# can $e^{j\phi_0}$ be incorporated into the sine and cosine terms?

This is from "communication system" by Carlson fifth edition page 109:

If transfer function of a channel be like this:

$$H(f)= Ae^{j\phi_0}e^{-j2\pi ft_g}$$

and input to this system be:

$$x(t)=x_1(t)\cos(\omega_c t) - x_2(t)\sin(\omega_c t)$$

will output be like this?

$$y(t) = Ax_1(t-t_g)\cos[\omega_c (t-t_g)+\phi_0] - Ax_2(t-t_g)\sin[\omega_c (t-t_g)+\phi_0]$$

my problem is with this sentence in the book:

$$e^{j\phi_0}$$ can be incorporated into the sine and cosine terms...

Edit:

picture from book: • Please check your equation specifying $y(t)$. The only question you have asked is "will output be like this?" to which the answer is No, it will not. May 21 at 19:27
• I agree with you, the answer must be no, but because it is from a well known book, I doubt.I have added a picture from the book, can you check it? May 21 at 20:15
• And notice that the $y(t)$ that you have typed into your question is not the same as the $y(t)$ in Prof. Carlson’s $(5)$ and $(6)$. What Prof. Carlson wrote in his book is correct; what you claim that Prof. Carlson wrote is incorrect. May 22 at 2:39
• Now that @MattL has chosen to fix the typographical errors in the OP's question so that they match what Prof.Carlson's book says, my comments above are no longer applicable to the current version of the question. May 22 at 12:31

Your problem might be the unmentioned fact that the given frequency response $$(4)$$ in Carlson's book is only valid for positive frequencies. The complete definition should be

$$H(f)=\begin{cases}Ae^{j\phi_0}e^{-j2\pi ft_g},&f>0\\Ae^{-j\phi_0}e^{-j2\pi ft_g},&f<0\end{cases}\tag{1}$$

Now we have $$H(f)=H^*(-f)$$, which is necessary for the corresponding system (channel) to be real-valued.

If you split the system into two subsystems $$H_1(f)=Ae^{j\textrm{sgn}(f)\phi_0}$$ and $$H_2(f)=e^{-j2\pi ft_g}$$, where $$H_1(f)$$ is a (scaled) phase shifter, and $$H_2(f)$$ is a pure delay, it's easy to see the impact of the total system on the input signal.

Consider the input signal $$x_1(t)\cos(\omega_ct)$$. The output of the scaled phase shifter is $$Ax_1(t)\cos(\omega_ct+\phi_0)$$, if the signal $$x_1(t)$$ is a lowpass signal with a maximum frequency smaller than the carrier frequency $$\omega_c$$. Delaying that signal by $$t_g$$ gives

$$y_1(t)=Ax_1(t-t_g)\cos[\omega_c(t-t_g)+\phi_0]\tag{2}$$

The result for the signal $$x_2(t)\sin(\omega_ct)$$ follows in a completely analogous way.

In sum, the two necessary conditions for the given formula to be true are

1. $$H(f)$$ corresponds to a real-valued system, i.e., $$H(f)=H^*(-f)$$, and
2. $$x_1(t)$$ and $$x_2(t)$$ are lowpass signals with maximum frequencies smaller than the carrier frequency $$\omega_c$$.