Inferring a response from the Analytic Signal

Question

I have a slowly varying sinusoidal system. The output is separated using methods that produce a real signal $x(t)$ and quadrature component $y(t)$ such that $x(t)$ represents the real physical value of the signal, $x^2 + y^2$ is instantaneous amplitude and $\mbox{atan}(y,x)$ is the instantaneous cosine argument with respect to a central frequency $\omega_c$. So you can think of it as $\omega_c + \phi(t)$ but those two pieces are bundled and the phase would have to be unwrapped to get at $\phi$.

I guess these qualities are unique enough to say that $z(t)=x+iy$ is the Analytic Signal representation of $x(t)$ for that frequency band and that $iy=\hat{x}$ is the Hilbert transform of $x$ in some bandpassed sense, but my knowledge is limited in this area.

Let's say this signal is the output of a linear and mostly time-invariant system with complex response $H(\omega)$. I am only interested in $H(\omega_c)$ at the widely spaced central frequencies and assume it is constant across the passed band around {\omega_c}. $H$ produces both amplitude and phase change but its characteristics are not known quantitatively. I can only probe it with the input and output I have, not probe it.

The idea I am exploring here is how to use the data I have to infer $H(\omega_c)$. If I take the complex output $Z$ as the Analytic Signal and input $W$ and do a complex regression $Z(\omega,t) \approx H(\omega) W(\omega,t) + \varepsilon(t)$ on the input using time to provide samples and yielding an estimate $\hat{H}$ is that an estimate of $H$? A good one? One concern I have is that $Z$ and $W$ likely have no negative frequency content, so I'm concerned that I'll be learning about the analytical analog of $H$ or something like that.

Answer 1

First to clarify- real signals include the positive and negative frequencies, so we would be looking at $H(-\omega)$ when we evaluate with real signals. If $H(\omega)$ itself was complex, then the resulting output even with a real input would be a complex output since the positive and negative frequencies would be processed with different responses. When $H(\omega)$ is real, then the resulting output for a real input would also be real.

What needs to be clarified in the question is if the output is the baseband equivalent signal (I assume so), in that the magnitude and phase represents the magnitude and phase changing versus time of some particular carrier for the passband signal. This way, given a real output as described in the first paragraph, we can by using the Analytic Signal representation, represent it as:

$$x(t)+j\hat{x}(t) = A(t)e^{j\phi(t)}e^{j\omega_c t}$$

Where $\hat{x}(t)$ is the Hilbert Transform of $x(t)$, and $\omega_c(t)$ is a carrier frequency of interest. The real waveform, $x(t)$ in this case would be $x(t)=A(t)\cos(\omega_c t + \phi(t)$) where $A(t)$ is the amplitude modulation and $\phi(t)$ is the phase modulation.

The "Analytic Signal" by definition has no negative frequency components so would be a single-sided (positive frequencies only) passband signal as represented above. What I suspect is the OP is looking at the complex baseband equivalent signal, which does have positive and negative frequency components, and is the frequency translated Analytic Signal as give by:

$$x(t) = A(t)e^{j\phi(t)}$$

In any event, what is desired here is a channel estimation where we know the input and output and from that we wish to estimate the channel response $H(\omega)$. If the channel is stationary and we only wish to know this response at discrete frequencies, then one option is to simply compare the input and output at each of these frequencies and from that derive the magnitude and phase for each given frequency. For more complicated cases where we wish to have a complete estimate of the channel over a given bandwidth, we can use the Wiener-Hopf equations for least squares channel estimate as detailed in this link.

Also related is this link showing the use of a frequency chirp (ramp) along with a Tukey windowed FFT for accurate frequency response measurement.