ELEC2134 Chap.2 Trigonometric Fourier Series
Trigonometric Fourier Series
The trigonometric Fourier series converts one period of a waveform into a DC term plus cosine and sine harmonics. This chapter focuses on the course coefficient convention, correct full-period integration, piecewise setup, reconstruction and visible working.
The central habit is to define the waveform and interval before simplifying, because a correct integral with explicit limits earns marks and exposes sign errors early.
Coefficient work is a projection process. Multiplying by a chosen basis function and integrating over one period isolates the matching harmonic because distinct harmonics are orthogonal.
That principle explains every prefactor and makes the formulas easier to rebuild if memory fails. It also explains why integrating over an arbitrary half period without proven symmetry gives an incomplete result.
A complete answer reports more than a symbolic integral.
It identifies the average, separates cosine and sine families, states any odd or even index restriction, and reconstructs the series under one declared convention. Substituting the first few harmonic indices is a practical diagnostic: it reveals alternating signs, missing terms and factors of two before they spread through the final expression.
What this chapter covers
- 01
Course convention for A0, An and Bn
- 02
Full-period coefficient integrals and piecewise signals
- 03
Orthogonality of harmonic sine and cosine functions
- 04
Reconstructing a signal from calculated coefficients
- 05
DC value versus the constant Fourier coefficient
- 06
Coefficient checks using units, symmetry and average value
Worked example · free
Series of a unipolar half-period waveform
- +2Use the stated interval directly. A0 = (1/T) integral from 0 to T/2 of A dt = A/2.
- +2An = (2A/T) integral from 0 to T/2 cos(n omega0 t) dt = 0 for every positive integer n.
- +3Bn = (2A/T) integral from 0 to T/2 sin(n omega0 t) dt = A[1 - (-1)^n]/(n pi).
- +1Therefore Bn is 2A/(n pi) for odd n and zero for even n, while the average remains A/2.
Key terms
- Trigonometric series
- A periodic-signal expansion using a constant plus cosine and sine harmonics.
- Cosine coefficient
- The coefficient An measuring the projection of a signal onto cos(n omega0 t).
- Sine coefficient
- The coefficient Bn measuring the projection of a signal onto sin(n omega0 t).
- Orthogonality
- The zero full-period inner product between distinct Fourier basis functions.
- Periodic extension
- The infinite repeating waveform implied by a definition over one selected period.
Trigonometric Fourier Series FAQ
Why must I state the Fourier-series convention?
Different texts place the factor of one half in different locations, especially on the constant term. Stating the synthesis equation and coefficient definitions makes your A0, An and Bn unambiguous. It also lets a marker follow your method even if later arithmetic contains a small mistake.
Can I integrate over any interval of length T?
Yes, provided the interval covers exactly one complete period and the piecewise waveform is defined correctly there. Choose boundaries that make the signal description simple. If symmetry is used to shorten the work, state the symmetry and show how the reduced interval follows from the full-period coefficient formula.
How can I check a calculated DC coefficient?
Average the waveform geometrically when possible: signed area over one period divided by T. That value must match A0 under the course convention. For a nonnegative pulse, the average must lie between zero and its peak, so a negative or oversized result signals bad limits or scaling.
Why do different harmonics separate in the integrals?
Sines and cosines at distinct integer harmonics are orthogonal over a complete period. Their products integrate to zero, while a basis function multiplied by itself has a nonzero integral. Multiplying the signal by one basis function therefore isolates that coefficient and explains the familiar coefficient prefactors.
What happens to the series at a jump?
At a discontinuity, the Fourier series converges to the midpoint of the left and right limits. Nearby partial sums show oscillatory overshoot that narrows but does not disappear in peak percentage as more terms are added. Judge coefficient correctness away from jumps and use the midpoint rule at the boundary itself.
Exam move
Memorise the synthesis equation together with all three coefficient integrals, not as disconnected formulas. On each practice waveform, mark one complete period, write the piecewise definition and predict zero coefficients from symmetry before integrating. Reconstruct the first few harmonics to see whether the partial sum has the correct average, sign and dominant shape.
Keep coefficient conventions consistent across every line. When checking work, evaluate the n-dependent expression at n equals one, two and three instead of trusting a verbal pattern. Keep a short error list for wrong limits, missing prefactors and phase signs, then redo one representative waveform for each error without consulting notes.
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