fourier transform - How do I convert a real baseband signal to a complex baseband signal?

I have radio telescope observations that have resulted in two real-valued signals (corresponding to the right- and left-handed circular polarizations).

The signals are sampled at rate $2B$, and provide a bandwidth of $B$.

I wish to get the full Stokes parameters from these signals, which requires me to convert these real signals into complex signals sampled at $B$ (half that of the real signals).

However, I don't know the procedure to do this. I just know that it is possible.

In general, how do I take a real-valued baseband signal and convert it to a complex-valued signal?

Answer

To convert a real signal sampled at rate $2B$ to its complex baseband representation (sampled at rate $B$), you want to map the frequency content in the range $[0, B)$ in the real signal to the range $[-\frac{B}{2}, \frac{B}{2})$ in the resulting complex signal. This can be done in a couple different ways:

1. 
   

   Design a linear filter to approximate a Hilbert transform. Run your real signal $r[n]$ through the filter to yield the transformed signal $\tilde{r}[n]$. Use this to form the analytic signal: $$ r_a[n] = r[n] +j\tilde{r}[n] $$ $r_a[n]$ will contain only the positive frequency components of the original real signal $r[n]$; all of the negative frequencies will be zero (assuming a perfect Hilbert transformer; in practice, the effect will not be perfect). Thus, you have isolated the desired frequency band $[0, B)$.

   
   

   Multiply $r_a[n]$ by $e^{-j\frac{\pi}{2}n}$ to effect a frequency shift of $-\frac{B}{2}$, shifting the desired frequency content to the range $[-\frac{B}{2}, \frac{B}{2})$. Then, decimate the signal by 2 by discarding every other sample. The result is a complex baseband signal sampled at rate $B$.

   
   


2. 
   

   My preferred approach is a more straightforward implementation of the shift that you're looking for:

   
   
   
   
   • Multiply $r[n]$ by $e^{-j\frac{\pi}{2}n}$ to effect a frequency shift of $-\frac{B}{2}$, shifting the desired frequency content to the range $[-\frac{B}{2}, \frac{B}{2})$. The result is a complex signal that is centered in the appropriate place (the center of the band is at zero frequency).
   
   
   • Apply a lowpass filter to pass only the content in the range $[-\frac{B}{2}, \frac{B}{2})$ (to the extent required while meeting your application's antialiasing requirements).
   
   
   
   • Decimate the signal by 2 by discarding every other sample. The result is a complex signal sampled at rate $B$.
   
   
   
   

In practice, I always use some variant of strategy #2. You can make it even more computationally efficient by implementing the decimation as part of the lowpass filtering process, for instance with a polyphase filter.