Ramp Pulse

Define a ramp time pulse.



Figure 1. Define a ramp time pulse.
Time axis unit
Specify the unit to be used for the time axis.
Total signal duration ( s d )
The total length of the signal in the specified units.
Amplitude ( u 0 )
The amplitude of the time signal.
Pulse delay ( t 0 )
The pulse delay is the time until the peak of the time signal envelope.
Pulse width ( p w )
This is the half-amplitude pulse width of the signal. The pulse width is the total length of time that the signal is above 50% of its peak value ( u 0 ).
Rise time ( τ 1 )
The time required for the pulse to reach its peak value ( u 0 ) from rest.
Fall time ( τ 2 )
The time required for the pulse to reach the rest value from its peak ( u 0 ).
Note: The discharge time will be determined by the pulse width ( p w ).
Number of samples
The number of samples taken from the signal’s analytical equation.
(1) u ( t ) = { 0 for  t t 1 + t 0 u 0 ( 1 | t t 2 t 0 | τ 1 ) for  t 1 + t 0 t t 2 + t 0 u 0 for  t 2 + t 0 t t 3 + t 0 u 0 ( 1 | t t 3 t 0 | τ 2 ) for  t 3 + t 0 t t 4 + t 0 0 for  t t 4 + t 0
(2) t 1 = p w + τ 1 2 t 2 = p w τ 1 2 t 3 = p w τ 2 2 t 4 = p w + τ 2 2

The Fourier transform is as follows:

(3) U ( f ) = { u 0 [ ( ( p w τ 1 + τ 2 2 ) s i n c ( ( p w τ 1 + τ 2 2 ) f ) ) + 1 τ 1 ( 2 π f ) 2 ( e j 2 π f τ 1 ( 1 + j 2 π f τ 1 ) 1 ) e + j 2 π f ( p w 2 + τ 1 ) + 1 τ 2 ( 2 π f ) 2 ( e + j 2 π f τ 2 ( 1 j 2 π f τ 2 ) 1 ) e j 2 π f ( p w 2 + τ 2 ) ] e j 2 π f t 0 for  f > 0 u 0 p w for  f = 0