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Re: [Discuss-gnuradio] why quadrature samples?

From: Matt Ettus
Subject: Re: [Discuss-gnuradio] why quadrature samples?
Date: Wed, 22 Jan 2003 19:42:41 -0000

> Allow me to throw my request into the hat for more technical detail
> on this subject, or at the very least, a good reference on where we
> can read up on it ourselves.  This is a new concept to me that I'd
> love to have more detail on.

Ok... the motivation is this --

Normally, to sample a BW of X, you need to sample at 2X.  So what can you do if
you want to increase the BW sampled, but can't increase the sample rate?  If,
instead, you can throw a second sampler at it, you use quadrature sampling.

It comes up often here, because soundcards are usually stereo, but their sample
rates are limited.

If your BW of interest is from w to w+2x (a BW of 2x), you mix the signal with a
sine of frequency w+x (i.e. the center frequency).  This will make the
components of the signal at w+x go to a frequency of 0.  The components from w
will mix down to 0-x, and those from w+2x will mix down to 0+x.  Since +x and -x
are really the same, the BW is from 0 to X, and all the components now will get
through a sampler sampling at rate 2x.

The problem is that 0-x and 0+x alias onto each other, since you can't tell the
difference between negative and positive frequencies.  So instead of just mixing
with a sine wave and sampling, you also mix with a cosine wave, and sample it
with the other sampler.  Now you have 2 sampled signals, each at rate 2x, each
with components from -x to +x.  Independently the negative and positive
components are indistinguishable.  By intelligently using complex numbers and
BOTH samples, the negative frequencies can now be separated from the positives.

Thus you have sampled a BW of 2X by using 2 samplers of rate 2x, instead of
using one at a rate of 4X.

For the math, there are many sources on the net.

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