RE: [Discuss-gnuradio] Recovering x(t) from IQ samples

[Bahn William L Civ USAFA/DFCS]

> From: Johnathan Corgan [mailto:address@hidden
>
> Bahn William L Civ USAFA/DFCS wrote:
>
> > Please tell me there is a better way.
>
> I didn't follow the start of this thread, but can you reiterate what
> you're trying to accomplish, vs. alternative ways of getting there?  It
> does seem like things are getting overcomplicated, and some times it's
> useful to go back to a high-level description of the problem.

Sure - that sounds like a good idea.

I'm asking the question for several reasons - some theoretical and others 
practical. The basic driving force is as follows:

I have the following processing chain:

BBC -> FILE -> GR -> USRP -> RF -> USRP -> GR -> FILE -> BBC

SENDER:
1) An external program (called BBC) takes messages from the user and produces 
the waveform data that we want to transmit from the sender to the receiver.
2) The data is written to a file in a format that GNU Radio can read.
3) GNU Radio reads the data from the file and sends it to the USRP.
4) The USRP upconverts it and transmits it via RF.
5) The (other) USRP receives the RF signal, downconverts it, and sends the data 
to GNU Radio.
6) GNU Radio receives the data from the USRP and writes it to a file.
7) The BBC program reads the data file and extracts the messages.

It would be very nice if the two files, the one being read by GNU Radio and the 
one being written by GNU Radio, used the same format. If I leverage the 
existing usrp_rx_cfile.py program, then the data written by GR is in IQ data 
pairs and BBC will need to convert this to a stream of real-valued data 
samples. Conversely, BBC will need to convert the real-valued data stream that 
it generates into IQ data before writing it to the file that GR will later read 
and transmit via the USRP.

This raises the following questions:

Q1) One of the formats in which I can send data to the USRP is as IQ data. What 
does the USRP do with IQ data pairs? In the USRP documentation there is a block 
diagram of the Digital Down Converter, but there is no diagram of the Digital 
Up Converter.

Q2) If I have x(t) data (just time-domain data samples that have been generated 
by an external program that is completely unrelated to GNU Radio) that I want 
to transmit, how do I convert it to the necessary IQ data? What frequency do I 
use in the calculations?

Q3) Given a stream of IQ data pairs, how do I convert them to plain-ole vanilla 
x(t) time-domain samples?

Supposedly, as near as I can tell, there is a direct equivalency between data 
represented as time-domain samples and the same data represented as IQ data 
pairs. Is this true? If so, then it should be possible to convert back and 
forth between the two representations? Is this the case?

Here is what I see from where I am sitting:

There is a time domain signal at the front of the Digital Down Converter (DDC) 
in the USRP that I will call x(t). This signal (assuming the same signal is 
applied to both DDC inputs) follows two paths to produce the i(t) and q(t) 
samples that are delivered to the user over the USB interface.

Therefore the relationship between x(t), the signal that I am interested in, 
and the i(t) and q(t) samples that I actually get to see are:

i(t) = x(t) * sin(wt)
q(t) = x(t) * cos(wt)

Now, I know that there is also the decimation low pass filter which, up to this 
point, I have been ignoring but may actually hold the key. I don't know.

Assuming that x(t) has no frequency components that are going to get aliased, 
how do I recover x(t) from the i(t) and q(t) samples?

Several people have responded, some to the list and some to me privately, that 
you just treat i(t) and q(t) and the components of a complex number and use the 
four-quadrant arctangent function. Typical of these responses has been:

=====================================================
Interpret I and Q as real and imaginary part of a complex number:

R = I + jQ.

Then |R| = sqrt(I^2 + Q^2) and ang(R) = atan2(imag(z),real(z)), where "atan2" 
(four-quadrant inverse tangent, a Matlab function, to get the signs right).

You're simply looking for magnitude and phase (angle) of a complex number.
======================================================

But that doesn't work because there is, as I am looking at things, a 
fundamental sign ambiguity involved. I tried to make that clear in my original 
post and obviously failed to do so, hence I will be explicit in my reasoning 
here. If I am wrong at some step, perhaps someone can spot it and point it out.

First off, R is not the x(t) I am looking for. R is a complex sequence. So 
unless there is an unambiguous mapping between R and x(t), then it doesn't help.

Second, the closest mapping is not R = I + jQ, but rather R = Q + jI. To see 
this, consider:

Z(t) = x(t) * e^(jwt)

Z(t) = x(t) * (cos(wt) + j*sin(wt))

Z(t) = x(t)*cos(wt) + j*x(t)*sin(wt)

Z(t) = q(t) + j*i(t)

Now, finding the magnitude of x(t) is very straightforward since:

|Z(t)| = sqrt(q^2(t)+i^2(t))

|Z(t)| = sqrt(x^2(t)*cos^2(wt) + x^2(t)*sin^2(t))

|Z(t)| = sqrt(x^2(t)*[cos^2(wt) + sin^2(t)])

|Z(t)| = sqrt(x^2(t)*[cos^2(wt) + sin^2(t)])

|Z(t)| = sqrt(x^2(t))

|Z(t)| = |x(t)|

Fine so far, but now I need to determine if x(t) is positive or negative.

IF I know what the phase of (wt) is, then I can trivially determine the sign of 
x(t). A couple of responses I have gotten basically tell me to use this 
information even though the original post specifically stated that all I have 
are the i(t) and q(t) data.

What people keep suggesting is to use the four-quadrant arctangent, but the 
problem is that knowing which quadrant an i(t),q(t) data pair is in does not 
tell me which quadrant (wt) is in. It only allows me to determine which of two 
possible quadrants (wt) is in and I need to know the sign of x(t) in order to 
distinguish between them.

phi(Z(t)) = atan2(q(t), i(t))

phi(Z(t)) = atan2(x(t)*cos(wt), x(t)*sin(wt))

As a specific example of this ambiguity, let's say that sample number 34765 
yields the following for i(t) and q(t) at that point:

i(34765) = 0.500
q(34765) = 0.866

What is x(34765)?

Well, if (w*34765) = 30 deg, then we have the following:

x(t) = 1.0
wt = 30 degrees
sin(wt) = 0.500
cos(wt) = 0.866
i(t) = 0.500
q(t) = 0.866

But if (w*34765) = 210 deg, then we have:

x(t) = -1.0
wt = 210 degrees
sin(wt) = -0.500
cos(wt) = -0.866
i(t) = 0.500
q(t) = 0.866

So the problem remains: Given only the i(t) and q(t) samples, which is all the 
USRP gives back, how do I reconstruct x(t)?