
\begin{nicebox}
\begin{align*}
 x(n) &= \sum \limits_{k=\langle p\rangle}X_ke^{ik\omega_0n}  \quad &\mbox{(synthesis equation)} \\
 X_k &= \frac{1}{p} \sum \limits_{n=\langle p\rangle}x(n)e^{-ik\omega_0n} \quad &\mbox{(analysis equation)}
\end{align*}
In DFS, $\omega_0 = 2\pi/p$. Both $X_k, x(n)$ are $p$-periodic in discrete-time.
\end{nicebox}

\begin{nicebox}
\begin{align*}
x(t) &= \sum \limits_{k=-\infty}^{\infty}X_ke^{ik\omega_0t} \quad &\mbox{(synthesis equation)} \\
X_k &= \frac{1}{p} \int_{\langle p\rangle}x(t)e^{-ik\omega_0t}dt \quad &\mbox{(analysis equation)}
\end{align*}

Because we can write $x(t)$ as an infinite number of $X_k$s, $X_k$ is not necessarily periodic, but $x(t)$ is $p$-periodic in continuous-time.
\end{nicebox}

\begin{nicebox}
\begin{align*}
x(n) &= \frac{1}{2\pi} \int_{\langle 2\pi\rangle} X(\omega)e^{i\omega n}d\omega \quad &\mbox{(synthesis equation)} \\
X(\omega) &= \sum \limits_{n=-\infty}^{\infty}x(n)e^{-i\omega n} \quad &\mbox{(analysis equation)}
\end{align*}
$X(\omega)$ is $2\pi$-periodic, but $x(n)$ is not necessarily periodic.
\end{nicebox}

DTFT

\begin{nicebox}
\begin{align*}
x(t) &= \frac{1}{2\pi}\int_{-\infty}^{\infty}X(\omega)e^{i\omega t}d\omega \quad &\mbox{(synthesis equation)} \\
X(\omega) &= \int_{-\infty}^{\infty}x(t)e^{-i\omega t}dt \quad &\mbox{(analysis equation)}
\end{align*}

Neither $x(t)$ or $X(\omega)$ has to be periodic.
\end{nicebox}

CTFT

The continuous-time fourier series lives in the universe of $p$-periodic functions of a continuous variable $t$.

\begin{nicebox}
$$ \langle f,g\rangle = \int_{\langle p\rangle}f(t)g^*(t)dt $$
\end{nicebox}

The discrete-time fourier transform lives in the universe of $2\pi$-periodic functions of a continuous variable $\omega$.

\begin{nicebox}
$$ \langle F,G\rangle = \int_{\langle 2\pi\rangle }F(\omega)G^*(\omega)d\omega$$
\end{nicebox}

\begin{align*}
X &= \frac{1}{p} \Psi^H x \\
x &= \Psi X
\end{align*}

\begin{align*}
\sum \limits_{n=A}^{B} \alpha^n &= \frac{\alpha^{B+1} - \alpha^A}{1-\alpha} \\ 
\sum \limits_{n=0}^{\infty} \alpha^n &= \frac{1}{1-\alpha} \\ 
\end{align*}

\begin{align*}
(x * h) (n) &= \sum \limits_{n = \langle p \rangle} X_\ell H(\ell \omega_0) e^{i\ell \omega_0 n} \\
\end{align*}

\begin{align*}
e^{i\omega_0 n} \to \fbox{H} \to \abs{H(\omega_0)}e^{i(\omega_0 n + \angle H(\omega_0))} 
\end{align*}

Useful Formulae

\begin{nicebox}
{\bf Time-Shifting (Translation) Property }
\begin{align*}
x(n - N) &\ztp z^{-N} \hat{X}(z)
\end{align*}
The $RoC$ is the same as $RoC(x)$ except possibly addition or deletion of 0 or $\infty$.
\end{nicebox}

\begin{nicebox}
{\bf Convolution in the Time Domain }
\begin{align*}
x(n) * h(n) &\ztp \hat{H}(z)\hat{X}(z)
\end{align*}
$RoC \supseteq RoC(x) \cap RoC(h)$.
\end{nicebox}

\begin{nicebox}
{\bf Modulation with a Complex Exponential }
\begin{align*}
z_0^n x(n) &\ztp \hat{X}\lr{\frac{z}{z_0}}
\end{align*}
$RoC:\abs{z_0} RoC(x)$ 
\end{nicebox}


\begin{nicebox}
{\bf Time-Reversal Property }
\begin{align*}
x(-n) &\ztp \hat{X}(z^{-1})
\end{align*}
$RoC : \frac{1}{RoC(x)}$
\end{nicebox}

\begin{nicebox}
{\bf Z-domain Differentiation Property }
\begin{align*}
n x(n) &\ztp -z \frac{d}{dz} \hat{X}(z)
\end{align*}
The $RoC$ is the same as $RoC(x)$ except possibly addition or deletion of 0 or $\infty$.
\end{nicebox}

\begin{nicebox}
{\bf Conjugation Property }
\begin{align*}
x^*(n) &\ztp \hat{X}^*(z^*)
\end{align*}
$RoC = RoC(x)$
\end{nicebox}

\begin{nicebox}
{\bf Dilation }
$$
\begin{cases}
x\lr{\frac{n}{N}} & n\bmod N=0 \\
0 & \mbox{otherwise} \\
\end{cases}
\ztp 
\hat{X}(z^N)
$$

$RoC = \sqrt[n]{RoC(x)}$
\end{nicebox}


Z-Transforms

\begin{nicebox}
{\bf Time-Shifting (Translation) Property }
\begin{align*}
x(t+T) &\ltp e^{sT}\hat{X}(s) \\
\end{align*}
The $RoC$ excludes $\infty$ if the signal is not causal, or excludes $-\infty$ if the signal is not anti-causal. 
\end{nicebox}

\begin{nicebox}
{\bf Convolution in the Time Domain }
\begin{align*}
h(t) &= f(t) * g(t) \\
\hat{H}(s) &= \hat{F}(s)\hat{G}(s) \\
\end{align*}
The $RoC(h) \supseteq RoC(f) \cap RoC(g)$.
\end{nicebox}

\begin{nicebox}
{\bf Modulation with a Complex Exponential }
\begin{align*}
e^{s_0 t} x(t) &\ltp \hat{X}(s-s_0) \\
\end{align*}
$RoC = RoC(x) + \Real{s_0} $
\end{nicebox}

\begin{nicebox}
{\bf Integration in the Time Domain }
\begin{align*}
 \int_{-\infty}^t x(\tau) d\tau &\ltp \frac{\hat{X}(s)}{s}
\end{align*}
$RoC(y) \supseteq RoC(x) \cap \{ s \st \Real{s} \gt 0 \} $
\end{nicebox}


\begin{nicebox}
{\bf Differentiation in the Time Domain }
\begin{align*}
\frac{d}{dt} x(t) \ltp s\hat{X}(s) \\
\end{align*}
$RoC \supseteq RoC(x)$
\end{nicebox}

\begin{nicebox}
{\bf Generalized Differentiation in the Time Domain }
\begin{align*}
\frac{d^k}{dt^k} x(t) \ltp s^k \hat{X}(s) \quad k \in \N \\
\end{align*}
$RoC \supseteq RoC(x)$
\end{nicebox}

\begin{nicebox}
{\bf Frequency Differentiation }
\begin{align*}
-tx(t) \ltp \frac{d}{ds}\hat{X}(s) \\
\end{align*}
$RoC = RoC(x)$
\end{nicebox}

\begin{nicebox}
{\bf Conjugation Property }
\begin{align*}
x^*(t) &\ltp \hat{X}^*(s^*)
\end{align*}
$RoC = RoC(x)$
\end{nicebox}

1 Equations