Spectral Inversion and swapping I and Q in DQPSK

Spectral Inversion and swapping I and Q in DQPSK

I am working with a DQPSK system  (differentially encoded QPSK).  As
you know,  the differential encoding makes it unnecessary to determine
the absolute phase of the carrier in the receiver.  The data is encoded
by the _relative_ rotation of the constellation points.  Sometimes the
DQPSK signal must pass through a frequency converter (mixer) and the
signal is spectrally inverted.  This also changes the direction of the
rotation of the constellation changes.   In the past I have  been told
that a spectral inversion is equivalent to a swapping of the I and Q
channels.  I can see how a reversal of the I and Q channels also
changes the direction of the constellation rotation so this makes sense
to me.   But I have also read that a spectral inversion is equivalent
to inversion of the Q channel i.e. Q=-Q and some experiments seem to
indicate this may be true.   Can someone clear this up?

So the question is:

A spectral inversion of a DQPSK signal is equivalent to:

1) a swapping of the channel I and Q data?
2) a swapping of the pre differentially encoded and post differentially
decoded I and Q data?
3) inversion of the Q data?
4 something else?

thanks
Mark

On 30 Mar 2006 14:59:11 -0800, "Mark" <makolber@yahoo.com> wrote:

>Spectral Inversion and swapping I and Q in DQPSK
>
>I am working with a DQPSK system  (differentially encoded QPSK).  As
>you know,  the differential encoding makes it unnecessary to determine
>the absolute phase of the carrier in the receiver.  The data is encoded
>by the _relative_ rotation of the constellation points.  Sometimes the
>DQPSK signal must pass through a frequency converter (mixer) and the
>signal is spectrally inverted.  This also changes the direction of the
>rotation of the constellation changes.   In the past I have  been told
>that a spectral inversion is equivalent to a swapping of the I and Q
>channels.  I can see how a reversal of the I and Q channels also
>changes the direction of the constellation rotation so this makes sense
>to me.   But I have also read that a spectral inversion is equivalent
>to inversion of the Q channel i.e. Q=-Q and some experiments seem to
>indicate this may be true.   Can someone clear this up?

Sure, it's pretty simple once you get the idea (as are many things).

>So the question is:
>
>A spectral inversion of a DQPSK signal is equivalent to:
>
>1) a swapping of the channel I and Q data?

Yup.

>2) a swapping of the pre differentially encoded and post differentially
>decoded I and Q data?

I don't know what you mean by that one.

>3) inversion of the Q data?

Yup.

>4 something else?

Yup, inversion of the I data.

Imagine it this way, if you will:  There is a piece of square
plexiglass or lexan or your favorite clear sheet material, suspended
in the air so that the sheet is vertical.   There is a phasor diagram
drawn on the sheet, with the orthogonal axes also drawn.  Some magical
apparatus makes the phasor rotate, let's say in the clockwise
direction when viewed from whichever side of the clear sheet that we
arbitrarily choose to be the "front".  From the "front" the axes are
labelled in the conventional fashion in that the I axis is vertical
with positive values increasing toward the top, and the Q axis is
horizontal with the positive values increasing toward the right.

Realize that if you look at the rotating phasor from the other side of
the sheet, it appears to be rotating counterclockwise.   So, when
viewed from the "front" it rotates clockwise, when viewed from the
rear it rotates counterclockwise.

In order to "invert" the rotation, all one needs to do is flip the
sheet from front to back.   Clearly there are an infinite number of
axes about which to flip the sheet around 180 degrees so that the
"front" and back of the sheet are reversed.   One way is to flip it
about the Q axes, which is done by inverting the I channel.   Another
way is to flip it about the I axis, which is done by inverting the Q
channel.  Swapping the channels is effectively flipping the sheet
about an axis at 45 degrees running from the first to third quadrants.
Any of these methods are easy to implement and accomplish the exact
same task of flipping the sheet from front to back, so that the phase
now appears to rotate counterclockwise by observers standing in the
original "front" location.

With a little trig the sheet can be flipped about any possible axis,
but this requires appropriate scaling and recombination of data from
each axis, which is naturally just way more complicated than the
previously discussed methods.
Eric Jacobsen
Minister of Algorithms, Intel Corp.
My opinions may not be Intel's opinions.
http://www.ericjacobsen.org

> >So the question is:
> >
> >A spectral inversion of a DQPSK signal is equivalent to:
> >
> >1) a swapping of the channel I and Q data?
>
> Yup.
>
> >2) a swapping of the pre differentially encoded and post differentially
> >decoded I and Q data?
>
> I don't know what you mean by that one.
>
> >3) inversion of the Q data?
>
> Yup.
>
> >4 something else?
>
> Yup, inversion of the I data.
>
> Imagine it this way, if you will:  There is a piece of square
> plexiglass or lexan or your favorite clear sheet material, suspended
> in the air so that the sheet is vertical.   There is a phasor diagram
> drawn on the sheet, with the orthogonal axes also drawn.  Some magical
> apparatus makes the phasor rotate, let's say in the clockwise
> direction when viewed from whichever side of the clear sheet that we
> arbitrarily choose to be the "front".  From the "front" the axes are
> labelled in the conventional fashion in that the I axis is vertical
> with positive values increasing toward the top, and the Q axis is
> horizontal with the positive values increasing toward the right.
>
> Realize that if you look at the rotating phasor from the other side of
> the sheet, it appears to be rotating counterclockwise.   So, when
> viewed from the "front" it rotates clockwise, when viewed from the
> rear it rotates counterclockwise.
>
> In order to "invert" the rotation, all one needs to do is flip the
> sheet from front to back.   Clearly there are an infinite number of
> axes about which to flip the sheet around 180 degrees so that the
> "front" and back of the sheet are reversed.   One way is to flip it
> about the Q axes, which is done by inverting the I channel.   Another
> way is to flip it about the I axis, which is done by inverting the Q
> channel.  Swapping the channels is effectively flipping the sheet
> about an axis at 45 degrees running from the first to third quadrants.
> Any of these methods are easy to implement and accomplish the exact
> same task of flipping the sheet from front to back, so that the phase
> now appears to rotate counterclockwise by observers standing in the
> original "front" location.
>
> With a little trig the sheet can be flipped about any possible axis,
> but this requires appropriate scaling and recombination of data from
> each axis, which is naturally just way more complicated than the
> previously discussed methods.
> Eric Jacobsen
> Minister of Algorithms, Intel Corp.
> My opinions may not be Intel's opinions.
> http://www.ericjacobsen.org


Eric,
thank you for the helpful reply.  I also found a similar  post of yours
 from 1998 describing the "Bullwinkle plane".  This is very helpful.
It is clear to me  now that negating Q or negating I or swapping I and
Q changes the perspective to the opposite side of the plane which is
equivalent to a spectral inversion.  Some of these operations also
result in a rotation of the plane and it is clear to me  that for the
case of DQPSK that the rotation of the axis representing the _CHANNEL_
 I/Q bits does not cause any errors because of the differential
encoding/decoding.

My question #2 above was referring to the fact that in the DQPSK system
there are several pairs of I/Q data.  Starting at the Tx, the payload
stream is broken into pairs.  Let me call these I/Q bits the DATA I/Q
bits.  These then pass through the differential encoder (where the
previous pair and the present pair are combined to create a new output
pair) and are applied to the QPSK modulator.  Let me call these bits
the Tx CHANNEL I/Q bits.  These bits are sent to the Rx but when the
are received, they are possibly different from the Tx Channel I/Q bits
due to the carrier recovery phase ambiguity that may have caused a
rotation about the origin of the plane.    The received Rx CHANNEL I/Q
bits are then fed to the differential decoder and (hopefully) the
original DATA I/Q bits are returned.

I understand from the "Bullwinkle plane"  that a spectral inversion is
equivalent to any of the following operations applied to the  CHANNEL
I/Q bits:  negation of Q, negation of I, and swapping I and Q.     This
is partly due to the differential encoding/ decoding that makes the
system ignore rotations about the origin.

Now lets look at the DATA I/Q bits instead of the CHANNEL I/Q bits.  I
(think) I  understand that a spectral inversion is also equivalent to a
swapping of DATA I and Q bits.  This can be seen from looking at the
differential encoding operation and swapping I and Q simply changes the
direction of the rotation which is the same as a spectral inversion.

The part that remains unclear to me is the negation of Q or the
negation of I relative to the DATA bits.  It would seem that this is
NOT equivalent to a spectral inversion?

To recap, there are really 6 questions relative to equivalence to
spectral inversion:

negation of I channel bits                  yes equivalent
negation of Q channel bits                 yes equivalent
swap of I and Q channel bits              yes equivalent

negation of I  data bits                         ???
negation of Q data bits                        ???
swap of I and Q  data bits                    yes equivalent

thank you again
Mark