An Impulse in Complex Envelope (i.e. baseband) Notation?

Question

In complex envelope notation, a modulated RF signal, say v(t), is represented in complex envelope notation as v(t) = Re{ g(t)e^(jwct)}, where wc is the carrier angular freq, g(t) = x(t) + jy(t), and g(t) is the complex baseband signal. Since x(t) and y(t) are inputs to an in-phase (cos) and quadrature phase (sine) mixers, how would I represent v(t) in baseband g(t) notation, where v(t) = an impulse? Since a mixer is technically nonlinear, and therefore NOT an LTI system, isn't this impossible to do in a mathematically correct way?

The issue is we have a MATLAB RF simulator for PHY signals, whereas the simulator is designed to work only with RF signals represented in baseband notation (and fc is metadata), and I want to send impulses through the RF channel to clearly see the fast fading reflections. My current method is to use impulses for x(t), with y(t) = 0, but not sure if this translates correctly, or if this even makes sense.

Answer 1

The OP commented that he has practical reason he would like to pursue a simple bandpass equivalent of an impulse test. Here are some considerations with that in mind:

An impulse in time represents all frequencies, which will ultimately be band-limited by the channel. We could implement the IQ representation of the band-limited channel that could then be transmitted at a real carrier if the carrier frequency itself far exceeds the bandwidth (which is often the case). In this case, the impulse could be implemented as a truncated Sinc function on I and Q to maximize the channel power, or even on just I if that isn't a concern in such a way that the Fourier Transform of the Sinc (which is flat in frequency) sufficiently exceeds the channel bandwidth, which would then be no different than an impulse itself. So this means the ideal solution would be to transmit Sinc shaped RF pulse envelopes in time such that the entire channel is occupied evenly in frequency, which can be well approximated from a truncated Sinc on I and Q at baseband. This approach would maximize the energy in the band of interest while providing the same result as an impulse.

Prior comments below on more typical channel estimation techniques:

Any waveform that occupies a particular bandwidth can be identically represented at any carrier including a carrier frequency of zero (which is DC) as long as we use complex representation to allow for different positive and negative frequencies in the overall bandwidth. This is often much much simpler for modeling and expression given we don't have to represent each cycle of some higher frequency carrier. A mixer as a pure multiplier is non-linear in that it's output frequencies do not match it's input frequencies but as such it is simply a frequency translation with no other modification to the signal. Actual mixers will have other non-linearities generating other frequencies that we control with good design practice as with any other analog component.

The second point is to consider NOT using an impulse for channel estimation. In practice any impulse that can be created will have so much less energy than other techniques often used, and as such will provide far less SNR in the answer. The two most common approaches for channel estimation are swept tones (such as that done in a network analyzer) or psuedo-random (PRN) sequences which emulate a white noise source and as such occupy the channel. The difference is both of these approaches can provide a sustained signal for as long as needed for desired SNR only limited by how stationary the channel is. The approach for channel estimation with a pseudo-random sequence would be ideal for a fast fading channel, with the duration of the test sequence set to be less than the fade rate of the channel. I detail channel estimation using PRN sequences further in these posts:

This shows an equalizer implementation but by swapping input and output you would instead get channel estimation:

Compensating Loudspeaker frequency response in an audio signal

and

How determine the delay in my signal practically

These approaches will return the impulse response in time, from which you see a single dominant impulse (for a channel with no reflections) or multiple impulses (from the multiple reflections, with their corresponding magnitude and phase).

Answer 2

"How does an impulse at the output of an IQ modulator translate into the baseband signals?"

I am going to interpret this question as: "What is the complex envelope of $s(t)$?", where $s(t)$ is gvien by: $$\delta(t)\cos(2\pi f_c t)-\delta(t)\sin(2\pi f_c t)$$

Since the product of an impulse and a signal is mathematically a suspicious concept, and impossible to generate in the real world, let's instead use a very narrow pulse $\Pi(t)$. The IQ modulator output is then $$\Pi(t)\cos(2\pi f_c t)-\Pi(t)\sin(2\pi f_c t)$$

This signal's complex envelope is easy to calculate: it is $\Pi(t) + j\Pi(t)$. Its amplitude is $|\Pi(t)|$ and its phase is zero outside the pulse's support, and $\arctan(1)=\pi/4$ inside it.

If the duration of the pulse $\Pi(t)$ is shorter than the propagation delay along each path, and assuming enough bandwidth at both transmitter and receiver, then you can use this signal to sound the channel: the receiver will see several replicas of the pulse, each with different amplitude and delay.