EC-4.4 The Fourier Transform

Work through these

• From series to transform: what happens when the period grows without bound

  As the period lengthens the harmonics crowd together until the discrete line spectrum becomes a continuous function of frequency, and the sum becomes an integral. Watching that limit is what makes the transform pair look inevitable rather than invented.

  NPTEL: Transform Techniques for Engineers · Course

• The transform pair, and the conventions that differ between books

  Where the factor sits and whether frequency is measured in radians or cycles varies between texts, so a formula copied from one source into another can be wrong by a constant. Fixing one convention and stating it is the practical defence.

• Linearity, shifting and scaling, and what each does to the spectrum

  A shift in time multiplies the transform by a phase term and leaves magnitude untouched, and compressing a signal in time stretches its spectrum by the same factor. These are used far more often than the defining integral.

  NPTEL: Transform Techniques for Engineers · Course

• The standard pairs: rectangle, impulse and Gaussian

  A rectangle transforms to a sinc, an impulse to a constant, and a Gaussian to another Gaussian. These three cover most of what appears in practice and are worth deriving once and then knowing.

• Convolution in one domain is multiplication in the other

  This single property is why the transform is central to engineering rather than merely elegant, because it converts an awkward integral into a product. The systems reading of it belongs to a later module and is deliberately not taken here.

• Parseval: the energy is the same counted in either domain

  Total energy computed by integrating the squared signal equals the energy computed from the squared spectrum, up to the convention's constant. It is both a physical statement and a useful check on a hand computation.