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## Main Question or Discussion Point

Hey guys, I'm trying to understand the complex envelope of a Frequency Modulated signal. According to the Textbook by Leon Couch, the complex envelope of an FM signal, g(t) modulated by the signal m(t) is given by:

g(t) = Ae[itex]^{jθ(t)}[/itex]

where

A = |m(t)|

and

θ(t) = D[itex]_{f}[/itex] [itex]\int[/itex][itex]^{t}_{-∞}[/itex] m(σ) dσ

Now if i want to determine the angle θ(t) of the complex envelope of the FM signal modulated by m(t) = A

i get:

θ(t) = D[itex]_{f}[/itex] [itex]\int[/itex][itex]^{t}_{-∞}[/itex] A

= D[itex]_{f}[/itex] A

= D[itex]_{f}[/itex] A

I though I knew how to deal with improper integrals, but I'm not sure where to go from here. Also I'm not sure why the lower limit of integration should be -∞, it seems to me like it should be zero? Any one know how I can resolve that integral? I would like to really understand this concept, but this part has me stuck.

Thanks a lot.

g(t) = Ae[itex]^{jθ(t)}[/itex]

where

A = |m(t)|

and

θ(t) = D[itex]_{f}[/itex] [itex]\int[/itex][itex]^{t}_{-∞}[/itex] m(σ) dσ

Now if i want to determine the angle θ(t) of the complex envelope of the FM signal modulated by m(t) = A

_{m}cos (ωt)i get:

θ(t) = D[itex]_{f}[/itex] [itex]\int[/itex][itex]^{t}_{-∞}[/itex] A

_{m}cos (ωσ) dσ= D[itex]_{f}[/itex] A

_{m}[itex]\frac{sin(ωσ)}{ω}[/itex]|[itex]^{t}_{-∞}[/itex]= D[itex]_{f}[/itex] A

_{m}( [itex]\frac{sin(ωt)}{ω}[/itex] - [itex]\frac{sin(ω(-∞)}{ω}[/itex] ) This is where i get stuck. I'm not sure what to do with the -∞.I though I knew how to deal with improper integrals, but I'm not sure where to go from here. Also I'm not sure why the lower limit of integration should be -∞, it seems to me like it should be zero? Any one know how I can resolve that integral? I would like to really understand this concept, but this part has me stuck.

Thanks a lot.