Summary

This paper provides complete, closed-form equations for expressing the sum of two equal-frequency real sinusoids as a single equivalent real sinusoid. Readers will find explicit derivations and special-case formulas that are often missing or scattered in signal processing literature, enabling accurate amplitude–phase conversion and practical handling of edge cases.

Key Takeaways

• Derive closed-form amplitude and phase expressions for the single equivalent sinusoid resulting from two equal-frequency real sinusoids.
• Convert sums of sine and cosine components into standard amplitude–phase (C cos(ωt+φ)) or alternative single-sinusoid forms.
• Apply the formulas to simplify spectral interpretation, phasor addition, and signal reconstruction in communications and audio contexts.
• Handle special cases and numerical pitfalls (phase wrapping, zero-amplitude conditions) using the provided conditional expressions.

Who Should Read This

Advanced DSP engineers, researchers, and graduate students working on signal representation, spectral analysis, communications, or audio/speech processing who need exact formulas for combining equal-frequency sinusoids.

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