# Rick's 3rd edition is out

Started by November 16, 2010
```rickman <gnuarm@gmail.com> wrote:
> On Nov 18, 7:47&#4294967295;pm, Greg Berchin <gjberc...@charter.net> wrote:
(snip)

>> Everybody seems to forget that a signal extending from DC to Fs/2 also
>> extends from DC to -Fs/2, so the bandwidth is actually Fs. &#4294967295;So Fs can
>> unambiguously represent a bandwidth incrementally less than Fs.

> That is only a way of interpreting the math... there is a reason why
> they call them "real" and "imaginary" numbers.  Negative frequency???
> Think about what that means in the physical world...  :^)

OK, but with a synchronous demodulator you can separate out
the I and Q, or sine and cosine parts, as the NTSC color
subcarrier demodulators do (did).

If you don't like imaginary numbers, use the Hartley transform instead.
Then there are still sine and cosine, or even and odd, components,
but with real input only real output.

> Actually, there is a discrepancy comparing imaginary number samples
> with real number samples.  First only real data can be generated by
> sampling a physical signal (or reconstructing it), one data sample
> per...  By the time you have complex data you have twice as much data
> per sample.  So in a sense with complex numbers you can represent the
> signal with half the sample rate, but twice the amount of data per.
> In reality, it is just a way of juggling the numbers.

As NTSC puts two parts of the color subcarrier into the bandwidth
needed for one.

Actually, NTSC is more complicated than that.  At the time the
standard was specified, they did research into the resolution of
the eye to different colors, and found that it varies.  In NTSC
terms, the axis of high resolution is not the same as that
for the red or blue components, but rotated in color space
(phase shifted relative to the reference burst).  For easier
demodulation, especially in the vacuum tube days, the reference
burst is such that it can be easily demodulated along the red and
blue axes, with reduced chrominance resolution.  Only recently,
just before HDTV came out, were receivers that demodulated the full
resolution available.

-- glen

```
```On Thu, 18 Nov 2010 19:52:57 -0800 (PST), rickman <gnuarm@gmail.com> wrote:

>I'm not sure where you are getting that.  It is correct if you are
>using complex numbers, but if you are sampling a real signal there is
>no imaginary part, so the only signal is real and no negative
>frequencies.

Incorrect.  Sample any real signal and you get a negative spectrum between 0 and
-Fs/2 that is the complex conjugate of the positive spectrum between 0 and
+Fs/2.  The bandwidth is Fs.

>But a sample rate of Fs can't represent a real signal
>with a wider bandwidth than Fs/2.  Try it, you will see that all
>frequencies outside of 0 to Fs/2 get aliased into that frequency
>range.

Given a real signal of one-sided bandwidth B (i.e., under the common definition
that excludes negative frequencies, the bandwidth extends from DC to B), it can
be unambiguously sampled at Fs > 2B.

Amplitude-modulate that same signal by a carrier of frequency C.  The resulting
AM spectrum will extend from +C-B to +C+B (and also from -C+B to -C-B).  Under
the common definition it will have a bandwidth of 2B, yet it can still be
unambiguously sampled by any Fs > 2B.

Now set C = 0.  The bandwidth hasn't changed, but the negative image now
overlaps the positive image exactly.  So why is the bandwidth now B instead of
2B?

The problem is in the definition of bandwidth.

>The only way to sample a real signal and end up with complex numbers
>is to perform a complex operation to convert the signal.

Sample a real sine wave (as opposed to a cosine wave), and you'll get an
imaginary spectrum.

>Are we talking about the same thing and just not understanding each
>other?

I think we're not understanding each other.

Greg
```
```On Nov 18, 11:52&#4294967295;pm, glen herrmannsfeldt <g...@ugcs.caltech.edu>
wrote:
> rickman <gnu...@gmail.com> wrote:
> > On Nov 18, 7:47&#4294967295;pm, Greg Berchin <gjberc...@charter.net> wrote:
>
> (snip)
>
> >> Everybody seems to forget that a signal extending from DC to Fs/2 also
> >> extends from DC to -Fs/2, so the bandwidth is actually Fs. &#4294967295;So Fs can
> >> unambiguously represent a bandwidth incrementally less than Fs.
> > That is only a way of interpreting the math... there is a reason why
> > they call them "real" and "imaginary" numbers. &#4294967295;Negative frequency???
> > Think about what that means in the physical world... &#4294967295;:^)
>
> OK, but with a synchronous demodulator you can separate out
> the I and Q, or sine and cosine parts, as the NTSC color
> subcarrier demodulators do (did). &#4294967295;
>
> If you don't like imaginary numbers, use the Hartley transform instead.
> Then there are still sine and cosine, or even and odd, components,
> but with real input only real output.

That's not the point.  The question is about what it takes to
represent a signal.  When you sample a signal (the only thing the
Nyquist criterion relates to) you must sample at twice the band width
to capture all information intact.  Then you can play games with that
data and reduce the "sample rate", although you just reformatting the
data to use two components per sample with half the sample rate. In
reality you have not really gotten around the Nyquist criterion.

Rick

> > Actually, there is a discrepancy comparing imaginary number samples
> > with real number samples. &#4294967295;First only real data can be generated by
> > sampling a physical signal (or reconstructing it), one data sample
> > per... &#4294967295;By the time you have complex data you have twice as much data
> > per sample. &#4294967295;So in a sense with complex numbers you can represent the
> > signal with half the sample rate, but twice the amount of data per.
> > In reality, it is just a way of juggling the numbers.
>
> As NTSC puts two parts of the color subcarrier into the bandwidth
> needed for one. &#4294967295;
>
> Actually, NTSC is more complicated than that. &#4294967295;At the time the
> standard was specified, they did research into the resolution of
> the eye to different colors, and found that it varies. &#4294967295;In NTSC
> terms, the axis of high resolution is not the same as that
> for the red or blue components, but rotated in color space
> (phase shifted relative to the reference burst). &#4294967295;For easier
> demodulation, especially in the vacuum tube days, the reference
> burst is such that it can be easily demodulated along the red and
> blue axes, with reduced chrominance resolution. &#4294967295;Only recently,
> just before HDTV came out, were receivers that demodulated the full
> resolution available. &#4294967295;
>
> -- glen

```
```
Greg Berchin wrote:
>
> On Nov 18, 7:08 pm, jim <"sjedgingN0Sp"@m...@mwt.net> wrote:
>
> > rickman wrote:
> > Wouldn't you name that the "Nyquist bandwidth"? The Nyquist frequency would be
> > the maximum frequency that can be represented by a given sample rate, in other
> > words Fs/2.
>
> No,

What is it you are saying No to? If someone is talking about bandwidth
why would they call it frequency?

Fs can unambiguously represent an analytic signal (no negative
> frequency content) extending from DC to incrementally less than Fs.

Sure you can multiply any digital signal by the Nyquist Frequency to
produce a new digital signal. And then you can do the same thing again.
Multiply that digital signal by the Nyquist Frequency and you end up
with the original signal. Now what you consider the bandwidth of those
two signals to be and how you name that bandwidth may not be the same,
but the Nyquist Frequency never changed.

-jim

>
> Greg
```
```On Nov 19, 7:55&#4294967295;am, Greg Berchin <gjberc...@chatter.net.invalid>
wrote:
> On Thu, 18 Nov 2010 19:52:57 -0800 (PST), rickman <gnu...@gmail.com> wrote:
> >I'm not sure where you are getting that. &#4294967295;It is correct if you are
> >using complex numbers, but if you are sampling a real signal there is
> >no imaginary part, so the only signal is real and no negative
> >frequencies. &#4294967295;
>
> Incorrect. &#4294967295;Sample any real signal and you get a negative spectrum between 0 and
> -Fs/2 that is the complex conjugate of the positive spectrum between 0 and
> +Fs/2. &#4294967295;The bandwidth is Fs.

That is what you aren't getting.  When you sample the signal, you only
get real data, no imaginary part and so no complex conjugate...  The
idea of negative frequency only appears out of the math once you
change the data to include a zero imaginary part.  This comes out of
the math *after* you change the representation of the signal.

> >But a sample rate of Fs can't represent a real signal
> >with a wider bandwidth than Fs/2. &#4294967295;Try it, you will see that all
> >frequencies outside of 0 to Fs/2 get aliased into that frequency
> >range.
>
> Given a real signal of one-sided bandwidth B (i.e., under the common definition
> that excludes negative frequencies, the bandwidth extends from DC to B), it can
> be unambiguously sampled at Fs > 2B.
>
> Amplitude-modulate that same signal by a carrier of frequency C. &#4294967295;The resulting
> AM spectrum will extend from +C-B to +C+B (and also from -C+B to -C-B). &#4294967295;Under
> the common definition it will have a bandwidth of 2B, yet it can still be
> unambiguously sampled by any Fs > 2B.

It only has a bandwidth of 2B because the original bandwidth was
doubled and the information in each half of the doubled signal is
equivalent.  You don't need both halves of the modulated signal to
carry the full content of the signal B.  Isn't that what single
sideband modulation is about?  This is just an artifact of inefficient
modulation.  An extreme example of this is CDMA where the bandwidth is
spread to be many times the original bandwidth, but the information
content is the same.

> Now set C = 0. &#4294967295;The bandwidth hasn't changed, but the negative image now
> overlaps the positive image exactly. &#4294967295;So why is the bandwidth now B instead of
> 2B?

Because the information content was always just B with a duplication
of the signal B into -B for want of a better way to express it if you
are working in the math domain.  I'm not sure what you are trying to
say with this.

> The problem is in the definition of bandwidth.

I won't argue that.  In your modulated example the bandwidth was
doubled from B to 2B, but in reality it wasn't.  The information is
still just B, but it was spread out over 2B by inefficient
modulation.  The rest is just how you interpret the math.

> >The only way to sample a real signal and end up with complex numbers
> >is to perform a complex operation to convert the signal. &#4294967295;
>
> Sample a real sine wave (as opposed to a cosine wave), and you'll get an
> imaginary spectrum.

I don't follow.  If you sample *any* real signal (and by real I mean
in the real world, not the theoretical world) and you will get a real
sequence.  You have to add data to get an imaginary component.

> >Are we talking about the same thing and just not understanding each
> >other?
>
> I think we're not understanding each other. &#4294967295;

I can't argue that either.  Every example of a violation of the
Nyquist criterion anyone else has mentioned appears to me to be an
artifact of the math being used and does not violate the criterion.

Rick
```
```On Thu, 18 Nov 2010 09:34:49 -0800 (PST), robert bristow-johnson
<rbj@audioimagination.com> wrote:

>On Nov 17, 6:13&#4294967295;am, Rick Lyons <R.Lyons@_BOGUS_ieee.org> wrote:
>> On Tue, 16 Nov 2010 06:20:44 GMT, Al Clark <acl...@danvillesignal.com>
>> wrote:
>>
>> >I just noticed that Rick Lyon's 3rd edition of Understanding Digital Signal
>> >Processing is out, so I guess I will have to get a copy to place next to the
>> >1st & 2nd editions on my bookshelf.
>>
>> >I've always considered Rick's book the definitive DSP primer.
>>
>> >There are two kinds of engineers, those who can't write and those who can
>> >barely write. I strive for the latter. Rick is the clear exception to the
>> >rule.
>>
>> &#4294967295; Yep, the 3rd Edition is *FINALLY* finished.
>> Whew!
>>
>> Thanks for the kind words Al.
>
>hay Rick, not an unkind word, just a question:  does your 3rd edition
>take a position of the meaning of "Nyquist frequency"?  3 different
>definitions i have seen:
>
>
> Nyquist frequency =   a)  Fs/2
>                       b)  B (bandlimit of signal to be sampled)
>                       c)  Fs (rarely)
>                       d)  2B (usually called the "Nyquist rate")
>
>
>several years ago, when this definition discussion came up, i was
>appalled to find that O&S themselves had the wrong definition (at
>least in my 1989 edition).  and surprized to see Rick was sorta
>agnostic about it (maybe that was smart).
>
>r b-j
Hi Robert,
I did my best to avoid using the phrase
"Nyquist frequency."  When I was referring
to half the sample rate I attempted by be
as clear as possible by using the term "Fs/2."

[-Rick-]

```
```On Fri, 19 Nov 2010 07:43:20 -0600, jim <"sjedgingN0Sp"@m@mwt,net> wrote:

>> > Wouldn't you name that the "Nyquist bandwidth"? The Nyquist frequency would be
>> > the maximum frequency that can be represented by a given sample rate, in other
>> > words Fs/2.
>>
>> No,
>
>What is it you are saying No to? If someone is talking about bandwidth
>why would they call it frequency?

I didn't trim enough of the quote.  I was saying "no" to "the maximum frequency
that can be represented by a given sample rate, in other words Fs/2."  The
maximum *bandwidth* that can be represented by Fs is incrementally less than Fs,
under conditions stated in some of my other posts.

Greg
```
```On Nov 19, 10:17&#4294967295;am, Rick Lyons <R.Lyons@_BOGUS_ieee.org> wrote:
> On Thu, 18 Nov 2010 09:34:49 -0800 (PST), robert bristow-johnson
>
>
>
> <r...@audioimagination.com> wrote:
> >On Nov 17, 6:13 am, Rick Lyons <R.Lyons@_BOGUS_ieee.org> wrote:
> >> On Tue, 16 Nov 2010 06:20:44 GMT, Al Clark <acl...@danvillesignal.com>
> >> wrote:
>
> >> >I just noticed that Rick Lyon's 3rd edition of Understanding Digital Signal
> >> >Processing is out, so I guess I will have to get a copy to place next to the
> >> >1st & 2nd editions on my bookshelf.
>
> >> >I've always considered Rick's book the definitive DSP primer.
>
> >> >There are two kinds of engineers, those who can't write and those who can
> >> >barely write. I strive for the latter. Rick is the clear exception to the
> >> >rule.
>
> >> Yep, the 3rd Edition is *FINALLY* finished.
> >> Whew!
>
> >> Thanks for the kind words Al.
>
> >hay Rick, not an unkind word, just a question: &#4294967295;does your 3rd edition
> >take a position of the meaning of "Nyquist frequency"? &#4294967295;3 different
> >definitions i have seen:
>
> > Nyquist frequency = &#4294967295; a) &#4294967295;Fs/2
> > &#4294967295; &#4294967295; &#4294967295; &#4294967295; &#4294967295; &#4294967295; &#4294967295; &#4294967295; &#4294967295; &#4294967295; &#4294967295; b) &#4294967295;B (bandlimit of signal to be sampled)
> > &#4294967295; &#4294967295; &#4294967295; &#4294967295; &#4294967295; &#4294967295; &#4294967295; &#4294967295; &#4294967295; &#4294967295; &#4294967295; c) &#4294967295;Fs (rarely)
> > &#4294967295; &#4294967295; &#4294967295; &#4294967295; &#4294967295; &#4294967295; &#4294967295; &#4294967295; &#4294967295; &#4294967295; &#4294967295; d) &#4294967295;2B (usually called the "Nyquist rate")
>
> >several years ago, when this definition discussion came up, i was
> >appalled to find that O&S themselves had the wrong definition (at
> >least in my 1989 edition). &#4294967295;and surprized to see Rick was sorta
> >agnostic about it (maybe that was smart).
>
...
> &#4294967295; I did my best to avoid using the phrase
> "Nyquist frequency."

i know you did in the first two editions.  as i said, it might have
been smart to avoid a tiff regarding semantics.

but i believe we have language and words for a reason (that is, of
efficiency of communication).  i don't say "my 4-wheeled personal
vehicle" every time i mean "my car".  nor do i say "the identity that
translates the natural exponential of an imaginary argument to
sinusoidal functions" when i mean "Euler's formula".  when someone
else takes issue with these semantics and says "'car' could mean
something that runs on rails" or "that's really Roger Cotes' formula",
i just want to tell them that the names have settled meaning and it's
efficient if we leave them as settled.

> &#4294967295;When I was referring
> to half the sample rate I attempted by be
> as clear as possible by using the term "Fs/2."

grrr.  i just wanted you (as another author) to make a stand on the
semantic issue (and take the *correct* stand).

L8r,

r b-j
```
```rickman <gnuarm@gmail.com> wrote:

(snip, someone wrote)

>> >> Everybody seems to forget that a signal extending from DC to Fs/2 also
>> >> extends from DC to -Fs/2, so the bandwidth is actually Fs. &#4294967295;So Fs can
>> >> unambiguously represent a bandwidth incrementally less than Fs.
>> > That is only a way of interpreting the math... there is a reason why
>> > they call them "real" and "imaginary" numbers. &#4294967295;Negative frequency???
>> > Think about what that means in the physical world... &#4294967295;:^)

(then I wrote)
>> OK, but with a synchronous demodulator you can separate out
>> the I and Q, or sine and cosine parts, as the NTSC color
>> subcarrier demodulators do (did). &#4294967295;

>> If you don't like imaginary numbers, use the Hartley transform instead.
>> Then there are still sine and cosine, or even and odd, components,
>> but with real input only real output.

> That's not the point.  The question is about what it takes to
> represent a signal.  When you sample a signal (the only thing the
> Nyquist criterion relates to) you must sample at twice the band width
> to capture all information intact.  Then you can play games with that
> data and reduce the "sample rate", although you just reformatting the
> data to use two components per sample with half the sample rate. In
> reality you have not really gotten around the Nyquist criterion.

I completely agree.  The comment had to do with signals being real,
and that the imaginary component couldn't be sampled.

Yes, with the ones I described you need two real values at each
sample interval, and that is not a lot different than one real
value at twice the rate.  It is comparable to real and imaginary
components at the lower rate.

(snip)

>> As NTSC puts two parts of the color subcarrier into
>> the bandwidth needed for one.

-- glen
```