I need a closed-form expression for the time derivative of the arctangent of a ratio of two time-dependent variables. I need advice because a credible reference I found doesn't produce the expected result.
After decomposing a real-valued signal into its in-phase ( I(t) ) and quadrature ( Q(t) ) components, I know that arctan(Q/I) is the instantaneous phase angle of the signal. I need to calculate the time derivative of arctan(Q/I) to obtain the instantaneous frequency of the signal.
A paper from MIT OpenCourseWare uses a straightforward derivation to reduce this problem to a simple result:
$$\frac{d}{dt}\arctan(x) = \frac{1}{1+x^2}$$
For a test signal comprising two tones of equal amplitude, simulation demonstrates that arctan(Q/I) increases linearly with time, implying a constant time derivative. The "MIT" expression for the time derivative produces a sine wave whose frequency is the sum of the frequencies of the two tones comprising the test signal.
I applied the result of each expression to the input terminal of a voltage-controlled oscillator. The constant time derivative produces the desired response: a sinusoid with "phase flips" at the zeros of the envelope and a spectrum comprising two "tones" at the frequencies of input signals. The "MIT" derivative produces a sinusoid without the "phase flips" and a spectrum that appears to be an FM signal with peaks corresponding to the sum of the input signal's component tones.
Can anyone explain what I have done wrong?
I think the problem might actually be not with the differentiation of $\arctan\frac{Q}{I}$, but your claim that this is the phase. It's a common mistake, but one easily identified with the following observation: multiplying both of $Q,\,I$ by $-1$ should change the phase by $\pi$. The phase is denoted $\operatorname{atan2}(Q,\,I):=2\arctan\frac{Q}{\sqrt{Q^2+I^2}+I}$. The partial derivatives are given here. One way to obtain these results is to note that the phase $\theta$ satisfies $\tan\theta=\frac{Q}{I}$. Differentiating both sides, then dividing by $\sec^2\theta=1+\frac{Q^2}{I^2}=\frac{Q^2+I^2}{I^2}$, gets the results I've linked to.