Double Exponential Piecewise Pulse

Define a double exponential piecewise time pulse.



Figure 1. Define a double exponential difference 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.
Charge duration ( c d )
The charge duration is the time from the pulse delay has ended until the signal begins to discharge.
Charge time constant ( τ 2 )
The time that would be required to discharge the signal down to 36.8% of its full potential ( u 0 ).
Charge time constant ( τ 1 )
The time that would be required to charge the signal up to 63.2% of its full potential ( u 0 ).
Number of samples
The number of samples taken from the signal’s analytical equation.
(1) u ( t ) = { 0 for  t t 0 u 1 ( 1 e t t 0 τ 1 ) for  t 0 t c d + t 0 u 2 e t t 0 τ 2 for  t c d + t 0
(2) u 1 = u 0 1 e c d τ 1 u 2 = u 0 e c d τ 2

The Fourier transform is as follows:

(3) U ( f ) = { [ u 1 ( 1 e j 2 π f c d j 2 π f + τ 1 ( e c d ( j 2 π f + 1 τ 1 ) 1 ) 1 + j 2 π f τ 1 ) + u 2 τ 2 ( 1 + j 2 π f τ 2 ) e c d ( j 2 π f + 1 τ 2 ) ] e j 2 π f t 0 for  f > 0 u 1 ( τ 1 ( e c d τ 1 1 ) + c d ) + u 2 τ 2 e c d τ 2 for  f = 0