains firmly stuck
at the value R
This analysis is at once
optimistic and pessimistic. It is optimistic in neglecting the noise contributions
of the transistor resistances, notably R Thus, in this example,
one-fifth of the collector shot noise, that is, 0.2√2 _{C} = 10 mA. This operates
on the total base impedance, thus at the very least the source impedance of
50 ohms (it need not be resistive), generating 566 pV/√Hz of VNSD. This is over twelve times the 46.3 pV/√Hz due to the _{e}-inducedBut these figures are
pessimistic in neglecting all the ingenious things that can be done using [Ed. Nonetheless, an NF as
low as 0.3 dB is practicable in a
absorb some source
power, accurately and completely, as does a resistor; but the heat that is
generated in this way must then be measured with corresponding accuracy.
When a resistor was included across the input terminals of our ideal voltage-mode
amplifier, the power supplied by the source heated up the resistor by a minuscule
amount. Just as an example, if the signal power were This
is a tiny temperature change; but some power detectors Some
TruPwr kV. This crucial first step is then
followed by averaging and a square-root operation—finally yielding
the _{SIG}^{2}root-mean-square (rms) value. In designing these products,
vigilant attention has to be paid to maintaining low-frequency accuracy
at every step, while using circuit techniques that are at the same time
accurate with microwave waveforms.Many
of the newer
An
earlier type of power detector, now universally known as a “logarithmic
amplifier” (although it usually performs only the
none of these detectors respond to the power of a signal being absorbed at their input. Rather, the response is strictly
to the voltage waveform of the signal. All of the signal’s power is absorbed
by the resistive component of the input impedance, which is in part internal
to the IC, and in part added externally to lower this impedance, commonly down
to 50 ohms. This casts doubt
on the value of an NF specification. Ideally, the sensitivity and measurement
range of log amps of these types ought never to be specified in “dBm”—which
refers to power in decibels above 1 mW—but always in “dBV,” the
decibel level of a voltage relative to 1 V rms. A signal of this amplitude
dissipates 20 mW in a re 50 ohms (“referred to a 50-ohm load”).Nevertheless, provided
the net shunt resistance at the log amp input is known, graphs of its amplitude
response may use a common horizontal axis scaled in both dBm and dBV,
offset by a fixed amount, which for 50 ohms is 13 dB, as illustrated in Figure 2. Unfortunately,
the RF community does not generally think in dBV terms, and this practice is
not rigorously followed. In many data sheets only a dBm scale is used, leading
to the Even
when a log amp’s input stage is designed to match the source impedance—which
makes better use of all the available power and usefully lowers the noise
floor—the response is still to the Recall that, for the case
of a 50-ohm source, loaded by a 50-ohm resistor, the degradation in noise figure, to 3 dB, was entirely due to the additional noise of the Suppose the latter is
stated as 1 nV/√Hz. Next, take the 300 K (27°C) value—the
typical operating temperature of a PC board—for the Johnson noise at 25 ohms (the 50-ohm source in shunt with the net 50 ohms of the external loading resistor and the log amp’s R
The more general form
for the case of a 50-ohm source and 50-ohm load is 20 log
noise figure is a relevant metric when the log amp being
quantified is a multistage limiting amplifier, providing signal output,
which may also operate as a detector, providing an RSSI output—for
example, the AD8309.
This part is specified as having an
input-referred noise (VNSD) of 1.28 nV/√Hz
when driven from a terminated 50-ohm source (that is, with a net 25 ohms
of resistance across its input port). From the expression provided above,
this amounts to an NF of 9.963 dB. The data sheet
value of NF (p. 1) is 6 dB lower, at 3 dB, based on taking the ratio 1.28 nV to the 50-ohm VNSD of 0.91 nV, with a decibel equivalent of 20 log_{10}(1.28/0.91)
= 2.96 dB.The baseline sensitivity
of a log amp is limited by its bandwidth. For example, assume a total VNSD at
the input of a log amp (whether the progressive compression or AGC type) of
1.68 nV/√Hz and an effective noise
bandwidth of 800 MHz. The integrated RTI noise over this bandwidth is 47.5 μV
rms (that is, 1.68 nV/√Hz This
“measurement floor” is a more useful metric than NF, since
it shows that measurements of signal power below this level will be inaccurate.
Here, it will be found that the
^{1}www.analog.com/library/analogdialogue/cd/vol23n3.pdf#page=3^{2}www.analog.com/library/analogdialogue/cd/vol26n2.pdf#page=3^{3}Paterson, W. L. “Multiplication and Logarithmic Conversion
by Operational-Amplifier-Transistor Circuits.” 34-12, Dec. 1963.Rev. Sci. Instr. ^{4}Gilbert, B. “Monolithic Logarithmic Amplifiers.” Lausanne, Switzerland.
Mead Education S.A. Course Notes. [1988?]^{5}Hughes, R. S. Logarithmic Amplification: with Application
to Radar and EW. Dedham, MA: Artech, 1986.^{6}Johnson, J. B. “Thermal Agitation of Electricity in Conductors.” Phys. Rev. 32, 1928, p. 97.^{7}Nyquist, H. “Thermal Agitation of Electronic Charge
in Conductors.” Phys. Rev. 32, 1928, p. 110.^{8}Van der Ziel, A. Noise. Prentice
Hall, 1954.*[Ed. Note—The two earliest papers in this series,
"The
Four Dees of Analog, †Information and data sheets on all products mentioned here may be found on the Analog Devices website, www.analog.com.
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