Tehospektri

Tehospektri, literally meaning “power spectrum” in Finnish (and similarly used in some other languages), is a concept in signal processing that describes how the power of a signal is distributed across different frequencies. It provides a frequency-domain representation of a signal’s energy content and is a fundamental tool for analyzing periodicities and spectral features.

Definition and theory: For a deterministic time-domain signal x(t), the power spectrum can be expressed as the squared magnitude of its Fourier transform, P(f) = |X(f)|^2, under suitable normalization. For stochastic or random processes, the power spectral density (often still called tehospektri in some contexts) is the Fourier transform of the autocorrelation function R_xx(τ): P_xx(f) = ∫ R_xx(τ) e^{-j 2π f τ} dτ. The spectrum summarizes how average power is distributed over frequency and is related to the time-domain properties of the signal through the Wiener–Khinchin relation.

Calculation and estimation: In practice, the periodogram (the squared magnitude of the discrete Fourier transform) is a basic estimator but can be noisy. Common improvements include the Welch method (averaging periodograms from overlapping segments) and multitaper methods (using multiple orthogonal windows). Challenges include spectral leakage, resolution versus variance trade-offs, and the influence of sampling and windowing.

Applications and interpretation: Tehospektri is widely used in audio engineering, vibration analysis, seismology, and neuroscience to identify dominant frequencies, periodic components, and noise characteristics. Peaks indicate strong frequency components; the overall shape reveals broadband content. Interpretation depends on data length, window choice, and stationarity assumptions.