# What phase rotation occurs when you take the derivative of an audio signal?

If you take the derivative of an audio signal, it provides a 6 dB/oct upward sloping filter (increasing high frequencies / cutting low frequencies) all the way across the spectrum.

What is the result in terms of the phase? Is there phase rotation? Is it uniform across the spectrum or does it vary with frequency? What would the plot of this phase rotation look like?

Thanks

• Hi Mike! You're probably familiar with the Fourier transform, and to make things a bit more interesting for you, just a hint: The Fourier transform of a signal, and its time-derivative, are linked in frequency domain. You only need to figure out how to convert that relationship to a phase! May 3 '21 at 16:35
• Thanks Marcus. So I presume taking the derivative of a sine wave gives a cosine wave and if you break the audio down into a series of sine waves, then take the cosine of each, you are rotating all phases of all frequencies by 90 degrees, which explains the answer.
– mike
May 3 '21 at 23:10
• love this! it definitiely works for all discrete spectra! Clever! May 3 '21 at 23:46

$$\mathcal{F}\{\frac{\partial}{\partial t}x(t)\}=j\omega \cdot\mathcal{F}\{x(t)\}$$
Differentiation in time corresponds to multiplication with $$j\omega$$ in frequency. Hence the +6dB/octave slope. The phase shift is a constant 90 degrees for all frequencies.