IQ modulation principle

Question

I am new in the signal processing. I hope I can get help. I want to ask a question regarding IQ modulation work .

Why can’t I simply multiply a complex base-band signal with a cosine to modulate it?

Answer 1

If you have a complex baseband signal and you modulate it with a cosine then you shift the signal in frequency but your passband signal is still complex-valued, so it cannot be transmitted over a single channel (you would need two channels, one for the real part and one for the imaginary part).

You can, however, transmit the same information over a single real-valued channel by modulating with a complex carrier and taking the real part of the result.

Let $x(t)$ be the complex baseband signal, and let $\omega_c$ be the desired carrier frequency (in radians). Your suggestion results in a passband signal

$$s_1(t)=x(t)\cos(\omega_ct)=\frac12 x(t)e^{j\omega_ct}+\frac12 x(t)e^{-j\omega_ct}\tag{1}$$

If $x(t)$ is complex-valued, then $s_1(t)$ is also complex-valued, and if $x(t)$ has bandwidth $B$, then for transmitting $s_1(t)$ you need two real-valued channels (for the real and imaginary part of $s_1(t)$) with bandwidth $2B$.

However, if you define a passband signal

$$s_2(t)=\text{Re}\left\{x(t)e^{j\omega_ct}\right\}=x_R(t)\cos(\omega_ct)-x_I(t)\sin(\omega_ct)\tag{2}$$

where $x_R(t)$ and $x_I(t)$ are the real and imaginary parts of $x(t)$, respectively, then you can transmit $s_2(t)$ over a single real-valued channel with bandwidth $2B$.