How does complex data of an FFT and IQ modulation differ?

Question

If I have an arbitrary time domain signal (e.g. a simple cosine) i can sample it with an ADC and get a time discrete representation. I can perform an FFT on the samples and get a complex output. The phase of the signal is:

$$\phi=\arctan\frac\Im\Re$$

Using an IQ down modulation of the same signal gives complex data, too, with the phase being:

$$\phi=\arctan\frac QI$$

Is there a difference between these two and when is one preferred over the other?

EDIT

Thanks for the input!

@Dilip Sarwate: Here is what I did: set $\phi=60°$ (see your example)

The results are exactly what I expected. I can see the frequency of 1 Hz in the spectrum (plot 3) and the initial phase of $60°$ in the phase spectrum (plot 4). I get the same result when I use a non-complex function, e.g. $x(t)=cos(2\pi t+\phi)$.

I read everywhere that phase information is lost if I don't use IQ down-modulation, but that does not seem to be the case, as I can see the phase information in the plots.

Or am I missing something here?

Answer 1

Why not try your hand at working out the differences for yourself?

Since you are interested in cosines, take, for example, the signal $x(t) = \exp(j(2\pi t + \theta))$ which is a complex sinusoid of period $1$ and sample it $16$ times per second to get $16$ samples $x[n]$, $n = 0, 1, \dots, 15$, where $x[n] = x\left(\frac{n}{16}\right) = \exp\left(j\left(\pi \frac{n}{8} + \theta\right)\right), n=0,1,\ldots, 15.$ Then, take the 16-point FFT of $\mathbf{x} =\big(x[0], x[1], \ldots, x[15]\big)$ which will give you $\mathbf{X} =\big(X[0], X[1], \ldots, X[15]\big)$. Report back to us what $\mathbf{x}$ and $\mathbf X$ are and whether you notice any differences between them.

Answer 2

If the signal frequency is constant and exactly integer periodic in the FFT width, then the transform basis vector for each FFT result bin is the same as a sinusoidal (complex exponential) IQ down-modulator mixer input that starts with a phase of zero at sample 0. Thus the atan2() results for a signal within that FFT result bin should be identical.

If you only need one result bin of a DFT, then a complex mixer (complex Goertzel) requires less computation than a full FFT result (assuming equivalent numerical precision, etc.). If you need O(logN) DFT result bins or more, then the FFT is more efficient.

If the frequency is not known to be exactly integer periodic in the DFT/FFT width, then an FFT might be useful as part of an initial frequency estimator algorithm.