Re: [Discuss-gnuradio] Signal coming from the USRP to the computer

[Patrick Sisterhen]

John,

A phase difference (phi) between the
frequency of the transmit carrier (f_c) and the receiver local oscillator
(f_r) will be exhibited as a rotation of your received symbols in the complex
plain.

I think that's what you mean to imply
in your equations, but to get a little more precise:

Let x_i and x_q be the in-phase and
quadrature components of your baseband message at the transmitter.
Let y_i and y_q be the same for the
baseband message at the receiver (after downconversion with an oscillator
with a phase offset).

y_i = (x_i * cos(phi)) + (x_q * sin(phi))
y_q = (x_i * cos(phi)) - (x_q * cos(phi))

If phi is 0, you recover the original
in-phase and quadrature components.  Otherwise, it works like a rotation
by phi.

If the receiver local oscillator has
a frequency offset from the transmitter oscillator (f_c != f_r), the received
symbols will continuously rotate over time.

(I may have reversed the sign in those
equations... it depends on the implementation of the downconverter, but
you can detect and correct for it in the same way.)

Patrick Sisterhen
National Instruments



From:      
 John Andrews <address@hidden>
To:      
 Patrick Sisterhen <address@hidden>
Cc:      
 address@hidden
Date:      
 05/31/2011 02:08 PM
Subject:    
   Re: [Discuss-gnuradio]
Signal coming from the USRP to the computer

---

Thanks Patrick. I was concerned with the received signal
path. Suppose, I have the receiver tuned to, let's say, GPS signal. What
will the received signal look like. Considering the GPS message signal
is m(t), then what would equation would best describe the received signal. 

If 'f_c' is the carrier frequency then the signal coming
over the USB bus on to the computer for baseband processing will be,
inphase(t) = m(t) cos(phi)
quadrature(t) = m(t)sin(phi)

where, 'phi' is the instantaneous offset. Remember, phi
here is a broad term which includes all kinds of offsets(frequency, phase
etc).

On Tue, May 31, 2011 at 11:47 AM, Patrick Sisterhen <address@hidden>
wrote:
I think a little more detailed precise
answer to John's question might help: 

John Andrews wrote: 

> each complex sample that enters the
> USB bus is the following,
> 
> x[i] = (inphase_component) + j (quadrature_component), and
> x[i] = m(t)cos( 2*pi*FREQ_OFFSET*t + PHI ) + jm(t)sin( 2*pi*FREQ_OFFSET*t
+
> PHI ), where m(t), is the actual message signal, FREQ_OFFSET is the
> frequency offset, and PHI is the phase.
> 
> Is that correct? 

I think you're confusing the baseband
and passband signals a little, and the equations aren't quite right.

The complex-baseband signal (your message) is the data that is transferred
across the USB channel. 
x[i] = (in-phase) + j*(quadrature) 
       = (x_i) + j*(x_q) 

These are samples of your message signal, after modulation (mapping to
a complex QAM-constellation, for example), coding, pulse-shaping, etc.

The signal is up/down converted on the USRP device such that the transmitted
RF signal is 

r(t) = x_i*cos(2*pi*f_c) - (x_q)*sin(2*pi*f_c) 

(where f_c is your RF carrier frequency, and I'm ignoring phase offsets
and noise) 

Notice the subtraction there (which comes from the trig identities) and
that all the terms are real (it's a real passband signal).

Hope that helps a little.

Patrick Sisterhen
National Instruments
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