> Jerry Avins <jya@ieee.org> writes:
>
>> Randy Yates wrote:
>>> Jerry Avins <jya@ieee.org> writes:
>>>
>>>> Randy Yates wrote:
>>>>> Jerry Avins <jya@ieee.org> writes:
>>>>>> I think we're saying the same thing.
>>>>> I don't think we are, Jerry.
>>>> We agree, I think, that neither imaginary numbers nor negative time
>>>> are needed to describe sinusoids of arbitrary phase. We agree that it
>>>> is convenient to use them.
>>> I agree with that, but that isn't the main point I'm trying to make.
>>>
>>> My point is that to fully describe some phenomena requires an "extended
>>> system," whatever the representation we choose to use for that extended
>>> system is. To recast your statement above in these terms, a specific
>>> representation may be convenient but not necessary---yes, I agree. But
>>> we need SOME representation of the extended system (e.g., the complex
>>> numbers) and CANNOT simply use the simple system (e.g., the reals) to
>>> accomplish certain tasks (e.g., represent all N roots of an Nth-order
>>> polynomial).
>> I completely agree about roots. In electrical theory, though, we use
>> complex numbers as ordered pairs, which is overkill (but we grow to
>> think it fundamental). x+iy, R=jX; it is rare that extracting a root
>> comes into play. Sin(r), cos(r) often serves where we use cos(r) +
>> j*sin(r) for convenience. (I don't remember the formulas for the sums
>> of sines and cosines. I use complex numbers to derive the relations
>> when I need them. I'm all for convenience!)
>
> Well once you agree there, it's the similar situation for negative
> frequencies, isn't it? If you're talking about a two-dimensional
> "wheel," then you need to specify whether it's turning CW or CCW. Now
> you don't HAVE to use negative numbers (that's just one representation)
> - you could use positive numbers plus the letters [CW | CCW] - but,
> somehow, you must represent the information, and any two representations
> are isomorphic.
CW and CCW are rather arbitrary and limited views of the world. The
notion can't adequately describe which way the wheels of your car turn.
Most of us would agree that all the wheels turn the same way when a car
moves forward, yet the "standard" engineering nomenclature has the right
wheels turning clockwise and the left ones, counterclockwise.*
That's all just an amusing side issue. A sinusoid is not a wheel.
Sin(-x) = -sin(x) and cos(-x) = cos(x). The former has an inversion.
When I run sin(ft) through a unity-gain inverting amplifier, would you
insist that sin(-ft) comes out? That wouldn't be wrong, but rather than
deal with negative frequency or time marching backward, I prefer to
write -sin(ft).
I don't have a problem using negative frequency when it's appropriate. I
*do* have a problem with letting the notion become so ingrained that it
seems to be indispensable.
Jerry
_________________________________
* The standard describes the direction when looking into the shaft. A
double-ended motor, such as those used on many grinders and polishers,
turns both ways at once.
--
Engineering is the art of making what you want from things you can get.
�����������������������������������������������������������������������
Reply by Rick Lyons●June 13, 20082008-06-13
On Mon, 09 Jun 2008 14:18:17 -0500, "BobTheDog"
<andrew.capon@zen.co.uk> wrote:
>Hi Guys,
>
>First please excuse my ignorance, I am just starting in DSP as a bit of a
>hobby.
>
>I Have Richard Lyons "understanding dsp" here and am working my way
>through the "Sampling Bandpass Signals" chapter. I understand the idea of
>Aliases and aliasing in the sampled signal spectrum but in the diagrams for
>the continuous signal spectrum there is always an inverse of the signal. So
>say there is an amplitude of 1 and 2MHz then there is also have an
>amplitude of 1 at -2MHz.
>
>This doesn't seem to be explained, probably as it is something basic that
>I should already know.
>
>Could someone direct me to some information that would explain this to
>me.
>
>Thanks for any help.
>
>Andy
Hello Andy,
Forgive me for taking so long to
reply to you.
You have pointed out an explanatory "gap" in my
Chapter 2 material. I'm so accustomed to showing
signal spectra having negative-frequency components
that I just went ahead and created figures such as
Figure 2-4 through Figure 2-7.
At the bottom of page 24 I very briefly mentioned
that the concept of negative-frequency was going to be
useful to us, and that the reader could go to Chapter
8 for more info on negative-frequency.
But I now see that this 'super-brief' Chapter 2
mention of negative-frequency may well leave the
reader in an "uncertain & uncomfortable" state
of puzzlement.
If I ever create a 3rd edition to my book, I'll
definitely add a bit more Chapter 2 explanation of
the reason for, and usefulness of, drawing signal
spectra having negative-frequency components.
So Andy, Chapter 3 shows how we obtain negative-frequency
results when we use the DFT to perform spectrum analysis,
and Chapter 8 gently, and thoroughly, describes how
negative-frequencies are related to real and complex signals.
By the way, please end me an E-mail and we'll
figure a way for me to send the book's errata
to you (if you're interested.)
[-Rick-]
Reply by Randy Yates●June 13, 20082008-06-13
Jerry Avins <jya@ieee.org> writes:
> Randy Yates wrote:
>> Jerry Avins <jya@ieee.org> writes:
>>
>>> Randy Yates wrote:
>>>> Jerry Avins <jya@ieee.org> writes:
>>>>> I think we're saying the same thing.
>>>> I don't think we are, Jerry.
>>> We agree, I think, that neither imaginary numbers nor negative time
>>> are needed to describe sinusoids of arbitrary phase. We agree that it
>>> is convenient to use them.
>>
>> I agree with that, but that isn't the main point I'm trying to make.
>>
>> My point is that to fully describe some phenomena requires an "extended
>> system," whatever the representation we choose to use for that extended
>> system is. To recast your statement above in these terms, a specific
>> representation may be convenient but not necessary---yes, I agree. But
>> we need SOME representation of the extended system (e.g., the complex
>> numbers) and CANNOT simply use the simple system (e.g., the reals) to
>> accomplish certain tasks (e.g., represent all N roots of an Nth-order
>> polynomial).
>
> I completely agree about roots. In electrical theory, though, we use
> complex numbers as ordered pairs, which is overkill (but we grow to
> think it fundamental). x+iy, R=jX; it is rare that extracting a root
> comes into play. Sin(r), cos(r) often serves where we use cos(r) +
> j*sin(r) for convenience. (I don't remember the formulas for the sums
> of sines and cosines. I use complex numbers to derive the relations
> when I need them. I'm all for convenience!)
Well once you agree there, it's the similar situation for negative
frequencies, isn't it? If you're talking about a two-dimensional
"wheel," then you need to specify whether it's turning CW or CCW. Now
you don't HAVE to use negative numbers (that's just one representation)
- you could use positive numbers plus the letters [CW | CCW] - but,
somehow, you must represent the information, and any two representations
are isomorphic.
--
% Randy Yates % "Remember the good old 1980's, when
%% Fuquay-Varina, NC % things were so uncomplicated?"
%%% 919-577-9882 % 'Ticket To The Moon'
%%%% <yates@ieee.org> % *Time*, Electric Light Orchestra
http://www.digitalsignallabs.com
Reply by Jerry Avins●June 12, 20082008-06-12
Randy Yates wrote:
> Jerry Avins <jya@ieee.org> writes:
>
>> Randy Yates wrote:
>>> Jerry Avins <jya@ieee.org> writes:
>>>> I think we're saying the same thing.
>>> I don't think we are, Jerry.
>> We agree, I think, that neither imaginary numbers nor negative time
>> are needed to describe sinusoids of arbitrary phase. We agree that it
>> is convenient to use them.
>
> I agree with that, but that isn't the main point I'm trying to make.
>
> My point is that to fully describe some phenomena requires an "extended
> system," whatever the representation we choose to use for that extended
> system is. To recast your statement above in these terms, a specific
> representation may be convenient but not necessary---yes, I agree. But
> we need SOME representation of the extended system (e.g., the complex
> numbers) and CANNOT simply use the simple system (e.g., the reals) to
> accomplish certain tasks (e.g., represent all N roots of an Nth-order
> polynomial).
I completely agree about roots. In electrical theory, though, we use
complex numbers as ordered pairs, which is overkill (but we grow to
think it fundamental). x+iy, R=jX; it is rare that extracting a root
comes into play. Sin(r), cos(r) often serves where we use cos(r) +
j*sin(r) for convenience. (I don't remember the formulas for the sums of
sines and cosines. I use complex numbers to derive the relations when I
need them. I'm all for convenience!)
Jerry
--
Engineering is the art of making what you want from things you can get.
�����������������������������������������������������������������������
Reply by Randy Yates●June 12, 20082008-06-12
Jerry Avins <jya@ieee.org> writes:
> Randy Yates wrote:
>> Jerry Avins <jya@ieee.org> writes:
>>> I think we're saying the same thing.
>>
>> I don't think we are, Jerry.
>
> We agree, I think, that neither imaginary numbers nor negative time
> are needed to describe sinusoids of arbitrary phase. We agree that it
> is convenient to use them.
I agree with that, but that isn't the main point I'm trying to make.
My point is that to fully describe some phenomena requires an "extended
system," whatever the representation we choose to use for that extended
system is. To recast your statement above in these terms, a specific
representation may be convenient but not necessary---yes, I agree. But
we need SOME representation of the extended system (e.g., the complex
numbers) and CANNOT simply use the simple system (e.g., the reals) to
accomplish certain tasks (e.g., represent all N roots of an Nth-order
polynomial).
--
% Randy Yates % "Midnight, on the water...
%% Fuquay-Varina, NC % I saw... the ocean's daughter."
%%% 919-577-9882 % 'Can't Get It Out Of My Head'
%%%% <yates@ieee.org> % *El Dorado*, Electric Light Orchestra
http://www.digitalsignallabs.com
Reply by Jerry Avins●June 12, 20082008-06-12
Jerry Avins wrote:
> glen herrmannsfeldt wrote:
>> Jerry Avins wrote:
>> (snip)
>>
>>> There's no need to assume negative frequencies for explaining what
>>> one sees on the spectrum analyzer, but as with exponential forms for
>>> sine and cosine, it is a convenient thing to do. Elevating a
>>> convenient construct to an immutable reality can occasionally lead
>>> one far astray.
>>
>> I would say somewhere in between. Not immutable, but more than
>> just a convenience. Using phasors and complex numbers to represent
>> phase shifted sin or cos is convenient. (Voltage and current are
>> still really real.)
>
> "More than just a convenience" and "is convenient". I think I missed the
> point you made.
>
>> and in another post Jerry wrote:
>>
>> > 2*sin(a)*cos(b) = sin(a+b) + sin(a-b). I see sum and difference
>> > frequencies, but no negative ones.
>>
>> Yes, but if a<b (and you add a t such that they are frequencies)
>> then you have negative frequency with no complex exponential
>> in sight.
>>
>> It isn't just a side effect of the exponential transform.
>
> Good point. But you still don't need to tune your receiver to a negative
> frequency to pick up sin(a-b)t signal. It's alias shows up (with
> appropriate phase).
Recall that sin(a-b)t = -sin(b-a)t.
Jerry
--
Engineering is the art of making what you want from things you can get.
�����������������������������������������������������������������������
Reply by glen herrmannsfeldt●June 12, 20082008-06-12
Jerry Avins wrote:
(snip)
> There's no need to assume negative frequencies for explaining what one
> sees on the spectrum analyzer, but as with exponential forms for sine
> and cosine, it is a convenient thing to do. Elevating a convenient
> construct to an immutable reality can occasionally lead one far astray.
I would say somewhere in between. Not immutable, but more than
just a convenience. Using phasors and complex numbers to represent
phase shifted sin or cos is convenient. (Voltage and current are
still really real.)
and in another post Jerry wrote:
> 2*sin(a)*cos(b) = sin(a+b) + sin(a-b). I see sum and difference
> frequencies, but no negative ones.
Yes, but if a<b (and you add a t such that they are frequencies)
then you have negative frequency with no complex exponential
in sight.
It isn't just a side effect of the exponential transform.
-- glen
Reply by Jerry Avins●June 12, 20082008-06-12
glen herrmannsfeldt wrote:
> Jerry Avins wrote:
> (snip)
>
>> There's no need to assume negative frequencies for explaining what one
>> sees on the spectrum analyzer, but as with exponential forms for sine
>> and cosine, it is a convenient thing to do. Elevating a convenient
>> construct to an immutable reality can occasionally lead one far astray.
>
> I would say somewhere in between. Not immutable, but more than
> just a convenience. Using phasors and complex numbers to represent
> phase shifted sin or cos is convenient. (Voltage and current are
> still really real.)
"More than just a convenience" and "is convenient". I think I missed the
point you made.
> and in another post Jerry wrote:
>
> > 2*sin(a)*cos(b) = sin(a+b) + sin(a-b). I see sum and difference
> > frequencies, but no negative ones.
>
> Yes, but if a<b (and you add a t such that they are frequencies)
> then you have negative frequency with no complex exponential
> in sight.
>
> It isn't just a side effect of the exponential transform.
Good point. But you still don't need to tune your receiver to a negative
frequency to pick up sin(a-b)t signal. It's alias shows up (with
appropriate phase).
Jerry
--
Engineering is the art of making what you want from things you can get.
�����������������������������������������������������������������������
Reply by Jerry Avins●June 12, 20082008-06-12
Randy Yates wrote:
> Jerry Avins <jya@ieee.org> writes:
>> [...]
>> If by extension you mean the inclusion of imaginary numbers, that
>> isn't required.
>
> No, that is not what I mean. Complex numbers are a representation.
>
> What I mean is that there is some system that is different than the
> plain old real number system along with the standard arithmetic
> operations, i.e., the field R(+,*) (to use a little abstract algebra),
> and that without it, you cannot accomplish certain things (like
> represent all N roots of any Nth-order polynomial with coefficients in
> R(+,*)).
Yes; for that, one needs complex numbers. Complex numbers close
arithmetic. By "close", I mean that every operation can be performed on
any number. Negative numbers close subtraction, fractions close
division. Imaginary numbers make possible the extraction of roots.
>> I think we're saying the same thing.
>
> I don't think we are, Jerry.
We agree, I think, that neither imaginary numbers nor negative time are
needed to describe sinusoids of arbitrary phase. We agree that it is
convenient to use them.
Jerry
--
Engineering is the art of making what you want from things you can get.
�����������������������������������������������������������������������
Reply by Randy Yates●June 12, 20082008-06-12
Jerry Avins <jya@ieee.org> writes:
> [...]
> If by extension you mean the inclusion of imaginary numbers, that
> isn't required.
No, that is not what I mean. Complex numbers are a representation.
What I mean is that there is some system that is different than the
plain old real number system along with the standard arithmetic
operations, i.e., the field R(+,*) (to use a little abstract algebra),
and that without it, you cannot accomplish certain things (like
represent all N roots of any Nth-order polynomial with coefficients in
R(+,*)).
> I think we're saying the same thing.
I don't think we are, Jerry.
--
% Randy Yates % "Remember the good old 1980's, when
%% Fuquay-Varina, NC % things were so uncomplicated?"
%%% 919-577-9882 % 'Ticket To The Moon'
%%%% <yates@ieee.org> % *Time*, Electric Light Orchestra
http://www.digitalsignallabs.com