How to estimate the delay in time domain when knowing the phase shift in frequency domain

Question

I am strugling with how to compute the exact value of delay in time-doamin when knowing phase shift in frequency domain. I have a analog circuit, I sweeped about 30 single tones frequency ranging from 250 MHz to 8 GHz, and achieved amplitude (A) and phase ($\phi$) response of the circuit. I then used that transfer curve (denoted H) (combining amplitudes and phases) to generate response in time-domain (denoted h) using iFFT. Finally, I want to confirm that the h is correctly constructed by taking convolution of a single tone with that h to compare the output power and phase. However, I observed some phase shifts. For sinstance, the phase rotated a value of [65 degree + 2.3*n], where n= [0, 1, 2,.. 30]. I know that that the phase rotation in frequency domain will induce a delay tin time domain, and I need to include that delay when doing verification process. But, I do not know how to estimate the delay in time-domain. Please help me with this issues.

Thank you verymuch.

Answer 1

Group delay is the negative derivative of phase with respect to frequency:

$$GD = \frac{-d\phi}{d\omega} \tag{1} \label{1}$$

When the delay in time is fixed, the phase in frequency will be linear versus frequency, increasing at a constant rate in the negative direction ("linear phase"), however when the phase versus frequency is non-linear, the delay will be different for each frequency component leading to group-delay distortion.

To estimate the time delay, compute the derivative of the frequency versus time as given in equation \ref{1}. This is a function available in Matlab, Octave as grpdelay and in Python scipy.signal as group_delay.

For example, the delay of 1 sample will have a phase versus frequency starting at $0$ radians at $f=0$ and going to $-2\pi$ radians at $f=f_s$ where $f_s$ is the sampling rate. When $f_s$ is in normalized units of radians/sample, the sampling rate is $2\pi $ radians/sample.

In the OP's case with phase linearly increasing $2.5n$ with $n$ from $0$ to $30$ over a frequency range of 250 MHz to 8 GHz, assuming the indices are linearly spaced over that frequency range, the delay would be:

$$GD = \frac{-\Delta \phi}{\Delta \omega}$$

$\Delta \phi$ in radians is $(30-0)\frac{2\pi}{360}$

$\Delta \omega$ in radians/sec is $2\pi (8E9-250E6)$

Therefore, the Group Delay in this case, in units of seconds, would be:

$$GD = \frac{-30/360}{8E9-250E6} \approx -10.8 \text{ ps}$$

Either the phase measurements were actually increasingly negative in contrast to what was provided in the OP, or the group delay is actually negative which is feasible without violating causality, as explained in this post.