intuitive understanding of magnitude and phase from STFT

In the context of audio signals, the magnitude and phase obtained from the Short-Time Fourier Transform (STFT) have intuitive meanings related to the perception and structure of sound:

Magnitude:

The magnitude represents the strength (or amplitude) of each frequency component in the signal at a given time. Intuitively, it tells you:

• How loud each frequency is: Higher magnitude values correspond to louder sounds at that particular frequency.
• Timbre and spectral content: The combination of different magnitudes across frequencies shapes the timbre of the sound, giving it its characteristic tone. For instance, a violin and a piano playing the same note will have different magnitudes in their harmonic overtones, leading to distinct sounds.

In essence:

• Magnitude controls volume and timbre. For example, a loud bass drum hit will have high magnitude values in low frequencies, whereas a violin playing a high-pitched note will have higher magnitudes at higher frequencies.

Phase:

The phase represents the timing or alignment of the frequency components in relation to one another. While it might be less directly perceptible to the human ear than magnitude, it is still important for:

• Waveform reconstruction: The phase ensures that the different frequency components combine at the correct points in time when reconstructing the signal. Incorrect phases can cause destructive interference, leading to a distorted or incorrect audio signal.
• Localization of sound: Phase differences between left and right ear signals help the brain localize where sound is coming from (e.g., stereo sound). This is especially true for low frequencies, where phase differences play a significant role in directional perception.

In essence:

• Phase controls how frequencies align to shape the final waveform. If two audio signals have the same magnitude spectra but different phase information, they can sound quite different when reconstructed.

Example:

• Magnitude affects how "big" a note or sound feels (loudness, timbre).
• Phase affects the fineness of timing and how frequencies combine to form the final sound. Without correct phase alignment, a note or sound could become muddled or hollow.

So, while magnitude gives you the energy of the sound at different frequencies, phase ensures that this energy is combined correctly to reconstruct the original sound accurately.