EC-4.3 Fourier Series, and Why a Periodic Signal Decomposes

Work through these

• Orthogonality, and why it makes the coefficients computable one at a time

  Two sinusoids at different multiples of the same fundamental integrate to zero over a period, so multiplying the signal by one of them and integrating isolates that coefficient alone. Without this property the coefficients would have to be solved for together.

  NPTEL: Transform Techniques for Engineers · Course

• The series in sine-cosine form and in exponential form, and why the second is preferred

  The two forms carry identical information, but the exponential form has one coefficient per frequency instead of two and makes the algebra of shifting and scaling far shorter. Later subjects use the exponential form almost exclusively.

• Compute the coefficients for a square wave and for a sawtooth

  These two waveforms appear constantly and their coefficient patterns are worth knowing by heart: one has only odd harmonics falling as one over the harmonic number, the other has all harmonics falling at the same rate. Deriving both once is the point.

  NPTEL: Transform Techniques for Engineers · Course

• Convergence, and the overshoot that never goes away at a jump

  Adding more terms improves the fit everywhere except immediately beside a discontinuity, where a fixed percentage overshoot persists and merely narrows. It is a property of the series, not an error in the computation.

• Symmetry, and the coefficients it removes before you integrate

  An even waveform has no sine terms, an odd one has no cosine terms, and half-wave symmetry removes every even harmonic. Checking symmetry first often halves the work or more.

• The line spectrum: a periodic signal occupies discrete frequencies only

  Plotting coefficient magnitude against frequency gives isolated lines at multiples of the fundamental rather than a continuous curve. That discreteness is what the next topic gives up when the period is allowed to grow without bound.