# Can complex envelope be writen in the form of quadrature components when it has symmetric spectrum?

I am reading a chapter on VCO noise in "Design of CMOS phase-locked loops from circuit level to architecture level by Behzad Razavi";I am confused when the upconverted noise is writen as $$N_I\cos\omega t - N_Q\sin\omega t$$, which means the baseband noise(complex envelope of the upconverted noise) have a form of $$N_I+jN_Q$$; I have this confusion because I think the complex envelope has symmetric spectrum which implies it is real signal so Nq should be 0; please help me understand this, thanks!

## 1 Answer

The noise is actually never upconverted, it was already there in the bandpass channel. The textbook is simply modeling the noise as a quadrature signal.

To recap:

• The transmitted passband signal is usually assumed to be noise-free.
• The passband signal is real, and it occupies frequencies from $$-W_2$$ to $$-W_1$$ and from $$W_1$$ to $$W_2$$.
• The receiver adds noise to the signal, usually in the first few stages of the analog front-end. This noise is passband: it occupies the same spectrum as the received signal.
• The passband noise goes through the quadrature receiver and its complex envelope is calculated. In the process, the spectrum from $$-W_2$$ to $$-W_1$$ is discarded, and the spectrum from $$W_1$$ to $$W_2$$ is shifted to baseband.
• As a result, the spectrum of the complex envelope of the noise is not symmetric.
• In conclusion, we can think of the passband noise as a quadrature signal.

Here's an illustration:

• Thanks for answering, I still have question: why baseband one can be writen as a complex number: ni+nq*j,even it has a symmetric spectrum(implies it is real signal) – Ziyuan Ning Apr 26 at 23:15
• The complex envelope of baseband noise is real; you are quite correct about that. What the text you're reading is doing, however, is bringing passband noise back to baseband, using a quadrature receiver. Then, each of the I and Q branches of the receiver will see different noise. The noise in the I branch is the real part of its complex envelope, and the noise in the Q branch is the complex part. – MBaz Apr 26 at 23:26
• I don't understand the confict between "complex envelope has complex part" and "The complex envelope of baseband noise is real" – Ziyuan Ning Apr 26 at 23:35
• Let me try a different approach. Consider a receiver that consists of a bandpass filter with center frequency $f_c$ and bandwidth $W$, followed by a quadrature receiver. The input is white Gaussian noise $n(t)$. Can you calculate the receiver's output? It is $n_I(t)$ in the I branch and $n_Q(t)$ in the Q branch; this can be written as a complex envelope $n_I(t)+jn_Q(t)$. $n_I(t)$ and $n_Q(t)$ are the result of filtering and downconverting the white noise at the input. – MBaz Apr 27 at 1:23
• Yet another approach, this time from the transmitter. You generate two different baseband Gaussian random noise signals, $n_I(t)$ and $n_Q(t)$, each with bandwidth $B = W/2$. You upconvert them using a quadrature transmitter to a center frequency $f_c$; the bandpass noise will have bandwidht $W$. The important point is: This bandpass noise is statistically indistinguishable from the noise after the receiver's BPF (assuming equal variance, zero mean, etc). – MBaz Apr 27 at 1:28