Signals and Systems Essentials

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Continuous-Time Fourier Transform (CTFT) Definition Forward Transform: $X(j\omega) = \mathcal{F}\{x(t)\} = \int_{-\infty}^{\infty} x(t) e^{-j\omega t} dt$ Inverse Transform: $x(t) = \mathcal{F}^{-1}\{X(j\omega)\} = \frac{1}{2\pi} \int_{-\infty}^{\infty} X(j\omega) e^{j\omega t} d\omega$ Properties of CTFT Linearity: $\mathcal{F}\{ax(t) + by(t)\} = aX(j\omega) + bY(j\omega)$ Time Shift: $\mathcal{F}\{x(t-t_0)\} = e^{-j\omega t_0} X(j\omega)$ Frequency Shift: $\mathcal{F}\{e^{j\omega_0 t} x(t)\} = X(j(\omega - \omega_0))$ Scaling: $\mathcal{F}\{x(at)\} = \frac{1}{|a|} X\left(j\frac{\omega}{a}\right)$ Time Reversal: $\mathcal{F}\{x(-t)\} = X(-j\omega)$ Differentiation in Time: $\mathcal{F}\left\{\frac{d}{dt} x(t)\right\} = j\omega X(j\omega)$ Differentiation in Frequency: $\mathcal{F}\{t x(t)\} = j \frac{d}{d\omega} X(j\omega)$ Integration: $\mathcal{F}\left\{\int_{-\infty}^{t} x(\tau) d\tau\right\} = \frac{1}{j\omega} X(j\omega) + \pi X(0) \delta(\omega)$ Convolution in Time: $\mathcal{F}\{x(t) * y(t)\} = X(j\omega) Y(j\omega)$ Multiplication in Time: $\mathcal{F}\{x(t) y(t)\} = \frac{1}{2\pi} [X(j\omega) * Y(j\omega)]$ Parseval's Relation: $\int_{-\infty}^{\infty} |x(t)|^2 dt = \frac{1}{2\pi} \int_{-\infty}^{\infty} |X(j\omega)|^2 d\omega$ CTFT Pairs (Common) Signal $x(t)$ Fourier Transform $X(j\omega)$ $\delta(t)$ $1$ $1$ $2\pi \delta(\omega)$ $e^{j\omega_0 t}$ $2\pi \delta(\omega - \omega_0)$ $\cos(\omega_0 t)$ $\pi [\delta(\omega - \omega_0) + \delta(\omega + \omega_0)]$ $\sin(\omega_0 t)$ $j\pi [\delta(\omega + \omega_0) - \delta(\omega - \omega_0)]$ $u(t)$ $\frac{1}{j\omega} + \pi \delta(\omega)$ $e^{-at} u(t)$, $a > 0$ $\frac{1}{a + j\omega}$ $rect(t/\tau)$ $\tau \text{sinc}(\frac{\omega \tau}{2})$ $e^{-a|t|}$, $a > 0$ $\frac{2a}{a^2 + \omega^2}$ Laplace Transform (LT) Definition Forward Transform (Unilateral): $X(s) = \mathcal{L}\{x(t)\} = \int_{0}^{\infty} x(t) e^{-st} dt$ Forward Transform (Bilateral): $X(s) = \mathcal{L}\{x(t)\} = \int_{-\infty}^{\infty} x(t) e^{-st} dt$ Inverse Transform: $x(t) = \mathcal{L}^{-1}\{X(s)\} = \frac{1}{2\pi j} \int_{\sigma - j\infty}^{\sigma + j\infty} X(s) e^{st} ds$ $s = \sigma + j\omega$ (complex frequency) Properties of LT (Unilateral) Linearity: $\mathcal{L}\{ax(t) + by(t)\} = aX(s) + bY(s)$ Time Shift: $\mathcal{L}\{x(t-t_0)u(t-t_0)\} = e^{-st_0} X(s)$ ($t_0 \ge 0$) Frequency Shift: $\mathcal{L}\{e^{at} x(t)\} = X(s-a)$ Scaling: $\mathcal{L}\{x(at)\} = \frac{1}{a} X\left(\frac{s}{a}\right)$, $a > 0$ Differentiation in Time: $\mathcal{L}\left\{\frac{d}{dt} x(t)\right\} = sX(s) - x(0^-)$ Differentiation in Frequency: $\mathcal{L}\{t x(t)\} = -\frac{d}{ds} X(s)$ Integration: $\mathcal{L}\left\{\int_{0}^{t} x(\tau) d\tau\right\} = \frac{1}{s} X(s)$ Convolution: $\mathcal{L}\{x(t) * y(t)\} = X(s) Y(s)$ Initial Value Theorem: $x(0^+) = \lim_{s \to \infty} sX(s)$ (if limit exists) Final Value Theorem: $\lim_{t \to \infty} x(t) = \lim_{s \to 0} sX(s)$ (if poles of $sX(s)$ are in LHP) LT Pairs (Common) Signal $x(t)$ Laplace Transform $X(s)$ ROC $\delta(t)$ $1$ All $s$ $u(t)$ $\frac{1}{s}$ $\text{Re}\{s\} > 0$ $e^{-at} u(t)$ $\frac{1}{s+a}$ $\text{Re}\{s\} > -a$ $t u(t)$ $\frac{1}{s^2}$ $\text{Re}\{s\} > 0$ $t^n u(t)$ $\frac{n!}{s^{n+1}}$ $\text{Re}\{s\} > 0$ $\sin(\omega_0 t) u(t)$ $\frac{\omega_0}{s^2 + \omega_0^2}$ $\text{Re}\{s\} > 0$ $\cos(\omega_0 t) u(t)$ $\frac{s}{s^2 + \omega_0^2}$ $\text{Re}\{s\} > 0$ $e^{-at} \sin(\omega_0 t) u(t)$ $\frac{\omega_0}{(s+a)^2 + \omega_0^2}$ $\text{Re}\{s\} > -a$ $e^{-at} \cos(\omega_0 t) u(t)$ $\frac{s+a}{(s+a)^2 + \omega_0^2}$ $\text{Re}\{s\} > -a$ Conversion between CTFT and LT If the Region of Convergence (ROC) of $X(s)$ includes the $j\omega$-axis, then $X(j\omega) = X(s)|_{s=j\omega}$. Discrete-Time Fourier Transform (DTFT) Definition Forward Transform: $X(e^{j\omega}) = \mathcal{F}\{x[n]\} = \sum_{n=-\infty}^{\infty} x[n] e^{-j\omega n}$ Inverse Transform: $x[n] = \mathcal{F}^{-1}\{X(e^{j\omega})\} = \frac{1}{2\pi} \int_{-\pi}^{\pi} X(e^{j\omega}) e^{j\omega n} d\omega$ Properties of DTFT Linearity: $\mathcal{F}\{ax[n] + by[n]\} = aX(e^{j\omega}) + bY(e^{j\omega})$ Time Shift: $\mathcal{F}\{x[n-n_0]\} = e^{-j\omega n_0} X(e^{j\omega})$ Frequency Shift: $\mathcal{F}\{e^{j\omega_0 n} x[n]\} = X(e^{j(\omega - \omega_0)})$ Scaling: Not simple in DTFT (corresponds to D-T resampling) Time Reversal: $\mathcal{F}\{x[-n]\} = X(e^{-j\omega})$ Differentiation in Frequency: $\mathcal{F}\{n x[n]\} = j \frac{d}{d\omega} X(e^{j\omega})$ Convolution in Time: $\mathcal{F}\{x[n] * y[n]\} = X(e^{j\omega}) Y(e^{j\omega})$ Multiplication in Time: $\mathcal{F}\{x[n] y[n]\} = \frac{1}{2\pi} [X(e^{j\omega}) * Y(e^{j\omega})]$ (convolution is periodic) Parseval's Relation: $\sum_{n=-\infty}^{\infty} |x[n]|^2 = \frac{1}{2\pi} \int_{-\pi}^{\pi} |X(e^{j\omega})|^2 d\omega$ DTFT Pairs (Common) Signal $x[n]$ DTFT $X(e^{j\omega})$ $\delta[n]$ $1$ $1$ (constant) $2\pi \sum_{k=-\infty}^{\infty} \delta(\omega - 2\pi k)$ $e^{j\omega_0 n}$ $2\pi \sum_{k=-\infty}^{\infty} \delta(\omega - \omega_0 - 2\pi k)$ $u[n]$ $\frac{1}{1 - e^{-j\omega}} + \pi \sum_{k=-\infty}^{\infty} \delta(\omega - 2\pi k)$ $a^n u[n]$, $|a| $\frac{1}{1 - a e^{-j\omega}}$ $\cos(\omega_0 n)$ $\pi \sum_{k=-\infty}^{\infty} [\delta(\omega - \omega_0 - 2\pi k) + \delta(\omega + \omega_0 - 2\pi k)]$ $\sin(\omega_0 n)$ $j\pi \sum_{k=-\infty}^{\infty} [\delta(\omega + \omega_0 - 2\pi k) - \delta(\omega - \omega_0 - 2\pi k)]$ Z-Transform Definition Forward Transform (Bilateral): $X(z) = \mathcal{Z}\{x[n]\} = \sum_{n=-\infty}^{\infty} x[n] z^{-n}$ Unilateral Z-Transform: $X(z) = \sum_{n=0}^{\infty} x[n] z^{-n}$ Inverse Transform: $x[n] = \frac{1}{2\pi j} \oint_C X(z) z^{n-1} dz$ (contour integral) $z = r e^{j\omega}$ (complex variable) Properties of Z-Transform (Unilateral) Linearity: $\mathcal{Z}\{ax[n] + by[n]\} = aX(z) + bY(z)$ Time Shift (Right): $\mathcal{Z}\{x[n-k]u[n-k]\} = z^{-k} X(z)$ ($k \ge 0$) Time Shift (Left): $\mathcal{Z}\{x[n+k]\} = z^k X(z) - z^k \sum_{m=0}^{k-1} x[m] z^{-m}$ ($k > 0$) Scaling (Multiplication by $a^n$): $\mathcal{Z}\{a^n x[n]\} = X(z/a)$ Time Reversal: $\mathcal{Z}\{x[-n]\}$ (ROC is inverted) Differentiation in Z: $\mathcal{Z}\{n x[n]\} = -z \frac{d}{dz} X(z)$ Convolution: $\mathcal{Z}\{x[n] * y[n]\} = X(z) Y(z)$ Initial Value Theorem: $x[0] = \lim_{z \to \infty} X(z)$ (for causal signals) Final Value Theorem: $\lim_{n \to \infty} x[n] = \lim_{z \to 1} (z-1)X(z)$ (if poles of $(z-1)X(z)$ are inside unit circle) Z-Transform Pairs (Common) Signal $x[n]$ Z-Transform $X(z)$ ROC $\delta[n]$ $1$ All $z$ $u[n]$ $\frac{1}{1 - z^{-1}}$ $|z| > 1$ $a^n u[n]$ $\frac{1}{1 - a z^{-1}}$ $|z| > |a|$ $-a^n u[-n-1]$ $\frac{1}{1 - a z^{-1}}$ $|z| $n u[n]$ $\frac{z^{-1}}{(1 - z^{-1})^2}$ $|z| > 1$ $n a^n u[n]$ $\frac{a z^{-1}}{(1 - a z^{-1})^2}$ $|z| > |a|$ $\cos(\omega_0 n) u[n]$ $\frac{1 - z^{-1} \cos(\omega_0)}{1 - 2z^{-1} \cos(\omega_0) + z^{-2}}$ $|z| > 1$ $\sin(\omega_0 n) u[n]$ $\frac{z^{-1} \sin(\omega_0)}{1 - 2z^{-1} \cos(\omega_0) + z^{-2}}$ $|z| > 1$ Conversion between DTFT and Z-Transform If the Region of Convergence (ROC) of $X(z)$ includes the unit circle ($|z|=1$), then $X(e^{j\omega}) = X(z)|_{z=e^{j\omega}}$.