Frequency representation of signals
I have a very basic question about representation of signals in the frequency domain. Let's say I have a cos which I want to represent in the frequency domain. I have usually read books where only the positive frequency domain is presented. However as far as I know, a cos is a sum of two exp components and therefore a e-jwt components also exists. My question is as follow: when I do mixing and modulation stuff, do I need to consider both positive and negative frequencies or can I limit myself to positive axis. I am not clear at all with this. My concern is let's say I mix a signal at a 2000MHz carrier with LO at 1800MHz, I get signal at 200MHz. But next I mix with 220MHz. What do I get? -20MHz or +20MHz? As I understand I do get my signal at -20MHz but there's also a component at +20MHz but from the signal component which was originally at -2GHz.
As I understand I should only pay attention to frequency components sign when components such as complex filters are used.
Sorry and Thanx in advance for your help
Fred
When I'm not working with complex signals, I find it convenient to simply ignore the frequency's sign, because positive and negative frequency are usually indistinguishable. When you mix 2000MHz with 1800MHz, you get the sum and difference frequencies, 3800Mhz and 200MHz. When you mix 200MHz with 220MHz, you get 420MHz and 20MHz.
If you have complex signals, then positive and negative frequencies really are different, so maintain the sign during all arithmetic.
Thank you for your help. How do you determine when you have complex or real signals? I have heard so much theory about complex representation of real signals that I am a little bit lost with this stuff. Or is it simply thinking about the representation of a signal on the real/imaginary axis (BPSK would be real and QPSK would be complex?)?
I think you should also comsider neg and pos axis different when complex processing is applied shouldn't you? Latest architectures use this kind of stuff a lot and that's why I would like to have a really clear understanding of the thing
I usually consider a complex signal to be two signals in 90 degree quadrature. For example, let's build a simple 2 GHz digital receiver. Your antenna receives a 2000 MHz sine-wave signal. You amplify it and feed it into a complex mixer (which contains two multipliers). It multiplies the input by a -2000 MHz complex local oscillator (that's a 2000 MHz sine-wave and a 2000 MHz cosine-wave). Now use two low-pass filters to eliminate the 4000 MHz stuff from those two mixed signals. You now have a complex output signal centered at 0 Hz. If your antenna input increases to 2001 MHz, then the output will be +1 MHz. If the input falls to 1999 MHz, then the output will be -1 MHz. You can easily distinguish positive from negative output frequencies because it's a complex value, so the vector spins in opposite directions. If your 2000 MHz antenna signal shifts by +43 degrees, then the complex output will rotate by +43 degrees.
This stuff probably seems confusing at first, but once you realize what's going on, and how useful it is, it'll be like a light bulb going on in your head. :)
In the exponential notation it is common practice to not put RE (for real part) in front of it even though it is required and understood to be there.
Here are some things to think about:
Physical electrical circuits respond independently of the phase going one direction or the other (frequency is the derivative of phase vs time).
You have to do extra circuitry to be able to keep track of the positive and negative frequency components. The Weaver method of SSB generation was done in the days before DSP and was done with analog hardware.
In full carrier AM, the two sidebands rotate in opposite phase.