Double Exponential Difference Pulse

Define a double exponential difference 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.
Time constant ( $τ_{1}$ ): The pulse is defined as the difference of two exponentially charging pulses. The value of $τ_{1}$ describes the time that would be required for the subtracted signal to reach 63.2% of its full potential ( $u_{0}$ ).
Time constant ( $τ_{2}$ ): The value of $τ_{2}$ describes the time that would be required for the base signal to reach 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_{0} (e^{- \frac{t - t_{0}}{τ_{1}}} - e^{- \frac{t - t_{0}}{τ_{2}}}) & for & t > t_{0} \end{array}

The Fourier transform is as follows:

(2)

U (f) = u_{0} (\frac{τ_{1}}{1 + j 2 π f τ_{1}} - \frac{τ_{2}}{1 + j 2 π f τ_{2}}) e^{- j 2 π f t_{0}}