3 added 870 characters in body edited Feb 1 at 22:29 robert bristow-johnson 11.8k33 gold badges1919 silver badges5252 bronze badges okay, for step 1, let's get the continuous time math down right and simplified to as much as possible. i am changing some notational convention slightly to be more consistent with Electrical Engineering practice: $$v(t) = \sum_{k=0}^{N-1} V_\mathrm{step}(1-e^{-(t-k\Delta t_\mathrm{step})/\tau})u(t-k\Delta t_\mathrm{step})$$ $$V(j\omega) =\mathscr{F}\Big\{v(t)\Big\} = V_\mathrm{step}\left(\sum_{k=0}^{N-1}e^{-j\omega k\Delta t_\mathrm{step}}\right)\left(\frac{1}{j\omega} +\pi\delta(\omega) - \frac{1}{\frac{1}{\tau}+j\omega}\right)$$ there is a well known closed form for the summation of a geometric series: $$\sum_{k=0}^{N-1} x^k = \frac{x^N -1}{x-1}$$ so \begin{align} \sum_{k=0}^{N-1} e^{-j\omega k\Delta t_\mathrm{step}} &= \sum_{k=0}^{N-1} (e^{-j\omega \Delta t_\mathrm{step}})^k \\ \\ &= \frac{(e^{-j\omega \Delta t_\mathrm{step}})^N -1}{e^{-j\omega \Delta t_\mathrm{step}}-1} \\ \\ &= \frac{e^{-j\omega N \Delta t_\mathrm{step}}-1}{e^{-j\omega \Delta t_\mathrm{step}}-1} \\ \\ &= \frac{e^{-j (\omega/2) N \Delta t_\mathrm{step}}(e^{-j (\omega/2) N \Delta t_\mathrm{step}}-e^{j (\omega/2) N \Delta t_\mathrm{step}})}{e^{-j(\omega/2) \Delta t_\mathrm{step}}(e^{-j(\omega/2) \Delta t_\mathrm{step}}-e^{j(\omega/2) \Delta t_\mathrm{step}})} \\ \\ &= e^{-j (\omega/2) (N-1) \Delta t_\mathrm{step}}\frac{(e^{j (\omega/2) N \Delta t_\mathrm{step}}-e^{-j (\omega/2) N \Delta t_\mathrm{step}})/(2j)}{(e^{j(\omega/2) \Delta t_\mathrm{step}}-e^{-j(\omega/2) \Delta t_\mathrm{step}})/(2j)} \\ \\ &= e^{-j (\omega/2) (N-1) \Delta t_\mathrm{step}}\frac{\sin(\omega N \Delta t_\mathrm{step}/2)}{\sin(\omega \Delta t_\mathrm{step}/2)} \\ \\ \end{align} hang on....so i think your final expression in continuous frequency of the spectrum should be $$V(j\omega) = V_\mathrm{step}\frac{\sin(\omega N \Delta t_\mathrm{step}/2)}{\sin(\omega \Delta t_\mathrm{step}/2)} \left(\frac{1}{j\omega} +\pi\delta(\omega) - \frac{\tau}{1+j\omega \tau}\right) e^{-j (\omega/2)(N-1)\Delta t_\mathrm{step}}$$ does that look right to you? okay, for step 1, let's get the continuous time math down right and simplified to as much as possible. i am changing some notational convention slightly to be more consistent with Electrical Engineering practice: $$v(t) = \sum_{k=0}^{N-1} V_\mathrm{step}(1-e^{-(t-k\Delta t_\mathrm{step})/\tau})u(t-k\Delta t_\mathrm{step})$$ $$V(j\omega) =\mathscr{F}\Big\{v(t)\Big\} = V_\mathrm{step}\left(\sum_{k=0}^{N-1}e^{-j\omega k\Delta t_\mathrm{step}}\right)\left(\frac{1}{j\omega} +\pi\delta(\omega) - \frac{1}{\frac{1}{\tau}+j\omega}\right)$$ there is a well known closed form for the summation of a geometric series: $$\sum_{k=0}^{N-1} x^k = \frac{x^N -1}{x-1}$$ so \begin{align} \sum_{k=0}^{N-1} e^{-j\omega k\Delta t_\mathrm{step}} &= \sum_{k=0}^{N-1} (e^{-j\omega \Delta t_\mathrm{step}})^k \\ \\ &= \frac{(e^{-j\omega \Delta t_\mathrm{step}})^N -1}{e^{-j\omega \Delta t_\mathrm{step}}-1} \\ \\ &= \frac{e^{-j\omega N \Delta t_\mathrm{step}}-1}{e^{-j\omega \Delta t_\mathrm{step}}-1} \\ \\ &= \frac{e^{-j (\omega/2) N \Delta t_\mathrm{step}}(e^{-j (\omega/2) N \Delta t_\mathrm{step}}-e^{j (\omega/2) N \Delta t_\mathrm{step}})}{e^{-j(\omega/2) \Delta t_\mathrm{step}}(e^{-j(\omega/2) \Delta t_\mathrm{step}}-e^{j(\omega/2) \Delta t_\mathrm{step}})} \\ \\ &= e^{-j (\omega/2) (N-1) \Delta t_\mathrm{step}}\frac{(e^{j (\omega/2) N \Delta t_\mathrm{step}}-e^{-j (\omega/2) N \Delta t_\mathrm{step}})/(2j)}{(e^{j(\omega/2) \Delta t_\mathrm{step}}-e^{-j(\omega/2) \Delta t_\mathrm{step}})/(2j)} \\ \\ &= e^{-j (\omega/2) (N-1) \Delta t_\mathrm{step}}\frac{\sin(\omega N \Delta t_\mathrm{step}/2)}{\sin(\omega \Delta t_\mathrm{step}/2)} \\ \\ \end{align} hang on.... okay, for step 1, let's get the continuous time math down right and simplified to as much as possible. i am changing some notational convention slightly to be more consistent with Electrical Engineering practice: $$v(t) = \sum_{k=0}^{N-1} V_\mathrm{step}(1-e^{-(t-k\Delta t_\mathrm{step})/\tau})u(t-k\Delta t_\mathrm{step})$$ $$V(j\omega) =\mathscr{F}\Big\{v(t)\Big\} = V_\mathrm{step}\left(\sum_{k=0}^{N-1}e^{-j\omega k\Delta t_\mathrm{step}}\right)\left(\frac{1}{j\omega} +\pi\delta(\omega) - \frac{1}{\frac{1}{\tau}+j\omega}\right)$$ there is a well known closed form for the summation of a geometric series: $$\sum_{k=0}^{N-1} x^k = \frac{x^N -1}{x-1}$$ so \begin{align} \sum_{k=0}^{N-1} e^{-j\omega k\Delta t_\mathrm{step}} &= \sum_{k=0}^{N-1} (e^{-j\omega \Delta t_\mathrm{step}})^k \\ \\ &= \frac{(e^{-j\omega \Delta t_\mathrm{step}})^N -1}{e^{-j\omega \Delta t_\mathrm{step}}-1} \\ \\ &= \frac{e^{-j\omega N \Delta t_\mathrm{step}}-1}{e^{-j\omega \Delta t_\mathrm{step}}-1} \\ \\ &= \frac{e^{-j (\omega/2) N \Delta t_\mathrm{step}}(e^{-j (\omega/2) N \Delta t_\mathrm{step}}-e^{j (\omega/2) N \Delta t_\mathrm{step}})}{e^{-j(\omega/2) \Delta t_\mathrm{step}}(e^{-j(\omega/2) \Delta t_\mathrm{step}}-e^{j(\omega/2) \Delta t_\mathrm{step}})} \\ \\ &= e^{-j (\omega/2) (N-1) \Delta t_\mathrm{step}}\frac{(e^{j (\omega/2) N \Delta t_\mathrm{step}}-e^{-j (\omega/2) N \Delta t_\mathrm{step}})/(2j)}{(e^{j(\omega/2) \Delta t_\mathrm{step}}-e^{-j(\omega/2) \Delta t_\mathrm{step}})/(2j)} \\ \\ &= e^{-j (\omega/2) (N-1) \Delta t_\mathrm{step}}\frac{\sin(\omega N \Delta t_\mathrm{step}/2)}{\sin(\omega \Delta t_\mathrm{step}/2)} \\ \\ \end{align} so i think your final expression in continuous frequency of the spectrum should be $$V(j\omega) = V_\mathrm{step}\frac{\sin(\omega N \Delta t_\mathrm{step}/2)}{\sin(\omega \Delta t_\mathrm{step}/2)} \left(\frac{1}{j\omega} +\pi\delta(\omega) - \frac{\tau}{1+j\omega \tau}\right) e^{-j (\omega/2)(N-1)\Delta t_\mathrm{step}}$$ does that look right to you? 2 added 870 characters in body edited Feb 1 at 22:21 robert bristow-johnson 11.8k33 gold badges1919 silver badges5252 bronze badges okay, for step 1, let's get the continuous time math down right and simplified to as much as possible. i am changing some notational convention slightly to be more consistent with Electrical Engineering practice: $$v(t) = \sum_{k=0}^{N-1} V_\mathrm{step}(1-e^{-(t-k\Delta t_\mathrm{step})/\tau})u(t-k\Delta t_\mathrm{step})$$ $$V(j\omega) =\mathscr{F}\Big\{v(t)\Big\} = V_\mathrm{step}\left(\sum_{k=0}^{N-1}e^{-j\omega k\Delta t_\mathrm{step}}\right)\left(\frac{1}{j\omega} +\pi\delta(\omega) - \frac{1}{\frac{1}{\tau}+j\omega}\right)$$ there is a well known closed form for the summation of a geometric series: $$\sum_{k=0}^{N-1} x^k = \frac{x^N -1}{x-1}$$ so \begin{align} \sum_{k=0}^{N-1} e^{-j\omega k\Delta t_\mathrm{step}} &= \sum_{k=0}^{N-1} (e^{-j\omega \Delta t_\mathrm{step}})^k \\ \\ &= \frac{(e^{-j\omega \Delta t_\mathrm{step}})^N -1}{e^{-j\omega \Delta t_\mathrm{step}}-1} \\ \\ &= \frac{e^{-j\omega N \Delta t_\mathrm{step}}-1}{e^{-j\omega \Delta t_\mathrm{step}}-1} \\ \\ &= \frac{e^{-j (\omega/2) N \Delta t_\mathrm{step}}(e^{-j (\omega/2) N \Delta t_\mathrm{step}}-e^{j (\omega/2) N \Delta t_\mathrm{step}})}{e^{-j(\omega/2) \Delta t_\mathrm{step}}(e^{-j(\omega/2) \Delta t_\mathrm{step}}-e^{j(\omega/2) \Delta t_\mathrm{step}})} \\ \\ &= e^{-j (\omega/2) (N-1) \Delta t_\mathrm{step}}\frac{(e^{j (\omega/2) N \Delta t_\mathrm{step}}-e^{-j (\omega/2) N \Delta t_\mathrm{step}})/(2j)}{(e^{j(\omega/2) \Delta t_\mathrm{step}}-e^{-j(\omega/2) \Delta t_\mathrm{step}})/(2j)} \\ \\ &= e^{-j (\omega/2) (N-1) \Delta t_\mathrm{step}}\frac{\sin(\omega N \Delta t_\mathrm{step}/2)}{\sin(\omega \Delta t_\mathrm{step}/2)} \\ \\ \end{align} hang on.... okay, for step 1, let's get the continuous time math down right and simplified to as much as possible. i am changing some notational convention slightly to be more consistent with Electrical Engineering practice: $$v(t) = \sum_{k=0}^{N-1} V_\mathrm{step}(1-e^{-(t-k\Delta t_\mathrm{step})/\tau})u(t-k\Delta t_\mathrm{step})$$ $$V(j\omega) =\mathscr{F}\Big\{v(t)\Big\} = V_\mathrm{step}\left(\sum_{k=0}^{N-1}e^{-j\omega k\Delta t_\mathrm{step}}\right)\left(\frac{1}{j\omega} +\pi\delta(\omega) - \frac{1}{\frac{1}{\tau}+j\omega}\right)$$ there is a well known closed form for the summation of a geometric series: $$\sum_{k=0}^{N-1} x^k = \frac{x^N -1}{x-1}$$ hang on.... okay, for step 1, let's get the continuous time math down right and simplified to as much as possible. i am changing some notational convention slightly to be more consistent with Electrical Engineering practice: $$v(t) = \sum_{k=0}^{N-1} V_\mathrm{step}(1-e^{-(t-k\Delta t_\mathrm{step})/\tau})u(t-k\Delta t_\mathrm{step})$$ $$V(j\omega) =\mathscr{F}\Big\{v(t)\Big\} = V_\mathrm{step}\left(\sum_{k=0}^{N-1}e^{-j\omega k\Delta t_\mathrm{step}}\right)\left(\frac{1}{j\omega} +\pi\delta(\omega) - \frac{1}{\frac{1}{\tau}+j\omega}\right)$$ there is a well known closed form for the summation of a geometric series: $$\sum_{k=0}^{N-1} x^k = \frac{x^N -1}{x-1}$$ so \begin{align} \sum_{k=0}^{N-1} e^{-j\omega k\Delta t_\mathrm{step}} &= \sum_{k=0}^{N-1} (e^{-j\omega \Delta t_\mathrm{step}})^k \\ \\ &= \frac{(e^{-j\omega \Delta t_\mathrm{step}})^N -1}{e^{-j\omega \Delta t_\mathrm{step}}-1} \\ \\ &= \frac{e^{-j\omega N \Delta t_\mathrm{step}}-1}{e^{-j\omega \Delta t_\mathrm{step}}-1} \\ \\ &= \frac{e^{-j (\omega/2) N \Delta t_\mathrm{step}}(e^{-j (\omega/2) N \Delta t_\mathrm{step}}-e^{j (\omega/2) N \Delta t_\mathrm{step}})}{e^{-j(\omega/2) \Delta t_\mathrm{step}}(e^{-j(\omega/2) \Delta t_\mathrm{step}}-e^{j(\omega/2) \Delta t_\mathrm{step}})} \\ \\ &= e^{-j (\omega/2) (N-1) \Delta t_\mathrm{step}}\frac{(e^{j (\omega/2) N \Delta t_\mathrm{step}}-e^{-j (\omega/2) N \Delta t_\mathrm{step}})/(2j)}{(e^{j(\omega/2) \Delta t_\mathrm{step}}-e^{-j(\omega/2) \Delta t_\mathrm{step}})/(2j)} \\ \\ &= e^{-j (\omega/2) (N-1) \Delta t_\mathrm{step}}\frac{\sin(\omega N \Delta t_\mathrm{step}/2)}{\sin(\omega \Delta t_\mathrm{step}/2)} \\ \\ \end{align} hang on.... 1 answered Feb 1 at 22:09 robert bristow-johnson 11.8k33 gold badges1919 silver badges5252 bronze badges okay, for step 1, let's get the continuous time math down right and simplified to as much as possible. i am changing some notational convention slightly to be more consistent with Electrical Engineering practice: $$v(t) = \sum_{k=0}^{N-1} V_\mathrm{step}(1-e^{-(t-k\Delta t_\mathrm{step})/\tau})u(t-k\Delta t_\mathrm{step})$$ $$V(j\omega) =\mathscr{F}\Big\{v(t)\Big\} = V_\mathrm{step}\left(\sum_{k=0}^{N-1}e^{-j\omega k\Delta t_\mathrm{step}}\right)\left(\frac{1}{j\omega} +\pi\delta(\omega) - \frac{1}{\frac{1}{\tau}+j\omega}\right)$$ there is a well known closed form for the summation of a geometric series: $$\sum_{k=0}^{N-1} x^k = \frac{x^N -1}{x-1}$$ hang on....