Documentation 
Response of singlerate, fixedpoint IIR filter
report = limitcycle(hd)
report = limitcycle(hd,ntrials,inputlengthfactor,stopcriterion)
report = limitcycle(hd) returns the structure report that contains information about how filter hd responds to a zerovalued input vector. By default, the input vector has length equal to twice the impulse response length of the filter.
limitcycle returns a structure whose elements contain the details about the limit cycle testing. As shown in this table, the report includes the following details.
Output Object Property  Description 

LimitCycleType  Contains one of the following results:

Zi  Contains the initial condition value(s) that caused the detected limit cycle to occur. 
Output  Contains the output of the filter in the steady state. 
Trial  Returns the number of the Monte Carlo trial on which the limit cycle testing stopped. For example, Trial = 10 indicates that testing stopped on the tenth Monte Carlo trial. 
Using an input vector longer than the filter impulse response ensures that the filter is in steadystate operation during the limit cycle testing. limitcycle ignores output that occurs before the filter reaches the steady state. For example, if the filter impulse length is 500 samples, limitcycle ignores the filter output from the first 500 input samples.
To perform limit cycle testing on your IIR filter, you must set the filter Arithmetic property to fixed and hd must be a fixedpoint IIR filter of one of the following forms:
df1 — directform I
df1t — directform I transposed
df1sos — directform I with secondorder sections
df1tsos — directform I transposed with secondorder sections
df2 — directform II
df2t — directform II transposed
df2sos — directform II with secondorder sections
df2tsos — directform II transposed with secondorder sections
When you use limitcycle without optional input arguments, the default settings are
Run 20 Monte Carlo trials
Use an input vector twice the length of the filter impulse response
Stop testing if the simulation process encounters either a granular or overflow limit cycle
To determine the length of the filter impulse response, use impzlength:
impzlength(hd)
During limit cycle testing, if the simulation runs reveal both overflow and granular limit cycles, the overflow limit cycle takes precedence and is the limit cycle that appears in the report.
Each time you run limitcycle, it uses a different sequence of random initial conditions, so the results can differ from run to run.
Each Monte Carlo trial uses a new set of randomly determined initial states for the filter. Test processing stops when limitcycle detects a zeroinput limit cycle in filter hd.
report = limitcycle(hd,ntrials,inputlengthfactor,stopcriterion) returns a report of how filter hd responds to a zerovalued input vector, using the following optional input arguments:
ntrials — Number of Monte Carlo trials (default is 20).
inputlengthfactor — integer factor used to calculate the length of the input vector. The length of the input vector comes from (impzlength(hd) * inputlengthfactor), where inputlengthfactor = 2 is the default value.
stopcriterion — the criterion for stopping the Monte Carlo trial processing. stopcriterion can be set to either (the default), granular, overflow. This table describes the results of each stop criterion.
stopcriterion Setting  Description 

either  Stop the Monte Carlo trials when limitcycle detects either a granular or overflow limit cycle. 
granular  Stop the Monte Carlo trials when limitcycle detects a granular limit cycle. 
overflow  Stop the Monte Carlo trials when limitcycle detects an overflow limit cycle. 
Note An important feature is that if you specify a specific limit cycle stop criterion, such as overflow, the Monte Carlo trials do not stop when testing encounters a granular limit cycle. You receive a warning that no overflow limit cycle occurred, but consider that a granular limit cycle might have occurred. 
In this example, there is a region of initial conditions in which no limit cycles occur and a region where they do. If no limit cycles are detected before the Monte Carlo trials are over, the state sequence converges to zero. When a limit cycle is found, the states do not end at zero. Each time you run this example, it uses a different sequence of random initial conditions, so the plot you get can differ from the one displayed in the following figure.
s = [1 0 0 1 0.9606 0.9849]; hd = dfilt.df2sos(s); hd.arithmetic = 'fixed'; greport = limitcycle(hd,20,2,'granular') oreport = limitcycle(hd,20,2,'overflow') figure, subplot(211),plot(greport.Output(1:20)), title('Granular Limit Cycle'); subplot(212),plot(oreport.Output(1:20)), title('Overflow Limit Cycle'); greport = LimitCycle: 'granular' Zi: [2x1 double] Output: [1303x1 embedded.fi] Trial: 1 oreport = LimitCycle: 'overflow' Zi: [2x1 double] Output: [1303x1 embedded.fi] Trial: 2
The plots shown in this figure present both limit cycle types — the first displays the small amplitude granular limit cycle, the second the larger amplitude overflow limit cycle.
As you see from the plots, and as is generally true, overflow limit cycles are much greater magnitude than granular limit cycles. This is why limitcycle favors overflow limit cycle detection and reporting.