Started by July 7, 2003
```Hi all,
What is quadrature sampling?It says that sampling frequency be greater
than signal BW.Does it not violate Nyquist criterion?What are its
applications?
Regards.
Raghavendra
```
```"Raghavendra" <raghurash@rediffmail.com> wrote in message
> Hi all,
> What is quadrature sampling?It says that sampling frequency be greater

As I understand the term, it means that the sampling freq is exactly 4 times
the IF frequency. Usually the BW of the signal is smaller than fs/2.
By picking this sampling freq, it allows you to perform quadrature mixing to
extract I,Q in a very easy fashion (multiply the signal with only a series
of 1s, -1s and zeros).

Cheers

> than signal BW.Does it not violate Nyquist criterion?What are its
> applications?
> Regards.
> Raghavendra

```
```In article <becb8p\$3khtp\$1@ID-82263.news.dfncis.de>, bhaskart@deja.com
says...
> "Raghavendra" <raghurash@rediffmail.com> wrote in message
> > Hi all,
> > What is quadrature sampling?It says that sampling frequency be greater
>
> As I understand the term, it means that the sampling freq is exactly 4 times
> the IF frequency. Usually the BW of the signal is smaller than fs/2.
> By picking this sampling freq, it allows you to perform quadrature mixing to
> extract I,Q in a very easy fashion (multiply the signal with only a series
> of 1s, -1s and zeros).
>
> Cheers
>
> > than signal BW.Does it not violate Nyquist criterion?What are its
> > applications?
> > Regards.
> > Raghavendra
>
>
>

heres an angle:

quad sampling exploits the mathematical property that comparing a test
signal (the information carrying signal) to a reference sin wave can be
done independently to comparing it to a reference cosine wave.  ie you
can extract 2 bits of information from a single symbols width of signal
by comparing it to 2 different reference points (sin and cos) at the
receiving end.  So for example, you quite literally extract symbols at
(eg) 1200 baud, but you end up with a bit rate of 2400 bps.

At the transmitting end, you start with a pair of bits (pull them out fo
the incoming stream in pairs), apply one to the sin reference, apply one
to the cos reference, then you add the two signals for transmission.
What you end up with after this addition is just a single sinusoid.  Its
strange to imagine that 2 bits can be extracted from it.  At the
receiving end, to extract the I channel bit, the signal is compared to
the cos reference wave, then to extract the Q channel bit, the same wave
is compared to the sin reference wave.  The curious part is that the 2
data bits can vary independently so that their combination for
transmission will not affect extraction of the 'other' bit at the
receiving end.

Spiro

--

http://www.mobilecomms.com.au
http://www.nexiondata.com
```
```"Spiro" <HOUSTON_we_have_a_problem@127.0.0.1> wrote in message
news:MPG.1974a224e288cf76989691@61.9.128.12...
> says...
> > "Raghavendra" <raghurash@rediffmail.com> wrote in message
> > > Hi all,
> > > What is quadrature sampling?It says that sampling frequency be greater
> >
> > As I understand the term, it means that the sampling freq is exactly 4
times
> > the IF frequency. Usually the BW of the signal is smaller than fs/2.
> > By picking this sampling freq, it allows you to perform quadrature
mixing to
> > extract I,Q in a very easy fashion (multiply the signal with only a
series
> > of 1s, -1s and zeros).

(snip)

> heres an angle:
>
> quad sampling exploits the mathematical property that comparing a test
> signal (the information carrying signal) to a reference sin wave can be
> done independently to comparing it to a reference cosine wave.  ie you
> can extract 2 bits of information from a single symbols width of signal
> by comparing it to 2 different reference points (sin and cos) at the
> receiving end.  So for example, you quite literally extract symbols at
> (eg) 1200 baud, but you end up with a bit rate of 2400 bps.
>
> At the transmitting end, you start with a pair of bits (pull them out fo
> the incoming stream in pairs), apply one to the sin reference, apply one
> to the cos reference, then you add the two signals for transmission.
> What you end up with after this addition is just a single sinusoid.  Its
> strange to imagine that 2 bits can be extracted from it.  At the
> receiving end, to extract the I channel bit, the signal is compared to
> the cos reference wave, then to extract the Q channel bit, the same wave
> is compared to the sin reference wave.  The curious part is that the 2
> data bits can vary independently so that their combination for
> transmission will not affect extraction of the 'other' bit at the
> receiving end.

It sounds pretty much the same as extracting both the signal and its
derivative, which you can also do at half the Nyquist rate.  As long as you
extract the appropriate amount of information, Nyquist doesn't really care
how you do it.

-- glen

```
```On Tue, 08 Jul 2003 00:00:07 GMT, "Glen Herrmannsfeldt"
<gah@ugcs.caltech.edu> wrote:

>
>"Spiro" <HOUSTON_we_have_a_problem@127.0.0.1> wrote in message
>news:MPG.1974a224e288cf76989691@61.9.128.12...
>> says...
>> > "Raghavendra" <raghurash@rediffmail.com> wrote in message
>> > > Hi all,
>> > > What is quadrature sampling?It says that sampling frequency be greater
>> >
>> > As I understand the term, it means that the sampling freq is exactly 4
>times
>> > the IF frequency. Usually the BW of the signal is smaller than fs/2.
>> > By picking this sampling freq, it allows you to perform quadrature
>mixing to
>> > extract I,Q in a very easy fashion (multiply the signal with only a
>series
>> > of 1s, -1s and zeros).
>
>(snip)
>
>> heres an angle:
>>
>> quad sampling exploits the mathematical property that comparing a test
>> signal (the information carrying signal) to a reference sin wave can be
>> done independently to comparing it to a reference cosine wave.  ie you
>> can extract 2 bits of information from a single symbols width of signal
>> by comparing it to 2 different reference points (sin and cos) at the
>> receiving end.  So for example, you quite literally extract symbols at
>> (eg) 1200 baud, but you end up with a bit rate of 2400 bps.
>>
>> At the transmitting end, you start with a pair of bits (pull them out fo
>> the incoming stream in pairs), apply one to the sin reference, apply one
>> to the cos reference, then you add the two signals for transmission.
>> What you end up with after this addition is just a single sinusoid.  Its
>> strange to imagine that 2 bits can be extracted from it.  At the
>> receiving end, to extract the I channel bit, the signal is compared to
>> the cos reference wave, then to extract the Q channel bit, the same wave
>> is compared to the sin reference wave.  The curious part is that the 2
>> data bits can vary independently so that their combination for
>> transmission will not affect extraction of the 'other' bit at the
>> receiving end.
>
>It sounds pretty much the same as extracting both the signal and its
>derivative, which you can also do at half the Nyquist rate.  As long as you
>extract the appropriate amount of information, Nyquist doesn't really care
>how you do it.
>
>-- glen

Yes, and lately I've seen Julius and a few folks use the term
"innovation rate" in papers rather than sample rate to clarify this
distinction (at least I think that's the intent).   This touches on
the real or complex sample debate that we have here occasionally as
well...

Eric Jacobsen
Minister of Algorithms, Intel Corp.
My opinions may not be Intel's opinions.
http://www.ericjacobsen.org
```
```Raghavendra wrote:
>
> Hi all,
> What is quadrature sampling?It says that sampling frequency be greater
> than signal BW.Does it not violate Nyquist criterion?What are its
> applications?
> Regards.
> Raghavendra

Hello Raghavendra,

The Fourier transform of a real signal g(t) is a complex function of
radian frequency, G(w) = Gr(w) + i*Gi(w). G(w) has the property that
its real part is an even function, i.e., Gr(w) = Gr(-w), and that its
imaginary part is an odd function, i.e., Gi(w) = -Gi(-w). This means
that you can construct the entire signal from just one half of the
bandwidth. For example, if g(t) had a bandwidth B, then, even though
G(w) would have non-zero components from -2*pi*B to +2*pi*B, only
half of that frequency range is unique. So why carry this redundant
information around?

In quadrature demodulation, you take the signal spectrum from 0 to
2*pi*B (or, more generally, from 2*pi*(fc - B/2) to 2*pi*(fc + B/2),
i.e., some bandwidth around a carrier fc) and shift it down so that it
is centered around 0 radians/second. This means that, since this new
signal only extends to B/2 Hz, we can halve the sampling rate and not
violate the Nyquist criterion. However, this also means that the
resulting time-domain signal is no longer real (because the symmetry
between positive and negative frequencies is no longer guaranteed).

That's the entire idea behind quadrature modulation and demodulation.
A quadrature demodulator takes a real signal and moves half of it
down to baseband (i.e., around 0 Hz), making it a complex signal.
A quadrature modulator does the inverse.
--
%% Fuquay-Varina, NC            %       'cause no one knows which side
%%% 919-577-9882                %                   the coin will fall."
%%%% <yates@ieee.org>           %  'Big Wheels', *Out of the Blue*, ELO
```
```A couple of extra points:

Randy Yates wrote:
> [...]
> In quadrature demodulation, you take the signal spectrum from 0 to
> 2*pi*B (or, more generally, from 2*pi*(fc - B/2) to 2*pi*(fc + B/2),
> i.e., some bandwidth around a carrier fc) and shift it down so that it
> is centered around 0 radians/second.

I should also add that you throw away the signal spectrum from -2*pi*B to 0.

I must inform you that you've been lied to all your life with this
Nyquist criterion thing. It says that sampling at a rate of Fs provides
a bandwidth of Fs/2. This is a lie! Sampling at a rate of Fs provides
a bandwidth of Fs, that is, from -Fs/2 to +Fs/2! The problem is that
real signals contain no useful information from -Fs/2 to 0, but that's
not the sampling process's fault! It does indeed provide you with a
bandwidth of Fs, and a quadrature demodulator is a device that allows
you to make use of this full bandwidth.
--
%% Fuquay-Varina, NC            %       'cause no one knows which side
%%% 919-577-9882                %                   the coin will fall."
%%%% <yates@ieee.org>           %  'Big Wheels', *Out of the Blue*, ELO
```
```"Randy Yates" <yates@ieee.org> wrote in message
news:3F0A2847.BA306D6C@ieee.org...
> A couple of extra points:
>
> Randy Yates wrote:
> > [...]
> > In quadrature demodulation, you take the signal spectrum from 0 to
> > 2*pi*B (or, more generally, from 2*pi*(fc - B/2) to 2*pi*(fc + B/2),
> > i.e., some bandwidth around a carrier fc) and shift it down so that it
> > is centered around 0 radians/second.
>
> I should also add that you throw away the signal spectrum from -2*pi*B to
0.
>
> I must inform you that you've been lied to all your life with this
> Nyquist criterion thing. It says that sampling at a rate of Fs provides
> a bandwidth of Fs/2. This is a lie! Sampling at a rate of Fs provides
> a bandwidth of Fs, that is, from -Fs/2 to +Fs/2! The problem is that
> real signals contain no useful information from -Fs/2 to 0, but that's
> not the sampling process's fault! It does indeed provide you with a
> bandwidth of Fs, and a quadrature demodulator is a device that allows
> you to make use of this full bandwidth.

Well, you can do it that way.

The way I like to think of it is that, for a signal of length T, there are T
Fs unknows, and T Fs equations are needed to solve for them.  Sampling the
real signal at the appropriate number of points works, sampling the signal
and its derivative at half that points works, and sampling the appropriate
complex function at half the number of points works.  The sample data must
be linearly independent which restricts it a little bit.

(Sampling the value, first, and second derivative at one third the number
of points works, too, and doesn't have any connection to complex math.)

-- glen

```
```"Glen Herrmannsfeldt" <gah@ugcs.caltech.edu> wrote in message news:<T5qOa.2362\$N7.679@sccrnsc03>...
> "Randy Yates" <yates@ieee.org> wrote in message
> news:3F0A2847.BA306D6C@ieee.org...
> > A couple of extra points:
> >
> > Randy Yates wrote:
> > > [...]
> > > In quadrature demodulation, you take the signal spectrum from 0 to
> > > 2*pi*B (or, more generally, from 2*pi*(fc - B/2) to 2*pi*(fc + B/2),
> > > i.e., some bandwidth around a carrier fc) and shift it down so that it
> > > is centered around 0 radians/second.
> >
> > I should also add that you throw away the signal spectrum from -2*pi*B to
>  0.
> >
> > I must inform you that you've been lied to all your life with this
> > Nyquist criterion thing. It says that sampling at a rate of Fs provides
> > a bandwidth of Fs/2. This is a lie! Sampling at a rate of Fs provides
> > a bandwidth of Fs, that is, from -Fs/2 to +Fs/2! The problem is that
> > real signals contain no useful information from -Fs/2 to 0, but that's
> > not the sampling process's fault! It does indeed provide you with a
> > bandwidth of Fs, and a quadrature demodulator is a device that allows
> > you to make use of this full bandwidth.
>
> Well, you can do it that way.
>
> The way I like to think of it is that, for a signal of length T, there are T
> Fs unknows, and T Fs equations are needed to solve for them.  Sampling the
> real signal at the appropriate number of points works, sampling the signal
> and its derivative at half that points works, and sampling the appropriate
> complex function at half the number of points works.  The sample data must
> be linearly independent which restricts it a little bit.
>
>  (Sampling the value, first, and second derivative at one third the number
> of points works, too, and doesn't have any connection to complex math.)

Glen,

I don't follow you. What are the T Fs equations?

I also don't see how the number of equations required to solve
for a certain number of unknowns has a relationship to the
spectrum of a signal.
```
```Randy Yates wrote:
>
...
>
> I also don't see how the number of equations required to solve
> for a certain number of unknowns has a relationship to the
> spectrum of a signal.

That part is easy. Given one period T of a periodic function bandlimited
so that no frequency higher than f exists in it, then its spectrum can
consist of no more than f*T sines and an equal number of cosines. We
know what they are -- harmonics of sin(2*pi/T) and cos(2*pi/T).
Determining the spectrum consists of solving for 2*f*T amplitudes, so
2*f*T measurements are needed. The signal and its first (2*f*T - 1)th
derivatives at t = zero, for example, suffice. When the measurements are
spread uniformly in time and consist of the requisite number either of
the waveform alone or the waveform and its derivative, a Fourier
transform becomes a convenient way to do the computation.

Jerry
--
Engineering is the art of making what you want from things you can get.
&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;
```