The asker has a pure sinusoid with amplitude 1 and wants to recover that amplitude from the output of MATLAB's pwelch(). They take the square root of the power spectral density (PSD), but get 0.605 instead of the expected 1.0. This is a deceptively simple question that touches on several DSP fundamentals that trip up even experienced engineers.

Why the naive approach fails: There are multiple scaling factors baked into pwelch() output that all conspire against a simple sqrt-and-done recovery:

• PSD vs power spectrum: pwelch() returns power spectral density by default, meaning the power is normalized per unit frequency (units of V²/Hz). To get a power spectrum (V²), you'd need to multiply by the frequency resolution df = fs/NFFT. Many people miss this distinction entirely.
• Window energy correction: The Hamming window attenuates the signal. The correct normalization requires dividing by the squared sum of the window coefficients for PSD, or the square of the sum for power spectrum. The asker divided by window length, which is neither of these.
• One-sided spectrum: By default, pwelch() returns a one-sided spectrum, which doubles the power at all frequencies except DC and Nyquist to conserve total energy. This factor of 2 must be accounted for when recovering amplitude.
• Sinusoid power: A sinusoid with amplitude A has power A²/2, not A². So even with perfect scaling, you need to account for this factor.

The correct approach: For a one-sided PSD estimate Pxx from pwelch() with frequency resolution df, the amplitude of a sinusoidal component at a spectral peak is:

amplitude = sqrt(2 * Pxx_peak * df)

The factor of 2 inside the sqrt corrects for sinusoidal power (A²/2). The multiplication by df converts PSD to power spectrum. Alternatively, call pwelch with the 'power' option (MATLAB R2013a+) to get the power spectrum directly, then simply take sqrt(2 * Pxx_peak).

Gotchas to watch for: Spectral leakage will spread energy across adjacent bins, so the peak bin alone may underestimate amplitude. For precise recovery, sum over the main lobe of the leaked peak. Also, the 0.605 value the asker reports is suspiciously close to 1/sqrt(2) × sqrt(sum(hamming²)/N), suggesting they're hitting the window energy correction issue directly. Finally, the overlap parameter (default 50% in MATLAB) affects the equivalent degrees of freedom but not the expected amplitude—though it's a common source of confusion.