Statistical Design and Optimization for Amplifiers

This chapter provides details on the statistical design and optimization schematics and data displays included in the Amplifier DesignGuide. They are accessed from the ADS Schematic window, as follows:
Design Guide > Amplifier > Amplifier Statistical Design

Your amplifier will never be constructed with exactly the same parameter values that you specify on the schematic. Furthermore, no two constructed amplifiers will have all the same parameter values. Statistical design puts your amplifier in this kind of parametrically varying environment, analyzing and optimizing the resulting performance statistics.

For a table of the available statistical design data displays, with cross-reference links to the pages where they are documented, refer to Amplifier Statistical Design in About Amplifier DesignGuides.

Note We assume you are familiar with the use of Advanced Design System and have successfully used the Amplifier DesignGuide to develop a working amplifier design. For complete information on statistical design features of ADS, refer to the Tuning, Optimization and Statistical Design documentation, Summary of Optimizers and Using Statistical Design.

Overview of Techniques

Statistical analysis is the basis for statistical design. Statistical analysis is the process of varying a set of parameter values within your amplifier design, using specified probability distributions, and determining how your amplifier's performance will vary as the parameters vary. Many possible combinations of parameters are analyzed in your amplifier and the resulting performance variations, or performance statistics, are determined.

Yield is an important unit of measure for statistical design. It is defined as the ratio of the number of amplifiers that pass the performance specifications to the total number of amplifiers that are analyzed during a statistical analysis. Yield also is the probability that a given amplifier design sample will pass the performance specifications.

Because the total number of amplifiers to be manufactured may be large or unknown, yield is usually estimated over a smaller number of samples, or trials, in the process known as yield estimation. As the number of trials becomes large, the yield estimate approaches the true design yield. Parameter values that have statistical variations are referred to as statistical variables or statistical parameters.

Note Yield cannot be calculated exactly, only estimated. This gives yield analysis and optimization a statistical nature that is not present in standard fixed-parameter performance analysis and optimization.

Two statistical design options are available:

• Yield analysis: This process involves simulating the amplifier over a given number of trials, with the statistical parameter values varying randomly about their nominal values according to specified probability distribution functions. The numbers of passing and failing trials are recorded and these numbers are used to compute an estimate of the yield.
• Yield optimization: Also known as design centering, this process involves multiple yield analyses with the goal of adjusting the statistical parameters' nominal values to maximize the yield estimate. During yield optimization, each yield improvement is referred to as a design iteration.

This section of the Amplifier DesignGuide simplifies the application of yield analysis and yield optimization to your amplifier design. After using this DesignGuide to statistically analyze and optimize your amplifier design, the result will be a design that is less sensitive to the types of parameter variations that will be encountered during manufacturing. This gives a higher yield design.

Yield Analysis

Yield analysis numerically estimates the sensitivity of your amplifier's performance to parameter variations that are defined by you on the amplifier's schematic. There are five yield analysis schematics included in this DesignGuide to allow you to easily accomplish yield analysis on your amplifier design. Yield analysis randomly varies circuit parameter values according to statistical distributions while comparing each amplifier's measurements to the user-specified performance criteria found in the YieldSpec block on the amplifier schematic.

Yield analysis is based on the Monte Carlo method. A series of trials is run in which random values are assigned to all of your design's statistical parameters, a simulation is performed, and the yield specifications are checked against the simulated performance. The number of passing and failing simulations is accumulated over the set of trials and used to compute the yield estimate. Yield analysis is shown graphically in Flow diagram for yield analysis.

Flow diagram for yield analysis

Other capabilities of yield analysis include the following:

• Accumulated sets of selected amplifier responses can be viewed or plotted.
• Performance histograms display the distribution of measured amplifier responses and statistical sensitivities
• Overall performance variation can be assessed.

  Note The only parameters included in a statistical analysis are the ones assigned as statistical parameters in the VAR block of your amplifier's schematic.

Yield Optimization

Yield optimization minimizes the sensitivity of your amplifier's performance to the component variations that are assigned on the amplifier schematic. There are five yield optimization schematics included in this DesignGuide to allow you to easily accomplish yield optimization on your amplifier design. Yield optimization (essentially) estimates yield and yield sensitivities and changes the circuit statistical parameter nominal values in order to simultaneously minimize statistical sensitivity and maximize circuit yield. (For information on statistical sensitivity, refer to Statistical Sensitivity.) This process is done in a step-wise fashion with each step called a design iteration. This can be a user specified parameter, although it is initially set for you in this DesignGuide.

Note The only parameters that are changed during yield optimization are those statistical parameters designated as optimization variables on your amplifier schematic.

Each design iteration will require many yield analyses (Monte Carlo trials). The number of yield analyses is a dynamic variable computed during yield optimization, varying with changing yield estimates and confidence levels. Therefore, the yield estimate derived from yield optimization often differs from that for a single yield analysis with a user-specified number of trials. To have control over the confidence level and hence the accuracy of the yield estimate, it is recommended that you perform a yield analysis after the yield optimization is completed, using the nominal parameter values obtained from the yield optimization. Choose an appropriate number of trials based upon your understanding of confidence intervals, which are explained later in this manual. For more information, refer to the ADS Tuning, Optimization and Statistical Design documentation, Chapters 3 and 6.

Yield Analysis Displays: YSH, MH, SRP

Following are descriptions of the yield analysis displays.

Yield Sensitivity Histogram, YSH

A key to understanding, communicating and performing statistical design and optimization is the Yield Sensitivity Histogram (YSH). The Yield Sensitivity Histogram is a graph of yield, on the vertical axis, versus a circuit parameter's (stepped) values on the horizontal axis.

Note Of all the statistical data displays, the YSH is usually the most helpful because it shows which parameters affect the amplifier's yield and how possibly to change the parameters to increase yield.

The Yield Sensitivity Histogram gives an indication of whether the design is at maximum yield (a centered design) or whether the design needs to be yield optimized (an uncentered design). The Yield Sensitivity Histograms also tell the designer which parameters in the design affect the design yield and need to be included in the yield optimization. An example YSH is shown in Yield Sensitivity Histogram. The vertical axis (0-100) is yield, and the horizontal axis (30-42) is the range of parameter values used for yield analysis.

Yield Sensitivity Histogram

The YSH is really a parametric study of yield versus one of your amplifier's parameter values. The parameter value being graphed is (virtually) not a statistical variable in a YSH but all other parameters are allowed to vary according to their assigned statistical distributions, and yield is calculated for each step as the (virtual) fixed parameter is stepped across its allowable range of variation. The YSH is the graph of the estimated yield versus each of the stepped parameter values. For example, looking at Yield Sensitivity Histogram, when curVar (the value of a given circuit element parameter, like a capacitance or inductance) is fixed at 32, the estimated circuit yield is approximately 95%. When curVar is fixed at 40, the estimated yield is 44%.

Note The lower limit (LL) and the upper limit (UL) used on the YSH plot axes are the upper and lower extent of the statistical parameter as defined on the amplifier schematic. Only statistical parameters may be plotted using a YSH.

If the YSH is essentially flat, then the parameter over the range from the lower limit (LL) to the upper limit (UL), does not affect the amplifier's yield. This is shown in the bottom right graph in How to Use the Yield Sensitivity Histogram. In this case we say the parameter is centered. It may not be necessary to include this parameter in yield optimization, as it (on its present range) has no effect on yield. It might also be possible to increase the tolerance of this parameter without decreasing the yield. We say that this parameter is centered.

How to Use the Yield Sensitivity Histogram

If the YSH slopes, as in the top two graphs in How to Use the Yield Sensitivity Histogram, the parameter affects the yield value, and we say the parameter is not centered. Moving the parameter's nominal value to a value of higher yield may increase the amplifier's overall yield.

Note Each rectangle in a YSH is called a bin. The height of each bin is a yield estimate using the measurements from the trials with parameter values within the interval covered by the bin's base. Confidence intervals can be given for each bin's height.

If the YSH is high in the center and lower on the extremes, like the lower left graph in How to Use the Yield Sensitivity Histogram, the upper and lower limits (UL and LL)must be brought in to decrease the statistical extent of the parameter. The extent of a parameter's variation is its tolerance, and in this case the parameter tolerance should be reduced.

Note You can reduce a parameter's tolerance by going to the amplifier schematic page and reducing the extent of the parameter's variation by changing its statistical definition.

Statistical Sensitivity

Statistical Sensitivity is a very important concept in statistical design. Looking at the Yield Sensitivity Histogram in Statistical Sensitivity., the statistical sensitivity is the slope of the Yield Sensitivity Histogram. A parameter whose YSH has a large slope, like shown in this figure, is said to be a statistically sensitive parameter.

Statistical Sensitivity.

Note Because each bin height represents a yield estimate, an YSH using a small number of trials can be rough and erratic. This is always due to numerical estimation errors. The true yield versus parameter plots will always be smooth functions.

The statistical sensitivity of your amplifier is only measured over the assigned tolerance range of the statistical parameters. Starting with wide tolerances will measure sensitivity over a wider parameter range. However, wide tolerances may reduce yield to too small a value.
The idea of statistical sensitivity reduction, which is central to design centering, is illustrated in Statistical Sensitivity Reduction, with YSH before design centering (left), and YSH after design centering (right).

Statistical Sensitivity Reduction

From the graph on the left, we see the parameter nominal value (the center of the YSH) is 35Ω and that when the value is above 36Ω, the yield is zero. As the parameter value decreases from 35Ω, the yield increases. From the slope of this YSH, we see that there is a large statistical sensitivity to this parameter value. The after statistical optimization, YSH for this parameter is shown in the graph on the right. After yield optimization (design centering) we see that this parameter's YSH has no dominant slope in either direction, and the parameter is therefore considered to be centered. From the graph on the right, we can be seen that the YSH decreases in both directions from the nominal value of 28Ω. Therefore reducing this parameter's tolerance may also increase the yield.) The estimated yield corresponding to the left graph is approximately 25%, while the estimated yield corresponding to the right graph is approximately 86%.

Note You will want to know the statistical sensitivity of every statistical parameter in your design. You can graph up to four YSH's in a data display at a time.

Measurement Histogram, MH

A measurement histogram is a histogram graph of the number (or percentage) of occurrences of a measurement versus the measurement values. An example is given in Measurement Histogram Example. The histogram gives the spread of measurement values that were encountered during yield analysis. The measurement (like dB(S11)) on the interval 4.0 to 3.5 occurred during 18% of the circuit simulation trials. Also the extent of this measurement over all the circuit simulation trials was from 7.0 to 2.5 with the most measurements occurring between 3.5 and 3.0.

Measurement Histogram Example

Interpreting the Measurement Histogram

The measurement histogram displays the measurement value variations that are possible and the number of each binned value that occurred, due to the statistical variations given to the amplifier parameters.

Note MH's can help you set the amplifier's specifications for an acceptable yield value. Just set the specification to include the desired percentage of measurements. This may be necessary in the beginning of yield optimization because a yield value around 50% is best when starting yield optimization.

Statistical Response Plot, SRP

A statistical response plot is a superimposed plot of the responses encountered during the yield analysis simulation due to parameter variations. Usually each individual response is plotted versus the independent variable, like S11 versus frequency. It gives a measure of the response variations that occur due to the defined statistical parameters. An example of a statistical response plot is shown in Statistical Response Plot.

Statistical Response Plot

Sigma plots present the mean response, +1 standard deviation response and -1 standard deviation response. An example of a Sigma Plot corresponding to the SRP is shown in Sigma Plot.

Sigma Plot

Similar to the measurement histogram, the Statistical Response Plot displays the type of measurement variations that are possible, due to the statistical variations given to the amplifier parameters.

Statistical Design Methodology

An approach to Statistical Design includes four major steps:

• Step 1 - Develop and single-point optimize the amplifier design.
• Step 2 - Perform yield analysis.
• Step 3 - Determine if the yield is acceptable and access any statistical sensitivity among the statistical variables.
• Step 4 - If necessary, yield optimize and re-analyze yield.

Developing and Single-Point Optimizing the Amplifier Design

This step of the process is referred to as single-point design, and is the classical amplifier design process. This Amplifier DesignGuide is an excellent aid to doing the single-point design. The end result of this step is

• A desired amplifier circuit structure
• A set of parameter values which define each element in your amplifier design with a single-valued number.
• A circuit structure and parameter values which give acceptable (or even optimal) performance for your amplifier.
  A proper single-point design is a requirement to begin the statistical design process.

Performing Yield Analysis

Next, we begin the statistical design process by assessing the statistical sensitivity of your single-point amplifier design. First, assign statistical variations (distributions) to all critical amplifier parameters. The distribution options presently available are Uniform and Gaussian. (See Choosing Parameter Statistics.)

Note The entire yield analysis and statistical sensitivity assessment process might be done first on the input and then on the output of your amplifier. For example assign statistical variables to the input of your amplifier and assess the sensitivity of your amplifier's input component parameters, then the bias, then the output, then finally to all the critical parameters together. It might enhance your understanding when analysis and optimization are performed in this manner.

To properly assess the statistical sensitivity of your amplifier, it is necessary that the yield not be zero, and not be 100%. A good yield to assess sensitivity is around 50% (30% to 70%). If your yield is not about 50%, then do one of these two things or a combination of:

• Change the performance specifications for the amplifier. Relaxing the performance specifications will almost always increase the yield. (Look at your amplifier's MH's to help you set specifications.)
• Change the tolerance of some of the critical statistical variables. Decreasing the tolerances will almost always increase the yield.
  As you adjust the specifications and tolerances on your amplifier, you will learn a lot about the statistical properties of your design, and how it will likely perform in a manufacturing environment.

Assessing Statistical Sensitivity among the Statistical Variables

After a successful yield analysis, the data displays, specifically the Yield Sensitivity Histogram, give the statistical sensitivity of each parameter in your amplifier. This is gotten by visually examining the YSH's for each statistical parameter. The slope of the YSH and the shape of the YSH give the statistical sensitivity as already explained. If no variable exhibits sensitivity, it is unlikely that yield optimization will increase your amplifier yield. (If no variable exhibits sensitivity, you might want to increase the tolerance of your parameters, or consider defining new statistical parameters to your design.) However, if sensitivity is observed, yield optimization will likely increase the amplifier yield and simultaneously decrease the amplifier's statistical sensitivities.

Note Using this DesignGuide, you can view four YSH's at a time. After viewing the four, change curVar1, curVar2, curVar3 and curVar4, to four new statistical parameter names and view their YSH's. The data displays are automatically updated when curVar is defined with a new statistical parameter name. Do this until all the statistical parameter YSH's have been viewed and recorded.

Optimizing and Re-Analyzing Yield (Optional)

Enter yield optimization with a design exhibiting less than 100% yield and more than 0% yield, with 50% yield a good starting point. Be sure to include the most sensitive statistical variables as optimization variables. This is done by assigning the optimization property to these variables on your amplifier schematic. This DesignGuide simplifies the optimization process.

Note Optimization will involve a large number of circuit simulations, so optimization will likely take 100 to 1000 times, or more, longer than a single circuit simulation.

The results of optimization are a new set of statistical variable nominal values which give increased circuit yield and reduced statistical sensitivities. Since we have not included data displays to directly plot the results of the yield optimizations, it is necessary to save the optimized parameter values (choose Simulate , then Update Optimization Parameters ), then choose the appropriate Statistical Analysis schematic, and perform statistical analysis. The data displays are usable from the analysis schematics. This is Step 2. Then proceed again with Step 3.

This whole process is iterative, stopping when acceptable statistical performance is achieved, or when no better statistical performance is achievable. If the latter is true, perhaps the circuit structure, or parameter tolerances, can be changed to give a better statistical performance.

Note The matching structure can have an effect on yield. For instance if both a series-C parallel-L series-C and a parallel-L series-C parallel-L matching structures will accomplish the match, one will likely give a higher yield.

Using the Statistical Simulations

For a detailed description of all the statistical design features, refer to the Tuning, Optimization and Statistical Design documentation, Chapters 3 and 6. The Amplifier DesignGuide's statistical section includes many useful simulation setups and data displays for amplifier statistical design. The simulation setups are characterized by:

• The type of measurement desired, and
• Whether analysis or optimization is desired.

Both linear and nonlinear simulations are possible. The data displays accompany the simulations and give many useful formats for viewing the results of your statistical analysis and optimization.

Note The Amplifier DesignGuide's statistical section is a helpful aid to performing statistical analysis and design on your amplifier, however it is by no means exhaustive in its scope. We have tried to include the most useful and frequently used schematics and displays.

The statistical features and content of the Amplifier DesignGuide are accessible from the ADS Schematic window by selecting DesignGuide > Amplifier DesignGuide > Amplifier Statistical Design . The Amplifier Statistical Design menu is arranged as follows:

The available statistical design data displays are grouped by their association to the schematics. For each analysis schematic, there are available Yield Sensitivity Histograms (YSH), Measurement Histograms (MH), and Statistical Response Plots (SRP). For a table of the available statistical design data displays, with cross-reference links to the pages where they are documented, refer to Amplifier Statistical Design in About Amplifier DesignGuide.

Note This DesignGuide's statistical data displays are not directly usable from the optimization schematics. After performing optimization the optimization parameters must be updated to the schematic, the optimized circuit must be analyzed, and then data displays are available, showing the optimization results.

Using the DesignGuide Schematics

After choosing your schematic from the Amplifier Statistical Design menu, prepare it for statistical analysis, as follows:

1. Step 1 - Insert your amplifier by first pushing into the sample amplifier subcircuit and entering or pasting your amplifier schematic, as you've done before when using the Amplifier DesignGuide.
2. Step 2 - Assign variable names to all the statistical variables on the amplifier schematic.
3. Step 3 - Assign statistical distributions to all named variables using the VAR block on the amplifier schematic.
4. Step 4 - Configure the YIELD and YIELD SPEC blocks.

Steps 2-4 are detailed in the ADS Tuning, Optimization and Statistical Design documentation.

Selecting the Appropriate Simulation Schematic

It is important to first perform statistical analysis on your amplifier design. Of the five analysis schematics, three are for linear analysis and two are for non-linear analysis. Associated with each are a set of specified measurements. After choosing between linear and nonlinear measurements, choose among the measurement options to determine the exact schematic to use. After analysis is complete, if optimization is necessary, use the corresponding (linear/nonlinear and measurement type) optimization schematic.

Selecting the Appropriate Data Display

After the statistical analysis is complete, open a data display corresponding to the measurements used in your analysis. For example, if your analysis calculated S-parameters, group delay and noise figure, then choose data displays that use these measurements. In this case there are two: a data display showing one YSH and a data display showing four YSH's.

Note As the data display is opened, the data is processed before the window completes opening. This data processing can sometimes take minutes to finish, especially with the SRP's. Therefore the data displays often do not open quickly.

Choosing Parameter Statistics

Assigning component statistics is important for both statistical analysis and optimization. There are two approaches here:

• Choose parameter statistical distributions that accurately model the manufacturing environment, with the goal of getting yield estimates that accurately predict the yield which will be encountered during manufacturing.
• Choose parameter statistics that sufficiently measure the statistical sensitivity of the amplifier design, with the goal of measuring and reducing statistical sensitivity. In this case the yield estimate is more a measure of statistical sensitivity and sensitivity reduction than it is of actual yield encountered during manufacturing.

It's clear that the second approach does not put as much emphasis on the statistical models used, or the ability of the yield estimate to predict actual yield during manufacture.
It's always best to use as much knowledge about the manufacturing environment as you have. For instance if you are sure that some parameters will only have variation of +/- 1%, use this variation in your statistical model. There is no need to check the sensitivity of your design to parameter variations that will not be encountered during manufacturing. Making the parameter tolerances match your understanding of the manufacturing environment will let the yield optimizer be more effective in determining a set of insensitive parameter values for your design.

Note In your initial use of statistical design, it will be helpful to think of the process as one of measuring and reducing statistical sensitivity, rather than one of accurately predicting and maximizing the actual manufacturing yield.

If statistical sensitivity measurement and reduction are the goals, it is usually best to use uniformly distributed parameter statistics. The uniform distribution will

• Effectively explore your amplifier's performance over all combinations of parameter values, with equal statistical weight on each possible combination. This makes for effective exploration of the parameter and performance spaces of your amplifier.
• Give the most accurate and easy to read YSH's, because each bin of the YSH will have approximately the same number of simulations. This gives nearly the same statistical confidence for each bin, and therefore the confidence for the YSH is essentially constant everywhere.

  Note If a YSH bin has only a few simulations in its calculation, the confidence interval associated with that bin's calculation is large, while if a YSH bin has a large number of simulations in its bin, the confidence interval associated with that bin's calculation is small. Having a similar number of simulations in each bin of the YSH, which is what happens when you use the uniform distribution, gives a similar confidence interval for the entire YSH.

  However, the uniform distribution will likely give lower yield estimates than when the Gaussian distribution is used.

  Note Be careful when interpreting the YSH when the parameter being graphed has a Gaussian distribution. The bins at the outer extent of the YSH can represent as few as one or two trials, and therefore the yield estimate for these outer bins can have very large errors.

Value Types for Statistical Design

As described in Specifying Component Parameters for Yield Analysis in the Tuning, Optimization and Statistical Design documentation, the Statistics tab of the Setup dialog box is used to enable or disable the yield analysis status of a parameter and to specify the type and format for the parameter range over which yield analysis is to take place. In the Statistics tab, the Type drop-down list includes the following options:
Gaussian. Denotes a Gaussian distributed statistical variable that can be one of two types, which are selected from the Format drop-down list, as follows:

• +/- Delta %. Specifies the +/- 1 sigma deviation range as a percentage of the nominal value.
• +/- Delta . Specifies the +/- 1 sigma deviation value as an absolute value.
  Uniform. Denotes a variable that can be one of three types, which are selected from the Format drop down list, as follows:
• min/max. Allows you to specify a nominal value, minimum value, and maximum value and to specify appropriate units for each
• +/- Delta %. Spe cifies the deviation range as a percentage of the nominal value.
• +/- Delta. Specifies the deviation value as an absolute value

Discrete. Denotes a discrete uniform statistical variable. The set of discrete values is directly specified when you enter nominal value, minimum value, maximum value, and a step value. Notice that for this option, the Format drop-down list only includes min/max/step.

Yield Analysis Schematics

This section contains a description of each of the five analysis schematics in this DesignGuide. Included also are descriptions of their associated data displays.