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) = {\begin{array}{crr} 0 & for & t \leq t_{0} \\ u_{1} (1 - e^{- \frac{t - t_{0}}{τ_{1}}}) & for & t_{0} \leq t \leq c_{d} + t_{0} \\ u_{2} e^{- \frac{t - t_{0}}{τ_{2}}} & for & t \geq c_{d} + t_{0} \end{array}

(2)

u_{1} = \frac{u_{0}}{1 - e^{- \frac{c_{d}}{τ_{1}}}} u_{2} = \frac{u_{0}}{e^{- \frac{c_{d}}{τ_{2}}}}

The Fourier transform is as follows:

(3)

U (f) = {\begin{array}{cr} [u_{1} (\frac{1 - e^{- j 2 π f c_{d}}}{j 2 π f} + \frac{τ_{1} (e^{- c_{d} (j 2 π f + \frac{1}{τ_{1}})} - 1)}{1 + j 2 π f τ_{1}}) \dots \\ + \frac{u_{2} τ_{2}}{(1 + j 2 π f τ_{2})} e^{- c_{d} (j 2 π f + \frac{1}{τ_{2}})}] e^{- j 2 π f t_{0}} & for f > 0 \\ u_{1} (τ_{1} (e^{\frac{- c_{d}}{τ_{1}}} - 1) + c_{d}) + u_{2} τ_{2} e^{\frac{- c_{d}}{τ_{2}}} & for f = 0 \end{array}