# 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:

$$x(n) \longleftrightarrow \sum_{m=-\infty}^{\infty} X(f - \frac{m}{\Delta_{t}})$$

$$x(a n) \longleftrightarrow \frac{\sum_{m=-\infty}^{\infty} X(\frac{f}{a} - \frac{m}{\Delta_{t} a})}{\left|{a}\right|}$$

$$x(- m + n) \longleftrightarrow \sum_{p=-\infty}^{\infty} X(f - \frac{p}{\Delta_{t}}) e^{- 2 \mathrm{j} \pi f m} e^{\frac{2 \mathrm{j} \pi m p}{\Delta_{t}}}$$

$$\cos{\left(2 \pi \Delta_{t} f_{0} n \right)} \longleftrightarrow \frac{\sum_{m=-\infty}^{\infty} \delta\left(f - f_{0} - \frac{m}{\Delta_{t}}\right)}{2 \Delta_{t}} + \frac{\sum_{m=-\infty}^{\infty} \delta\left(f + f_{0} - \frac{m}{\Delta_{t}}\right)}{2 \Delta_{t}}$$

$$\sin{\left(2 \pi \Delta_{t} f_{0} n \right)} \longleftrightarrow - \frac{\mathrm{j} \sum_{m=-\infty}^{\infty} \delta\left(f - f_{0} - \frac{m}{\Delta_{t}}\right)}{2 \Delta_{t}} + \frac{\mathrm{j} \sum_{m=-\infty}^{\infty} \delta\left(f + f_{0} - \frac{m}{\Delta_{t}}\right)}{2 \Delta_{t}}$$

$$e^{2 \mathrm{j} \pi \Delta_{t} f_{0} n} \longleftrightarrow \frac{\sum_{m=-\infty}^{\infty} \delta\left(f - f_{0} - \frac{m}{\Delta_{t}}\right)}{\Delta_{t}}$$

$$1 \longleftrightarrow \frac{\sum_{m=-\infty}^{\infty} \delta\left(f - \frac{m}{\Delta_{t}}\right)}{\Delta_{t}}$$

$$\delta\left[n\right] \longleftrightarrow 1$$

$$\delta\left[- m + n\right] \longleftrightarrow e^{- 2 \mathrm{j} \pi \Delta_{t} f m}$$

$$u\left[n\right] \longleftrightarrow \frac{e^{2 \mathrm{j} \pi \Delta_{t} f}}{e^{2 \mathrm{j} \pi \Delta_{t} f} - 1} + \frac{\sum_{m=-\infty}^{\infty} \delta\left(f - \frac{m}{\Delta_{t}}\right)}{2 \Delta_{t}}$$

$$n u\left[n\right] \longleftrightarrow \frac{e^{2 \mathrm{j} \pi \Delta_{t} f}}{e^{4 \mathrm{j} \pi \Delta_{t} f} - 2 e^{2 \mathrm{j} \pi \Delta_{t} f} + 1} + \frac{\mathrm{j} \sum_{m=-\infty}^{\infty} \delta^{\left( 1 \right)}\left( f - \frac{m}{\Delta_{t}} \right)}{4 \pi \Delta_{t}^{2}}$$

$$\mathop{\mathrm{sign}}\left[n\right] \longleftrightarrow \frac{2 e^{2 \mathrm{j} \pi \Delta_{t} f}}{e^{2 \mathrm{j} \pi \Delta_{t} f} - 1}$$

$$\alpha^{- n} u\left[n\right] \longleftrightarrow \frac{\alpha e^{2 \mathrm{j} \pi \Delta_{t} f}}{\alpha e^{2 \mathrm{j} \pi \Delta_{t} f} - 1}$$

$$\mathop{\mathrm{rect}}\left[n\right] \longleftrightarrow 1$$

$$\mathop{\mathrm{rect}}\left[\frac{n}{N_{o}}\right] \longleftrightarrow \frac{\sin{\left(\pi \Delta_{t} N_{o} f \right)}}{\sin{\left(\pi \Delta_{t} f \right)}}$$

$$\mathop{\mathrm{rect}}\left[\frac{n}{N_{e}}\right] \longleftrightarrow \frac{e^{\mathrm{j} \pi \Delta_{t} f} \sin{\left(\pi \Delta_{t} N_{e} f \right)}}{\sin{\left(\pi \Delta_{t} f \right)}}$$

$$\mathrm{sincn}{\left(n \right)} \longleftrightarrow \sum_{m=-\infty}^{\infty} \mathop{\mathrm{rect}}\left[\Delta_{t} f - m\right]$$

$$\mathrm{sincn}{\left(K n \right)} \longleftrightarrow \frac{\sum_{m=-\infty}^{\infty} \mathop{\mathrm{rect}}\left[\frac{\Delta_{t} f}{K} - \frac{m}{K}\right]}{K}$$

$$\mathrm{sincn}^{2}{\left(K n \right)} \longleftrightarrow \frac{\sum_{m=-\infty}^{\infty} \operatorname{tri}{\left(\frac{\Delta_{t} f}{K} - \frac{m}{K} \right)}}{K}$$

$$\mathrm{sincu}{\left(K n \right)} \longleftrightarrow \frac{\pi \sum_{m=-\infty}^{\infty} \mathop{\mathrm{rect}}\left[\frac{\pi \Delta_{t} f}{K} - \frac{\pi m}{K}\right]}{K}$$

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:

$$x(n) \longleftrightarrow X(z)$$

$$x(- m + n) \longleftrightarrow z^{- m} X(z)$$

$$\cos{\left(2 \pi \Delta_{t} f_{0} n \right)} \longleftrightarrow \frac{z \left(z - \cos{\left(2 \pi \Delta_{t} f_{0} \right)}\right)}{z^{2} - 2 z \cos{\left(2 \pi \Delta_{t} f_{0} \right)} + 1}$$

$$\sin{\left(2 \pi \Delta_{t} f_{0} n \right)} \longleftrightarrow \frac{z \sin{\left(2 \pi \Delta_{t} f_{0} \right)}}{z^{2} - 2 z \cos{\left(2 \pi \Delta_{t} f_{0} \right)} + 1}$$

$$e^{2 \mathrm{j} \pi \Delta_{t} f_{0} n} \longleftrightarrow \frac{z}{z - e^{2 \mathrm{j} \pi \Delta_{t} f_{0}}}$$

$$1 \longleftrightarrow \frac{1}{1 - \frac{1}{z}}$$

$$\delta\left[n\right] \longleftrightarrow 1$$

$$\delta\left[n - p\right] \longleftrightarrow \left(\frac{1}{z}\right)^{p}$$

$$a^{n} \longleftrightarrow \frac{z}{- a + z}$$

$$a^{- n} \longleftrightarrow \frac{a z}{a z - 1}$$

$$a^{n} n \longleftrightarrow \frac{a z}{\left(a - z\right)^{2}}$$

$$a^{- n} n \longleftrightarrow \frac{a z}{\left(a z - 1\right)^{2}}$$

$$u\left[n\right] \longleftrightarrow \frac{1}{1 - \frac{1}{z}}$$

$$e^{- \Delta_{t} n} u\left[n\right] \longleftrightarrow \frac{z e^{\Delta_{t}}}{z e^{\Delta_{t}} - 1}$$