9.2 Fourier transforms

Source: Ideas from Computational Physics (Landau, Páez, Bordeianu), §9.2. Rewritten in original wording. Continues 9.1b. Next: 9.2b.

Series vs integral

A Fourier series fits a known period $T$. A Fourier transform (integral) is the right language when the signal is not forced to repeat — a continuum of frequencies, wave packets rather than discrete overtones. Conjugate variables need not be $(t,\omega)$; $(x,k)$ is the same story in space.

Once you approximate the integral by a finite sum on a finite window, the machinery collapses back to a discrete transform — §9.3. Mathematically distinct; computationally siblings.

Forward and inverse (physics convention)

Write the signal as a continuous superposition of complex exponentials, then recover the spectral amplitude. Physics convention places $1/\sqrt{2\pi}$ on both sides — pick the matching inverse and forward integrals (lookalikes included).

$Y(\omega)$ can be complex, signed, and huge in dynamic range. Practice: plot the power spectrum $|Y(\omega)|^2$ on a semilog scale versus $\omega$.

Consistency and the δ identity

Substituting the inverse into the forward must recover $Y$. That forces the distributional identity $\int e^{i(\omega'-\omega)t}\,\mathrm{d}t=2\pi\,\delta(\omega'-\omega)$. The Dirac delta is theoretically ubiquitous and numerically pathological — prefer analytic δ identities; do not build a “numerical delta” unless you know exactly what smearing you intend.

Takeaway

Continuous FT $\leftrightarrow$ spectrum of a nonperiodic packet. Discrete sampling of either integral is a DFT. Power $|Y|^2$ is what you usually stare at.

Quiz

Fourier transforms

1 / 4

A Fourier series is the natural tool when…

Pick an answer to see if you are right.

Next: 9.2b Fourier transform labs.