From: Johnathan C. [mailto:[email protected]]

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:

- 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.
- The data is written to a file in a format that GNU Radio can read.
- GNU Radio reads the data from the file and sends it to the USRP.
- The USRP upconverts it and transmits it via RF.
- The (other) USRP receives the RF signal, downconverts it, and sends

the data to GNU Radio.
- GNU Radio receives the data from the USRP and writes it to a file.
- 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® = 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)?