# Complex Envelope

Two signals have arrived with different phases. The sum signal is given by

$$x(t) = A\cos(2\pi f_ct) + B\cos(2\pi f_ct + \theta)$$

What is the complex envelope of $$x(t)$$? Need advice on how to get started.

What is the analytic signal $$x_+(t)$$? For this, can I not apply the $$x(t) + \mathscr{H}\big\{x(t)\big\}$$ to both $$A\cos(2\pi f_ct)$$ and $$B\cos(2\pi f_ct + \theta)$$, and just add the result?

• Hi and welcome. A rule here is that any question with homework problems needs to contain the attempts at solving them. Please edit your question with what you've tried so far and where you're stuck, and I'm sure you'll get help and guidance!
– Jdip
Commented Feb 1, 2023 at 2:53
• Homework? Please edit your question with the work you've done so far, or if you're utterly stuck, tell what you do know. Commented Feb 1, 2023 at 2:54
• Are $A$ and $B$ constant? Doesn't look like your envelope has much shape. Commented Feb 1, 2023 at 3:24
• @robertbristow-johnson yes, A & B are constants... same signals, just the 2nd one comes in with a phase Commented Feb 1, 2023 at 3:30
• There are trig identities that you should make use of. This is hardly a complex envelope problem. Not a complex-valued signal (i suppose $A$ and $B$ could be complex?) and the envelope is a constant function. It's just a sinusoid with an amplitude and a phase. Commented Feb 1, 2023 at 3:44

Using the fact that

$$\cos(\omega_0t)=\textrm{Re}\left\{e^{j\omega_0t}\right\}$$

it is straightforward to write down the analytic signal of $$x(t)$$.

The analytic signal has the form

$$x_c(t)e^{j\omega_0t}$$

and you have

$$x(t)=\textrm{Re}\left\{x_c(t)e^{j\omega_0t}\right\}$$

where $$x_c(t)$$ is the complex envelope. Judging from the comments under the question, you know everything you need in order to obtain the analytic signal as well as the complex envelope.

• what if I have two different fc values ie f1 and f2? Commented Feb 24, 2023 at 21:30
• @Gotz2bril: You could better ask that as a new question. This question has been answered as far as I can tell. Commented Feb 26, 2023 at 17:41

Perhaps the third row of Table 1 at:

https://www.dsprelated.com/showarticle/635.php