7.4: Cascaded Module Design Using the Contribution Method

The contribution method is an approach to the design of cascade RF systems for maximum SFDR [6] rather than separate treatment of noise and nonlinear distortion. The contribution method provides a good initial assignment of the noise figure, gain, and required linearity to individual stages and enables informed assessment of trade-offs during system design.

In the budget method, noise performance is largely determined by the first stage and the nonlinear performance by latter stages. This often results in suboptimum system design. One of the costs is often the use of more expensive stages than necessary or higher linearity requirements of some stages leading to increased supply power.

The contribution method assigns to each active stage a percentage contribution to the overall SFDR. Contributions of the stages are balanced as no single stage dominates noise or distortion, and all contributions to SFDR are equally weighted. In realizing a cascaded system, typically one or more offending stages either dominate the SNR or significantly degrade overall distortion. In some cases a redistribution of stage parameters (such as gain) reduces the degradation. In other cases a change in architecture is required.

The mathematics behind the contribution approach is based on an extension of the noise and distortion analyses of cascaded stages. The noise performance of a stage is characterized by its noise factor \(F = N_{o}/(GN_{i})\), where \(G\) is the available power gain of the stage, \(N_{o}\) is the output noise power, and \(N_{i}\) is the input noise power of a resistor held at standard temperature (\(290\text{ K}\)). Distortion is characterized by the amount of intermodulation distortion (IMD) produced in a two-tone test and specifically the third-order intercept point (\(\text{IP3}\)) (see Figure 7.3.1). In the following development a passive linear stage such as a filter or attenuator is combined with the following active stage.

7.4.1 Noise Contribution

Friis’s formula yields the noise factor of a cascaded system and was derived in Section 4.3.1. This section presents an alternative development of Friis’s formula with interim steps that can be used in calculating the dynamic range contributions of each stage. The analysis here was originally presented in [6].

First consider the contribution of individual stages to system noise, and assume that all stages are matched. The \(n\)th stage, with all ports having equal bandwidth \(B\), has output noise power (from Equation (4.3.10)

\[\label{eq:1}N_{no}=F_{n}G_{n}kT_{0}B \]

where \(k\) is the Boltzmann constant, and \(F_{n}\) and \(G_{n}\) are the noise factor and available power gain of the \(n\)th stage, respectively. The excess noise power of successive stages is additional to that of the first. So, while the output noise power contribution of the first stage is

\[\label{eq:2}N_{1o}=F_{1}G_{1}kT_{0}B \]

the excess output noise power of the second stage is

\[\label{eq:3}N_{2o}=(F_{2}-1)kT_{0}BG_{2} \]

Note that the output noise of the first stage includes the noise contribution of the source resistance held at the standard temperature. However, this is not included in the noise contributed by the second or latter stages. The total noise power at the output of a two-stage cascade is

\[\label{eq:4}N_{2o}^{\text{T}}=(F_{2}-1)kT_{0}BG_{2}+N_{1o}G_{2} \]

Then the total noise power at the output of the \(m\)th stage is

\[\label{eq:5}N_{mo}^{\text{T}}=\sum_{n=2}^{m}\left[ (F_{n}-1)kT_{0}B\prod_{i=2}^{n}G_{i}\right] +F_{1}kT_{0}B\prod_{n=1}^{m}G_{n} \]

Thus an \(m\)-stage cascade has total cascaded system noise factor \(F^{T} = N_{mo}^{T}/(G^{T}N_{1i})\), with \(G^{T}\) being the total cascaded available gain and \(N_{1i}\) the noise power input to the first stage. In terms of the parameters of individual stages

\[\label{eq:6}F^{\text{T}}=F_{1}+\sum_{n=2}^{m}\frac{F_{n}-1}{\prod_{i=n}^{m}G_{i-1}} \]

The link between Equations \(\eqref{eq:5}\) and \(\eqref{eq:6}\) enables the noise contribution of each stage to be determined. Treated separately, Equation \(\eqref{eq:5}\) provides the output noise power of a cascade such as a receiver. Although the extension of Equation \(\eqref{eq:5}\) to Equation \(\eqref{eq:6}\) is normally associated with the derivation of the receiver noise factor, it can also be used in transmitter noise analysis.

Now the noise contribution of a stage can be defined. The gain accumulated at the \(j\)th stage (the total cascade gain up to and including the \(j\)th stage) is

\[\label{eq:7}G_{j}^{\text{A}}=\prod_{n=1}^{j}G_{n} \]

(and so \(G^{T} = G_{m}^{A}\) for an \(m\)-stage cascade). The fractional noise contribution of the \(j\)th stage to the total output noise is then defined as

\[\label{eq:8}C_{j}^{N}=\text{ stage noise contribution }=\left[\frac{F_{j}-1}{(G_{j}^{\text{A}}/G_{j})F^{\text{T}}}\right] \]

This is one component of the SFDR contribution of a stage.

7.4.2 Intermodulation Contribution

The intermodulation distortion of a cascade is assessed using the cascade intercept method (see Section 7.2.2). The usual design approach of establishing the SNR in the early stages of the cascade (this is the budget method) results in stages further along the cascade needing to have higher intercept values. Consequently these stages will generally have additional power consumption, as this is usually required to increase \(\text{IIIP3}\) (or \(\text{OIP3}\)). Trade-offs of the contributions of individual stages to distortion and noise will lead to constrained total power consumption while still achieving the required system SFDR.

For receiver systems the contribution to nonlinear distortion is captured by \(\text{IIP3}\) and for all stages this will be referred to the input of the cascaded system. Keeping with the nomenclature of the previous section, the accumulated \(\text{IP3}\) of the \(j\)th stage referred to the system input is

\[\label{eq:9}\text{IIP3}_{j}^{\text{A}}=\text{IIP3}_{j}/G_{j-1}^{\text{A}} \]

and is \(\text{IIP3}_{\text{dBm },j}^{A}\) when expressed in \(\text{dBm}\). Note that all prior gain and loss up to the \(j\)th stage modifies the intercept when it is referred to the input of the cascaded stages. Combining the contribution of individual stages using Equation (7.2.16) yields the total system \(\text{IIP3}\):

\[\label{eq:10}\text{IIP3}^{\text{T}}=\left(\sum_{n=1}^{m}\frac{1}{\text{IIP3}_{n}^{\text{A}}}\right)^{-1} \]

This leads to the definition of the fractional contribution of the \(j\)th stage to the system \(\text{IIP3}\):

\[\label{eq:11}C_{j}^{\text{IIP3}}=\text{ IIP3 contribution }≡\frac{\text{IIP3}^{\text{T}}}{\text{IIP3}_{j}^{\text{A}}} \]

This is the second and final component of the SFDR contribution of a stage.

7.4.3 Design Methodology for Maximizing Dynamic Range

In this section a design methodology is developed for trading off the performance of each stage in maximizing the SFDR of the RF down-converter module shown in Figure 7.3.2. A central component of the methodology is the use of a stage contribution graph that indicates performance of each stage. This graph enables visualization of the contribution of each stage to system dynamic range, identifying which stage or stages dominate performance. For example, the initial contributions for the example used in the case study that follows are shown in Table \(\PageIndex{1}\). The noise and distortion contributions indicate the stage that tends to have the most impact on noise factor or distortion. Redistribution of gain, noise, or distortion alters this relationship and leads to changes in system SFDR. Passive stages generally do not introduce distortion, but they do contribute noise. So, to simplify the following discussion, passive stages immediately preceding an active stage and the active stage itself will be considered as a single stage.

First of all, consider the traditional budget method approach in which the noise figure of a receiver cascade is established by maximizing the gain of the first stage, and thus the first stage establishes the SNR of the system since the noise contributions of subsequent stages are assumed to be negligible. With the system SNR fixed, a target system SFDR determines the overall \(\text{IP3}\) performance described by the total system \(\text{IIP3}\), \(\text{IIP3}^{\text{T}}\). Also, in the budget method, the first stage has negligible impact on overall \(\text{IIP3}\). Then a reasonable choice in the design process is to select stages following the first as contributing equally to the reduction in \(\text{IIP3}^{\text{T}}\). Thus, in the budget method, the minimum acceptable \(\text{IIP3}\) (in \(\text{dBm}\)) of the \(j\)th stage (i.e., \(\text{IIP3}_{\text{dBm, }j}\)) in an \(m\)-stage cascade that is required to meet the target total \(\text{IIP3}\) in \(\text{dBm}\)

Stage \((i)\)	Gain \((\text{dB})\)	Gain \(\text{G}_{i}\)	\(\text{NF}_{i}\) \((\text{dB})\)	\(\text{F}_{i}\)	\(\text{IIP3}_{i}\) \((\text{dBm})\)
\(1\) filter	\(-3.98\)	\(0.4\)	\(3.98\)	\(2.5\)	\(-\)
\(2\) LNA	\(1.76\)	\(1.5\)	\(3.01\)	\(2.0\)	\(-0.969\)
\(3\) filter	\(-3.98\)	\(0.4\)	\(3.98\)	\(2.5\)	\(-\)
\(3\) mixer	\(-\)	\(-\)	\(2.04\)	\(1.6\)	\(-3.18\)

Table \(\PageIndex{1}\): Stage assignments based on balanced contributions to the SFDR of the cascade. Initial assignment, \(i = 1\).

\((\overline{\text{IIPS}}^{\text{T}}_{\text{dBm}})\) is obtained from Equations \(\eqref{eq:9}\) and \(\eqref{eq:10}\) as

\[\label{eq:12}\text{IIP3}_{\text{dBm, }j}=\text{IIP3}_{\text{dBm, }j}^{\text{A}} +G_{\text{dB, }(j-1)}^{\text{A}} \]

where

\[\label{eq:13}\text{IIP3}_{\text{dBm, }j}^{\text{A}}=\overline{\text{IIP3}}_{\text{dBm}}^{\text{T}}+10\log m \]

(Note that the overline in \(\overline{\text{IIP3}}_{\text{dBm}}^{\text{T}}\) identifies the target \(\text{IIP3}_{\text{dBm}}^{\text{T}}\) and not the actual \(\text{IIP3}\).) In addition, the gain or loss of preceding stages will modify this \(\text{IIP3}\) as indicated in Equation \(\eqref{eq:9}\). That is, with this assignment each stage makes an equal contribution to the overall intermodulation distortion. That is, each stage has the same \(\text{IP3}\) referred to the system input. For example, in a cascade of three stages, each stage would need to have a minimum system \(\text{IIP3}\) of \(4.8\text{ dB}\) \((= 10 \log 3)\) in excess of the target system \(\text{IIP3}\) (i.e., \(\text{IIP3}_{\text{dBm}}^{\text{T}}\)).

If instead, using the contribution method, the cascade system was designed so that each stage contributed equally to the overall noise (\(C_{j}^{N}\) being the same for all stages), and each stage had the same gain (\(G_{j}\) being the same for all stages), then the required individual \(\text{IIP3}\) values tend to be minimum. Maximizing system dynamic range becomes an exercise in maintaining the lowest noise power and highest \(\text{IIP3}\) value throughout the cascade. The link between these parameters values is the distribution of gain and loss [7, 8].

For a transmitter, the SFDR is most commonly referred to the output and (repeating Equation (4.6.8))

\[\label{eq:14}\text{SFDR}_{\text{dB, }o}=\frac{2}{3}(\text{OIP3}_{\text{dBm}}-N_{\text{dBm, }o}) \]

and \(\text{SFDR}_{i} = \text{SFDR}_{o}\). In the above, \(N_{i}\) and \(N_{o}\) are the total input and output noise powers and are assigned a noise floor value dependent on the cascade noise factor target and linear system gain.

7.4.4 Summary

The contribution method for designing systems of cascaded modules focuses on assigning the same dynamic range to each module. Of course linear modules such as a filter easily meet any system dynamic range requirement so the dynamic range assignments should focus on the nonlinear modules. Very often the dynamic range of an active module can be increased by either increasing the biasing of the module or switching to an alternative module but with high power requirements. Sometimes modules with different technology could provide increased dynamic range without increasing power consumption but such modules could be more expensive. The contribution method and the previously considered budget method only provide an initial starting point for system design. The system must still be optimized and often manually optimized as there are too many hard to quantify design goals. For example minimizing time-to-market and design cost are goals that cannot be parameterized.