ELEC2134 Chap.4 Amplitude, Phase and Exponential Series
Amplitude, Phase and Exponential Series
A real periodic signal can be written with separate cosine and sine coefficients, with one amplitude and phase per harmonic, or with complex exponential coefficients at positive and negative indices. These are equivalent descriptions.
This chapter builds reliable conversions between them, uses atan2 to preserve the phase quadrant, and explains why real signals produce conjugate-symmetric exponential spectra.
The cosine and sine coefficients behave like Cartesian coordinates, while harmonic amplitude and phase behave like polar coordinates. That geometric view makes the conversion systematic and exposes why a one-argument arctangent is unsafe.
Both signs matter when selecting a quadrant. Expanding the resulting phase-shifted cosine is the quickest way to verify that the original sine sign has been preserved.
Complex coefficients split one real sinusoid into positive- and negative-frequency rotations. For real time signals the two sides are conjugates, so neither half can be chosen independently.
Even real signals give real exponential coefficients, and odd real signals give purely imaginary ones. These special cases provide stronger checks than conjugate symmetry alone and help diagnose phase or factor-of-two mistakes.
What this chapter covers
- 01
Cosine-sine coefficients as Cartesian components
- 02
Amplitude and phase as polar coordinates
- 03
Complex exponential coefficient definition
- 04
Positive and negative frequency components
- 05
Conjugate symmetry for real time signals
- 06
Phase quadrants and the atan2 convention
Worked example · free
Convert one harmonic between forms
- +1The amplitude is M = sqrt(3 squared plus negative 4 squared) = 5.
- +2For M cos(n omega0 t + phi), A = M cos phi and B = -M sin phi, so phi = atan2(4, 3) = 53.13 degrees.
- +2For positive n, Cn = (A - jB)/2 = (3 + j4)/2 = 1.5 + j2.
- +1Because the signal is real, C minus n = Cn conjugate = 1.5 - j2.
Key terms
- Amplitude-phase form
- A harmonic representation using one nonnegative magnitude and one phase angle.
- Exponential coefficient
- The complex coefficient Cn multiplying exp(jn omega0 t).
- Conjugate symmetry
- For a real signal, the relation C minus n equals the complex conjugate of Cn.
- Phase quadrant
- The correct angular sector determined jointly by the signs of Cartesian components.
- Negative frequency
- The exponential-series component indexed below zero that pairs with a positive-frequency component for real signals.
Amplitude, Phase and Exponential Series FAQ
Why should I use atan2 instead of ordinary arctangent?
Ordinary arctangent sees only a ratio and cannot distinguish opposite quadrants. atan2 uses both signed components, so it returns the phase consistent with the actual cosine and sine coefficients. This matters when one or both coefficients are negative and a 180-degree correction would otherwise be missed.
What does a negative frequency mean physically?
Negative frequency is part of the complex exponential representation, not a separate sinusoid rotating in ordinary time measurements. A real cosine is built from equal positive and negative rotating exponentials. Their paired coefficients combine into a real oscillation and make conjugate symmetry visible.
How are An and Bn related to Cn?
For positive n under the course convention, Cn = (An - jBn)/2. Therefore An = 2 Re(Cn) and Bn = -2 Im(Cn). The zero-index coefficient equals the DC value. Expanding the exponential pair is a dependable way to verify signs.
Why is each exponential magnitude half the cosine amplitude?
Euler's identity writes one cosine as the sum of positive- and negative-frequency exponentials, each carrying half the real sinusoid amplitude. The pair recombines into the original cosine. Forgetting the half factor doubles the reconstructed waveform and breaks the relationship between trigonometric and exponential coefficients.
Can two phase answers both be correct?
Yes, when they differ by an integer multiple of 360 degrees or use explicitly different cosine conventions. A plus-phase form and a minus-phase form report opposite angle signs. State the convention, keep amplitude nonnegative and expand the result; matching the original cosine and sine coefficients settles equivalence.
Exam move
Treat the three series forms as coordinate systems for the same harmonic. Practise conversions in both directions and expand your answer after every phase calculation. Sketch Cn in the complex plane so magnitude and phase have a geometric meaning. For real signals, fill the negative-frequency side by conjugation rather than recalculating it, and confirm that the reconstructed pair is real.
State the cosine phase convention before giving an angle, because an alternative minus-phase convention reverses the reported sign. Use a short checklist: magnitude nonnegative, quadrant correct, exponential amplitude halved, and negative index conjugated.
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