ELEC2134 Chap.3 Symmetry and Harmonic Elimination
Symmetry and Harmonic Elimination
Symmetry is a calculation shortcut and a diagnostic tool, but only when each condition is tested precisely. Even symmetry removes sine coefficients, odd symmetry removes the DC and cosine coefficients, and half-wave symmetry removes even harmonic indices.
This chapter separates those rules, shows how they combine, and corrects the tempting but false claim that odd symmetry alone removes every even harmonic.
The selected time origin controls even and odd tests. A shifted waveform may acquire a cleaner parity description, but its coefficient phases are then referenced to the shifted origin.
Half-wave symmetry is different: it compares samples separated by T/2 and does not follow automatically from oddness. Writing all three defining equations before deleting coefficients prevents one visual impression from being mistaken for another property.
Symmetry can also audit a proposed answer. A real even waveform should not produce sine coefficients, while an odd waveform must have zero average.
Half-wave symmetry forbids every even harmonic index regardless of whether the remaining family is sine, cosine or both. If a claimed coefficient violates one of these consequences, revisit the waveform definition, origin or integration limits before continuing.
What this chapter covers
- 01
Even symmetry about the selected time origin
- 02
Odd symmetry and the zero average condition
- 03
Half-wave symmetry across half a period
- 04
Quarter-wave reductions when additional structure exists
- 05
Combining symmetry tests without overclaiming
- 06
Diagnosing forbidden lines in a proposed spectrum
Worked example · free
Odd sawtooth coefficient pattern
- +2Because x(-t) = -x(t), the signal is odd. Therefore A0 = 0 and An = 0 for all n.
- +2The product x(t)sin(n omega0 t) is even, so Bn = (4/T) integral from 0 to T/2 of kt sin(n omega0 t) dt.
- +2Integration by parts gives Bn = kT(-1)^(n+1)/(n pi). Even and odd indices both remain, with alternating sign.
Key terms
- Even symmetry
- The relation x(-t) = x(t), which eliminates sine coefficients.
- Odd symmetry
- The relation x(-t) = -x(t), which eliminates DC and cosine coefficients.
- Half-wave symmetry
- The relation x(t + T/2) = -x(t), which eliminates even harmonic indices.
- Quarter-wave symmetry
- Additional waveform structure that can reduce coefficient integrals to one quarter period.
- Forbidden harmonic
- A spectral line forced to zero by a proven waveform symmetry.
Symmetry and Harmonic Elimination FAQ
Does odd symmetry remove even harmonics?
No. Odd symmetry removes the constant term and all cosine coefficients because their integrands have the required parity. Even-indexed sine coefficients can remain. They disappear only when another condition, such as half-wave symmetry, forces every even harmonic index to zero.
Can a signal be both odd and half-wave symmetric?
Yes. Many standard waveforms satisfy both conditions. Odd symmetry then leaves only sine coefficients, while half-wave symmetry removes the even-indexed sine terms. The combined result is a series containing only odd sine harmonics, but each conclusion should be tied to its own condition.
What if shifting the time origin reveals symmetry?
A time shift can make the waveform even or odd and simplify coefficient work. The physical signal is unchanged, but the coefficient phases change relative to the original origin. State the shifted variable clearly, derive coefficients in that coordinate and convert phase back if the question requires the original reference.
Can one point prove that a signal is symmetric?
No. A symmetry identity must hold for every relevant time, so one matching pair cannot prove it. However, one counterexample can reject it immediately. Use carefully selected mirrored or half-period-separated points to disprove a claim, then use the waveform definition to establish any property you retain.
Does half-wave symmetry force zero average?
Yes. Every value in one half period is cancelled by its negative counterpart in the next half, so signed area over a full period is zero. This removes the DC coefficient as well as every even harmonic index. It does not decide whether surviving odd-index terms are sine, cosine or both.
Exam move
Use a fixed decision sequence: test even, test odd, then test half-wave. Write each equation explicitly rather than judging from appearance. On practice spectra, mark which lines each rule forbids and invent a counterexample to every overgeneralised rule. When reducing an integral, show the full-period formula first so the factor of two or four has a visible justification.
Test claims with one carefully chosen pair of points: a single counterexample can reject a symmetry. Finish by stating both what vanishes and what may remain, because absence rules do not determine the amplitudes of surviving coefficients.
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