ELEC 6218 · Lecture 2 companion
| Laplace transform | Fourier transform | |
|---|---|---|
| Forward | \(X(s)=\displaystyle\int_{-\infty}^{\infty}x(t)\,e^{-st}\,dt\) Unilateral: lower limit \(0^-\) | \(X(j\Omega)=\displaystyle\int_{-\infty}^{\infty}x(t)\,e^{-j\Omega t}\,dt\) |
| Inverse | \(x(t)=\dfrac{1}{2\pi j}\displaystyle\int_{\sigma-j\infty}^{\sigma+j\infty}X(s)\,e^{st}\,ds\), with \(\sigma\) inside the ROC | \(x(t)=\dfrac{1}{2\pi}\displaystyle\int_{-\infty}^{\infty}X(j\Omega)\,e^{j\Omega t}\,d\Omega\) |
| Basis | \(e^{st}=e^{\sigma t}e^{j\Omega t}\): growing, decaying or constant-envelope sinusoids | \(e^{j\Omega t}\): constant-envelope sinusoids only |
| Where it lives | A region of the \(s\)-plane (the ROC), a vertical strip \(\sigma_1<\operatorname{Re}\{s\}<\sigma_2\) | The \(j\Omega\)-axis |
| Exists when | \(x(t)e^{-\sigma t}\) is absolutely integrable for some \(\sigma\) | \(x(t)\) is absolutely integrable (or finite energy); periodic and step-like signals only with impulses in \(\Omega\) |
| Must be stated | \(X(s)\) and its ROC; the same expression with different ROCs gives different signals | \(X(j\Omega)\) alone; no ROC |
| Link | If the ROC contains the \(j\Omega\)-axis, \(X(j\Omega)=X(s)\big|_{s=j\Omega}\). If it does not, the Laplace transform exists but the Fourier transform does not (e.g. unstable signals), or exists only with impulses (e.g. \(u(t)\)). | |
| System reading | Poles and zeros of \(H(s)\); ROC gives causality (right of the rightmost pole) and stability (ROC contains the \(j\Omega\)-axis) | Frequency response \(H(j\Omega)\): magnitude and phase seen by a sinusoid in steady state |
| Typical use | Solving LCCDEs with initial conditions (unilateral), transients, stability, filter design in the \(s\)-plane | Spectra, bandwidth, filtering, modulation, sampling |
| Laplace: \(x(t)\leftrightarrow X(s)\), ROC \(R\) | Fourier: \(x(t)\leftrightarrow X(j\Omega)\) | |
|---|---|---|
| Linearity | \(aX_1(s)+bX_2(s)\), ROC contains \(R_1\cap R_2\) | \(aX_1(j\Omega)+bX_2(j\Omega)\) |
| Time shift \(x(t-t_0)\) | \(e^{-st_0}X(s)\), ROC \(R\) | \(e^{-j\Omega t_0}X(j\Omega)\) |
| Frequency shift | \(e^{s_0t}x(t)\leftrightarrow X(s-s_0)\), ROC shifted by \(\operatorname{Re}\{s_0\}\) | \(e^{j\Omega_0t}x(t)\leftrightarrow X(j(\Omega-\Omega_0))\) |
| Time scaling \(x(at)\) | \(\dfrac{1}{|a|}X\!\left(\dfrac{s}{a}\right)\), ROC scaled by \(a\) | \(\dfrac{1}{|a|}X\!\left(\dfrac{j\Omega}{a}\right)\) |
| Differentiation \(\dfrac{dx}{dt}\) | \(sX(s)\); unilateral: \(sX(s)-x(0^-)\) | \(j\Omega\,X(j\Omega)\) |
| Integration \(\displaystyle\int_{-\infty}^{t}x(\tau)d\tau\) | \(\dfrac{X(s)}{s}\), ROC contains \(R\cap\{\operatorname{Re}\{s\}>0\}\) | \(\dfrac{X(j\Omega)}{j\Omega}+\pi X(0)\,\delta(\Omega)\) |
| Convolution \(x*h\) | \(X(s)H(s)\), ROC contains \(R_x\cap R_h\) | \(X(j\Omega)H(j\Omega)\) |
| Multiplication \(x(t)y(t)\) | Complex convolution in \(s\); rarely used | \(\dfrac{1}{2\pi}\,X(j\Omega)*Y(j\Omega)\) |
| Initial / final value | Unilateral: \(x(0^+)=\lim_{s\to\infty}sX(s)\); \(x(\infty)=\lim_{s\to0}sX(s)\) if the poles of \(sX(s)\) are in the left-half plane | No direct equivalent |
| Energy | No direct equivalent | Parseval: \(\displaystyle\int|x(t)|^2dt=\frac{1}{2\pi}\int|X(j\Omega)|^2d\Omega\) |
| Duality | No direct equivalent | \(X(jt)\leftrightarrow 2\pi\,x(-\Omega)\) |
| \(x(t)\) | Laplace \(X(s)\), ROC | Fourier \(X(j\Omega)\) |
|---|---|---|
| \(\delta(t)\) | \(1\), all \(s\) | \(1\) |
| \(u(t)\) | \(\dfrac{1}{s}\), \(\operatorname{Re}\{s\}>0\) | \(\pi\delta(\Omega)+\dfrac{1}{j\Omega}\): ROC edge on the axis, so an impulse appears |
| \(e^{-at}u(t)\), \(a>0\) | \(\dfrac{1}{s+a}\), \(\operatorname{Re}\{s\}>-a\) | \(\dfrac{1}{a+j\Omega}\): ROC contains the axis, so substitute \(s=j\Omega\) |
| \(e^{at}u(t)\), \(a>0\) | \(\dfrac{1}{s-a}\), \(\operatorname{Re}\{s\}>a\) | Does not exist: ROC excludes the axis |
| \(e^{-a|t|}\), \(a>0\) | \(\dfrac{-2a}{(s-a)(s+a)}\), \(-a<\operatorname{Re}\{s\}<a\) | \(\dfrac{2a}{a^2+\Omega^2}\) |
| \(\operatorname{rect}(t/T)\) | \(\dfrac{e^{sT/2}-e^{-sT/2}}{s}\), all \(s\) | \(T\,\dfrac{\sin(\Omega T/2)}{\Omega T/2}\) |
| \(1\) (all \(t\)) | Does not exist: no common ROC | \(2\pi\,\delta(\Omega)\) |
| \(\cos(\Omega_0t)\) (all \(t\)) | Does not exist: no common ROC | \(\pi\bigl[\delta(\Omega-\Omega_0)+\delta(\Omega+\Omega_0)\bigr]\) |
| \(e^{-at}\cos(\Omega_0t)\,u(t)\), \(a>0\) | \(\dfrac{s+a}{(s+a)^2+\Omega_0^2}\), \(\operatorname{Re}\{s\}>-a\) | \(\dfrac{a+j\Omega}{(a+j\Omega)^2+\Omega_0^2}\) |
Reading the pairs. The Laplace transform handles signals that grow, because \(e^{-\sigma t}\) can tame them; the Fourier transform handles two-sided periodic signals, which have no ROC, by allowing impulses in \(\Omega\). Where both exist as ordinary functions, the Fourier transform is the Laplace transform evaluated on the \(j\Omega\)-axis.
ELEC 6218 · University of Southampton