Tables of transforms¶
Fourier Transforms¶
Lcapy can compute many Fourier transforms including:
\(x(t) \longleftrightarrow X(f)\)
\(x(a t) \longleftrightarrow \frac{X(\frac{f}{a})}{\left|{a}\right|}\)
\(x(t - \tau) \longleftrightarrow X(f) e^{2 \mathrm{j} \pi f \tau}\)
\(\cos{\left(2 \pi f_{0} t \right)} \longleftrightarrow \frac{\delta\left(f - f_{0}\right)}{2} + \frac{\delta\left(f + f_{0}\right)}{2}\)
\(\sin{\left(2 \pi f_{0} t \right)} \longleftrightarrow \frac{\mathrm{j} \left(- \delta\left(f - f_{0}\right) + \delta\left(f + f_{0}\right)\right)}{2}\)
\(e^{2 \mathrm{j} \pi f_{0} t} \longleftrightarrow \delta\left(f - f_{0}\right)\)
\(1 \longleftrightarrow \delta\left(f\right)\)
\(t \longleftrightarrow \frac{\mathrm{j} \delta^{\left( 1 \right)}\left( f \right)}{2 \pi}\)
\(t^{2} \longleftrightarrow - \frac{\delta^{\left( 2 \right)}\left( f \right)}{4 \pi^{2}}\)
\(\frac{1}{t} \longleftrightarrow - \mathrm{j} \pi \operatorname{sign}{\left(f \right)}\)
\(\delta\left(t\right) \longleftrightarrow 1\)
\(\delta\left(t - t_{0}\right) \longleftrightarrow e^{2 \mathrm{j} \pi f t_{0}}\)
\(u\left(t\right) \longleftrightarrow \frac{\delta\left(f\right)}{2} - \frac{\mathrm{j}}{2 \pi f}\)
\(t u\left(t\right) \longleftrightarrow \frac{\mathrm{j} \delta^{\left( 1 \right)}\left( f \right)}{4 \pi} - \frac{1}{4 \pi^{2} f^{2}}\)
\(\mathrm{sign}{\left(t \right)} \longleftrightarrow - \frac{\mathrm{j}}{\pi f}\)
\(\mathrm{rect}{\left(t \right)} \longleftrightarrow \mathrm{sincn}{\left(f \right)}\)
\(\mathrm{sincn}{\left(t \right)} \longleftrightarrow \mathrm{rect}{\left(f \right)}\)
\(\mathrm{tri}{\left(t \right)} \longleftrightarrow \mathrm{sincn}^{2}{\left(f \right)}\)
\(\mathrm{trap}{\left(t,\alpha \right)} \longleftrightarrow \alpha \operatorname{sincn}{\left(f \right)} \operatorname{sincn}{\left(\alpha f \right)}\)
\(e^{- \left|{t}\right|} \longleftrightarrow \frac{2}{4 \pi^{2} f^{2} + 1}\)
\(e^{- t} u\left(t\right) \longleftrightarrow \frac{1}{2 \mathrm{j} \pi f + 1}\)
Discrete-Time Fourier Transforms (DTFT)¶
Lcapy can compute many discrete-time Fourier transforms including:
\(x(n) \longleftrightarrow \sum_{m=-\infty}^{\infty} X(f - \frac{m}{\Delta_{t}})\)
\(x(a n) \longleftrightarrow \frac{\sum_{m=-\infty}^{\infty} X(\frac{f}{a} - \frac{m}{\Delta_{t} a})}{\left|{a}\right|}\)
\(x(- m + n) \longleftrightarrow \sum_{p=-\infty}^{\infty} X(f - \frac{p}{\Delta_{t}}) e^{- 2 \mathrm{j} \pi f m} e^{\frac{2 \mathrm{j} \pi m p}{\Delta_{t}}}\)
\(\cos{\left(2 \pi \Delta_{t} f0 n \right)} \longleftrightarrow \frac{\sum_{m=-\infty}^{\infty} \delta\left(- f + f0 + \frac{m}{\Delta_{t}}\right)}{2 \Delta_{t}} + \frac{\sum_{m=-\infty}^{\infty} \delta\left(f + f0 - \frac{m}{\Delta_{t}}\right)}{2 \Delta_{t}}\)
\(\sin{\left(2 \pi \Delta_{t} f0 n \right)} \longleftrightarrow - \frac{\mathrm{j} \sum_{m=-\infty}^{\infty} \delta\left(- f + f0 + \frac{m}{\Delta_{t}}\right)}{2 \Delta_{t}} + \frac{\mathrm{j} \sum_{m=-\infty}^{\infty} \delta\left(f + f0 - \frac{m}{\Delta_{t}}\right)}{2 \Delta_{t}}\)
\(e^{2 \mathrm{j} \pi \Delta_{t} f0 n} \longleftrightarrow \frac{\sum_{m=-\infty}^{\infty} \delta\left(- f + f0 + \frac{m}{\Delta_{t}}\right)}{\Delta_{t}}\)
\(1 \longleftrightarrow \frac{\sum_{m=-\infty}^{\infty} \delta\left(f - \frac{m}{\Delta_{t}}\right)}{\Delta_{t}}\)
\(\delta\left[n\right] \longleftrightarrow 1\)
\(\delta\left[- m + n\right] \longleftrightarrow e^{- 2 \mathrm{j} \pi \Delta_{t} f m}\)
\(u\left[n\right] \longleftrightarrow \frac{e^{2 \mathrm{j} \pi \Delta_{t} f}}{e^{2 \mathrm{j} \pi \Delta_{t} f} - 1} + \frac{\sum_{m=-\infty}^{\infty} \delta\left(f - \frac{m}{\Delta_{t}}\right)}{2 \Delta_{t}}\)
\(n u\left[n\right] \longleftrightarrow \frac{e^{2 \mathrm{j} \pi \Delta_{t} f}}{e^{4 \mathrm{j} \pi \Delta_{t} f} - 2 e^{2 \mathrm{j} \pi \Delta_{t} f} + 1} + \frac{\mathrm{j} \sum_{m=-\infty}^{\infty} \delta^{\left( 1 \right)}\left( f - \frac{m}{\Delta_{t}} \right)}{4 \pi \Delta_{t}^{2}}\)
\(\mathop{\mathrm{sign}}\left[n\right] \longleftrightarrow \frac{2 e^{2 \mathrm{j} \pi \Delta_{t} f}}{e^{2 \mathrm{j} \pi \Delta_{t} f} - 1}\)
\(\alpha^{- n} u\left[n\right] \longleftrightarrow \frac{\alpha e^{2 \mathrm{j} \pi \Delta_{t} f}}{\alpha e^{2 \mathrm{j} \pi \Delta_{t} f} - 1}\)
\(\mathop{\mathrm{rect}}\left[n\right] \longleftrightarrow 1\)
\(\mathop{\mathrm{rect}}\left[\frac{n}{N_{o}}\right] \longleftrightarrow \frac{\sin{\left(\pi \Delta_{t} N_{o} f \right)}}{\sin{\left(\pi \Delta_{t} f \right)}}\)
\(\mathop{\mathrm{rect}}\left[\frac{n}{N_{e}}\right] \longleftrightarrow \frac{e^{\mathrm{j} \pi \Delta_{t} f} \sin{\left(\pi \Delta_{t} N_{e} f \right)}}{\sin{\left(\pi \Delta_{t} f \right)}}\)
\(\mathrm{sincn}{\left(n \right)} \longleftrightarrow \sum_{m=-\infty}^{\infty} \mathop{\mathrm{rect}}\left[\Delta_{t} f - m\right]\)
\(\mathrm{sincn}{\left(K n \right)} \longleftrightarrow \frac{\sum_{m=-\infty}^{\infty} \mathop{\mathrm{rect}}\left[\frac{\Delta_{t} f}{K} - \frac{m}{K}\right]}{K}\)
\(\mathrm{sincn}^{2}{\left(K n \right)} \longleftrightarrow \frac{\sum_{m=-\infty}^{\infty} \operatorname{tri}{\left(\frac{\Delta_{t} f}{K} - \frac{m}{K} \right)}}{K}\)
\(\mathrm{sincu}{\left(K n \right)} \longleftrightarrow \frac{\pi \sum_{m=-\infty}^{\infty} \mathop{\mathrm{rect}}\left[\frac{\pi \Delta_{t} f}{K} - \frac{\pi m}{K}\right]}{K}\)
Note, \(N_e\) is even and \(N_o\) is odd.
Discrete-Time Fourier Transforms (DTFT) (normalized angular frequency)¶
Lcapy can compute many discrete-time Fourier transforms in the \(\Omega\) form, including:
\(x(n) \longleftrightarrow \sum_{m=-\infty}^{\infty} X(\frac{\Omega}{2 \pi \Delta_{t}} - \frac{m}{\Delta_{t}})\)
\(x(a n) \longleftrightarrow \frac{\sum_{m=-\infty}^{\infty} X(\frac{\Omega}{2 \pi \Delta_{t} a} - \frac{m}{\Delta_{t} a})}{\left|{a}\right|}\)
\(x(- m + n) \longleftrightarrow \sum_{p=-\infty}^{\infty} X(\frac{\Omega}{2 \pi \Delta_{t}} - \frac{p}{\Delta_{t}}) e^{- \frac{\mathrm{j} \Omega m}{\Delta_{t}}} e^{\frac{2 \mathrm{j} \pi m p}{\Delta_{t}}}\)
\(\cos{\left(\Omega_{0} n \right)} \longleftrightarrow \frac{\sum_{m=-\infty}^{\infty} \delta\left(\frac{\Omega}{2 \pi \Delta_{t}} - \frac{\Omega_{0}}{2 \pi \Delta_{t}} - \frac{m}{\Delta_{t}}\right)}{2 \Delta_{t}} + \frac{\sum_{m=-\infty}^{\infty} \delta\left(\frac{\Omega}{2 \pi \Delta_{t}} + \frac{\Omega_{0}}{2 \pi \Delta_{t}} - \frac{m}{\Delta_{t}}\right)}{2 \Delta_{t}}\)
\(\sin{\left(\Omega_{0} n \right)} \longleftrightarrow - \frac{\mathrm{j} \sum_{m=-\infty}^{\infty} \delta\left(\frac{\Omega}{2 \pi \Delta_{t}} - \frac{\Omega_{0}}{2 \pi \Delta_{t}} - \frac{m}{\Delta_{t}}\right)}{2 \Delta_{t}} + \frac{\mathrm{j} \sum_{m=-\infty}^{\infty} \delta\left(\frac{\Omega}{2 \pi \Delta_{t}} + \frac{\Omega_{0}}{2 \pi \Delta_{t}} - \frac{m}{\Delta_{t}}\right)}{2 \Delta_{t}}\)
\(e^{\mathrm{j} \Omega_{0} n} \longleftrightarrow \frac{\sum_{m=-\infty}^{\infty} \delta\left(\frac{\Omega}{2 \pi \Delta_{t}} - \frac{\Omega_{0}}{2 \pi \Delta_{t}} - \frac{m}{\Delta_{t}}\right)}{\Delta_{t}}\)
\(1 \longleftrightarrow \frac{\sum_{m=-\infty}^{\infty} \delta\left(\frac{\Omega}{2 \pi \Delta_{t}} - \frac{m}{\Delta_{t}}\right)}{\Delta_{t}}\)
\(\delta\left[n\right] \longleftrightarrow 1\)
\(\delta\left[- m + n\right] \longleftrightarrow e^{- \mathrm{j} \Omega m}\)
\(u\left[n\right] \longleftrightarrow \frac{e^{\mathrm{j} \Omega}}{e^{\mathrm{j} \Omega} - 1} + \frac{\sum_{m=-\infty}^{\infty} \delta\left(\frac{\Omega}{2 \pi \Delta_{t}} - \frac{m}{\Delta_{t}}\right)}{2 \Delta_{t}}\)
\(n u\left[n\right] \longleftrightarrow \frac{e^{\mathrm{j} \Omega}}{e^{2 \mathrm{j} \Omega} - 2 e^{\mathrm{j} \Omega} + 1} + \frac{\mathrm{j} \sum_{m=-\infty}^{\infty} \delta^{\left( 1 \right)}\left( \frac{\Omega}{2 \pi \Delta_{t}} - \frac{m}{\Delta_{t}} \right)}{4 \pi \Delta_{t}^{2}}\)
\(\mathop{\mathrm{sign}}\left[n\right] \longleftrightarrow \frac{2 e^{\mathrm{j} \Omega}}{e^{\mathrm{j} \Omega} - 1}\)
\(\alpha^{- n} u\left[n\right] \longleftrightarrow \frac{\alpha e^{\mathrm{j} \Omega}}{\alpha e^{\mathrm{j} \Omega} - 1}\)
\(\mathop{\mathrm{rect}}\left[n\right] \longleftrightarrow 1\)
\(\mathop{\mathrm{rect}}\left[\frac{n}{N_{o}}\right] \longleftrightarrow \frac{\sin{\left(\frac{N_{o} \Omega}{2} \right)}}{\sin{\left(\frac{\Omega}{2} \right)}}\)
\(\mathop{\mathrm{rect}}\left[\frac{n}{N_{e}}\right] \longleftrightarrow \frac{e^{\frac{\mathrm{j} \Omega}{2}} \sin{\left(\frac{N_{e} \Omega}{2} \right)}}{\sin{\left(\frac{\Omega}{2} \right)}}\)
\(\mathrm{sincn}{\left(n \right)} \longleftrightarrow \sum_{m=-\infty}^{\infty} \mathop{\mathrm{rect}}\left[\frac{\Omega}{2 \pi} - m\right]\)
\(\mathrm{sincn}{\left(K n \right)} \longleftrightarrow \frac{\sum_{m=-\infty}^{\infty} \mathop{\mathrm{rect}}\left[\frac{\Omega}{2 \pi K} - \frac{m}{K}\right]}{K}\)
\(\mathrm{sincn}^{2}{\left(K n \right)} \longleftrightarrow \frac{\sum_{m=-\infty}^{\infty} \operatorname{tri}{\left(\frac{\Omega}{2 \pi K} - \frac{m}{K} \right)}}{K}\)
\(\mathrm{sincu}{\left(K n \right)} \longleftrightarrow \frac{\pi \sum_{m=-\infty}^{\infty} \mathop{\mathrm{rect}}\left[\frac{\Omega}{2 K} - \frac{\pi m}{K}\right]}{K}\)
Note, \(N_e\) is even and \(N_o\) is odd.
Discrete Fourier Transforms (DFT)¶
Lcapy can compute some discrete Fourier transforms including:
\(x(n) \longleftrightarrow X(k)\)
\(x(a n) \longleftrightarrow \frac{X(\frac{k}{a})}{\left|{a}\right|}\)
\(x(- m + n) \longleftrightarrow X(k) e^{- \frac{2 \mathrm{j} \pi k m}{N}}\)
\(1 \longleftrightarrow N \delta\left[k\right]\)
\(\delta\left[n\right] \longleftrightarrow 1\)
\(\delta\left[- m + n\right] \longleftrightarrow e^{- \frac{2 \mathrm{j} \pi k m}{N}}\)
\(u\left[n\right] \longleftrightarrow N \delta\left[k\right]\)
\(n u\left[n\right] \longleftrightarrow \frac{N \left(N - 1\right) \delta\left[k\right]}{2} + \frac{N \left(\delta\left[k\right] - 1\right)}{1 - e^{- \frac{2 \mathrm{j} \pi k}{N}}}\)
\(\alpha^{- n} u\left[n\right] \longleftrightarrow \frac{\alpha^{- N} \left(- \alpha + \alpha^{N + 1}\right) e^{\frac{2 \mathrm{j} \pi k}{N}}}{\alpha e^{\frac{2 \mathrm{j} \pi k}{N}} - 1}\)
\(e^{\frac{2 \mathrm{j} \pi n}{N}} \longleftrightarrow N \delta\left[k - 1\right]\)
\(\cos{\left(\frac{2 \pi n}{N} \right)} \longleftrightarrow \frac{N \left(\delta\left[k - 1\right] + \delta\left[- N + k + 1\right]\right)}{2}\)
\(\sin{\left(\frac{2 \pi n}{N} \right)} \longleftrightarrow \frac{\mathrm{j} N \left(- \delta\left[k - 1\right] + \delta\left[- N + k + 1\right]\right)}{2}\)
Laplace Transforms¶
Lcapy can compute many Laplace transforms including:
\(x(t) \longleftrightarrow X(s)\)
\(x(a t) \longleftrightarrow \frac{X(\frac{s}{a})}{\left|{a}\right|}\)
\(x(t - \tau) \longleftrightarrow X(s) e^{- s \tau}\)
\(\cos{\left(2 \pi f0 t \right)} \longleftrightarrow \frac{s}{4 \pi^{2} f0^{2} + s^{2}}\)
\(\sin{\left(2 \pi f0 t \right)} \longleftrightarrow \frac{2 \pi f0}{4 \pi^{2} f0^{2} + s^{2}}\)
\(e^{2 \mathrm{j} \pi f0 t} \longleftrightarrow \frac{1}{s \left(- \frac{2 \mathrm{j} \pi f0}{s} + 1\right)}\)
\(1 \longleftrightarrow \frac{1}{s}\)
\(t \longleftrightarrow \frac{1}{s^{2}}\)
\(t^{2} \longleftrightarrow \frac{2}{s^{3}}\)
\(\delta\left(t\right) \longleftrightarrow 1\)
\(\delta\left(t - t0\right) \longleftrightarrow e^{- s t0}\)
\(u\left(t\right) \longleftrightarrow \frac{1}{s}\)
\(t u\left(t\right) \longleftrightarrow \frac{1}{s^{2}}\)
\(\mathrm{sign}{\left(t \right)} \longleftrightarrow \frac{1}{s}\)
\(e^{- \left|{t}\right|} \longleftrightarrow \frac{1}{s + 1}\)
\(e^{- t} u\left(t\right) \longleftrightarrow \frac{1}{s + 1}\)
\(\mathrm{rect}{\left(t - \frac{1}{2} \right)} \longleftrightarrow \frac{1}{s} - \frac{e^{- s}}{s}\)
\(\mathrm{tri}{\left(t - 1 \right)} \longleftrightarrow \frac{2 e^{- s}}{s} - \frac{2 e^{- 2 s}}{s} + \frac{4 \left(\frac{s}{2} + \frac{1}{4}\right) e^{- 2 s}}{s^{2}} - \frac{2 \left(s + 1\right) e^{- s}}{s^{2}} + \frac{1}{s^{2}}\)
\(\mathrm{ramp}{\left(t \right)} \longleftrightarrow \frac{1}{s^{2}}\)
\(\mathrm{rampstep}{\left(t \right)} \longleftrightarrow \frac{1 - e^{- s}}{s^{2}}\)
\(\frac{d}{d t} x(t) \longleftrightarrow s X(s) - x(0)\)
\(\frac{d^{2}}{d t^{2}} x(t) \longleftrightarrow s^{2} X(s) - s x(0) - \left. \frac{d}{d t} x(t) \right|_{\substack{ t=0 }}\)
\(\int\limits_{0}^{t} x(t)\, dt \longleftrightarrow \frac{X(s)}{s}\)
Z Transforms¶
Lcapy can compute many Z transforms including:
\(x(n) \longleftrightarrow X(z)\)
\(x(- m + n) \longleftrightarrow z^{- m} X(z)\)
\(\cos{\left(2 \pi \Delta_{t} f0 n \right)} \longleftrightarrow \frac{z \left(z - \cos{\left(2 \pi \Delta_{t} f0 \right)}\right)}{z^{2} - 2 z \cos{\left(2 \pi \Delta_{t} f0 \right)} + 1}\)
\(\sin{\left(2 \pi \Delta_{t} f0 n \right)} \longleftrightarrow \frac{z \sin{\left(2 \pi \Delta_{t} f0 \right)}}{z^{2} - 2 z \cos{\left(2 \pi \Delta_{t} f0 \right)} + 1}\)
\(e^{2 \mathrm{j} \pi \Delta_{t} f0 n} \longleftrightarrow \frac{z}{z - e^{2 \mathrm{j} \pi \Delta_{t} f0}}\)
\(1 \longleftrightarrow \frac{1}{1 - \frac{1}{z}}\)
\(\delta\left[n\right] \longleftrightarrow 1\)
\(\delta\left[n - p\right] \longleftrightarrow \left(\frac{1}{z}\right)^{p}\)
\(a^{n} \longleftrightarrow \frac{z}{- a + z}\)
\(a^{- n} \longleftrightarrow \frac{a z}{a z - 1}\)
\(a^{n} n \longleftrightarrow \frac{a z}{\left(a - z\right)^{2}}\)
\(a^{- n} n \longleftrightarrow \frac{a z}{\left(a z - 1\right)^{2}}\)
\(u\left[n\right] \longleftrightarrow \frac{1}{1 - \frac{1}{z}}\)
\(e^{- \Delta_{t} n} u\left[n\right] \longleftrightarrow \frac{z e^{\Delta_{t}}}{z e^{\Delta_{t}} - 1}\)