ELEC2134 Chap.1 Periodic Signals and Harmonic Spectra
Periodic Signals and Harmonic Spectra
This chapter establishes the vocabulary that every later ELEC2134 transform calculation uses. A periodic signal repeats after a smallest positive period, which fixes its fundamental frequency and the spacing of its harmonic lines.
You will separate DC from oscillating components, identify harmonic indices in composite waveforms, and read line spectra without confusing frequency in hertz with angular frequency in radians per second.
The smallest-period requirement matters whenever a waveform appears to repeat on several scales. A proposed interval may reproduce the graph yet still be an integer multiple of the fundamental period.
Using that longer interval changes the apparent frequency grid and gives misleading harmonic labels. The safest check is to test whether every component completes an integer number of cycles and then ask whether a shorter common interval also works.
Spectral reading should move in both directions. From a formula, identify the constant, frequencies, amplitudes and phases before drawing lines.
From a spectrum, recover the fundamental spacing and decide which harmonics are present or missing. For real signals, remember that a one-sided amplitude plot and a two-sided complex spectrum use different coefficient conventions even though they describe the same waveform.
What this chapter covers
- 01
Fundamental period and the smallest valid repetition interval
- 02
Frequency in hertz and angular frequency in radians per second
- 03
Harmonic indices and integer multiples of the fundamental
- 04
Average value as the zero-frequency or DC component
- 05
One-sided and two-sided line-spectrum interpretation
- 06
Periodicity tests for sums of commensurate sinusoids
Worked example · free
Read a composite periodic signal
- +1The angular frequencies are 40 pi and 80 pi radians per second. Their greatest common angular frequency is 40 pi.
- +2Thus omega0 = 40 pi, f0 = omega0/(2 pi) = 20 Hz and T = 1/f0 = 0.05 s.
- +2The constant 3 is the DC component. The cosine is harmonic n = 1 and the sine is harmonic n = 2.
Key terms
- Fundamental period
- The smallest positive time shift after which a periodic signal repeats exactly.
- Angular frequency
- Frequency measured in radians per second, related to hertz by omega = 2 pi f.
- Harmonic index
- The integer multiplier n locating a component at n times the fundamental frequency.
- DC component
- The zero-frequency component equal to the signal average over one period.
- Line spectrum
- A plot of discrete frequency lines representing the components of a periodic signal.
Periodic Signals and Harmonic Spectra FAQ
Can a periodic signal have more than one valid period?
Yes. Every integer multiple of a valid period is also a period, but the fundamental period is the smallest positive one. Fourier-series spacing must use that smallest interval. Choosing a larger repeating interval produces an unnecessarily fine frequency grid and can hide the true harmonic indices.
How do I test whether several sinusoids form a periodic sum?
Express all angular frequencies as multiples of a common positive value. If every ratio is rational, a common period exists; use the greatest common angular frequency as omega0. Then verify that each sinusoid completes an integer number of cycles during T = 2 pi divided by omega0.
Why does a periodic signal have spectral lines?
A periodic signal can contain only integer multiples of one fundamental frequency in its Fourier series. Each allowed sinusoid contributes at a discrete location, so the spectrum appears as separated lines rather than a continuous curve. The spacing between neighbouring harmonic locations is the fundamental frequency.
How should I label a two-sided spectrum?
Put negative frequencies to the left, positive frequencies to the right and zero frequency at the origin. State whether line heights are complex coefficients, magnitudes or amplitudes, because those conventions differ. For a real signal, use conjugate symmetry to check that the negative side mirrors the positive side correctly.
What does a missing harmonic tell me?
A missing line means the corresponding Fourier coefficient is zero. One isolated zero may result from waveform geometry or cancellation, while a repeated pattern can reveal symmetry or a sinc-envelope zero. Do not claim a symmetry from one absent line; test the time-domain identity or inspect the complete coefficient pattern first.
Exam move
Begin every problem by writing the candidate period and both forms of fundamental frequency. Label each component with its harmonic index before calculating amplitudes or phases. Practise moving between a waveform, a formula and a line spectrum, and always include the DC line when the average is nonzero. Use unit checks to prevent mixing hertz with radians per second.
After solving, test the result by shifting the waveform one proposed period and by checking that every nonzero spectral line lies on the same integer grid. Build speed with short drills that alternate between time plots, equations and spectra rather than repeating only one representation.
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