Help understanding quadrature sampling and effective bandwidth

Question

Let me start off by saying I'm at least a little embarrassed that there's something simple that I'm fundamentally not understanding here. Not sure if the blame is on me, or the prof that taught me DSP in grad school, but either way...

I have a very simple flowgraph in GNU Radio. LimeSDR block, (quadrature) sampling at the full 61.44MHz - so it's actually capturing 61.44MHz of spectrum - 30.72 above the center frequency, and 30.72 below the center frequency. So the effective bw at this point is 61.44MHz

Next a Frequency XLating FIR block, being used as a channel selector - no decimation (decimation=1, complex to complex, real taps). So let's say I take a 34 MHz chunk of the incoming spectrum and shift that to be centered around 0Hz and 34MHz bandwidth.

Do a very simple complex operation - multiply conjugate, the signal by itself to a 1 sample delayed version of itself.

Take the output of the multiply conjugate, and feed it into a QT GUI Sink to look at the spectral components. In the resulting FFT, I have a symmetric spectrum from -30.72MHz to +30.72MHz, and any spectral component that's above 30.72MHz is aliased around 30.72, so for instance if I have a spectral component at 34MHz, it ends up at 30.72 - (34-30.72) = 27.44 Mhz.

So, with a sampling frequency of 61.44MHz and everything being quadrature sampled, how do I end up with only half of that bandwidth in the end? Is it the multiply conjugate that halves the bandwidth? Do I need to 2x upsample before that operation? Do I need to use complex taps on the FIR filter instead of real taps?

Answer 1

When you multiply a signal by it's conjugate you would get a real result given the phase terms cancel in the conjugate multiplication:

$$A[n]e^{j\phi[n]}A[n]e^{-j\phi[n]} = A^2[n]$$

Thus we see we get a real result and specifically the square of the magnitude. However to be very clear, this will only occur reliably when the signal and its conjugate product are synchronized in time. The OP has stated that the conjugate product is the signal with a delayed version of itself, as:

$$A[n]e^{j\phi[n]}A[n-1]e^{-j\phi[n-1]} = A[n]A[n-1]e^{j(\phi[n]-\phi[n-1])}$$

This is a classic conjugate product frequency discriminator that can be used to correct for carrier offsets. The result gives us the phase between the two successive samples. It will only be real when there is no phase variation between adjacent samples. The two samples are separated in time, and frequency is given as the derivative of phase versus time (change in phase versus a change in time).

If there was no carrier offset for a baseband signal, the result could be real if the phase is not changing from sample to sample (other than the modulation itself). Typically with bandlimited complex waveforms (QPSK, QAM etc) we would still see phase variations between every sample. For equiprobable data, these variations would average to zero and the long term average out of the complex conjugate product with delay would still provide a reasonable error metric for carrier offset used in carrier tracking loops.

For carrier recovery applications where we drive the error term (the longer term average out of the cross product) to zero, we can use the small angle approximation and therefore extract the phase measurement from the imaginary portion of the product alone, which then reduces the frequency discriminator two two multiplies for each complex pair of samples as shown in the diagram below (this is a common frequency discrimination technique):