throbber
Signal-to-Noise Ratio,
`Dynamic Range, and
`Power Dissipation
`
`Paying attention to their interrelation
`can greatly benefit analog circuit design
`
`Yannis Tsividis
`
`ow power dissipation is paramount in an increasing number of ap ­
`plications. These notably include the Internet of Things and wire­
`less sensor networks, in which a battery must last a very long time
`(even ten years in some cases) or in which energy harvesting is used.
`Battery life is also a key consideration in more traditional appli­
`cations such as hearing aids and communication devices. All of these applications
`involve analog signals and require careful micropower circuit design. Such design
`can be more efficient if guided by an understanding of the fundamental limits [1]–[4],
`
`Digital Object Identifier 10.1109/MSSC.2018.2867246
`Date of publication: 16 November 2018
`
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`

`In this article, we review the relation between
`power dissipation and signal-to-noise ratio for
`analog circuits.
`
`components in that bandwidth are
`2D and
`2D respectively.
`,in
`denoted by vn
`The ratios in Figure 2 approach the cor­
`responding one­sided power spectral
`fD is allowed
`densities (PSDs) [9] if
`to approach zero. EN is the so­called
`excess noise factor because of the extra
`noise added by the various devices in
`the transconductor (e.g., 1 to 5).
`To find the contribution of a noise
`source with a known PSD to the output
`,v2
`on of a circuit, we
`mean square noise,
`make use of the transfer function from
`.)
`(H fn
`the noise source to the output,
`The resulting output mean square
`,N is given by [9]
`noise, denoted by
`
`
`
`N v
`=
`
`2
`on
`
`=
`
`3
`
`#
`
`0
`
`PSD
`
`#
`
`H
`
`n
`
`
`
` 2f( )
`
`df
`
`.
`
`
`
`(2)
`
`N is often called the noise power
`,V 2 not W. N
`although its units are
`
`ply voltage and the other for supply
`current, as in Figure 1. We want both
`excess factors to be as close as pos­
`sible to 1.
`We can now express the supply pow ­
`er dissipation as follows:
`
`.
`
`m
`
`(1)
`
`E I
`
`I
`
`P r
`
`
`
`P V I
`=
`DD DD
`
`=
`
`(
`E V
`V
`
`PP
`
`)
`
`c
`
`Noise
`We use the well­known noise models
`shown in Figure 2 [5]–[8], where T is
`the absolute temperature and k is the
`23
`
`( .1 38 10
`J/K
`.
`)
`
`Boltzmann constant
`-
`#
`We assume that we are not interes­
`ted in extremely high frequencies or
`extremely low temperatures, wherein
`quantum corrections become neces­
`sary. fD denotes a very small measure­
`ment bandwidth, and the mean square
`noise voltage and current with spectral
`
`VDD
`
`Average Supply
`Current: IDD
`
`v
`
`v
`
`VDD
`
`Peak Output Current: IP
`
`0
`
`VPP
`
`t
`
`Supply Voltage Excess Factor: EV =
`
`Supply Current Excess Factor:
`
`EI =
`
`VDD
`VPP
`IDD
`(Ip /)
`
`FIGURE 1: An active element with supply voltage and current excess factors defined.
`
`beyond which nature does not allow
`us to go. In this article, we review the
`relation between power dissipation
`and signal­to­noise ratio (SNR) for ana­
`log circuits. We make a crucial distinc­
`tion between SNR and usable dynamic
`range (UDR) and review ways to allow
`the SNR to vary as needed, thus ex ­
`tending the UDR and resulting in ada ­
`ptive power dissipation.
`To be able to provide some depth,
`we focus on only conventional, analog
`continuous­time, single­ended circuits
`using active elements, resistors, and
`capacitors. References to discussions
`of other circuits are provided. The cir ­
`cuits are assumed to be in or near
`thermal equilibrium. All signals are
`supposed to be held below a maximum
`value, beyond which unacceptable dis­
`tortion occurs. We consider first­order
`effects only, which allows us to make
`several simplifying assumptions.
`
`Supply Excess Factors
`and Power Dissipation
`A key to efficient micropower design
`is to be able to use supply voltage and
`current as efficiently as possible. For
`simplicity, signals will be assumed
`to be sinusoidal. Later in this article,
`we will use a transconductor as an
`example, but the definitions apply to
`other circuits as well. The ideal supply
`,VDD would be just equal to
`voltage,
`the peak­to­peak value of the signal,
`;VPP then no voltage would be wasted.
`,IDD
`
`The ideal average supply current,
`would be that for class B operation,
`which is / ,
`IP r with IP being the peak
`output current [5]. To quantify how
`far we are from those ideal values, we
`define two excess factors, one for sup­
`
`R
`
`vn
`
`G
`
`2
`+ ∆vn
`∆f
`
`= 4kTR
`
`(a)
`
`in
`
`(b)
`
`2
`∆in
`∆f
`
`= 4kTG
`
`vn
`
`+
`
`Gm
`
`Gm
`
`in
`
`2
`∆vn
`∆f
`
`= EN × 4kT
`
`1
`Gm
`
`(c)
`
`2
`∆in
`∆f
`
`= EN × 4kTGm
`
`(d)
`
`FIGURE 2: White noise models for low- and medium-frequency work. (a) and (b) Equivalent models for a resistor and (c) and (d) equivalent
`models for a transconductor.
`
`
`
`
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`The capacitance value required
`in this circuit to achieve a given SNR
`can be determined from (4), (6), and
`the equation for N in Figure 3; the
`result is shown by the third equation
`in the same figure. The smaller the
`signal and the larger the required
`SNR, the larger the capacitance (and
`area) needed to keep noise down. Note
`that each 3­dB improvement in SNR
`requires doubling the capacitor area
`(recall that SNR is defined as a pow ­
`er ratio).
`
`First-Order Gm-C Circuit Example
`Consider now the first­order low­pass
`Gm­C circuit in Figure 4 [4]. From
`the output’s point of view, this cor­
`responds to a resistor /G1
`m driving
`C but with noise power multiplied by
`the excess noise factor. Working as
`before, we find the noise power given
`by the second equation in the fig­
`ure; the equation after that gives the
`required capacitance for a given SNR.
`We see that these results are similar
`to those in Figure 3, only multiplied
`.EN
`by the excess noise factor,
`We can determine the power dissi­
`pation from the supply as follows. The
`,Ip is given by
`peak capacitor current,
`# r ) Using in this C from
`(
`
`/ )2
`
`(2
`.
`V
`fC
`pp
`the third equation in Figure 4 and the
`result in (1), we obtain the power dis­
`sipation as given by the last equation
`in the same figure, where we have
`assumed that the signal frequency
`is equal to the cutoff frequency of
`.fc Note that the first three
`the filter,
`factors in the right­hand side depend
`on specifications and design prow­
`ess; the rest of the factors, though,
`are there due to the laws of nature.
`To double the SNR, we need to use
`twice the capacitance and thus twice
`the transconductance to leave the cor­
`/( C2m )
`
`
`r intact. This
`ner frequency
`G
`halving of impedance levels requires
`twice the current for the same swing,
`thus doubling the power. Each 3­dB
`improvement in SNR (meaning, dou­
`bling this power ratio) requires dou­
`bling the power dissipation and
`ca pacitor area. In this respect, life is
`easier with digital circuits. For them,
`adding just one more bit improves the
`
`S
`
`=
`
`v
`
`2
`os
`
`=
`
`V
`
`2
`,
`os
`
`rms
`
`=
`
`
`
`
`c
`
`V
`
`2
`/
`PP
`2
`
`2
`
`m
`
`=
`
`2
`V
`PP
`8
`
`.
`(6)
`
`The maximum value of
`,S denoted
`Smax is that beyond which distor­
`,
`by
`tion becomes unacceptable.
`
`First-Order Resistance–Capacitance
`Circuit Example
`We now calculate the mean square
`voltage noise at the output of the resis­
`tance–capacitance (RC) low­pass filter
`in Figure 3. With the input signal set
`to zero, we use (2) with the PSD from
`1
`
`f(
`)
`
`(1
`2
`-)
`;
`Figure 2(a) and
`H
`j RC
`r
`= +
`n
`this gives the value in Figure 3.
`Another way to obtain this result is
`to use the concept of equivalent noise
`bandwidth [5], which for the above cir­
` ;RC) when this is multiplied cuit is /(1 4
`
`
`
`kTR4
`,
` we again obtain
`by the PSD of
`the previously given result. This makes
`clear why the noise generated by the
`resistor is independent of the resis­
`tance value: increasing the resistance
`increases the noise PSD but decreases
`the bandwidth in proportion.
`Yet another approach is to equate the
`,2
`capacitor’s average energy, ( /) C v1 2
`
`on
`to the thermal energy expected for this
`system in thermal equilibrium from
`/kT 2 [10], [11]. This
`statistical physics,
`gives again the same result.
`
`Each 3-dB improvement in SNR requires doubling
`the capacitor area.
`
`includes all noise; for a frequency­
`selective circuit, noise in both the pass­
`band and the stopband is included.
`
`Signal-to-Noise Ratio
`The so­called signal power, ,S is the
`mean­square value of the output sig­
`nal voltage vos
`
`Note that this quantity is defined
`as a ratio of mean square quanti­
`ties. [The reader is cautioned that in
`some treatments, root mean square
`(rms) quantities are used instead.]
`The corresponding ratio in deci­
`bels is
`
`(3)
`
`(4)
`
`
`
`SNR
`
`dB =
`
`10
`
`log
`
`10
`
`(SNR)
`
`(dB).
`
` (5)
`
`We stress that, in the above defini­
`tions, SNR is a power ratio. Thus, a
`3­dB increase in SNRdB means dou­
`bling the SNR.
`In this article, we will assume sinu­
`soidal signals for simplicity, so that
`
`
`
`S
`
`2=
`vos
`
`.
`
`
`
`The SNR is defined as
`
`.
`
`
`
`NS
`
`SNR =
`
`
`
`+ –v
`
`on
`
`vn
`+
`
`In
`
`R
`
`C
`
`Cutoff Frequency: fc =
`
`1
`2RC
`
`2 = kT
`N = von
`C
`SNR
`2
`Vpp
`
`C = 8kT
`
`FIGURE 3: A first-order RC low-pass filter.
`
`C
`
`VDD
`
`– +
`
`Gm
`
`Gm
`2C
`
`2 =
`N = von
`
`EN ×
`
`C = EN × 8kT
`
`Cutoff Frequency: fc =
`kT
`C
`SNR
`2
`Vpp
`P = EVEIEN × 8kTfc(SNR)
`
`FIGURE 4: A first-order Gm-C low-pass
`filter.
`
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`

`Plower bound
`f
`
`(J)
`
`10–9
`10–12
`10–15
`10–18
`
`T = 300 K 33 W/MHz
`33 nW/MHz
`33 pW/MHz
`
`30 60 90
`SNRdB (dB)
`
`Gm
`
`–Gm
`
`C
`
`In
`
`Gm
`Q
`
`–
`
`Gm
`Q
`
`C
`
`Center Frequency: f0 =
`
`Gm
`2C
`
`With Quality Factor Q >> 1:
`
`Out
`
`2 =
`N = von
`
`EN × 2Q
`
`C = EN × (16Q)kT
`
`kT
`C
`SNR
`2
`Vpp
`
`P = EVEIEN (32Q)kTf0(SNR)
`
`FIGURE 5: The power-to-frequency ratio for
`an ideal first-order Gm-C circuit [4].
`
`FIGURE 6: A bandpass Gm-C filter. A quality factor Q much larger than 1 is assumed.
`
`we obtain the results shown next to
`the circuit.
`Although these results are derived
`analytically [4], [7], [13], [14], intuition
`can be gained by comparing them to
`the results in Figure 4. We see that
`the noise in the second­order circuit
`is larger by a factor of Q2 . This noise
`is dominated by the two large trans­
`conductors, for which large internal
`gains exist from their equivalent
`input noise voltage to the output. The
`power dissipation is dominated by
`the two large transconductors, too.
`Thus, it is not surprising that the 8
`in the power equation in Figure 4 is
`, resulting in the
`2 2#
`multiplied by
`Q
`equation shown [4]. Distortion com­
`ponents can be thought of as distur­
`bances at internal points, which are
`also amplified by high gains. This
`
`signal­to­quantization­error ratio by
`6 dB, yet it does not increase power
`dissipation by a large factor in multi­
`bit systems.
`Assume now an ideal world in
`V =
`I = and
`N =
`
`,1
`,
`1
`.1
`
`which
`E
`E
`E
`Then, from the power equation in
`Figure 4, we obtain the lowest pos­
`sible power dissipation per unit
`of frequency:
`
`
`
`P
`ower boundl
`
`
`f
`
`=
`
`kT8
`
`
`
`(SNR
`
`).
`
`
`
`(7)
`
`This is plotted in Figure 5 [4]. This
`result, although it cannot be achieved
`in practice, can serve as a sanity check;
`if the specifications call for power
`lower than this bound, we know that
`this is not possible (under our stated
`assumptions). The lower bound also
`allows us to see how much room there
`is, theoretically, for improvement.
`According to the previous equa­
`tion, the minimum required power
`can be expected to scale linearly with
`SNR. Compare this to digital [1]–[4],
`where the power is logarithmic with
`SNR [however, analog­to­digital con­
`verters (ADCs) and digital­to­analog
`converters do not allow taking full
`advantage of this because they are
`governed by analog limitations].
`
`Highly Selective Bandpass
`Filter Example
`Figure 6 shows a bandpass filter cir­
`cuit [12], in which we assume that
`the quality factor, Q, is much larger
`than 1. Proceeding in the same way,
`
`makes small­amplitude signals nec­
`essary and, thus, requires smaller
`amounts of noise, further increasing
`the required capacitance; power dis­
`sipation increases as well because of
`.EI Simi­
`the resulting large EV and/or
`lar equations hold for resistive noise
`in the corresponding active RC filter
`in Figure 7.
`
`Other Circuits
`Detailed calculations give similar
`results as those already given for
`a variety of circuits [4], in the gen­
`eral form
`
`
`
`P
`
`=
`
`
`
`[E E E M kTf SNR] (
`#
`V I N
`
`
`
`
`
`),
`
`
`
`(8)
`
`where M is a circuit­specific quantity
`[4], and f is a characteristic frequ­
`ency depending on the circuit, e.g., a
`
`–1
`
`C
`
`R
`
`– +
`
`R
`
`QR
`
`C
`
`– +
`
`In
`
`QR
`
`Center Frequency: f0 =
`
`BP Out
`
`1
`
`2RC
`
`FIGURE 7: A bandpass RC filter. A quality factor Q much larger than 1 is assumed.
`
`
`
`
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`

`For the second­order circuit, we assume
` Using the equations in Fig ­
` .Q 20=
`
`ures 4 and 6, we obtain the results
`shown in Figure 8. These results exem­
`plify the fact that high SNR, highly se ­
`lective active filters are impractical.
`If inductors can be used, the picture
`is very different. For a parallel resistor­
`inductor­capacitor circuit, the noise is
`/ ,kT C but the power dissipation is
`
`still
`much lower [4]. If inductors with lim­
`ited Q are used with active loss can­
`celation, savings are still possible.
`
`Technology-Dependent Issues
`There are several technology­related
`issues not addressed in the above pre ­
`sented first­order estimates. They in ­
`clude the following:
` ■ Parasitic capacitances: These can
`limit speed; thus, extra currents are
`needed to drive them.
` ■ Flicker noise: Flicker noise can be
`lowered by using larger gate areas,
`but this results in larger parasitics,
`as discussed above. Flicker noise
`can be addressed through autoze­
`roing/correlated double sampling/
`chopper stabilization [19], [20] (but
`with the overhead of a clock).
` ■ Mismatches: These can be lowered
`by using larger dimensions, thus
`larger parasitics and larger pow­
`er [21], as discussed above. Mis­
`matches can be addressed through
`digital calibration, with an ensuing
`overhead that is sometimes quite
`significant.
` ■ Leakage currents: To be able to ignore
`leakage currents, bias currents must
`be kept well above them.
` ■ Voltage headroom: Headroom is­
`sues become more severe with low­
`er supply voltages [22], [23].
`The large power dissipation needed
`even for a modest SNR, exemplified in
`Figure 8, appears to be bad news. Does
`this mean that a high dynamic range
`with low power dissipation is an impos­
`sible dream? The answer is fortunately
`not, as discussed next.
`
`Maximum Signal-to-Noise Ratio
`and Usable Dynamic Range
`To understand the reason for the
`large amount of power required for
`
`Does this mean that a high dynamic range with
`low power dissipation is an impossible dream?
`
`band­edge or center frequency for a
`filter or the noise bandwidth for cur­
`rent mirrors and amplifiers. Results
`for oscillators are highly dependent
`on type and assumptions [4], [15]–
`[17]. A qualitative increase with fre­
`quency and SNR is still observed.
`Switched­capacitor circuits, when well
`designed, obey similar laws [2], [18].
`The product in brackets in (8) de ­
`pends strongly on the circuit type, spec­
`ifications, and a number of assumptions
`[4], of which there are too many to list
`here for each circuit. The voltage excess
`factor EV is large if VPP must be limited
`to avoid distortion. Similarly, the cur­
`rent excess factor EI is large for practical
`
`class A circuits; M has a minimum value
`of 8 (as in Figure 4) for known circuits,
`and it is much larger for highly selec­
`tive circuits (as in Figure 6). Thus, the
`product in brackets can be very large
`for practical circuits, depending on the
`details; values of several thousand are
`not uncommon [4].
`
`Numerical Example
`To get an order­of­magnitude idea
`of what we are talking about, let us
`consider two Gm­C circuits: the first­
`order one in Figure 4 and the second­
`order one in Figure 6. We will as sume
`
`
`
`.0 5 V,
`
`
`2
`,
`,1 V
`
`V
`E
`V
`r=
`PP =
`DD =
`I
`N = T
` K, and
`
`300=
`
`,3
`.
`10 MHz
`E
`f
`=
`
`Total
`Capacitance
`400 aF
`400 fF
`400 pF
`32 fF
`32 pF
`32 nF
`
`Power
`Dissipation
`12 nW
`12 W
`12 mW
`1 W
`1 mW
`1 W
`
`Signal (Do minant)
`S N R = S N R S P E C
`
`N oise
`
`SNRdB
`
`30 dB
`60 dB
`90 dB
`30 dB
`60 dB
`90 dB
`
`Out (dB)
`
`In (dB)
`
`In (dB)
`
`UDR
`(b)
`
`Type of Circuit
`
`Q
`
`First-Order Gm-C
`
`– – – 2
`
`Second-Order Gm-C
`
`20
`0
`20
`FIGURE 8: A numerical example.
`
`Signal (Do minant)
`
`SNR
`
`Waste
`
`Out (dB)
`
`SNRSPEC
`
`Noise
`
`UDR
`(a)
`
`FIGURE 9: SNR in (a) a conventional circuit with constant noise and (b) an adaptive circuit
`in which the noise is allowed to vary with input level. The desired signal is assumed to be
`dominant at the input of the filter.
`
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`

`Power
`
`Adaptive
`
`Conventional
`
`Average for
`Adaptive
`
`Time
`
`unit­cell capacitors, resistors, opera­
`tional transconductance amplifiers
`(OTAs), transconductors, and so on,
`in parallel, as illustrated in Figure 12
`[8] (internal nodes may also have to
`be strapped together). Note that we
`differentiate between OTAs and trans­
`conductors: the former are just opera­
`tional amplifiers with high output
`impedance, used closed­loop, with
`their input driven to a virtual short;
`the latter can be used open loop and
`need to remain linear when large sig­
`nals are applied to their input.
`We now present several examples.
`These examples are idealized; only
`first­order effects are discussed.
`
`Bias-Current Scaling
`A simple example of bias­current scal­
`ing is shown in Figure 13 for a class A
`circuit. Rather than keeping the bias
`current constant, as in Figure 13(a), we
`allow it to decrease when the signal is
`
`Impedance Control
`
`Amp
`
`Function
`Block
`
`Gain Control
`
`Bias Control
`
`FIGURE 11: The three adaptive power
`strategies: adaptive gain, adaptive bias,
`and adaptive impedance.
`
`FIGURE 10: Fixed power versus adaptive power.
`
`Adaptive Power Dissipation
`If the attributes of a circuit are al ­
`lowed to vary, so that when the signal
`is large the noise is also allowed to
`be large, as in Figure 9(b), SNR never
`,SNRSPEC
`
`needs to be much larger than
`and the power dissipation needed
`can be drastically lower than that of
`a fixed circuit; yet the UDR remains
`the same as in Figure 9(a). (We stress
`that, for now, we are assuming that
`the desired signal at the input of the
`filter is dominant; if there are strong
`interferers, the situation is more com­
`plicated, as will be discussed shortly.)
`With proper design, one can use the
`minimum power needed at each point
`in time, as illustrated in Figure 10. This
`approach is widely used today, e.g.,
`in smartphones.
`Adaptive power dissipation can be
`accomplished by allowing noise and
`signal­handling capability to vary de ­
`pending on signal and interference
`strengths (for the case of strong inter­
`ferers, see the “Handling Interferers”
`section). This can be done by mak­
`ing gain, bias, and impedance levels
`controllable [24]–[34], as indicated
`in Figure 11.
`Gain adaptation is a classical tech­
`nique that was first used in telephony
`and is widely employed in wireless
`receivers. We will combine it with two
`power adaptation techniques: bias­
`current scaling and bias­impedance
`scaling. Bias and impedance can be
`varied by adjusting a current, and/or
`activating/deactivating a number of
`
`conventional circuits with high SNR,
`we use the following definitions:
`
`(9)
`
`(10)
`
`,
`
`
`
`.
`
`
`
`S
`max
`N
`
`max
`
`min
`
`SS
`
`SNRmax
`
`=
`
`UDR
`
`=
`
`
`
`
`
`These two quantities are not the
`same. SNRmax is often called dynamic
`range, but this can cause confusion and
`lead to wrong conclusions. We use the
`above given definition of UDR [24] to
`avoid the possibility of such confusion.
`As will be seen, UDR can be very differ­
`SNRmax
`.
`ent from
`
`Power Waste in
`Conventional Circuits
`To understand how conventional cir ­
`cuits can waste large amounts of power,
`consider a simplified example of a fil­
`ter, in which the desired signal at its
`input is dominant; the filter is sup­
`posed to filter out small amounts of
`interference and noise in its stopband.
`Noise and the specification for SNR are
`assumed to be independent of signal
`level. Then, the situation is as shown
`in Figure 9(a). To satisfy the specifica­
`,SNRSPEC for SNR at the minimum
`
`tion,
`signal level, we end up greatly exceed­
`ing that specification at the maximum
`signal level. This can result in a large
`waste of power dissipation and area,
`as exemplified by the equations in
`Figures 4 and 6. The problem can be
`traced to the fixed nature (time­invari­
`ance) of conventional circuits, which
`is discussed next.
`
`
`
`
`
` IEEE SOLID-STATE CIRCUITS MAGAZINE
`
`fa ll 2 0 18
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`
`

`

`small, as in Figure 13(b). Then not only
`does the power go down, but current
`noise also decreases. This is shown
`in the log­log plot in Figure 13(c),
`where the subscript i indicates current­
`based quantities.
`Another example is shown in Fig­
`ure  14. The left side of the figure
`shows an OTA, composed by a number
`of unit OTA cells in parallel, as was done
`with transconductors in Figure 12.
`When the situation allows it, a number
`of those cells are deactivated [30], [31],
`resulting in the situation shown on the
`right side of the figure. The power dis­
`sipation decreases n times but so does
`the maximum current swing. Assuming
`the transconductance also decreases n
`times, the mean square noise current
`at the OTA output also decreases by
`the same factor [see Figure 2(d)]; this
`means that the mean square equiv­
`alent input noise increases n times
`[see Figure 2(c)].
`
`Bias-Impedance Scaling
`Figure 15 shows a first­order Gm­C
`filter and its bias­impedance­scaled
`version. We divide all biases, trans­
`,n
`conductances, and capacitances by
`e.g., using the approach in Figure 12.
`Then, all voltage transfer functions
`
`remain unchanged. Voltage swings
`remain unchanged, and linearity is
`not affected. The output mean square
`voltage noise increases n times, and
`SNR decreases n times. The power
`dissipation decreases n times.
`The corresponding strategy for
`an active RC circuit using an OTA is
`shown in Figure 16. The OTA bias and
`;n all
`transconductance are divided by
`.n Then,
`impedances are multipli ed by
`all voltage transfer functions remain
`unchanged. Voltage swings also remain
`unchanged, and linearity is not affected.
`The output mean square voltage noise
`increases n times, and the power dissi­
`pation goes down n times. The effect on
`the SNR is shown at the bottom of the
`figure (if the scaling is done in discrete
`steps, then the line for the noise would
`instead look like a staircase).
`
`Handling Interferers
`To put these techniques together,
`consider a simple wireless receiver
`ar chitecture as shown in Figure 17.
`Signal­strength indicators provide infor­
`mation about the size of the desired
`signal and the out­of­band interferers.
`The control signals adapt gain (which
`for simplicity is shown in front of the
`filter, rather than being distributed),
`
`IO,max
`n
`
`Ibias
`n
`
`+ –
`
`2
`nvn
`
`– +
`
`2
`vn
`–
`+
`
`+
`
`–
`
`Ibias
`
`IO,max
`
`FIGURE 14: Bias-current scaling for an OTA.
`
`VDD
`
`Isupp
`
`VDD
`
`Isupp/n
`
`vOut
`
`C/n
`
`+–
`
`vIn
`
`Gm /n
`
`vOut
`
`C
`
`+–
`
`Gm
`
`vIn
`
`FIGURE 15: Bias-impedance scaling for a Gm-C circuit.
`
`–+
`
`–+
`
`–+
`
`FIGURE 12: A transconductor composed
`of unit transconductors in parallel, with
`total transconductance Gm. Internal nodes
`may also have to be strapped together. A
`smaller transconductance can be imple-
`mented by disconnecting some of the units
`and shutting them down.
`
`i
`
`0
`
`i
`
`0
`
`Ibias
`
`(a)
`
`Ibias(t )
`
`t
`
`t
`
`(b)
`
`Si,max
`
`Ni
`
`(c)
`
`SNRi,max
`
`Ibias
`
`FIGURE 13: (a) A current bias and
`signal in a conventional class A circuit;
`(b) the corresponding quantities in an
`adaptively biased circuit; and (c) the
`log-log plot of current signal, current noise,
`and current SNR versus the bias current
`for the case in (b).
`
`66
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`
`

`

`Adaptive power dissipation can be accomplished
`by allowing noise and signal-handling
`capability to vary.
`
`biasing, and impedance levels as need­
` ed. Depending on the strength of the
`in­band signals and those that are out
`of band (interferers or blockers), dif­
`ferent strategies can be adopted, as is
`now illustrated using the baseband fil­
`ter as an example. (Similar techniques
`can be used for other blocks, e.g., mix­
`ers and oscillators.)
`First, consider the worst case of mi ­
`nimum desired signal and maximum
`interference at the input of the filter,
`shown in Figure 18. The horizontal di ­
`mension is frequency; the line labeled
`noise is the noise power in a given, small
`bandwidth. Thus, this figure is an ideal­
`ized version of what would be observed
`with a spectrum analyzer.
`The circuit must be designed for
`maximum signal­handling capability
`
`and minimum noise, thus a large SNR.
`This results in maximum power dissipa­
`tion. But for any other case, the power
`dissipation can be decreased. Several
`cases are now considered [28].
` ■ Small desired signal, small interfer­
`ence (Figure 19). In this case, we can
`amplify the input and allow the base­
`band filter noise floor to increase,
`as shown, via bias­impe dance scal­
`ing. SNR remains adequate, but
`it does not need to be made large.
`Thus, the achievable power dissipa­
`tion is small. As the signal is varied
`
`over the desired UDR, adaptation
`makes sure that the SNR remains
`small, thus keeping the power dissi­
`pation small, too. Another possibility
`is shown in Figure 20; bias scaling is
`used to reduce the maximum sig­
`nal handling capability. The power
`again decreases, at the expense of
`decreased linearity and decreased
`(but adequate) SNR.
` ■ Large desired signal, small interfer­
`ence (Figure 21). Here bias­impedance
`scaling is used to allow the noise to
`increase relative to the worst­case
`scenario. The linearity remains the
`same, and the power goes down. If
`the signal is now decreased over its
`UDR, the noise floor is correspond­
`ingly decreased to keep the SNR within
`the specification, thus requiring large
`power dissipation only when needed.
` ■ Large desired signal, large interfer­
`ence (Figure 22). Again bias­imped­
`ance scaling is used to allow the noise
`to increase. The linearity remains the
`same, and the power goes down.
`Again, if the signal decreases over
`its UDR, the noise floor is decreased
`to keep the SNR within the specifica­
`tion, thus increasing the power dissi­
`pation only when needed.
` ■ Medium desired signal, large inter­
`ference (Figure 23). This case is trick­
`ier and must be handled with care.
`Here bias scaling is used to re duce
`the signal­handling capability, and
`
`Smax
`
`Desired
`Signal
`
`Interferers
`
`Noise
`
`nRf
`
`C/n
`
`nRL
`vOut
`gm/n
`
`+–
`
`Ibias/n
`
`nR
`
`vIn
`
`C
`
`gm
`
`RL
`vOut
`
`Rf
`
`+–
`
`Ibias
`
`R
`
`vIn
`
`Smax
`
`N
`
`SNR
`
`P
`
`FIGURE 16: Bias-impedance scaling for an active RC circuit.
`
`Mixer
`
`Gain
`
`LNA
`
`Baseband
`(Incl. Filter and
`ADC)
`
`OSC.
`
`Control
`Circuits
`
`SSI
`
`SSI
`
`SSI
`
`Computational Circuits
`
`FIGURE 17: A wireless receiver with adaptive power control. SSI: signal-strength indicator;
`LNA: low-noise amplifier.
`
`FIGURE 18: The worst-case scenario:
`minimum desired signal with maximum
`interference.
`
`
`
`
`
` IEEE SOLID-STATE CIRCUITS MAGAZINE
`
`fa ll 2 0 18
`
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`
`IPR2023-00817
`Theta EX2004
`
`

`

`Smax
`
`Noise
`
`Smax
`
`Noise
`
`Before Adaptation
`
`After Adaptation
`
`FIGURE 19: Power adaptation in the baseband filter in the case of a small desired signal
`and small interference.
`
`Smax
`
`Noise
`
`Smax
`
`Noise
`
`FIGURE 20: Alternative power adaptation in the case of a small desired signal and small
`interference.
`
`Smax
`
`Noise
`
`Smax
`
`Noise
`
`FIGURE 21: Power adaptation in the case of a large desired signal and small interference.
`
`Smax
`
`Noise
`
`Smax
`
`Noise
`
`FIGURE 22: Power adaptation in the case of a large desired signal and large interference.
`
`68
`
`fa ll 2 0 18
`
`IEEE SOLID-STATE CIRCUITS MAGAZINE
`
`bias­impedance scaling is used to
`increase the noise floor. The power
`goes down, but the linearity decreas­
`es, which may result in signal con­
`tamination, e.g., by intermodulation
`products resulting from the block­
`ers. However, as long as the signal
`is still well above those products, it
`can be decoded.
`This discussion is limited to first­
`order effects. During design, one must,
`of course, consider second­order effects
`including the presence of parasitic
`capacitances. Still, practical designs
`are possible and have been well tested.
`Some are widely used. For example, a
`careful use of the techniques presented
`can decrease the average power dissi­
`pation of a cell phone receiver by a sig­
`nificant factor.
`
`Avoiding Output Disturbances
` During Scaling
`Gain, bias, and impedance adaptation
`can be applied during intervals in which
`no signal service is needed, to avoid dis­
`turbances during the transitions, e.g.,
`in between frames in certain wireless
`applications. However, techniques for
`applying the discussed techniques even
`in the presence of signals, without caus­
`ing disturbances at the output, have
`been developed [24], [29]–[34].
`
`Conclusions
`We have reviewed ways to estimate
`the power dissipation expected of
`simple circuits from fundamental
`considerations. For a given circuit, a
`lower bound can be determined for
`power dissipation, which is propor­
`tional to frequency and SNR (as a
`power ratio). Various practical limi­
`tations contribute to exceeding this
`lower bound; this has been quanti­
`fied by defining three excess factors:
`one for supply voltage, one for supply
`current, and one for noise. High­SNR
`circuits can be prohibitive in terms of
`power dissipation, especially if they
`are highly selective. However, in most
`cases, high SNR is used just to pro­
`vide a large UDR. In these cases, as we
`have shown, we can design for a large
`UDR, while the SNR is only as large as
`the specifications dictate. Gain, bias,
`
`IPR2023-00817
`Theta EX2004
`
`

`

`Smax
`
`Noise
`
`Smax
`
`Noise
`
`FIGURE 23: Power adaptation in the case of a medium desired signal and large interference.
`
`and impedance scaling techniques
`have been reviewed for making high­
`UDR circuits with low power dissipa­
`tion possible.
`
`Acknowledgments
`The author would like to thank Jacob
`Baker, Peter Kinget, and the anony­
`mous reviewers for useful comments.
`The material in this article was devel­
`oped based on a presentation by the
`author at the Forum on Energy Effi­
`cient Analog Design, IEEE Solid­State
`Circuits Conference, 2018. The author
`thanks Prof. Man­Kay Law for his invi­
`tation to the forum and his comments
`on the presentation draft.
`
`References
`[1] B. J. Hosticka, “Performance comparison
`of analog and digital circuits,” Proc. IEEE,
`vol. 73, no. 1, pp. 25–29, 1985.
`[2] R. Castello and P. Gray, “Performance li ­
`mitations in switched­capacitor filters,”
`IEEE Trans. Circuits Syst., vol. 32, no. 9,
`pp. 865–876, 1985.
`[3] E. A. Vittoz, “Future of analog in the VLSI
`environment,” in Proc. IEEE 1990 Int. Symp.
`Circuits and Systems, 1990, pp. 1372–1375.
`[4] E. A. Vittoz and Y. P. Tsividis, “Frequency­dy­
`namic range­power,” in Trade-Offs in Analog
`Circuit Design, C. Toumazou, G. Moschytz,
`and B. Gilbert, Eds. New York: Springer, 2002,
`pp. 283–313.
`[5] P. R. Gray, P. J. Hurst, R. G. Meyer, and
`S. H. Lewis, Analysis and Design of Analog
`Integrated Circuits. Hoboken, NJ: Wiley,
`2008.
`[6] H. Nyquist, “Thermal agitation of electric
`charge in conductors,” Phys. Rev., vol. 32,
`no. 1, p. 110, 1928.
`[7] D. Blom and J. Voorman, “Noise and dissi­
`pation of electronic gyrators,” Philips Res.
`Rep., vol. 26, pp. 103–113, 1971.
`[8] E. A. Klumperink and B. Nauta, “System­
`atic comparison of HF CMOS transconduc­
`tors,” IEEE Trans. Circuits Syst. II: Analog
`Digit. Signal Process., vol. 50, no. 10, pp.
`728–741, 2003.
`[9] A. Papoulis and S. U. Pillai, Probability,
`Ran dom Variables, and Stochastic Process-
`
`es. New York: Tata McGraw­Hill Educa­
`tion, 2002.
`[10] N. G. Van Kampen, Stochastic Processes in
`Physics and Chemistry, vol. 1. New York:
`Elsevier, 1992.
`[11] R. Sarpeshkar, T. Delbruck, and C. A. Mead,
`“White noise in MOS transistors and resis­
`tors,” IEEE Circuits Devices Mag., vol. 9, no. 6,
`pp. 23–29, 1993.
`[12] R. Schaumann, H. Xiao, and V. V. Mac, De-
`sign of Analog Filters, 2nd ed. New York:
`Oxford Univ. Press, 2009.
`[13] A. A. Abidi, “Noise in active resonators and
`the available dynamic range,” IEEE Trans.
`Circuits Syst. I: Fundam. Theory Appl., vol.
`39, no. 4, pp. 296–299, 1992.
`[14] G. Groenewold, “The design of high dynamic
`range continuous­time integratable band­
`pass filters,” IEEE Trans. Circuits Syst., vol.

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