Documentation

# dutycycle

Duty cycle of pulse waveform

## Syntax

```D = dutycycle(X) D = dutycycle(X,FS) D = dutycycle(X,T) D = dutycycle(TAU,PRF) [D,INITCROSS] = dutycycle(X,...) [D,INITCROSS,FINALCROSS] = dutycycle(X,...) [D,INITCROSS,FINALCROSS,NEXTCROSS] = dutycycle(X,...) [D,INITCROSS,FINALCROSS,NEXTCROSS,MIDLEV] = dutycycle(X,...) [D,INITCROSS,FINALCROSS,NEXTCROSS] = dutycycle(X,...,Name,Value) dutycycle(X,...) ```

## Description

`D = dutycycle(X)` returns the ratio of pulse width to pulse period for each positive-polarity pulse. `D` has length equal to the number of pulse periods in `X`. The sample instants of `X` correspond to the indices of `X`. To determine the transitions that define each pulse, `dutycycle` estimates the state levels of the input waveform by a histogram method. `dutycycle` identifies all regions, which cross the upper-state boundary of the low state and the lower-state boundary of the high state. The low-state and high-state boundaries are expressed as the state level plus or minus a multiple of the difference between the state levels. See State-Level Tolerances.

`D = dutycycle(X,FS)` specifies the sample rate, `FS`, in hertz as a positive scalar. The first sample instant of `X` corresponds to t=0.

`D = dutycycle(X,T)` specifies the sample instants, `T`, as a vector with the same number of elements as `X`.

`D = dutycycle(TAU,PRF)` returns the ratio of pulse width to pulse period for a pulse width of `TAU` seconds and a pulse repetition frequency of `PRF`. The product of `TAU` and `PRF` must be less than or equal to 1.

```[D,INITCROSS] = dutycycle(X,...)``` returns a vector, `INITCROSS`, whose elements correspond to the mid-crossings (mid-reference level instants) of the initial transition of each pulse with a corresponding `NEXTCROSS`.

```[D,INITCROSS,FINALCROSS] = dutycycle(X,...)``` returns a vector, `FINALCROSS`, whose elements correspond to the mid-crossings (mid-reference level instants) of the final transition of each pulse with a corresponding `NEXTCROSS`.

```[D,INITCROSS,FINALCROSS,NEXTCROSS] = dutycycle(X,...)``` returns a vector, `NEXTCROSS`, whose elements correspond to the mid-crossings (mid-reference level instants) of the next detected transition for each pulse.

```[D,INITCROSS,FINALCROSS,NEXTCROSS,MIDLEV] = dutycycle(X,...)``` returns the mid-reference level, `MIDLEV`. Because in a bilevel pulse waveform the state levels are constant, `MIDLEV` is a scalar.

```[D,INITCROSS,FINALCROSS,NEXTCROSS] = dutycycle(X,...,Name,Value)``` returns the ratio of pulse width to pulse period with additional options specified by one or more `Name,Value` pair arguments.

`dutycycle(X,...)` plots the waveform, `X`, and marks the location of the mid-reference level instants and the associated reference levels. The state levels and associated lower and upper state boundaries are also plotted.

## Input Arguments

 `X` Bilevel waveform. `X` is a real-valued row or column vector. `FS` Sample rate in hertz. `T` Vector of sample instants. The length of `T` must equal the length of the bilevel waveform, `X`. `TAU` Pulse width in seconds. The product of `TAU` and `PRF` must be less than or equal to 1. `PRF` Pulse repetition frequency in pulses/second. The product of `TAU` and `PRF` must be less than or equal to 1.

### Name-Value Pair Arguments

 `'MidPercentReferenceLevel'` Mid-reference level as a percentage of the waveform amplitude. Default: `50` `'Polarity'` Pulse polarity. Specify the polarity as `'positive'` or `'negative'`. If you specify `'positive'`, `dutycycle` looks for pulses with positive-going (positive polarity) initial transitions. If you specify `'negative'`, `dutycycle` looks for pulses with negative-going (negative polarity) initial transitions. See Pulse Polarity for examples of positive and negative-polarity pulses. Default: `'positive'` `'StateLevels'` Low- and high-state levels. `StateLevels` is a 1-by-2 real-valued vector. The first element is the low-state level. The second element is the high-state level. If you do not specify low- and high-state levels, `dutycycle` estimates the state levels from the input waveform using the histogram method. `'Tolerance'` Tolerance levels (lower- and upper-state boundaries) expressed as a percentage. See State-Level Tolerances. Default: `2`

## Output Arguments

 `D` Duty cycle. Duty cycle is the ratio of the pulse width to the pulse period. Because the pulse width cannot exceed the pulse period, 0≤`D` ≤1. `INITCROSS` Mid-reference level instant of initial transition. Because the duty cycle is defined as the ratio of pulse width to pulse period, initial transitions are only reported when `dutycycle` finds a corresponding `NEXTCROSS`. `FINALCROSS` Mid-reference level instant of final transition. The duty cycle is defined as the ratio of pulse width to pulse period. Thus, final transitions are only reported when `dutycycle` finds a corresponding `NEXTCROSS`. `NEXTCROSS` Mid-reference level instant of the first initial transition after the final transition of the preceding pulse. `MIDLEV` Mid-reference level. The waveform value that corresponds to the mid-reference level.

## Examples

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Determine the duty cycle of a bilevel waveform. Use the vector indices as the sample instants.

```load('pulseex.mat','x') d = dutycycle(x)```
```d = 0.3001 ```

Annotate the result on a plot of the waveform.

`dutycycle(x);`

Determine the duty cycle of a bilevel waveform. The sample rate is 4 MHz.

```load('pulseex.mat','x','t') fs = 1/(t(2)-t(1)); d = dutycycle(x,fs)```
```d = 0.3001 ```

Annotate the result on a plot of the waveform.

`dutycycle(x,fs);`

Create a pulse waveform with three pulses. The sample rate is 4 MHz. Determine the initial and final mid-reference level instants. Plot the result.

```load('pulseex.mat','x') fs = 4e6; pulse = x(1:30); wavef = [pulse;pulse;pulse]; t = (0:length(wavef)-1)/fs; [~,initcross,finalcross,~,midlev] = dutycycle(wavef,t)```
```initcross = 2×1 10-4 × 0.0312 0.1062 ```
```finalcross = 2×1 10-4 × 0.0463 0.1213 ```
```midlev = 2.5177 ```

Even though there are three pulses, only two pulses have corresponding subsequent transitions. Plot the result.

```plot(t,wavef) hold on plot([initcross finalcross],midlev*ones(2),'x','markersize',10) hold off legend('Waveform','Initial','Final','Location','best')```

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### Duty Cycle

The energy in a bilevel, or rectangular, pulse is equal to the product of the peak power, Pt, and the pulse width, τ. Devices to measure energy in a waveform operate on time scales longer than the duration of a single pulse. Therefore, it is common to measure the average power

`${P}_{\text{av}}=\frac{{P}_{t}\tau }{T},$`

where T is the pulse period.

The ratio of average power to peak power is the duty cycle:

`$D=\frac{{P}_{t}\tau /T}{{P}_{t}}$`

### Pulse Polarity

If the pulse has a positive-going initial transition, the pulse has positive polarity. The following figure shows a positive polarity pulse.

Equivalently, a positive-polarity (positive-going) pulse has a terminating state more positive than the originating state.

If the pulse has a negative-going initial transition, the pulse has negative polarity. The following figure shows a negative-polarity pulse.

Equivalently, a negative-polarity (negative-going) pulse has a originating state more positive than the terminating state.

### State-Level Tolerances

Each state level can have associated lower- and upper-state boundaries. These state boundaries are defined as the state level plus or minus a scalar multiple of the difference between the high state and the low state. To provide a useful tolerance region, the scalar is typically a small number such as 2/100 or 3/100. In general, the region for the low state is defined as

where is the low-state level and is the high-state level. Replace the first term in the equation with to obtain the tolerance region for the high state.

The following figure illustrates lower and upper 2% state boundaries (tolerance regions) for a positive-polarity bilevel waveform. The red dashed lines indicate the estimated state levels.

## References

[1] Skolnik, M. I. Introduction to Radar Systems. New York, NY: McGraw-Hill, 1980.

[2] IEEE® Standard on Transitions, Pulses, and Related Waveforms. IEEE Standard 181, 2003.