Fourier: one idea, two industries

If you only take one thing from the maths layer, take this: the Fourier transform describes a signal as weighted sinusoidal frequency components. For suitable signals, the inverse transform reconstructs the signal; periodic signals are often introduced through a Fourier series. The same idea applies to sound over time and image brightness over space.

X(f) = ∫ x(t) e^(-2πi f t) dt          # continuous, time domain

X[k] = Σ_{n=0}^{N-1} x[n] e^(-2πi k n / N)   # DFT, what you actually run

# spots where the two fields are literally the same operation:
#   x(t)  time-domain waveform      <->  x(u,v)  image brightness
#   f     frequency in Hz           <->  spatial frequency (cycles per pixel)
#   low-pass filter                 <->  blur
#   high-pass filter                <->  edge detection
#   harmonic (a multiple of f0)     <->  a periodic texture / grid pattern

Two practical issues follow from analysing a finite sampled segment. They are distinct from aliasing:

Window leakage. The DFT treats a finite chunk as one period repeated forever. If the endpoints do not join smoothly, energy spreads across bins. A Hann window tapers the ends and reduces sidelobes, at the cost of a wider main lobe; windowing reduces leakage but does not eliminate it.

Bin spacing. DFT bins are spaced `sampleRate / N`. A short observation has wider bins and makes nearby tones harder to separate. Zero-padding can interpolate the displayed spectrum, but it does not add information or improve the underlying ability to resolve tones.