Q:

RF Divider function in SDR

I have what may be an odd question for the SDR gurus out there.
What would be the physical implementation (in software) of a broadband frequency divider?
For example, say I want to capture a signal at 1 GHz, with a 10 MHz bandwidth, then divide it by a factor of 10.
I would expect to get a down-sampled signal at 100 MHz with a 1 MHz bandwidth.
Yes, I know I would lose information, but assume this would be presented as a spectrum analysis, not full audio, video, etc.
Conceptually, could this be accomplished by sampling the RF at 2+times the highest frequency components, say at 2.5 GHz, then discarding 9 out of 10 samples - decimating the input stream?
Thanks,
Dave

A:

Well, as soon as you've digitized your signal it loses the property "bandwidth", which is a real-world concept (and not one attached to the inherently meaningless stream of numbers that we're talking about in DSP and SDR). So, there's no signal with a bandwidth of 10MHz (without looking at the contents of the samples), but only a stream of numbers that we remember being produced by sampling an analog signal with a sampling rate of 20MS/s (if you're doing real sampling; if you have an I/Q downconverter and sample I and Q simultaneously, you'll get complex samples, of which 10MS/s will be enough to represent 10MHz of bandwidth). 
Now, if you just throw away 9 out of 10 samples, which is decimation, you'll get aliasing, because now you can't tell whether a sine that took 10 samples in the original signal is actually a sine or just a constant; the same goes for any sine with a frequency higher than your new sampling rate's Nyquist bandwidth. That is a loss of information, so yes, that would work.
I think however, you have something specific in mind, which is scaling the signal in frequency direction. Let's have a quick excourse in to Fourier analysis:
there is the well known correspondence for frequency scaling.
let G be the Fourier transform of g, then 
g(at) <--> 1/|a| G(t/a)
As you can see, compressing something in frequency domain actually means "speeding it up" in time domain, ie. decimation! 
So, in order to do this meaningfully, you could imagine taking the DFT of length N of your signal, and set 9 out of 10 bins to zero, by multiplying it with a comb of 1's. Now, multiplication with a signal in frequency domain is convolution with the fourier transform of that signal in time domain. The fourier transform of such a Comb is, to little surprise, the complement of a Nyquist-M filter, and thus a filter itself; you will thus end up with a multi-band-passed version of your signal, which you then can decimate without aliasing.
Hope that was what you're after!

