Question for the crowd: The definition of an "analytical" signal is a complex signal that only has a positive frequency. Would you still call a signal with only a negative frequency an analytical signal? Or do you call a signal with only a negative frequency an analytical conjugate? Or is it common to just call any signal with only a positive or negative component analytical?

A general complex-valued signal is just a "complex-valued signal" or something simple like that, which can have any mix of tones or composite signals with any frequencies, both positive and negative.   A subset of "complex-valued signals" that has only positive frequencies is an "analytic signal", sometimes made from a real-valued signal using a Hilbert transform.

I don't personally know of a specific name for a signal with only negative frequencies, but it is still a "complex-valued signal".

Thanks, Slartibartfast.

Slartibart, maybe an "imaginary signal", hehe

Yes by definition the "Analytic Signal" has only positive frequency components and DC, and all negative frequency components are zero valued. If you take the complex conjugate of the analytic signal, you will get a signal that has only negative frequency components and DC - so I would call it the complex conjugate Analytic Signal but I haven't seen that term anywhere. 

Note we also commonly see the analytic signal translated to baseband, such as a real passband waveform which is then passed through a 90 degree quadrature hybrid (creating the analytic signal!) and then having that resulting complex passband waveform as the analytic signal multiplied with a complex local oscillator resulting in the complex baseband signal (so that would be a frequency translated Analytic Signal, the result of which is no longer an Analytic Signal since at baseband it would have positive and negative frequency components).

First, thank you for the reply. I feel as if I've just completed a rite of passage because I understood everything you just said. Second, it sounds as if Slartibartfast's comment called it dead-on. A complex signal can be positive-only, negative-only, or both, with a positive-only being a subset with the special designator "analytical".

The reason I'm asking is that I'm working on some more tutorials for the Gnu Radio wiki, and I want to ensure I'm using terms correctly. I've found that newbies get all KINDS of confused if they hear or read terms described differently in different places.

Here is how I view the significance of the analytic signal at basic level:

Imagine one cosine waveform then neither its amplitude (or power), nor its phase is readily available.

If you create a sine wave alongside the cosine at same frequency/amplitude then amplitude becomes constant and phase rotates.

This pair of sine wave with its cosine represents an analytic signal.

Going back to original question: adding a sine waveform will do the job and so there is no need to go opposite and add inverted sine waveform which represents negative frequency only. Though can be done and can be as useful.

If the Hilbert transform could be simplified that would lead to a nice trick.  Rotate +Fs/4 to baseband and the original Nyquist zone is now over-sampled by 2 with extremely sharp frequency transitions.  This would normally require a high order filter.

Mark Napier

I believe that the notion of "no negative frequencies" is a bit constrained.  Basically an analytic signal has no frequency "image", whereas all real-valued signals have symmetric images.  Since signals are mixed and basebanded, the negative frequency concept can be confusing.

Here is a diagram I made many moons ago.  It shows an analytic signal formed by sampling a signal w/ frequency > fs/2.  Here it has a negative frequency and no positive frequency.  So the true definition of analytic doesnt include all cases.  Here an analytic signal is formed using a cosine i(t) and sin q(t) in frequency domain.  The rotation of the sine by multiplying with "j" puts it back on the real axis, and the sum i(t) + jq(t) gets rid of the "image" frequency.

If you want to know more, here is a paper I wrote long ago:

http://www.hyperdynelabs.com/dspdude/papers/quadra...

On another note, something I dont see covered often is showing the mechanism of complex-valued vs. analytic signals.  One example, if you have a slight imbalance in amplitude or phase in your analytic signal, the resulting signal is no longer analytic but just complex valued. You will actually see a small image that starts to pop up that has an amplitude equal to

abs(|I| - |Q|exp(1i*phi))

where phi is the phase imbalance error (from 90 deg).  The nice thing is you can readily see if |I| = |Q| and phi = zero, the image amplitude goes to zero and you have an analytic signal.  But any imbalance and you are stuck with an image, and your signal is no longer analytic.  I have another paper going over this if interested.  It kind of ties it all together...

Thank you, James. I understand the concept of analytical signal. I just wanted to ensure that it *only* referred to one without the negative image (as you've discussed above). Oh, and I found that paper of yours awhile back! Thank you VERY much for posting it!

And BELIEVE ME I understand the idea of "if the signals are not ABSOLUTELY, COMPLETELY, TOTALLY, UTTERLY in quadrature, that image signal will rear its ugly, little head". I've been dealing with trying to generate signals using various SDRs without those images popping up and it has... not been going well.