# Fourier Transform (spectrum) of QPSK and BPSK?

Started by November 12, 2006
```Hi All,

Does anyone have a mathematical derivation
of the expected spectrum of a QPSK signal, given a
known symbol rate (which is 1/2 the data rate for QPSK)?

The signal i am measuring doesn't have any
sideband carrier suppression at all, which is supposed
to be a figure of merit for phase and amplitude imbalance
of your IQ mixer (how orthogonal or in-quadrature the I and Q
channels are).

The bottom line is I'm trying to measure phase
and amplitude imbalance with just a spectrum analyzer,
without having to buy an expensive VSA which can display
the constellation, and give me EVM, etc.

Thanks for any REAL help (which is Info from someone
who isn't pretending to know more than they actually do!).

Slick

```
```radio913@aol.com wrote:
> Hi All,
>
>         Does anyone have a mathematical derivation
>  of the expected spectrum of a QPSK signal, given a
> known symbol rate (which is 1/2 the data rate for QPSK)?
>
>         The signal i am measuring doesn't have any
>  sideband carrier suppression at all, which is supposed
> to be a figure of merit for phase and amplitude imbalance
> of your IQ mixer (how orthogonal or in-quadrature the I and Q
> channels are).
>
>          The bottom line is I'm trying to measure phase
> and amplitude imbalance with just a spectrum analyzer,
> without having to buy an expensive VSA which can display
> the constellation, and give me EVM, etc.
>
>           Thanks for any REAL help (which is Info from someone
> who isn't pretending to know more than they actually do!).
>
>
> Slick

It will be harder to discern the imbalances from a QPSK SA display than
with an ordinary tone. Try changing the modulation to a complex
exponential: sin(wt) + j*cos(wt). You get a three-tone display on the
SA. From left to right, the components are: opposite sideband
(undesired), carrier (undesired), and desired signal. You can directly
measure the levels in dBc of the undesired components. I'm not sure if
you can relate them back mathematically to imbalances in the modulator,
but what you can do is tweak the modulator until they are minimized.

John

```
```"radio913@aol.com" <radio913@aol.com> writes:

> Hi All,
>
>         Does anyone have a mathematical derivation
>  of the expected spectrum of a QPSK signal, given a
> known symbol rate (which is 1/2 the data rate for QPSK)?

From [proakiscomm], the spectrum expression for any linear modulation
(such as QPSK) is

Phi(f) = (1/T) * | G(f) |^2 * Phi_ii(f),

where G(f) is the Fourier transform of the transmit pulse shape g(t)
and Phi_ii(f) is the power spectral density of the information
sequence I(n). T is the symbol period.

So without knowing your pulse shape, we can't really give you a
specific spectrum.

--Randy

@BOOK{proakiscomm,
title = "{Digital Communications}",
author = "John~G.~Proakis",
publisher = "McGraw-Hill",
edition = "fourth",
year = "2001"}
--
%  Randy Yates                  % "Though you ride on the wheels of tomorrow,
%% Fuquay-Varina, NC            %  you still wander the fields of your
%%% 919-577-9882                %  sorrow."
%%%% <yates@ieee.org>           % '21st Century Man', *Time*, ELO
```
```

>         Does anyone have a mathematical derivation
>  of the expected spectrum of a QPSK signal, given a
> known symbol rate (which is 1/2 the data rate for QPSK)?

If the data is random, then the spectum is the Fourier transform of a
single pulse. A textbook like Proakis or Sklar should have that.

>
>         The signal i am measuring doesn't have any
>  sideband carrier suppression at all,  which is supposed
> to be a figure of merit for phase and amplitude imbalance
> of your IQ mixer (how orthogonal or in-quadrature the I and Q
> channels are).
>
>          The bottom line is I'm trying to measure phase
> and amplitude imbalance with just a spectrum analyzer,
> without having to buy an expensive VSA which can display
> the constellation, and give me EVM, etc.

1. Run 01 10 01 10 01 10 ... and 11 00 11 00 ... patterns to measure the
carrier suppression in I and Q channels. There should be no peak at the
carrier frequency.

2. Run 00 01 11 10 00 01 11 10 .... pattern to measure the
amplitude/phase balance between I and Q. There should be a symmetrical
spectrum shifted up from the center frequency.

>
>           Thanks for any REAL help (which is Info from someone
> who isn't pretending to know more than they actually do!).

Your majesty's "Thank you" means soo much. You don't have to thank me,
\$100 will be just all right.

DSP and Mixed Signal Design Consultant

http://www.abvolt.com
```
```
On Nov 12, 6:36=A0am, Randy Yates <y...@ieee.org> wrote:
> > Hi All,
>
> > =A0 =A0 =A0 =A0 Does anyone have a mathematical derivation
> > =A0of the expected spectrum of a QPSK signal, given a
> > known symbol rate (which is 1/2 the data rate for QPSK)?From [proakisco=
mm], the spectrum expression for any linear modulation
> (such as QPSK) is
>
> =A0 Phi(f) =3D (1/T) * | G(f) |^2 * Phi_ii(f),
>
> where G(f) is the Fourier transform of the transmit pulse shape g(t)
> and Phi_ii(f) is the power spectral density of the information
> sequence I(n). T is the symbol period.
>
> So without knowing your pulse shape, we can't really give you a
> specific spectrum.
>

The data on the I and Q channels is just
a square wave from a LVTTL source.  The symbol
rate is 200kHz, which is 400kBits/sec.

The above equation doesn't look correct, because
if you have an infinitely long symbol period, which is the
same as staying in one quadrant forever, then we would
still have amplitude at the carrier frequency.  The above
equation goes to zero for an infinitely long T.

Slick

```
```radio913@aol.com skrev:
> Hi All,

>           Thanks for any REAL help (which is Info from someone
> who isn't pretending to know more than they actually do!).

Do you have any particular person in mind? If not, how do you
tell the difference between pretenders and others?

Rune

```
```
On Nov 12, 11:18=A0am, Vladimir Vassilevsky <antispam_bo...@hotmail.com>
wrote:
> > =A0 =A0 =A0 =A0 Does anyone have a mathematical derivation
> > =A0of the expected spectrum of a QPSK signal, given a
> > known symbol rate (which is 1/2 the data rate for QPSK)?If the data is =
random, then the spectum is the Fourier transform of a
> single pulse. A textbook like Proakis or Sklar should have that.

A single pulse would have a continuous spectrum,
which is not what i'm measuring at all.

And although i agree the data stream will affect
the spectrum (i.e., a bunch of "00"s or "11"s will be just
the carrier freq.), making the data even psuedo-
random will not give you a continuus spectrum.

>
>
> > =A0 =A0 =A0 =A0 The signal i am measuring doesn't have any
> > =A0sideband carrier suppression at all, =A0which is supposed
> > to be a figure of merit for phase and amplitude imbalance
> > of your IQ mixer (how orthogonal or in-quadrature the I and Q
> > channels are).
>
> > =A0 =A0 =A0 =A0 =A0The bottom line is I'm trying to measure phase
> > and amplitude imbalance with just a spectrum analyzer,
> > without having to buy an expensive VSA which can display
> > the constellation, and give me EVM, etc.1. Run 01 10 01 10 01 10 ... an=
d 11 00 11 00 ... patterns to measure the
> carrier suppression in I and Q channels. There should be no peak at the
> carrier frequency.

That would be BPSK in the cases you mention.

>
> 2. Run 00 01 11 10 00 01 11 10 .... pattern to measure the
> amplitude/phase balance between I and Q. There should be a symmetrical
> spectrum shifted up from the center frequency.

That's the pattern i'm using.  The center carrier
is suppressed, but the sidebands are very close in amplitude,
so unless my orthogonality is way off, I don't see any
indication of amplitude/phase imbalance.

>
>
>
> > =A0 =A0 =A0 =A0 =A0 Thanks for any REAL help (which is Info from someone
> > who isn't pretending to know more than they actually do!).Your majesty'=
s "Thank you" means soo much. You don't have to thank me,
> \$100 will be just all right.

You didn't earn it!

My remark refers to people just like you,
but I don't expect a C++ programmer to know
the Fourier of QPSK, PhD or not.

Slick

```
```
On Nov 12, 12:21=A0pm, "Rune Allnor" <all...@tele.ntnu.no> wrote:
>
> > Hi All,
> > =A0 =A0 =A0 =A0 =A0 Thanks for any REAL help (which is Info from someone
> > who isn't pretending to know more than they actually do!).Do you have a=
ny particular person in mind? If not, how do you
> tell the difference between pretenders and others?
>=20

posts!

```
```radio913@aol.com skrev:
> On Nov 12, 12:21?pm, "Rune Allnor" <all...@tele.ntnu.no> wrote:
> >
> > > Hi All,
> > > ? ? ? ? ? Thanks for any REAL help (which is Info from someone
> > > who isn't pretending to know more than they actually do!).Do you have any
particular person in mind? If not, how do you
> > tell the difference between pretenders and others?
> >
>
> posts!

Well, that's usenet for you; you get exactly what you pay for.
As others already hinted at, you might be better off contacting
people whose knowledge you acknowledge, and pay them.

Rune

```
```"radio913@aol.com" <radio913@aol.com> writes:

> On Nov 12, 11:18&#2013266080;am, Vladimir Vassilevsky
<antispam_bo...@hotmail.com>
> wrote:
>> > &#2013266080; &#2013266080; &#2013266080; &#2013266080; Does anyone have a
mathematical derivation
>> > &#2013266080;of the expected spectrum of a QPSK signal, given a
>> > known symbol rate (which is 1/2 the data rate for QPSK)?If the data is random,
then the spectum is the Fourier transform of a
>> single pulse. A textbook like Proakis or Sklar should have that.
>
>        A single pulse would have a continuous spectrum,
> which is not what i'm measuring at all.

Hi,

Vladimir said, "*IF* the data is random, then the spectrum is the
Fourier transform of a single pulse." Your data isn't random - it's
highly correlated. His statement agrees 100 percent with the equation
I posted from Proakis.

>        And although i agree the data stream will affect
> the spectrum (i.e., a bunch of "00"s or "11"s will be just
> the carrier freq.), making the data even psuedo-
> random will not give you a continuus spectrum.

I'm curious why you think so. It is pretty basic knowledge that the
power spectrum of a random signal is the transform of its
autocorrelation function (the Wiener-Khinchine theorem), and it's
pretty easy to see that a *continuous* flat spectrum is produced by a
sequence with a Kronecker delta, and that it requires an uncorrelated
sequence to produce a Kronecker delta autocorrelation.

So Proakis's expression makes a lot of sense: the total spectrum is
the cascade of the information sequence spectrum and the transmit
pulse spectrum.

In any case, you must argue the point with John Proakis, who has
written a textbook on the subject, because that is what he claims.

>> > &#2013266080; &#2013266080; &#2013266080; &#2013266080; &#2013266080; Thanks
for any REAL help (which is Info from someone
>> > who isn't pretending to know more than they actually do!).Your majesty's "Thank
you" means soo much. You don't have to thank me,
>> \$100 will be just all right.
>
>
>       You didn't earn it!
>
>       My remark refers to people just like you,
> but I don't expect a C++ programmer to know
> the Fourier of QPSK, PhD or not.

Vladimir is no mere C++ programmer. From what I've seen of him though
his posts over the past months/years, he is brilliant in many topics
on DSP and digital communications.
--
%  Randy Yates                  % "I met someone who looks alot like you,
%% Fuquay-Varina, NC            %             she does the things you do,
%%% 919-577-9882                %                     but she is an IBM."
%%%% <yates@ieee.org>           %        'Yours Truly, 2095', *Time*, ELO