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# Documentation

## Rectangular Pulse Waveforms

### Definition of Rectangular Pulse Waveform

Define the following function of time:

$a\left(t\right)=\left\{\begin{array}{ll}1\hfill & 0\le t\le \tau \hfill \\ 0\hfill & \text{otherwise}\hfill \end{array}$

Assume that a radar transmits a signal of the form:

$x\left(t\right)=a\left(t\right)\mathrm{sin}\left({\omega }_{c}t\right)$

where ωc denotes the carrier frequency. Note that a(t) represents an on-off rectangular amplitude modulation of the carrier frequency. After demodulation, the complex envelope of x(t) is the real-valued rectangular pulse a(t) of duration τ seconds.

### How to Create Rectangular Pulse Waveforms

To create a rectangular pulse waveform, use phased.RectangularWaveform. You can customize certain characteristics of the waveform, including:

• Sampling rate

• Pulse duration

• Pulse repetition frequency

• Number of samples or pulses in each vector that represents the waveform

### Rectangular Waveform Plot

This example shows how to create a rectangular pulse waveform variable using phased.RectangularWaveform. The example also plots the pulse and finds the bandwidth of the pulse.

Construct a rectangular pulse waveform with a duration of 50 μs, a sample rate of 1 MHz, and a pulse repetition frequency (PRF) of 10 kHz.

hrect = phased.RectangularWaveform('SampleRate',1e6,...
'PulseWidth',5e-5,'PRF',1e4);

Plot a single rectangular pulse by calling plot directly on the rectangular waveform variable.

figure;
plot(hrect)

plot is a method of phased.RectangularWaveform. This plot method produces an annotated graph of your pulse waveform.

Find the bandwidth of the rectangular pulse.

bw = bandwidth(hrect);

The bandwidth of a rectangular pulse in hertz is approximately the reciprocal of that pulse's duration. That is, bw is approximately 1/hrect.PulseWidth.

### Pulses of Rectangular Waveform

This example shows how to create rectangular pulse waveform signals having different durations. The example plots two pulses of each waveform.

Create a rectangular pulse with a duration of 100 μs and a PRF of 1 kHz. Set the number of pulses in the output equal to two.

hrect = phased.RectangularWaveform('PulseWidth',100e-6,...
'PRF',1e3,'OutputFormat','Pulses','NumPulses',2);

Make a copy of your rectangular pulse and change the pulse width in your original waveform to 10 μs.

hrect1 = clone(hrect);
hrect.PulseWidth = 10e-6;

hrect1 and hrect now specify different rectangular pulses because you changed the pulse width of hrect.

Use the step method to return two pulses of your rectangular pulse waveforms.

y = step(hrect);
y1 = step(hrect1);

Plot the real part of the waveforms.

totaldur = 2*1/hrect.PRF;
totnumsamp = totaldur*hrect.SampleRate;
t = unigrid(0,1/hrect.SampleRate,totaldur,'[)');
subplot(2,1,1)
plot(t.*1000,real(y)); axis([0 totaldur*1e3 0 1.5]);
title('Two 10-\musec duration pulses (PRF = 1 kHz)');
set(gca,'XTick',0:0.2:totaldur*1e3)
subplot(2,1,2);
plot(t.*1000,real(y1)); axis([0 totaldur*1e3 0 1.5]);
xlabel('Milliseconds');
title('Two 100-\musec duration pulses (PRF = 1 kHz)');
set(gca,'XTick',0:0.2:totaldur*1e3)