Spectrogram

The spectrogram maps a one-dimensional time series into a two-dimensional heatmap, where the x-axis represents time, the y-axis represents frequency, and color intensity represents magnitude. The resolution of this map is governed by the Heisenberg-Gabor uncertainty principle, which dictates an inverse trade-off between time and frequency localization. A narrow analysis window provides sharp temporal precision but smears frequency content, while a wide window yields fine frequency resolution at the cost of temporal smearing.

The spectrogram is mathematically defined as the squared magnitude of the STFT: Spectrogram(t, f) = |STFT(t, f)|^2. The computation involves segmenting the signal into overlapping frames, applying a window function (e.g., Hamming, Hann, or Gaussian) to each segment to mitigate spectral leakage, and then computing the Discrete Fourier Transform (DFT). The overlap between consecutive frames controls the temporal smoothness of the resulting image.

The choice of window function critically shapes the spectrogram's visual fidelity and measurement accuracy. A Gaussian window produces the Gabor transform, which achieves the optimal joint time-frequency concentration. A Hann window offers a good balance of side-lobe roll-off and main-lobe width for general-purpose analysis. A rectangular window, while having the narrowest main lobe, introduces severe spectral leakage that can mask weak signal components.

Spectrograms typically display logarithmic power, converting the linear squared magnitude to a decibel (dB) scale to compress the vast dynamic range of real-world signals into a visually interpretable format. The mapping from dB values to colors is defined by a colormap (e.g., 'inferno', 'viridis', or 'jet'). The dynamic range setting clips the lower bound of the display, setting the noise floor to black and ensuring that only the most energetic spectral components are prominently visible.

Two primary artifacts degrade spectrogram accuracy. Spectral leakage occurs when signal energy spreads into adjacent frequency bins because the analysis window does not contain an integer number of signal periods. Scalloping loss is the amplitude modulation observed when a signal's frequency falls exactly between two DFT bins. These effects are mitigated by applying smooth window functions and by using zero-padding to interpolate the frequency axis, though zero-padding does not increase true resolution.

While a spectrogram uses a fixed window size, resulting in a uniform time-frequency grid, a scalogram (derived from the Continuous Wavelet Transform) uses a variable window size that scales inversely with frequency. This provides a multi-resolution analysis: high temporal resolution for high-frequency transients and high frequency resolution for low-frequency tones. The spectrogram is computationally efficient via the Fast Fourier Transform (FFT), whereas the scalogram offers a more physiologically relevant constant-Q analysis.