Chapter 2. Basic Concepts in RF Design

RF design draws upon many concepts from a variety of fields, including signals and systems, electromagnetics and microwave theory, and communications. Nonetheless, RF design has developed its own analytical methods and its own language. For example, while the nonlinear behavior of analog circuits may be characterized by “harmonic distortion,” that of RF circuits is quantified by very different measures.

This chapter deals with general concepts that prove essential to the analysis and design of RF circuits, closing the gaps with respect to other fields such as analog design, microwave theory, and communication systems. The outline is shown below.

2.1 General Considerations

2.1.1 Units in RF Design

RF design has traditionally employed certain units to express gains and signal levels. It is helpful to review these units at the outset so that we can comfortably use them in our subsequent studies.

The voltage gain, V_out/V_in, and power gain, P_out/P_in, are expressed in decibels (dB):

(2.1)

(2.2)

These two quantities are equal (in dB) only if the input and output voltages appear across equal impedances. For example, an amplifier having an input resistance of R₀ (e.g., 50 Ω) and driving a load resistance of R₀ satisfies the following equation:

(2.3)-(2.5)

where V_out and V_in are rms values. In many RF systems, however, this relationship does not hold because the input and output impedances are not equal.

The absolute signal levels are often expressed in dBm rather than in watts or volts. Used for power quantities, the unit dBm refers to “dB’s above 1 mW.” To express the signal power, P_sig, in dBm, we write

(2.6)

The reader may wonder why the output voltage of the amplifier is of interest in the above example. This may occur if the circuit following the amplifier does not present a 50-Ω input impedance, and hence the power gain and voltage gain are not equal in dB. In fact, the next stage may exhibit a purely capacitive input impedance, thereby requiring no signal “power.” This situation is more familiar in analog circuits wherein one stage drives the gate of the transistor in the next stage. As explained in Chapter 5, in most integrated RF systems, we prefer voltage quantities to power quantities so as to avoid confusion if the input and output impedances of cascade stages are unequal or contain negligible real parts.

The reader may also wonder why we were able to assume 0 dBm is equivalent to 632 mV_pp in the above example even though the signal is not a pure sinusoid. After all, only for a sinusoid can we assume that the rms value is equal to the peak-to-peak value divided by . Fortunately, for a narrowband 0-dBm signal, it is still possible to approximate the (average) peak-to-peak swing as 632 mV.

Although dBm is a unit of power, we sometimes use it at interfaces that do not necessarily entail power transfer. For example, consider the case shown in Fig. 2.1(a), where the LNA drives a purely-capacitive load with a 632-mV_pp swing, delivering no average power. We mentally attach an ideal voltage buffer to node X and drive a 50-Ω load [Fig. 2.1(b)]. We then say that the signal at node X has a level of 0 dBm, tacitly meaning that if this signal were applied to a 50-Ω load, then it would deliver 1 mW.

Figure 2.1 (a) LNA driving a capacitive impedance, (b) use of fictitious buffer to visualize the signal level in dBm.

2.1.2 Time Variance

A system is linear if its output can be expressed as a linear combination (superposition) of responses to individual inputs. More specifically, if the outputs in response to inputs x₁(t) and x₂(t) can be respectively expressed as

(2.9)

(2.10)

then,

(2.11)

for arbitrary values of a and b. Any system that does not satisfy this condition is nonlinear. Note that, according to this definition, nonzero initial conditions or dc offsets also make a system nonlinear, but we often relax the rule to accommodate these two effects.

Another attribute of systems that may be confused with nonlinearity is time variance. A system is time-invariant if a time shift in its input results in the same time shift in its output. That is, if y(t) = f [x(t)], then y(t − τ) = f [x(t − τ)] for arbitrary τ.

As an example of an RF circuit in which time variance plays a critical role and must not be confused with nonlinearity, let us consider the simple switching circuit shown in Fig. 2.2(a). The control terminal of the switch is driven by v_in1(t) = A₁ cosω₁t and the input terminal by v_in2(t) = A₂ cos ω₂t. We assume the switch is on if v_in1 > 0 and off otherwise. Is this system nonlinear or time-variant? If, as depicted in Fig. 2.2(b), the input of interest is v_in1 (while v_in2 is part of the system and still equal to A₂ cos ω₂t), then the system is nonlinear because the control is only sensitive to the polarity of v_in1 and independent of its amplitude. This system is also time-variant because the output depends on v_in2. For example, if v_in1 is constant and positive, then v_out(t) = v_in2(t), and if v_in1 is constant and negative, then v_out(t) = 0 (why?).

Figure 2.2 (a) Simple switching circuit, (b) system with V_in1 as the input, (c) system with V_in2 as the input.

Now consider the case shown in Fig. 2.2(c), where the input of interest is v_in2 (while v_in1 remains part of the system and still equal to A₁ cos ω₁t). This system is linear with respect to v_in2. For example, doubling the amplitude of v_in2 directly doubles that of v_out. The system is also time-variant due to the effect of v_in1.

The circuit of Fig. 2.2(a) is an example of RF “mixers.” We will study such circuits in Chapter 6 extensively, but it is important to draw several conclusions from the above study. First, statements such as “switches are nonlinear” are ambiguous. Second, a linear system can generate frequency components that do not exist in the input signal—the system only need be time-variant. From Example 2.3,

(2.12)

where S(t) denotes a square wave toggling between 0 and 1 with a frequency of f₁ = ω₁/(2π). The output spectrum is therefore given by the convolution of the spectra of v_in2(t) and S(t). Since the spectrum of a square wave is equal to a train of impulses whose amplitudes follow a sinc envelope, we have

(2.13)-(2.14)

where T₁ = 2π/ω₁. This operation is illustrated in Fig. 2.4 for a V_in2 spectrum located around zero frequency.¹

Figure 2.4 Multiplication in the time domain and corresponding convolution in the frequency domain.

2.1.3 Nonlinearity

A system is called “memoryless” or “static” if its output does not depend on the past values of its input (or the past values of the output itself). For a memoryless linear system, the input/output characteristic is given by

(2.15)

where α is a function of time if the system is time-variant [e.g., Fig. 2.2(c)]. For a memoryless nonlinear system, the input/output characteristic can be approximated with a polynomial,

(2.16)

where α_j may be functions of time if the system is time-variant. Figure 2.5 shows a common-source stage as an example of a memoryless nonlinear circuit (at low frequencies). If M₁ operates in the saturation region and can be approximated as a square-law device, then

(2.17)-(2.18)

In this idealized case, the circuit displays only second-order nonlinearity.

Figure 2.5 Common-source stage.

The system described by Eq. (2.16) has “odd symmetry” if y(t) is an odd function of x(t), i.e., if the response to − x(t) is the negative of that to + x(t). This occurs if α_j = 0 for even j. Such a system is sometimes called “balanced,” as exemplified by the differential pair shown in Fig. 2.6(a). Recall from basic analog design that by virtue of symmetry, the circuit exhibits the characteristic depicted in Fig. 2.6(b) if the differential input varies from very negative values to very positive values.

Figure 2.6 (a) Differential pair and (b) its input/output characteristic.

A system is called “dynamic” if its output depends on the past values of its input(s) or output(s). For a linear, time-invariant, dynamic system,

(2.23)

where h(t) denotes the impulse response. If a dynamic system is linear but time-variant, its impulse response depends on the time origin; if δ(t) yields h(t), then δ(t − τ) produces h(t, τ). Thus,

(2.24)

Finally, if a system is both nonlinear and dynamic, then its impulse response can be approximated by a Volterra series. This is described in Section 2.8.

2.2 Effects of Nonlinearity

While analog and RF circuits can be approximated by a linear model for small-signal operation, nonlinearities often lead to interesting and important phenomena that are not predicted by small-signal models. In this section, we study these phenomena for memoryless systems whose input/output characteristic can be approximated by²

(2.25)

The reader is cautioned, however, that the effect of storage elements (dynamic nonlinearity) and higher-order nonlinear terms must be carefully examined to ensure (2.25) is a plausible representation. Section 2.7 deals with the case of dynamic nonlinearity. We may consider α₁ as the small-signal gain of the system because the other two terms are negligible for small input swings. For example, in Eq. (2.22).

The nonlinearity effects described in this section primarily arise from the third-order term in Eq. (2.25). The second-order term too manifests itself in certain types of receivers and is studied in Chapter 4.

2.2.1 Harmonic Distortion

If a sinusoid is applied to a nonlinear system, the output generally exhibits frequency components that are integer multiples (“harmonics”) of the input frequency. In Eq. (2.25), if x(t) = A cos ωt, then

(2.26)-(2.28)

In Eq. (2.28), the first term on the right-hand side is a dc quantity arising from second-order nonlinearity, the second is called the “fundamental,” the third is the second harmonic, and the fourth is the third harmonic. We sometimes say that even-order nonlinearity introduces dc offsets.

From the above expansion, we make two observations. First, even-order harmonics result from α_j with even j, and vanish if the system has odd symmetry, i.e., if it is fully differential. In reality, however, random mismatches corrupt the symmetry, yielding finite even-order harmonics. Second, in (2.28) the amplitudes of the second and third harmonics are proportional to A² and A³, respectively, i.e., we say the nth harmonic grows in proportion to Aⁿ.

In many RF circuits, harmonic distortion is unimportant or an irrelevant indicator of the effect of nonlinearity. For example, an amplifier operating at 2.4 GHz produces a second harmonic at 4.8 GHz, which is greatly suppressed if the circuit has a narrow bandwidth. Nonetheless, harmonics must always be considered carefully before they are dismissed. The following examples illustrate this point.

2.2.2 Gain Compression

The small-signal gain of circuits is usually obtained with the assumption that harmonics are negligible. However, our formulation of harmonics, as expressed by Eq. (2.28), indicates that the gain experienced by A cos ωt is equal to α₁ + 3α₃A²/4 and hence varies appreciably as A becomes larger.⁴ We must then ask, do α₁ and α₃ have the same sign or opposite signs? Returning to the third-order polynomial in Eq. (2.25), we note that if α₁α₃ > 0, then α₁x + α₃x³ overwhelms α₂x² for large x regardless of the sign of α₂, yielding an “expansive” characteristic [Fig. 2.9(a)]. For example, an ideal bipolar transistor operating in the forward active region produces a collector current in proportion to exp(V_BE/V_T), exhibiting expansive behavior. On the other hand, if α₁α₃ < 0, the term α₃x³ “bends” the characteristic for sufficiently large x [Fig. 2.9(b)], leading to “compressive” behavior, i.e., a decreasing gain as the input amplitude increases. For example, the differential pair of Fig. 2.6(a) suffers from compression as the second term in (2.22) becomes comparable with the first. Since most RF circuits of interest are compressive, we hereafter focus on this type.

Figure 2.9 (a) Expansive and (b) compressive characteristics.

With α₁α₃ < 0, the gain experienced by A cos ωt in Eq. (2.28) falls as A rises. We quantify this effect by the “1-dB compression point,” defined as the input signal level that causes the gain to drop by 1 dB. If plotted on a log-log scale as a function of the input level, the output level, A_out, falls below its ideal value by 1 dB at the 1-dB compression point, A_in,1dB (Fig. 2.10). Note that (a) A_in and A_out are voltage quantities here, but compression can also be expressed in terms of power quantities; (b) 1-dB compression may also be specified in terms of the output level at which it occurs, A_out,1dB. The input and output compression points typically prove relevant in the receive path and the transmit path, respectively.

Figure 2.10 Definition of 1-dB compression point.

To calculate the input 1-dB compression point, we equate the compressed gain, , to 1 dB less than the ideal gain, α₁:

(2.33)

It follows that

(2.34)

Note that Eq. (2.34) gives the peak value (rather than the peak-to-peak value) of the input. Also denoted by P_1dB, the 1-dB compression point is typically in the range of −20 to −25 dBm (63.2 to 35.6 mV_pp in 50-Ω system) at the input of RF receivers. We use the notations A_1dB and P_1dB interchangeably in this book. Whether they refer to the input or the output will be clear from the context or specified explicitly. While gain compression by 1 dB seems arbitrary, the 1-dB compression point represents a 10% reduction in the gain and is widely used to characterize RF circuits and systems.

Why does compression matter? After all, it appears that if a signal is so large as to reduce the gain of a receiver, then it must lie well above the receiver noise and be easily detectable. In fact, for some modulation schemes, this statement holds and compression of the receiver would seem benign. For example, as illustrated in Fig. 2.11(a), a frequency-modulated signal carries no information in its amplitude and hence tolerates compression (i.e., amplitude limiting). On the other hand, modulation schemes that contain information in the amplitude are distorted by compression [Fig. 2.11(b)]. This issue manifests itself in both receivers and transmitters.

Figure 2.11 Effect of compressive nonlinearity on (a) FM and (b) AM waveforms.

Another adverse effect arising from compression occurs if a large interferer accompanies the received signal [Fig. 2.12(a)]. In the time domain, the small desired signal is superimposed on the large interferer. Consequently, the receiver gain is reduced by the large excursions produced by the interferer even though the desired signal itself is small [Fig. 2.12(b)]. Called “desensitization,” this phenomenon lowers the signal-to-noise ratio (SNR) at the receiver output and proves critical even if the signal contains no amplitude information.

Figure 2.12 (a) Interferer accompanying signal, (b) effect in time domain.

To quantify desensitization, let us assume x(t) = A₁ cos ω₁t + A₂ cos ω₂t, where the first and second terms represent the desired component and the interferer, respectively. With the third-order characteristic of Eq. (2.25), the output appears as

(2.35)

Note that α₂ is absent in compression. For A₁ A₂, this reduces to

(2.36)

Thus, the gain experienced by the desired signal is equal to , a decreasing function of A₂ if α₁α₃ < 0. In fact, for sufficiently large A₂, the gain drops to zero, and we say the signal is “blocked.” In RF design, the term “blocking signal” or “blocker” refers to interferers that desensitize a circuit even if they do not reduce the gain to zero. Some RF receivers must be able to withstand blockers that are 60 to 70 dB greater than the desired signal.

2.2.3 Cross Modulation

Another phenomenon that occurs when a weak signal and a strong interferer pass through a nonlinear system is the transfer of modulation from the interferer to the signal. Called “cross modulation,” this effect is exemplified by Eq. (2.36), where variations in A₂ affect the amplitude of the signal at ω₁. For example, suppose that the interferer is an amplitude-modulated signal, A₂(1 + m cos ω_mt) cos ω₂t, where m is a constant and ω_m denotes the modulating frequency. Equation (2.36) thus assumes the following form:

(2.37)

In other words, the desired signal at the output suffers from amplitude modulation at ω_m and 2ω_m. Figure 2.14 illustrates this effect.

Figure 2.14 Cross modulation.

Cross modulation commonly arises in amplifiers that must simultaneously process many independent signal channels. Examples include cable television transmitters and systems employing “orthogonal frequency division multiplexing” (OFDM). We examine OFDM in Chapter 3.

2.2.4 Intermodulation

Our study of nonlinearity has thus far considered the case of a single signal (for harmonic distortion) or a signal accompanied by one large interferer (for desensitization). Another scenario of interest in RF design occurs if two interferers accompany the desired signal. Such a scenario represents realistic situations and reveals nonlinear effects that may not manifest themselves in a harmonic distortion or desensitization test.

If two interferers at ω₁ and ω₂ are applied to a nonlinear system, the output generally exhibits components that are not harmonics of these frequencies. Called “intermodulation” (IM), this phenomenon arises from “mixing” (multiplication) of the two components as their sum is raised to a power greater than unity. To understand how Eq. (2.25) leads to intermodulation, assume x(t) = A₁ cos ω₁t + A₂ cos ω₂t. Thus,

(2.40)

Expanding the right-hand side and discarding the dc terms, harmonics, and components at ω₁ ± ω₂, we obtain the following “intermodulation products”:

(2.41)

(2.42)

and these fundamental components:

(2.43)

Figure 2.15 illustrates the results. Among these, the third-order IM products at 2ω₁ − ω₂ and 2ω₂ − ω₁ are of particular interest. This is because, if ω₁ and ω₂ are close to each other, then 2ω₁ − ω₂ and 2ω₂ − ω₁ appear in the vicinity of ω₁ and ω₂. We now explain the significance of this statement.

Figure 2.15 Generation of various intermodulation components in a two-tone test.

Suppose an antenna receives a small desired signal at ω₀ along with two large interferers at ω₁ and ω₂, providing this combination to a low-noise amplifier (Fig. 2.16). Let us assume that the interferer frequencies happen to satisfy 2ω₁ − ω₂ = ω₀. Consequently, the intermodulation product at 2ω₁ − ω₂ falls onto the desired channel, corrupting the signal.

Figure 2.16 Corruption due to third-order intermodulation.

The reader may raise several questions at this point: (1) In our analysis of intermodulation, we represented the interferers with pure (unmodulated) sinusoids (called “tones”) whereas in Figs. 2.16 and 2.17, the interferers are modulated. Are these consistent? (2) Can gain compression and desensitization (P_1dB) also model intermodulation, or do we need other measures of nonlinearity? (3) Why can we not simply remove the interferers by filters so that the receiver does not experience intermodulation? We answer the first two here and address the third in Chapter 4.

For narrowband signals, it is sometimes helpful to “condense” their energy into an impulse, i.e., represent them with a tone of equal power [Fig. 2.18(a)]. This approximation must be made judiciously: if applied to study gain compression, it yields reasonably accurate results; on the other hand, if applied to the case of cross modulation, it fails. In intermodulation analyses, we proceed as follows: (a) approximate the interferers with tones, (b) calculate the level of intermodulation products at the output, and (c) mentally convert the intermodulation tones back to modulated components so as to see the corruption.⁵ This thought process is illustrated in Fig. 2.18(b).

Figure 2.18 (a) Approximation of modulated signals by impulses, (b) application to intermodulation.

We now deal with the second question: if the gain is not compressed, then can we say that intermodulation is negligible? The answer is no; the following example illustrates this point.

The two-tone test is versatile and powerful because it can be applied to systems with arbitrarily narrow bandwidths. A sufficiently small difference between the two tone frequencies ensures that the IM products also fall within the band, thereby providing a meaningful view of the nonlinear behavior of the system. Depicted in Fig. 2.19(a), this attribute stands in contrast to harmonic distortion tests, where higher harmonics lie so far away in frequency that they are heavily filtered, making the system appear quite linear [Fig. 2.19(b)].

Figure 2.19 (a) Two-tone and (b) harmonic tests in a narrowband system.

Third Intercept Point

Our thoughts thus far indicate the need for a measure of intermodulation. A common method of IM characterization is the “two-tone” test, whereby two pure sinusoids of equal amplitudes are applied to the input. The amplitude of the output IM products is then normalized to that of the fundamentals at the output. Denoting the peak amplitude of each tone by A, we can write the result as

(2.45)

where the unit dBc denotes decibels with respect to the “carrier” to emphasize the normalization. Note that, if the amplitude of each input tone increases by 6 dB (a factor of two), the amplitude of the IM products (∝ A³) rises by 18 dB and hence the relative IM by 12 dB.⁶

The principal difficulty in specifying the relative IM for a circuit is that it is meaningful only if the value of A is given. From a practical point of view, we prefer a single measure that captures the intermodulation behavior of the circuit with no need to know the input level at which the two-tone test is carried out. Fortunately, such a measure exists and is called the “third intercept point” (IP₃).

The concept of IP₃ originates from our earlier observation that, if the amplitude of each tone rises, that of the output IM products increases more sharply (∝ A³). Thus, if we continue to raise A, the amplitude of the IM products eventually becomes equal to that of the fundamental tones at the output. As illustrated in Fig. 2.20 on a log-log scale, the input level at which this occurs is called the “input third intercept point” (IIP₃). Similarly, the corresponding output is represented by OIP₃. In subsequent derivations, we denote the input amplitude as A_IIP3.

Figure 2.20 Definition of IP₃ (for voltage quantities).

To determine the IIP₃, we simply equate the fundamental and IM amplitudes:

(2.46)

obtaining

(2.47)

Interestingly,

(2.48)-(2.49)

This ratio proves helpful as a sanity check in simulations and measurements.⁷ We sometimes write IP₃ rather than IIP₃ if it is clear from the context that the input is of interest.

Upon further consideration, the reader may question the consistency of the above derivations. If the IP₃ is 9.6 dB higher than P_1dB, is the gain not heavily compressed at A_in = A_IIP3?! If the gain is compressed, why do we still express the amplitude of the fundamentals at the output as α₁A? It appears that we must instead write this amplitude as [α₁ + (9/4)α₃A²]A to account for the compression.

In reality, the situation is even more complicated. The value of IP₃ given by Eq. (2.47) may exceed the supply voltage, indicating that higher-order nonlinearities manifest themselves as A_in approaches A_IIP3 [Fig. 2.21(a)]. In other words, the IP₃ is not a directly measureable quantity.

Figure 2.21 (a) Actual behavior of nonlinear circuits, (b) definition of IP₃ based on extrapolation.

In order to avoid these quandaries, we measure the IP₃ as follows. We begin with a very low input level so that (and, of course, higher order nonlinearities are also negligible). We increase A_in, plot the amplitudes of the fundamentals and the IM products on a log-log scale, and extrapolate these plots according to their slopes (one and three, respectively) to obtain the IP₃ [Fig. 2.21(b)]. To ensure that the signal levels remain well below compression and higher-order terms are negligible, we must observe a 3-dB rise in the IM products for every 1-dB increase in A_in. On the other hand, if A_in is excessively small, then the output IM components become comparable with the noise floor of the circuit (or the noise floor of the simulated spectrum), thus leading to inaccurate results.

Since extrapolation proves quite tedious in simulations or measurements, we often employ a shortcut that provides a reasonable initial estimate. As illustrated in Fig. 2.22(a), suppose hypothetically that the input is equal to A_IIP3, and hence the (extrapolated) output IM products are as large as the (extrapolated) fundamental tones. Now, the input is reduced to a level A_in1. That is, the change in the input is equal to 20 log A_IIP3 − 20 log A_in1. On a log-log scale, the IM products fall with a slope of 3 and the fundamentals with a slope of unity. Thus, the difference between the two plots increases with a slope of 2. We denote 20 log A_f − 20 log A_IM by Δ P and write

(2.55)

obtaining

(2.56)

Figure 2.22 (a) Relationships among various power levels in a two-tone test, (b) illustration of shortcut technique.

In other words, for a given input level (well below P_1dB), the IIP₃ can be calculated by halving the difference between the output fundamental and IM levels and adding the result to the input level, where all values are expressed as logarithmic quantities. Figure 2.22(b) depicts an abbreviated notation for this rule. The key point here is that the IP₃ is measured without extrapolation.

Why do we consider the above result an estimate? After all, the derivation assumes third-order nonlinearity. A difficulty arises if the circuit contains dynamic nonlinearities, in which case this result may deviate from that obtained by extrapolation. The latter is the standard and accepted method for measuring and reporting the IP₃, but the shortcut method proves useful in understanding the behavior of the device under test.

We should remark that second-order nonlinearity also leads to a certain type of intermodulation and is characterized by a “second intercept point,” (IP₂).⁸ We elaborate on this effect in Chapter 4.

2.2.5 Cascaded Nonlinear Stages

Since in RF systems, signals are processed by cascaded stages, it is important to know how the nonlinearity of each stage is referred to the input of the cascade. The calculation of P_1dB for a cascade is outlined in Problem 2.1. Here, we determine the IP₃ of a cascade. For the sake of brevity, we hereafter denote the input IP₃ by A_IP3 unless otherwise noted.

Consider two nonlinear stages in cascade (Fig. 2.23). If the input/output characteristics of the two stages are expressed, respectively, as

(2.57)

(2.58)

then

(2.59)

Figure 2.23 Cascaded nonlinear stages.

Considering only the first- and third-order terms, we have

(2.60)

Thus, from Eq. (2.47),

(2.61)

Equation (2.61) leads to more intuitive results if its two sides are squared and inverted:

(2.63)-(2.65)

where A_IP3,1 and A_IP3,2 represent the input IP₃’s of the first and second stages, respectively. Note that A_IP3, A_IP3,1, and A_IP3,2 are voltage quantities.

The key observation in Eq. (2.65) is that to “refer” the IP₃ of the second stage to the input of the cascade, we must divide it by α₁. Thus, the higher the gain of the first stage, the more nonlinearity is contributed by the second stage.

IM Spectra in a Cascade

To gain more insight into the above results, let us assume x(t) = A cos ω₁t + A cos ω₂t and identify the IM products in a cascade. With the aid of Fig. 2.24, we make the following observations:⁹

1. The input tones are amplified by a factor of approximately α₁ in the first stage and β₁ in the second. Thus, the output fundamentals are given by α₁β₁A(cos ω₁t + cos ω₂t).

2. The IM products generated by the first stage, namely, (3α₃/4)A³[cos(2ω₁ − ω₂)t + cos(2ω₂ − ω₁)t], are amplified by a factor of β₁ when they appear at the output of the second stage.

3. Sensing α₁A(cos ω₁t + cos ω₂t) at its input, the second stage produces its own IM components: (3β₃/4)(α₁A)³cos(2ω₁ − ω₂)t + (3β₃/4)(α₁A)³cos(2ω₂ − ω₁)t.

4. The second-order nonlinearity in y₁(t) generates components at ω₁ − ω₂, 2ω₁, and 2ω₂. Upon experiencing a similar nonlinearity in the second stage, these components are mixed with those at ω₁ and ω₂ and translated to 2ω₁ − ω₂ and 2ω₂ − ω₁. Specifically, as shown in Fig. 2.24, y₂(t) contains terms such as 2β₂[α₁A cos ω₁t × α₂A²cos(ω₁ − ω₂)t] and 2β₂(α₁A cos ω₁t × 0.5α₂A²cos 2ω₂t). The resulting IM products can be expressed as (3α₁α₂β₂A³/2)[cos(2ω₁ − ω₂)t + cos(2ω₂ − ω₁)t]. Interestingly, the cascade of two second-order nonlinearities can produce third-order IM products.

Figure 2.24 Spectra in a cascade of nonlinear stages.

Adding the amplitudes of the IM products, we have

(2.66)

obtaining the same IP₃ as above. This result assumes zero phase shift for all components.

Why did we add the amplitudes of the IM₃ products in Eq. (2.66) without regard for their phases? Is it possible that phase shifts in the first and second stages allow partial cancellation of these terms and hence a higher IP₃? Yes, it is possible but uncommon in practice. Since the frequencies ω₁, ω₂, 2ω₁ − ω₂, and 2ω₂ − ω₁ are close to one another, these components experience approximately equal phase shifts.

But how about the terms described in the fourth observation? Components such as ω₁ − ω₂ and 2ω₁ may fall well out of the signal band and experience phase shifts different from those in the first three observations. For this reason, we may consider Eqs. (2.65) and (2.66) as the worst-case scenario. Since most RF systems incorporate narrowband circuits, the terms at ω₁ ± ω₂, 2ω₁, and 2ω₂ are heavily attenuated at the output of the first stage. Consequently, the second term on the right-hand side of (2.65) becomes negligible, and

(2.67)

Extending this result to three or more stages, we have

(2.68)

Thus, if each stage in a cascade has a gain greater than unity, the nonlinearity of the latter stages becomes increasingly more critical because the IP₃ of each stage is equivalently scaled down by the total gain preceding that stage.

In the simulation of a cascade, it is possible to determine which stage limits the linearity more. As depicted in Fig. 2.25, we examine the relative IM magnitudes at the output of each stage (Δ₁ and Δ₂, expressed in dB.) If Δ₂ ≈ Δ₁, the second stage contributes negligible nonlinearity. On the other hand, if Δ₂ is substantially less than Δ₁, then the second stage limits the IP₃.

Figure 2.25 Growth of IM components along the cascade.

2.2.6 AM/PM Conversion

In some RF circuits, e.g., power amplifiers, amplitude modulation (AM) may be converted to phase modulation (PM), thus producing undesirable effects. In this section, we study this phenomenon.

AM/PM conversion (APC) can be viewed as the dependence of the phase shift upon the signal amplitude. That is, for an input V_in(t) = V₁ cos ω₁t, the fundamental output component is given by

(2.71)

where φ(V₁) denotes the amplitude-dependent phase shift. This, of course, does not occur in a linear time-invariant system. For example, the phase shift experienced by a sinusoid of frequency ω₁ through a first-order low-pass RC section is given by − tan⁻¹(RCω₁) regardless of the amplitude. Moreover, APC does not appear in a memoryless nonlinear system because the phase shift is zero in this case.

We may therefore surmise that AM/PM conversion arises if a system is both dynamic and nonlinear. For example, if the capacitor in a first-order low-pass RC section is nonlinear, then its “average” value may depend on V₁, resulting in a phase shift, − tan⁻¹ (RCω₁), that itself varies with V₁. To explore this point, let us consider the arrangement shown in Fig. 2.26 and assume

(2.72)

Figure 2.26 RC section with nonlinear capacitor.

This capacitor is considered nonlinear because its value depends on its voltage. An exact calculation of the phase shift is difficult here as it requires that we write V_in = R₁C₁dV_out/dt + V_out and hence solve

(2.73)

We therefore make an approximation. Since the value of C₁ varies periodically with time, we can express the output as that of a first-order network but with a time-varying capacitance, C₁(t):

(2.74)

If R₁C₁(t)ω₁ 1 rad,

(2.75)

We also assume that (1 + αV_out)C₀ ≈ (1 + αV₁ cos ω₁t)C₀, obtaining

(2.76)

Does the output fundamental contain an input-dependent phase shift here? No, it does not! The reader can show that the third term inside the parentheses produces only higher harmonics. Thus, the phase shift of the fundamental is equal to −R₁C₀ω₁ and hence constant.

The above example entails no AM/PM conversion because of the first-order dependence of C₁ upon V_out. As illustrated in Fig. 2.27, the average value of C₁ is equal to C₀ regardless of the output amplitude. In general, since C₁ varies periodically, it can be expressed as a Fourier series with a “dc” term representing its average value:

(2.77)

Figure 2.27 Time variation of capacitor with first-order voltage dependence for small and large swings.

Thus, if C_avg is a function of the amplitude, then the phase shift of the fundamental component in the output voltage becomes input-dependent. The following example illustrates this point.

What is the effect of APC? In the presence of APC, amplitude modulation (or amplitude noise) corrupts the phase of the signal. For example, if V_in(t) = V₁(1 + m cos ω_mt) cos ω₁t, then Eq. (2.79) yields a phase corruption equal to . We will encounter examples of APC in Chapters 8 and 12.

2.3 Noise

The performance of RF systems is limited by noise. Without noise, an RF receiver would be able to detect arbitrarily small inputs, allowing communication across arbitrarily long distances. In this section, we review basic properties of noise and methods of calculating noise in circuits. For a more complete study of noise in analog circuits, the reader is referred to [1].

2.3.1 Noise as a Random Process

The trouble with noise is that it is random. Engineers who are used to dealing with well-defined, deterministic, “hard” facts often find the concept of randomness difficult to grasp, especially if it must be incorporated mathematically. To overcome this fear of randomness, we approach the problem from an intuitive angle.

By “noise is random,” we mean the instantaneous value of noise cannot be predicted. For example, consider a resistor tied to a battery and carrying a current [Fig. 2.29(a)]. Due to the ambient temperature, each electron carrying the current experiences thermal agitation, thus following a somewhat random path while, on the average, moving toward the positive terminal of the battery. As a result, the average current remains equal to V_B/R but the instantaneous current displays random values.¹⁰

Figure 2.29 (a) Noise generated in a resistor, (b) effect of higher temperature.

Since noise cannot be characterized in terms of instantaneous voltages or currents, we seek other attributes of noise that are predictable. For example, we know that a higher ambient temperature leads to greater thermal agitation of electrons and hence larger fluctuations in the current [Fig. 2.29(b)]. How do we express the concept of larger random swings for a current or voltage quantity? This property is revealed by the average power of the noise, defined, in analogy with periodic signals, as

(2.80)

where n(t) represents the noise waveform. Illustrated in Fig. 2.30, this definition simply means that we compute the area under n²(t) for a long time, T, and normalize the result to T, thus obtaining the average power. For example, the two scenarios depicted in Fig. 2.29 yield different average powers.

Figure 2.30 Computation of noise power.

If n(t) is random, how do we know that P_n is not?! We are fortunate that noise components in circuits have a constant average power. For example, P_n is known and constant for a resistor at a constant ambient temperature.

How long should T in Eq. (2.80) be? Due to its randomness, noise consists of different frequencies. Thus, T must be long enough to accommodate several cycles of the lowest frequency. For example, the noise in a crowded restaurant arises from human voice and covers the range of 20 Hz to 20 kHz, requiring that T be on the order of 0.5 s to capture about 10 cycles of the 20-Hz components.¹¹

2.3.2 Noise Spectrum

Our foregoing study suggests that the time-domain view of noise provides limited information, e.g., the average power. The frequency-domain view, on the other hand, yields much greater insight and proves more useful in RF design.

The reader may already have some intuitive understanding of the concept of “spectrum.” We say the spectrum of human voice spans the range of 20 Hz to 20 kHz. This means that if we somehow measure the frequency content of the voice, we observe all components from 20 Hz to 20 kHz. How, then, do we measure a signal’s frequency content, e.g., the strength of a component at 10 kHz? We would need to filter out the remainder of the spectrum and measure the average power of the 10-kHz component. Figure 2.31(a) conceptually illustrates such an experiment, where the microphone signal is applied to a band-pass filter having a 1-Hz bandwidth centered around 10 kHz. If a person speaks into the microphone at a steady volume, the power meter reads a constant value.

Figure 2.31 Measurement of (a) power in 1 Hz, and (b) the spectrum.

The scheme shown in Fig. 2.31(a) can be readily extended so as to measure the strength of all frequency components. As depicted in Fig. 2.31(b), a bank of 1-Hz band-pass filters centered at f₁ ... f_n measures the average power at each frequency.¹² Called the spectrum or the “power spectral density” (PSD) of x(t) and denoted by S_x(f), the resulting plot displays the average power that the voice (or the noise) carries in a 1-Hz bandwidth at different frequencies.¹³

It is interesting to note that the total area under S_x(f) represents the average power carried by x(t):

(2.81)

The spectrum shown in Fig. 2.31(b) is called “one-sided” because it is constructed for positive frequencies. In some cases, the analysis is simpler if a “two-sided” spectrum is utilized. The latter is an even-symmetric of the former scaled down vertically by a factor of two (Fig. 2.32), so that the two carry equal energies.

Figure 2.32 Two-sided and one-sided spectra.

2.3.3 Effect of Transfer Function on Noise

The principal reason for defining the PSD is that it allows many of the frequency-domain operations used with deterministic signals to be applied to random signals as well. For example, if white noise is applied to a low-pass filter, how do we determine the PSD at the output? As shown in Fig. 2.33, we intuitively expect that the output PSD assumes the shape of the filter’s frequency response. In fact, if x(t) is applied to a linear, time-invariant system with a transfer function H(s), then the output spectrum is

(2.86)

where H(f) = H(s = j2πf) [2]. We note that |H(f)| is squared because S_x(f) is a (voltage or current) squared quantity.

Figure 2.33 Effect of low-pass filter on white noise.

2.3.4 Device Noise

In order to analyze the noise performance of circuits, we wish to model the noise of their constituent elements by familiar components such as voltage and current sources. Such a representation allows the use of standard circuit analysis techniques.

Thermal Noise of Resistors

As mentioned previously, the ambient thermal energy leads to random agitation of charge carriers in resistors and hence noise. The noise can be modeled by a series voltage source with a PSD of [Thevenin equivalent, Fig. 2.34(a)] or a parallel current source with a PSD of [Norton equivalent, Fig. 2.34(b)]. The choice of the model sometimes simplifies the analysis. The polarity of the sources is unimportant (but must be kept the same throughout the calculations of a given circuit).

Figure 2.34 (a) Thevenin and (b) Norton models of resistor thermal noise.

If a resistor converts the ambient heat to a noise voltage or current, can we extract energy from the resistor? In particular, does the arrangement shown in Fig. 2.36 deliver energy to R₂? Interestingly, if R₁ and R₂ reside at the same temperature, no net energy is transferred between them because R₂ also produces a noise PSD of 4kTR₂ (Problem 2.8). However, suppose R₂ is held at T = 0 K. Then, R₁ continues to draw thermal energy from its environment, converting it to noise and delivering the energy to R₂. The average power transferred to R₂ is equal to

(2.88)-(2.90)

Figure 2.36 Transfer of noise from one resistor to another.

This quantity reaches a maximum if R₂ = R₁:

(2.91)

Called the “available noise power,” kT is independent of the resistor value and has the dimension of power per unit bandwidth. The reader can prove that kT = −173.8 dBm/Hz at T = 300 K.

For a circuit to exhibit a thermal noise density of , it need not contain an explicit resistor of value R₁. After all, Eq. (2.86) suggests that the noise density of a resistor may be transformed to a higher or lower value by the surrounding circuit. We also note that if a passive circuit dissipates energy, then it must contain a physical resistance¹⁵ and must therefore produce thermal noise. We loosely say “lossy circuits are noisy.”

A theorem that consolidates the above observations is as follows: If the real part of the impedance seen between two terminals of a passive (reciprocal) network is equal to Re{Z_out}, then the PSD of the thermal noise seen between these terminals is given by (Fig. 2.37) [8]. This general theorem is not limited to lumped circuits. For example, consider a transmitting antenna that dissipates energy by radiation according to the equation , where R_rad is the “radiation resistance” [Fig. 2.38(a)]. As a receiving element [Fig. 2.38(b)], the antenna generates a thermal noise PSD of¹⁶

(2.92)

Figure 2.37 Output noise of a passive (reciprocal) circuit.

Figure 2.38 (a) Transmitting antenna, (b) receiving antenna producing thermal noise.

Noise in MOSFETs

The thermal noise of MOS transistors operating in the saturation region is approximated by a current source tied between the source and drain terminals [Fig. 2.39(a)]:

(2.93)

where γ is the “excess noise coefficient” and g_m the transconductance.¹⁷ The value of γ is 2/3 for long-channel transistors and may rise to even 2 in short-channel devices [4]. The actual value of γ has other dependencies [5] and is usually obtained by measurements for each generation of CMOS technology. In Problem 2.10, we prove that the noise can alternatively be modeled by a voltage source in series with the gate [Fig. 2.39(b)].

Figure 2.39 Thermal channel noise of a MOSFET modeled as a (a) current source, (b) voltage source.

Another component of thermal noise arises from the gate resistance of MOSFETs, an effect that becomes increasingly more important as the gate length is scaled down. Illustrated in Fig. 2.40(a) for a device with a width of W and a length of L, this resistance amounts to

(2.94)

where denotes the sheet resistance (resistance of one square) of the polysilicon gate. For example, if W = 1 μm, L = 45 nm, and = 15 Ω, then R_G = 333 Ω. Since R_G is distributed over the width of the transistor [Fig. 2.40(b)], its noise must be calculated carefully. As proved in [6], the structure can be reduced to a lumped model having an equivalent gate resistance of R_G/3 with a thermal noise PSD of 4kTR_G/3 [Fig. 2.40(c)]. In a good design, this noise must be much less than that of the channel:

(2.95)

Figure 2.40 (a) Gate resistance of a MOSFET, (b) equivalent circuit for noise calculation, (c) equivalent noise and resistance in lumped model.

The gate and drain terminals also exhibit physical resistances, which are minimized through the use of multiple fingers.

At very high frequencies the thermal noise current flowing through the channel couples to the gate capacitively, thus generating a “gate-induced noise current” [3] (Fig. 2.41). This effect is not modeled in typical circuit simulators, but its significance has remained unclear. In this book, we neglect the gate-induced noise current.

Figure 2.41 Gate-induced noise, .

MOS devices also suffer from “flicker” or “1/f” noise. Modeled by a voltage source in series with the gate, this noise exhibits the following PSD:

(2.96)

where K is a process-dependent constant. In most CMOS technologies, K is lower for PMOS devices than for NMOS transistors because the former carry charge well below the silicon-oxide interface and hence suffer less from “surface states” (dangling bonds) [1]. The 1/f dependence means that noise components that vary slowly assume a large amplitude. The choice of the lowest frequency in the noise integration depends on the time scale of interest and/or the spectrum of the desired signal [1].

For a given device size and bias current, the 1/f noise PSD intercepts the thermal noise PSD at some frequency, called the “1/f noise corner frequency,” f_c. Illustrated in Fig. 2.43, f_c can be obtained by converting the flicker noise voltage to current (according to the above example) and equating the result to the thermal noise current:

(2.98)

It follows that

(2.99)

Figure 2.43 Flicker noise corner frequency.

The corner frequency falls in the range of tens or even hundreds of megahertz in today’s MOS technologies.

While the effect of flicker noise may seem negligible at high frequencies, we must note that nonlinearity or time variance in circuits such as mixers and oscillators may translate the 1/f-shaped spectrum to the RF range. We study these phenomena in Chapters 6 and 8.

Noise in Bipolar Transistors

Bipolar transistors contain physical resistances in their base, emitter, and collector regions, all of which generate thermal noise. Moreover, they also suffer from “shot noise” associated with the transport of carriers across the base-emitter junction. As shown in Fig. 2.44, this noise is modeled by two current sources having the following PSDs:

(2.100)

(2.101)

where I_B and I_C are the base and collector bias currents, respectively. Since g_m =I_C/(kT/q) for bipolar transistors, the collector current shot noise is often expressed as

(2.102)

in analogy with the thermal noise of MOSFETs or resistors.

Figure 2.44 Noise sources in a bipolar transistor.

In low-noise circuits, the base resistance thermal noise and the collector current shot noise become dominant. For this reason, wide transistors biased at high current levels are employed.

2.3.5 Representation of Noise in Circuits

With the noise of devices formulated above, we now wish to develop measures of the noise performance of circuits, i.e., metrics that reveal how noisy a given circuit is.

Input-Referred Noise

How can the noise of a circuit be observed in the laboratory? We have access only to the output and hence can measure only the output noise. Unfortunately, the output noise does not permit a fair comparison between circuits: a circuit may exhibit high output noise because it has a high gain rather than high noise. For this reason, we “refer” the noise to the input.

In analog design, the input-referred noise is modeled by a series voltage source and a parallel current source (Fig. 2.45) [1]. The former is obtained by shorting the input port of models A and B and equating their output noises (or, equivalently, dividing the output noise by the voltage gain). Similarly, the latter is computed by leaving the input ports open and equating the output noises (or, equivalently, dividing the output noise by the transimpedance gain).

Figure 2.45 Input-referred noise.

From the above example, it may appear that the noise of M₁ is “counted” twice. It can be shown that [1] the two input-referred noise sources are necessary and sufficient, but often correlated.

The computation and use of input-referred noise sources prove difficult at high frequencies. For example, it is quite challenging to measure the transimpedance gain of an RF stage. For this reason, RF designers employ the concept of “noise figure” as another metric of noise performance that more easily lends itself to measurement.

Noise Figure

In circuit and system design, we are interested in the signal-to-noise ratio (SNR), defined as the signal power divided by the noise power. It is therefore helpful to ask, how does the SNR degrade as the signal travels through a given circuit? If the circuit contains no noise, then the output SNR is equal to the input SNR even if the circuit acts as an attenuator.¹⁸ To quantify how noisy the circuit is, we define its noise figure (NF) as

(2.109)

such that it is equal to 1 for a noiseless stage. Since each quantity in this ratio has a dimension of power (or voltage squared), we express NF in decibels as

(2.110)

Note that most texts call (2.109) the “noise factor” and (2.110) the noise figure. We do not make this distinction in this book.

Compared to input-referred noise, the definition of NF in (2.109) may appear rather complicated: it depends on not only the noise of the circuit under consideration but the SNR provided by the preceding stage. In fact, if the input signal contains no noise, then SNR_in = ∞ and NF = ∞, even though the circuit may have finite internal noise. For such a case, NF is not a meaningful parameter and only the input-referred noise can be specified.

Calculation of the noise figure is generally simpler than Eq. (2.109) may suggest. For example, suppose a low-noise amplifier senses the signal received by an antenna [Fig. 2.48(a)]. As predicted by Eq. (2.92), the antenna “radiation resistance,” R_S, produces thermal noise, leading to the model shown in Fig. 2.48(b). Here, represents the thermal noise of the antenna, and the output noise of the LNA. We must compute SNR_in at the LNA input and SNR_out at its output.

Figure 2.48 (a) Antenna followed by LNA, (b) equivalent circuit.

If the LNA exhibits an input impedance of Z_in, then both V_in and V_RS experience an attenuation factor of α = Z_in/(Z_in + R_S) as they appear at the input of the LNA. That is,

(2.111)

where V_in denotes the rms value of the signal received by the antenna.

To determine SNR_out, we assume a voltage gain of A_v from the LNA input to the output and recognize that the output signal power is equal to . The output noise consists of two components: (a) the noise of the antenna amplified by the LNA, , and (b) the output noise of the LNA, . Since these two components are uncorrelated, we simply add the PSDs and write

(2.112)

It follows that

(2.113)-(2.115)

This result leads to another definition of the NF: the total noise at the output divided by the noise at the output due to the source impedance. The NF is usually specified for a 1-Hz bandwidth at a given frequency, and hence sometimes called the “spot noise figure” to emphasize the small bandwidth.

Equation (2.115) suggests that the NF depends on the source impedance, not only through but also through (Example 2.19). In fact, if we model the noise by input-referred sources, then the input noise current, , partially flows through R_S, generating a source-dependent noise voltage of at the input and hence a proportional noise at the output. Thus, the NF must be specified with respect to a source impedance—typically 50 Ω.

For hand analysis and simulations, it is possible to reduce the right-hand side of Eq. (2.114) to a simpler form by noting that the numerator is the total noise measured at the output:

(2.116)

where includes both the source impedance noise and the LNA noise, and A₀ = |α|A_v is the voltage gain from V_in to V_out (rather than the gain from the LNA input to its output). We loosely say, “to calculate the NF, we simply divide the total output noise by the gain from V_in to V_out and normalize the result to the noise of R_S.” Alternatively, we can say from (2.115) that “we calculate the output noise due to the amplifier (), divide it by the gain, normalize it to 4kTR_S, and add 1 to the result.”

It is important to note that the above derivations are valid even if no actual power is transferred from the antenna to the LNA or from the LNA to a load. For example, if Z_in in Fig. 2.48(b) goes to infinity, no power is delivered to the LNA, but all of the derivations remain valid because they are based on voltage (squared) quantities rather than power quantities. In other words, so long as the derivations incorporate noise and signal voltages, no inconsistency arises in the presence of impedance mismatches or even infinite input impedances. This is a critical difference in thinking between modern RF design and traditional microwave design.

Noise Figure of Cascaded Stages

Since many stages appear in a receiver chain, it is desirable to determine the NF of the overall cascade in terms of that of each stage. Consider the cascade depicted in Fig. 2.51(a), where A_v1 and A_v2 denote the unloaded voltage gain of the two stages. The input and output impedances and the output noise voltages of the two stages are also shown.¹⁹

Figure 2.51 (a) Noise in a cascade of stages, (b) simplified diagram.

We first obtain the NF of the cascade using a direct method; according to (2.115), we simply calculate the total noise at the output due to the two stages, divide by (V_out/V_in)², normalize to 4kTR_S, and add one to the result. Taking the loadings into account, we write the overall voltage gain as

(2.123)

The output noise due to the two stages, denoted by , consists of two components: (a) , and (b) amplified by the second stage. Since V_n1 sees an impedance of R_out1 to its left and R_in2 to its right, it is scaled by a factor of R_in2/(R_in2 + R_out1) as it appears at the input of the second stage. Thus,

(2.124)

The overall NF is therefore expressed as

(2.125)-(2.126)

The first two terms constitute the NF of the first stage, NF₁, with respect to a source impedance of R_S. The third term represents the noise of the second stage, but how can it be expressed in terms of the noise figure of this stage?

Let us now consider the second stage by itself and determine its noise figure with respect to a source impedance of R_out1 [Fig. 2.51(b)]. Using (2.115) again, we have

(2.127)

It follows from (2.126) and (2.127) that

(2.128)

What does the denominator represent? This quantity is in fact the “available power gain” of the first stage, defined as the “available power” at its output, P_out,av (the power that it would deliver to a matched load) divided by the available source power, P_S,av (the power that the source would deliver to a matched load). This can be readily verified by finding the power that the first stage in Fig. 2.51(a) would deliver to a load equal to R_out1:

(2.129)

Similarly, the power that V_in would deliver to a load of R_S is given by

(2.130)

The ratio of (2.129) and (2.130) is indeed equal to the denominator in (2.128).

With these observations, we write

(2.131)

where A_P1 denotes the “available power gain” of the first stage. It is important to bear in mind that NF₂ is computed with respect to the output impedance of the first stage. For m stages,

(2.132)

Called “Friis’ equation” [7], this result suggests that the noise contributed by each stage decreases as the total gain preceding that stage increases, implying that the first few stages in a cascade are the most critical. Conversely, if a stage suffers from attenuation (loss), then the NF of the following circuits is “amplified” when referred to the input of that stage.

The foregoing example represents a typical situation in modern RF design: the interface between the two stages does not have a 50-Ω impedance and no attempt has been made to provide impedance matching between the two stages. In such cases, Friis’ equation becomes cumbersome, making direct calculation of the NF more attractive.

While the above example assumes an infinite input impedance for the second stage, the direct method can be extended to more realistic cases with the aid of Eq. (2.126). Even in the presence of complex input and output impedances, Eq. (2.126) indicates that (1) must be divided by the unloaded gain from V_in to the output of the first stage; (2) the output noise of the second stage, , must be calculated with this stage driven by the output impedance of the first stage;²⁰ and (3) must be divided by the total voltage gain from V_in to V_out.

Noise Figure of Lossy Circuits

Passive circuits such as filters appear at the front end of RF transceivers and their loss proves critical (Chapter 4). The loss arises from unwanted resistive components within the circuit that convert the input power to heat, thereby producing a smaller signal power at the output. Furthermore, recall from Fig. 2.37 that resistive components also generate thermal noise. That is, passive lossy circuits both attenuate the signal and introduce noise.

We wish to prove that the noise figure of a passive (reciprocal) circuit is equal to its “power loss,” defined as L = P_in/P_out, where P_in is the available source power and P_out the available power at the output. As mentioned in the derivation of Friis’ equation, the available power is the power that a given source or circuit would deliver to a conjugate-matched load. The proof is straightforward if the input and output are matched (Problem 2.20). We consider a more general case here.

Consider the arrangement shown in Fig. 2.54(a), where the lossy circuit is driven by a source impedance of R_S while driving a load impedance of R_L.²¹ From Eq. (2.130), the available source power is . To determine the available output power, we construct the Thevenin equivalent shown in Fig. 2.54(b), obtaining . Thus, the loss is given by

(2.138)

Figure 2.54 (a) Lossy passive network, (b) Thevenin equivalent, (c) simplified diagram.

To calculate the noise figure, we utilize the theorem illustrated in Fig. 2.37 and the equivalent circuit shown in Fig. 2.54(c) to write

(2.139)

Note that R_L is assumed noiseless so that only the noise figure of the lossy circuit can be determined. The voltage gain from V_in to V_out is found by noting that, in response to V_in, the circuit produces an output voltage of V_out = V_ThevR_L/(R_L + R_out) [Fig. 2.54(b)]. That is,

(2.140)

The NF is equal to (2.139) divided by the square of (2.140) and normalized to 4kTR_S:

(2.141)-(2.142)

2.4 Sensitivity and Dynamic Range

The performance of RF receivers is characterized by many parameters. We study two, namely, sensitivity and dynamic range, here and defer the others to Chapter 3.

2.4.1 Sensitivity

The sensitivity is defined as the minimum signal level that a receiver can detect with “acceptable quality.” In the presence of excessive noise, the detected signal becomes unintelligible and carries little information. We define acceptable quality as sufficient signal-to-noise ratio, which itself depends on the type of modulation and the corruption (e.g., bit error rate) that the system can tolerate. Typical required SNR levels are in the range of 6 to 25 dB (Chapter 3).

In order to calculate the sensitivity, we write

(2.146)-(2.147)

where P_sig denotes the input signal power and P_RS the source resistance noise power, both per unit bandwidth. Do we express these quantities in V²/Hz or W/Hz? Since the input impedance of the receiver is typically matched to that of the antenna (Chapter 4), the antenna indeed delivers signal power and noise power to the receiver. For this reason, it is common to express both quantities in W/Hz (or dBm/Hz). It follows that

(2.148)

Since the overall signal power is distributed across a certain bandwidth, B, the two sides of (2.148) must be integrated over the bandwidth so as to obtain the total mean squared power. Assuming a flat spectrum for the signal and the noise, we have

(2.149)

Equation (2.149) expresses the sensitivity as the minimum input signal that yields a given value for the output SNR. Changing the notation slightly and expressing the quantities in dB or dBm, we have²²

(2.150)

where P_sen is the sensitivity and B is expressed in Hz. Note that (2.150) does not directly depend on the gain of the system. If the receiver is matched to the antenna, then from (2.91), P_RS = kT = −174 dBm/Hz and

(2.151)

Note that the sum of the first three terms is the total integrated noise of the system (sometimes called the “noise floor”).

2.4.2 Dynamic Range

Dynamic range (DR) is loosely defined as the maximum input level that a receiver can “tolerate” divided by the minimum input level that it can detect (the sensitivity). This definition is quantified differently in different applications. For example, in analog circuits such as analog-to-digital converters, the DR is defined as the “full-scale” input level divided by the input level at which SNR = 1. The full scale is typically the input level beyond which a hard saturation occurs and can be easily determined by examining the circuit.

In RF design, on the other hand, the situation is more complicated. Consider a simple common-source stage. How do we define the input “full scale” for such a circuit? Is there a particular input level beyond which the circuit becomes excessively nonlinear? We may view the 1-dB compression point as such a level. But, what if the circuit senses two interferers and suffers from intermodulation?

In RF design, two definitions of DR have emerged. The first, simply called the dynamic range, refers to the maximum tolerable desired signal power divided by the minimum tolerable desired signal power (the sensitivity). Illustrated in Fig. 2.56(a), this DR is limited by compression at the upper end and noise at the lower end. For example, a cell phone coming close to a base station may receive a very large signal and must process it with acceptable distortion. In fact, the cell phone measures the signal strength and adjusts the receiver gain so as to avoid compression. Excluding interferers, this “compression-based” DR can exceed 100 dB because the upper end can be raised relatively easily.

Figure 2.56 Definitions of (a) DR and (b) SFDR.

The second type, called the “spurious-free dynamic range” (SFDR), represents limitations arising from both noise and interference. The lower end is still equal to the sensitivity, but the upper end is defined as the maximum input level in a two-tone test for which the third-order IM products do not exceed the integrated noise of the receiver. As shown in Fig. 2.56(b), two (modulated or unmodulated) tones having equal amplitudes are applied and their level is raised until the IM products reach the integrated noise.²³ The ratio of the power of each tone to the sensitivity yields the SFDR. The SFDR represents the maximum relative level of interferers that a receiver can tolerate while producing an acceptable signal quality from a small input level.

Where should the various levels depicted in Fig. 2.56(b) be measured, at the input of the circuit or at its output? Since the IM components appear only at the output, the output port serves as a more natural candidate for such a measurement. In this case, the sensitivity—usually an input-referred quantity—must be scaled by the gain of the circuit so that it is referred to the output. Alternatively, the output IM magnitudes can be divided by the gain so that they are referred to the input. We follow the latter approach in our SFDR calculations.

To determine the upper end of the SFDR, we rewrite Eq. (2.56) as

(2.152)

where, for the sake of brevity, we have denoted 20 log A_x as P_x even though no actual power may be transferred at the input or output ports. Also, P_IM,out represents the level of IM products at the output. If the circuit exhibits a gain of G (in dB), then we can refer the IM level to the input by writing P_IM,in = P_IM,out − G. Similarly, the input level of each tone is given by P_in = P_out − G. Thus, (2.152) reduces to

(2.153)-(2.154)

and hence

(2.155)

The upper end of the SFDR is that value of P_in which makes P_IM,in equal to the integrated noise of the receiver:

(2.156)

The SFDR is the difference (in dB) between P_in,max and the sensitivity:

(2.157)-(2.158)

For example, a GSM receiver with NF = 7 dB, P_IIP3 = − 15 dBm, and SNR_min = 12 dB achieves an SFDR of 54 dB, a substantially lower value than the dynamic range in the absence of interferers.

2.5 Passive Impedance Transformation

At radio frequencies, we often employ passive networks to transform impedances—from high to low and vice versa, or from complex to real and vice versa. Called “matching networks,” such circuits do not easily lend themselves to integration because their constituent devices, particularly inductors, suffer from loss if built on silicon chips. (We do use on-chip inductors in many RF building blocks.) Nonetheless, a basic understanding of impedance transformation is essential.

2.5.1 Quality Factor

In its simplest form, the quality factor, Q, indicates how close to ideal an energy-storing device is. An ideal capacitor dissipates no energy, exhibiting an infinite Q, but a series resistance, R_S [Fig. 2.57(a)], reduces its Q to

(2.163)

where the numerator denotes the “desired” component and the denominator, the “undesired” component. If the resistive loss in the capacitor is modeled by a parallel resistance [Fig. 2.57(b)], then we must define the Q as

(2.164)

because an ideal (infinite Q) results only if R_P = ∞. As depicted in Figs. 2.57(c) and (d), similar concepts apply to inductors

(2.165)

(2.166)

Figure 2.57 (a) Series RC circuit, (b) equivalent parallel circuit, (c) series RL circuit, (d) equivalent parallel circuit.

While a parallel resistance appears to have no physical meaning, modeling the loss by R_P proves useful in many circuits such as amplifiers and oscillators (Chapters 5 and 8). We will also introduce other definitions of Q in Chapter 8.

2.5.2 Series-to-Parallel Conversion

Before studying transformation techniques, let us consider the series and parallel RC sections shown in Fig. 2.58. What choice of values makes the two networks equivalent?

Figure 2.58 Series-to-parallel conversion.

Equating the impedances,

(2.167)

and substituting jω for s, we have

(2.168)

and hence

(2.169)-(2.170)

Equation (2.169) implies that Q_S = Q_P.

Of course, the two impedances cannot remain equal at all frequencies. For example, the series section approaches an open circuit at low frequencies while the parallel section does not. Nevertheless, an approximation allows equivalence for a narrow frequency range. We first substitute for R_PC_P in (2.169) from (2.170), obtaining

(2.171)

Utilizing the definition of Q_S in (2.163), we have

(2.172)

Substitution in (2.169) thus yields

(2.173)

So long as (which is true for a finite frequency range),

(2.174)

(2.175)

That is, the series-to-parallel conversion retains the value of the capacitor but raises the resistance by a factor of . These approximations for R_P and C_P are relatively accurate because the quality factors encountered in practice typically exceed 4. Conversely, parallel-to-series conversion reduces the resistance by a factor of . This statement applies to RL sections as well.

2.5.3 Basic Matching Networks

A common situation in RF transmitter design is that a load resistance must be transformed to a lower value. The circuit shown in Fig. 2.59(a) accomplishes this task. As mentioned above, the capacitor in parallel with R_L converts this resistance to a lower series component [Fig. 2.59(b)]. The inductance is inserted to cancel the equivalent series capacitance.

Figure 2.59 (a) Matching network, (b) equivalent circuit.

Writing Z_in from Fig. 2.59(a) and replacing s with jω, we have

(2.176)

Thus,

(2.177)-(2.178)

indicating that R_L is transformed down by a factor of . Also, setting the imaginary part to zero gives

(2.179)-(2.180)

If , then

(2.181)-(2.182)

The following example illustrates how the component values are chosen.

In order to transform a resistance to a higher value, the capacitive network shown in Fig. 2.60(a) can be used. The series-parallel conversion results derived previously provide insight here. If Q² 1, the parallel combination of C₁ and R_L can be converted to a series network [Fig. 2.60(b)], where R_S ≈ [R_L(C₁ω)²]⁻¹ and C_S ≈ C₁. Viewing C₂ and C₁ as one capacitor, C_eq, and converting the resulting series section to a parallel circuit [Fig. 2.60(c)], we have

(2.183)-(2.184)

Figure 2.60 (a) Capacitive matching circuit, (b) simplified circuit with parallel-to-series conversion, (c) simplified circuit with series-to-parallel conversion.

That is, the network “boosts” the value of R_L by a factor of (1 + C₁/C₂)². Also,

(2.185)

Note that the capacitive component must be cancelled by placing an inductor in parallel with the input.

For low Q values, the above derivations incur significant error. We thus compute the input admittance (1/Y_in) and replace s with jω,

(2.186)

The real part of Y_in yields the equivalent resistance seen to ground if we write

(2.187)-(2.188)

In comparison with Eq. (2.184), this result contains an additional component, .

The intuition gained from our analysis of matching networks leads to the four “L-section” topologies²⁴ shown in Fig. 2.62. In Fig. 2.62(a), C₁ transforms R_L to a smaller series value and L₁ cancels C₁. Similarly, in Fig. 2.62(b), L₁ transforms R_L to a smaller series value while C₁ resonates with L₁. In Fig. 2.62(c), L₁ transforms R_L to a larger parallel value and C₁ cancels the resulting parallel inductance. A similar observation applies to Fig. 2.62(d).

Figure 2.62 Four L sections used for matching.

How do these networks transform voltages and currents? As an example, consider the circuit in Fig. 2.62(a). For a sinusoidal input voltage with an rms value of V_in, the power delivered to the input port is equal to , and that delivered to the load, . If L₁ and C₁ are ideal, these two powers must be equal, yielding

(2.191)

This result, of course, applies to any lossless matching network whose input impedance contains a zero imaginary part. Since P_in = V_inI_in and P_out = V_outI_out, we also have

(2.192)

For example, a network transforming R_L to a lower value “amplifies” the voltage and attenuates the current by the above factor.

Transformers can also transform impedances. An ideal transformer having a turns ratio of n “amplifies” the input voltage by a factor of n (Fig. 2.64). Since no power is lost, and hence R_in = R_L/n². The behavior of actual transformers, especially those fabricated monolithically, is studied in Chapter 7.

Figure 2.64 Impedance transformation by a physical transformer.

The networks studied here operate across only a narrow bandwidth because the transformation ratio, e.g., 1 + Q², varies with frequency, and the capacitance and inductance approximately resonate over a narrow frequency range. Broadband matching networks can be constructed, but they typically suffer from a high loss.

2.5.4 Loss in Matching Networks

Our study of matching networks has thus far neglected the loss of their constituent components, particularly, that of inductors. We analyze the effect of loss in a few cases here, but, in general, simulations are necessary to determine the behavior of complex lossy networks.

Consider the matching network of Fig. 2.62(a), shown in Fig. 2.65 with the loss of L₁ modeled by a series resistance, R_S. We define the loss as the power provided by the input divided by that delivered to R_L. The former is equal to

(2.195)

and the latter,

(2.196)

because the power delivered to R_in1 is entirely absorbed by R_L. It follows that

(2.197)-(2.198)

Figure 2.65 Lossy matching network with series resistence.

For example, if R_S = 0.1 R_in1, then the (power) loss reaches 0.41 dB. Note that this network transforms R_L to a lower value, , thereby suffering from loss even if R_S appears small.

As another example, consider the network of Fig. 2.62(b), depicted in Fig. 2.66 with the loss of L₁ modeled by a parallel resistance, R_P. We note that the power delivered by V_in, P_in, is entirely absorbed by R_P||R_L:

(2.199)-(2.200)

Figure 2.66 Lossy matching network with parallel resistence.

Recognizing as the power delivered to the load, P_L, we have

(2.201)

For example, if R_P = 10R_L, then the loss is equal to 0.41 dB.

2.6 Scattering Parameters

Microwave theory deals mostly with power quantities rather than voltage or current quantities. Two reasons can explain this approach. First, traditional microwave design is based on transfer of power from one stage to the next. Second, the measurement of high-frequency voltages and currents in the laboratory proves very difficult, whereas that of average power is more straightforward. Microwave theory therefore models devices, circuits, and systems by parameters that can be obtained through the measurement of power quantities. They are called “scattering parameters” (S-parameters).

Before studying S-parameters, we introduce an example that provides a useful viewpoint. Consider the L₁–C₁ series combination depicted in Fig. 2.67. The circuit is driven by a sinusoidal source, V_in, having an output impedance of R_S. A load resistance of R_L = R_S is tied to the output port. At an input frequency of , L₁ and C₁ form a short circuit, providing a conjugate match between the source and the load. In analogy with transmission lines, we say the “incident wave” produced by the signal source is absorbed by R_L. At other frequencies, however, L₁ and C₁ attenuate the voltage delivered to R_L. Equivalently, we say the input port of the circuit generates a “reflected wave” that returns to the source. In other words, the difference between the incident power (the power that would be delivered to a matched load) and the reflected power represents the power delivered to the circuit.

Figure 2.67 Incident wave in a network.

The above viewpoint can be generalized for any two-port network. As illustrated in Fig. 2.68, we denote the incident and reflected waves at the input port by and , respectively. Similar waves are denoted by and , respectively, at the output. Note that denotes a wave generated by V_in as if the input impedance of the circuit were equal to R_S. Since that may not be the case, we include the reflected wave, , so that the actual voltage measured at the input is equal to + . Also, denotes the incident wave traveling into the output port or, equivalently, the wave reflected from R_L. These four quantities are uniquely related to one another through the S-parameters of the network:

(2.202)

(2.203)

Figure 2.68 Illustration of incident and reflected waves at the input and output.

With the aid of Fig. 2.69, we offer an intuitive interpretation for each parameter:

1. For S₁₁, we have from Fig. 2.69(a)

(2.204)

Thus, S₁₁ is the ratio of the reflected and incident waves at the input port when the reflection from R_L (i.e., ) is zero. This parameter represents the accuracy of the input matching.

2. For S₁₂, we have from Fig. 2.69(b)

(2.205)

Thus, S₁₂ is the ratio of the reflected wave at the input port to the incident wave into the output port when the input port is matched. In this case, the output port is driven by the signal source. This parameter characterizes the “reverse isolation” of the circuit, i.e., how much of the output signal couples to the input network.

3. For S₂₂, we have from Fig. 2.69(c)

(2.206)

Thus, S₂₂ is the ratio of reflected and incident waves at the output when the reflection from R_S (i.e., ) is zero. This parameter represents the accuracy of the output matching.

4. For S₂₁, we have from Fig. 2.69(d)

(2.207)

Figure 2.69 Illustration of four S-parameters.

Thus, S₂₁ is the ratio of the wave incident on the load to that going to the input when the reflection from R_L is zero. This parameter represents the gain of the circuit.

We should make a few remarks at this point. First, S-parameters generally have frequency-dependent complex values. Second, we often express S-parameters in units of dB as follows

(2.208)

Third, the condition in Eqs. (2.204) and (2.207) requires that the reflection from R_L be zero, but it does not mean that the output port of the circuit must be conjugate-matched to R_L. This condition simply means that if, hypothetically, a transmission line having a characteristic impedance equal to R_S carries the output signal to R_L, then no wave is reflected from R_L. A similar note applies to the requirement in Eqs. (2.205) and (2.206). The conditions at the input or at the output facilitate high-frequency measurements while creating issues in modern RF design. As mentioned in Section 2.3.5 and exemplified by the cascade of stages in Fig. 2.53, modern RF design typically does not strive for matching between stages. Thus, if S₁₁ of the first stage must be measured with R_L = R_S at its output, then its value may not represent the S₁₁ of the cascade.

In modern RF design, S₁₁ is the most commonly-used S parameter as it quantifies the accuracy of impedance matching at the input of receivers. Consider the arrangement shown in Fig. 2.70, where the receiver exhibits an input impedance of Z_in. The incident wave is given by V_in/2 (as if Z_in were equal to R_S). Moreover, the total voltage at the receiver input is equal to V_inZ_in/(Z_in + R_S), which is also equal to . Thus,

(2.209)-(2.210)

Figure 2.70 Receiver with incident and reflected waves.

It follows that

(2.211)

Called the “input reflection coefficient” and denoted by Γ_in, this quantity can also be considered to be S₁₁ if we remove the condition in Eq. (2.204).

2.7 Analysis of Nonlinear Dynamic Systems²⁵

In our treatment of systems in Section 2.2, we have assumed a static nonlinearity, e.g., in the form of y(t) = α₁x(t) + α₂x²(t) + α₃x³(t). In some cases, a circuit may exhibit dynamic nonlinearity, requiring a more complex analysis. In this section, we address this task.

2.7.1 Basic Considerations

Let us first consider a general nonlinear system with an input given by x(t) = A₁ cos ω₁t + A₂ cos ω₂t. We expect the output, y(t), to contain harmonics at nω₁, mω₂, and IM products at kω₁ ± qω₂, where, n, m, k, and q are integers. In other words,

(2.218)

In the above equation, a_n, b_n, c_m,n, and the phase shifts are frequency-dependent quantities. If the differential equation governing the system is known, we can simply substitute for y(t) from this expression, equate the like terms, and compute a_n, b_n, c_m,n, and the phase shifts. For example, consider the simple RC section shown in Fig. 2.72, where the capacitor is nonlinear and expressed as C₁ = C₀(1 + αV_out). Adding the voltages across R₁ and C₁ and equating the result to V_in, we have

(2.219)

Figure 2.72 RC circuit with nonlinear capacitor.

Now suppose V_in(t) = V₀ cos ω₁t+V₀ cos ω₂t (as in a two-tone test) and assume the system is only “weakly” nonlinear, i.e., only the output terms at ω₁, ω₂, ω₁ ± ω₂, 2ω₁ ± ω₂, and 2ω₂ ± ω₁ are significant. Thus, the output assumes the form

(2.220)

where, for simplicity, we have used c_m and φ_m. We must now substitute for V_out(t) and V_in(t) in (2.219), convert products of sinusoids to sums, bring all of the terms to one side of the equation, group them according to their frequencies, and equate the coefficient of each sinusoid to zero. We thus obtain a system of 16 nonlinear equations and 16 knowns (a₁, b₁, c₁, ..., c₆, φ₁, ..., φ₈).

This type of analysis is called “harmonic balance” because it predicts the output frequencies and attempts to “balance” the two sides of the circuit’s differential equation by including these components in V_out(t). The mathematical labor in harmonic balance makes hand analysis difficult or impossible. The “Volterra series” approach, on the other hand, prescribes a recursive method that computes the response more accurately in successive steps without the need for solving nonlinear equations. A detailed treatment of the concepts described below can be found in [10–14].

2.8 Volterra Series

In order to understand how the Volterra series represents the time response of a system, we begin with a simple input form, V_in(t) = V₀ exp(jω₁t). Of course, if we wish to obtain the response to a sinusoid of the form V₀ cos ω₁t = Re{V₀ exp(jω₁t)}, we simply calculate the real part of the output.²⁶ (The use of the exponential form greatly simplifies the manipulation of the product terms.) For a linear, time-invariant system, the output is given by

(2.221)

where H(ω₁) is the Fourier transform of the impulse response. For example, if the capacitor in Fig. 2.72 is linear, i.e., C₁ = C₀, then we can substitute for V_out and V_in in Eq. (2.219):

(2.222)

It follows that

(2.223)

Note that the phase shift introduced by the circuit is included in H(ω₁) here.

As our next step, let us ask, how should the output response of a dynamic nonlinear system be expressed? To this end, we apply two tones to the input, V_in(t) = V₀ exp(jω₁t) + V₀ exp(jω₂t), recognizing that the output consists of both linear and nonlinear responses. The former are of the form

(2.224)

and the latter include exponentials such as exp[j(ω₁ + ω₂)t], etc. We expect that the coefficient of such an exponential is a function of both ω₁ and ω₂. We thus make a slight change in our notation: we denote H(ω_j) in Eq. (2.224) by H₁(ω_j) [to indicate first-order (linear) terms] and the coefficient of exp[j(ω₁ + ω₂)t] by H₂(ω₁, ω₂). In other words, the overall output can be written as

(2.225)

How do we determine the terms at 2ω₁, 2ω₂, and ω₁ − ω₂? If H₂(ω₁, ω₂) exp[j(ω₁ + ω₂)t] represents the component at ω₁ + ω₂, then H₂(ω₁, ω₁) exp[j(2ω₁)t] must model that at 2ω₁. Similarly, H₂(ω₂, ω₂) and H₂(ω₁, −ω₂) serve as coefficients for exp[j(2ω₂)t] and exp[j(ω₁ − ω₂)t], respectively. In other words, a more complete form of Eq. (2.225) reads

(2.226)

Thus, our task is simply to compute H₂(ω₁, ω₂).

The foregoing examples point to a methodical approach that allows us to compute the second harmonic or second-order IM components with a moderate amount of algebra. But how about higher-order harmonics or IM products? We surmise that for Nth-order terms, we must apply the input V_in(t) = V₀ exp(jω₁t) + ... + V₀ exp(jω_Nt) and compute H_n(ω₁,..., ω_n) as the coefficient of the exp[j(ω₁ + ... + ω_n)t] terms in the output. The output can therefore be expressed as

(2.236)

The above representation of the output is called the Volterra series. As exemplified by (2.230), H_m(ω₁, ..., ω_m) can be computed in terms of H₁, ..., H_m−1 with no need to solve nonlinear equations. We call H_m the m-th “Volterra kernel.”

The reader may wonder if the Volterra series can be used with inputs other than exponentials. This is indeed possible [14] but beyond the scope of this book.

The approach described in this section is called the “harmonic” method of kernel calculation. In summary, this method proceeds as follows:

1. Assume V_in(t) = V₀ exp(jω₁t) and V_out(t) = H₁(ω₁)V₀ exp(jω₁t). Substitute for V_out and V_in in the system’s differential equation, group the terms that contain exp(jω₁t), and compute the first (linear) kernel, H₁(ω₁).

2. Assume V_in(t) = V₀ exp(jω₁t)+V₀ exp(jω₂t) and . Make substitutions in the differential equation, group the terms that contain exp[j(ω₁ + ω₂)t], and determine the second kernel, H₂(ω₁, ω₂).

3. Assume V_in(t) = V₀ exp(jω₁t) + V₀ exp(jω₂t) + V₀ exp(jω₃t) and V_out(t) is given by Eq. (2.237). Make substitutions, group the terms that contain exp[j(ω₁ + ω₂ + ω₃)t], and calculate the third kernel, H₃(ω₁, ω₂, ω₃).

4. To compute the amplitude of harmonics and IM components, choose ω₁, ω₂, ... properly. For example, H₂(ω₁, ω₁) yields the transfer function for 2ω₁ and H₃(ω₁, −ω₂, ω₁) the transfer function for 2ω₁ − ω₂.

2.8.1 Method of Nonlinear Currents

As seen in Example 2.34, the harmonic method becomes rapidly more complex as n increases. An alternative approach called the method of “nonlinear currents” is sometimes preferred as it reduces the algebra to some extent. We describe the method itself here and refer the reader to [13] for a formal proof of its validity.

The method of nonlinear currents proceeds as follows for a circuit that contains a two-terminal nonlinear device [13]:

1. Assume V_in(t) = V₀ exp(jω₁t) and determine the linear response of the circuit by ignoring the nonlinearity. The “response” includes both the output of interest and the voltage across the nonlinear device.

2. Assume V_in(t) = V₀ exp(jω₁t) + V₀ exp(jω₂t) and calculate the voltage across the nonlinear device, assuming it is linear. Now, compute the nonlinear component of the current flowing through the device, assuming the device is nonlinear.

3. Set the main input to zero and place a current source equal to the nonlinear component found in Step 2 in parallel with the nonlinear device.

4. Ignoring the nonlinearity of the device again, determine the circuit’s response to the current source applied in Step 3. Again, the response includes the output of interest and the voltage across the nonlinear device.

5. Repeat Steps 2, 3, and 4 for higher-order responses. The overall response is equal to the output components found in Steps 1, 4, etc.

The following example illustrates the procedure.

The procedure described above applies to two-terminal nonlinear devices. For transistors, a similar approach can be taken. We illustrate this point with the aid of an example.

References

[1] B. Razavi, Design of Analog CMOS Integrated Circuits, Boston: McGraw-Hill, 2001.

[2] L. W. Couch, Digital and Analog Communication Systems, Fourth Edition, New York: Macmillan Co., 1993.

[3] A. van der Ziel, “Thermal Noise in Field Effect Transistors,” Proc. IRE, vol. 50, pp. 1808–1812, Aug. 1962.

[4] A. A. Abidi, “High-Frequency Noise Measurements on FETs with Small Dimensions,” IEEE Trans. Electron Devices, vol. 33, pp. 1801–1805, Nov. 1986.

[5] A. J. Sholten et al., “Accurate Thermal Noise Model of Deep-Submicron CMOS,” IEDM Dig. Tech. Papers, pp. 155–158, Dec. 1999.

[6] B. Razavi, “Impact of Distributed Gate Resistance on the Performance of MOS Devices,” IEEE Trans. Circuits and Systems-Part I, vol. 41, pp. 750–754, Nov. 1994.

[7] H. T. Friis, “Noise Figure of Radio Receivers,” Proc. IRE, vol. 32, pp. 419–422, July 1944.

[8] A. Papoulis, Probability, Random Variables, and Stochastic Processes, Third Edition, New York: McGraw-Hill, 1991.

[9] R. W. Bennet, “Methods of Solving Noise Problems,” Proc. IRE, vol. 44, pp. 609–638, May 1956.

[10] S. Narayanan, “Application of Volterra Series to Intermodulation Distortion Analysis of Transistor Feedback Amplifiers,” IEEE Tran. Circuit Theory, vol. 17, pp. 518–527, Nov. 1970.

[11] P. Wambacq et al., “High-Frequency Distortion Analysis of Analog Integrated Circuits,” IEEE Tran. Circuits and Systems, II, vol. 46, pp. 335–334, March 1999.

[12] P. Wambaq and W. Sansen, Distortion Analysis of Analog Integrated Circuits, Norwell, MA: Kluwer, 1998.

[13] J. Bussganag, L. Ehrman, and J. W. Graham, “Analysis of Nonlinear Systems with Multiple Inputs,” Proc. IEEE, vol. 62, pp. 1088–1119, Aug. 1974.

[14] E. Bedrosian and S. O. Rice, “The Output Properties of Volterra Systems (Nonlinear Systems with Memory) Driven by Harmonic and Gaussian Inputs,” Proc. IEEE, vol. 59, pp. 1688–1707, Dec. 1971.

Problems

2.1. Two nonlinear stages are cascaded. If the input/output characteristic of each stage is approximated by a third-order polynomial, determine the P_1dB of the cascade in terms of the P_1dB of each stage.

2.2. Repeat Example 2.11 if one interferer has a level of −3 dBm and the other, −35 dBm.

2.3. If cascaded, stages having only second-order nonlinearity can yield a finite IP₃. For example, consider the cascade identical common-source stages shown in Fig. 2.75.

Figure 2.75 Cascade of CS stages.

If each transistor operates in saturation and follows the ideal square-law behavior, determine the IP₃ of the cascade.

2.4. Determine the IP₃ and P_1dB for a system whose characteristic is approximated by a fifth-order polynomial.

2.5. Consider the scenario shown in Fig. 2.76, where ω₃ − ω₂ = ω₂ − ω₃ and the bandpass filter provides an attenuation of 17 dB at ω₂ and 37 dB at ω₃.

Figure 2.76 Cascade of BPF and amplifier.

(a) Compute the IIP₃ of the amplifier such that the intermodulation product falling at ω₁ is 20 dB below the desired signal.

(b) Suppose an amplifier with a voltage gain of 10 dB and IIP₃ = 500 mV_p precedes the band-pass filter. Calculate the IIP₃ of the overall chain. (Neglect second-order nonlinearities.)

2.6. Prove that the Fourier transform of the autocorrelation of a random signal yields the spectrum, i.e., the power measured in a 1-Hz bandwidth at each frequency.

2.7. A broadband circuit sensing an input V₀ cos ω₀t produces a third harmonic V₃ cos(3ω₀t). Determine the 1-dB compression point in terms of V₀ and V₃.

2.8. Prove that in Fig. 2.36, the noise power delivered by R₁ to R₂ is equal to that delivered by R₂ to R₁ if the resistors reside at the same temperature. What happens if they do not?

2.9. Explain why the channel thermal noise of a MOSFET is modeled by a current source tied between the source and drain terminals (rather than, say, between the gate and source terminals).

2.10. Prove that the channel thermal noise of a MOSFET can be referred to the gate as a voltage given by 4kTγ/g_m. As shown in Fig. 2.77, the two circuits must generate the same current with the same terminal voltages.

Figure 2.77 Equivalent circuits for noise of a MOSFET.

2.11. Determine the NF of the circuit shown in Fig. 2.52 using Friis’ equation.

2.12. Prove that the output noise voltage of the circuit shown in Fig. 2.46(c) is given by .

2.13. Repeat Example 2.23 if the CS and CG stages are swapped. Does the NF change? Why?

2.14. Repeat Example 2.23 if R_D1 and R_D2 are replaced with ideal current sources and channel-length modulation is not neglected.

2.15. The input/output characteristic of a bipolar differential pair is given by V_out = −2R_CI_EE tanh[V_in/(2V_T)], where R_C denotes the load resistance, I_EE is the tail current, and V_T = kT/q. Determine the IP₃ of the circuit.

2.16. What happens to the noise figure of a circuit if the circuit is loaded by a noiseless impedance Z_L at its output?

2.17. The noise figure of a circuit is known for a source impedance of R_S1. Is it possible to compute the noise figure for another source impedance R_S2? Explain in detail.

2.18. Equation (2.122) implies that the noise figure falls as R_S rises. Assuming that the antenna voltage swing remains constant, explain what happens to the output SNR as R_S increases.

2.19. Repeat Example 2.21 for the arrangement shown in Fig. 2.78, where the transformer amplifies its primary voltage by a factor of n and transforms R_S to a value of n²R_S.

Figure 2.78 CS stage driven by a transformer.

2.20. For matched inputs and outputs, prove that the NF of a passive (reciprocal) circuit is equal to its power loss.

2.21. Determine the noise figure of each circuit in Fig. 2.79 with respect to a source impedance R_S. Neglect channel-length modulation and body effect.

Figure 2.79 CS stages for NF calculation.

2.22. Determine the noise figure of each circuit in Fig. 2.80 with respect to a source impedance R_S. Neglect channel-length modulation and body effect.

Figure 2.80 CG stages for NF calculation.

2.23. Determine the noise figure of each circuit in Fig. 2.81 with respect to a source impedance R_S. Neglect channel-length modulation and body effect.

Figure 2.81 Stages for NF calculation.