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I recall a prof or two arm-waving over that one. However, I think if
you formulate your definition of linearity properly, the transfer
function y=mx+b will still satisfy linearity. Specifically, if the
_response_ is the _change_ that occurs in the output going from x=0 to
x=x1, then the response for x1 is (m*x1+b)-(m*0+b) = m*x1, and of
course for x2, it's m*x2. The response for x=x1+x2 is m*(x1+x2), which
is exactly the sum of the responses for x1 and x2.

Similarly, for a mixer/LO system with RF input and IF output, if the
mixer is unbalanced and lets LO get through, it is still a linear
system if the change in output when go from zero input to input x1(t)
plus the change in output when you go from zero input to input x2(t) is
equal to the change in output when you go from zero input to input
(x1(t)+x2(t)).

But note that a mixer/LO system is NOT time invariant, because the
output for x1(t+delta) is in general NOT the same as the output shifted
in time by delta for input x1(t).

You can most certainly find text books that define linearity
differently than I did above. I find the definition above to be a more
useful one, however, and it seems to be the one generally accepted in
practice, even if it's not stated accurately in words.

Cheers,
Tom



Roy Lewallen wrote:
Richard Harrison wrote:
Richard Clark, KB7QHC wrote:
"Who. in your estimation, does qualify to discuss it?"

If it`s about antennas, I nominate Kraus. If it`s about mathematics,
many marhematicians qualify.

In algebra, y = mx + b, (the point slope formula), is called linear
because it is the graph of a straight line.
. . .

But of course you realize that the function y = mx + b doesn't meet the
requirements of a linear function when applied to network theory.

Roy Lewallen, W7EL

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It's not clear to me whether you're proposing an alternative definition
for linearity or for superposition. I've never seen superposition
defined as other than that the sum of responses to individual
excitations be equal to the response to the sum of the excitations --
that's the definition in Pearson & Maler's _Introductory Circuit
Analysis_, Van Valkenburg's _Network Analysis_, and the rather old
edition of the _IEEE Standard Dictonary of Electrical and Electronic
Terms_ I have. Do you have a reference that gives the definition you
propose for superposition?

If on the other hand the alternative definition is only for linearity,
we'd then be faced with the possibility of having a linear (and
time-invariant) circuit which doesn't satisfy superposition. That's not
a pleasant circumstance to ponder.

Roy Lewallen, W7EL

K7ITM wrote:
I recall a prof or two arm-waving over that one. However, I think if
you formulate your definition of linearity properly, the transfer
function y=mx+b will still satisfy linearity. Specifically, if the
_response_ is the _change_ that occurs in the output going from x=0 to
x=x1, then the response for x1 is (m*x1+b)-(m*0+b) = m*x1, and of
course for x2, it's m*x2. The response for x=x1+x2 is m*(x1+x2), which
is exactly the sum of the responses for x1 and x2.

Similarly, for a mixer/LO system with RF input and IF output, if the
mixer is unbalanced and lets LO get through, it is still a linear
system if the change in output when go from zero input to input x1(t)
plus the change in output when you go from zero input to input x2(t) is
equal to the change in output when you go from zero input to input
(x1(t)+x2(t)).

But note that a mixer/LO system is NOT time invariant, because the
output for x1(t+delta) is in general NOT the same as the output shifted
in time by delta for input x1(t).

You can most certainly find text books that define linearity
differently than I did above. I find the definition above to be a more
useful one, however, and it seems to be the one generally accepted in
practice, even if it's not stated accurately in words.

Cheers,
Tom

FIGHT! FIGHT! FIGHT!
 

Steve N. wrote:

So you *were* questioning the linearity of antennas as we understand the
term?

No. I was claiming antennas were very linear in the electrical
superposition sense.

73,
Glenn

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Roy Lewallen wrote:
It's not clear to me whether you're proposing an alternative definition
for linearity or for superposition.

Lest anyone think that the addition of an offset term (the b in y=mx+b)
violating superposition is merely of academic interest, analyze a
direct conversion receiver. At my shack I have a Softrock 40, which
tunes about thirty KHz of the 40 meter band. At the center of the
spectrum (which corresponds to the mixer converting to very low
frequencies and DC) is a 1/f bump that the designers have attributed to
mixer 1/f noise. But it is not noise at all. When tuned in, it is
full of strong signals and strange sounds. These are distortion
products, and are caused mainly by the b term in the detector used on
the receiver.

They wipe out about 3% of the receivable band.

Sometimes the b term is benign, but not in this case.

73,
Glenn Dixon AC7ZN

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I would not propose to change the definition for either linearity or
superposition, at least as I saw them a moment ago in one text, the
"Linear Circuit Analysis" chapter of "The Electrical Engineering
Handbook," Richard C. Dorf, editor. Rather I propose we think more
carefully about just what "response" to a stimulus means. If you say
that in the system y=mx+b that b is the response to zero input (x=0)
then you will conclude that the system is nonlinear. On the other
hand, if the "response" to a "stimulus" only has meaning as the
_change_ (or difference) in output for a given _change_ (or difference)
in input, then it is a linear system.
Roy Lewallen wrote:
It's not clear to me whether you're proposing an alternative definition
for linearity or for superposition. I've never seen superposition
defined as other than that the sum of responses to individual
excitations be equal to the response to the sum of the excitations --
that's the definition in Pearson & Maler's _Introductory Circuit
Analysis_, Van Valkenburg's _Network Analysis_, and the rather old
edition of the _IEEE Standard Dictonary of Electrical and Electronic
Terms_ I have. Do you have a reference that gives the definition you
propose for superposition?

If on the other hand the alternative definition is only for linearity,
we'd then be faced with the possibility of having a linear (and
time-invariant) circuit which doesn't satisfy superposition. That's not
a pleasant circumstance to ponder.

Roy Lewallen, W7EL

K7ITM wrote:
I recall a prof or two arm-waving over that one. However, I think if
you formulate your definition of linearity properly, the transfer
function y=mx+b will still satisfy linearity. Specifically, if the
_response_ is the _change_ that occurs in the output going from x=0 to
x=x1, then the response for x1 is (m*x1+b)-(m*0+b) = m*x1, and of
course for x2, it's m*x2. The response for x=x1+x2 is m*(x1+x2), which
is exactly the sum of the responses for x1 and x2.

Similarly, for a mixer/LO system with RF input and IF output, if the
mixer is unbalanced and lets LO get through, it is still a linear
system if the change in output when go from zero input to input x1(t)
plus the change in output when you go from zero input to input x2(t) is
equal to the change in output when you go from zero input to input
(x1(t)+x2(t)).

But note that a mixer/LO system is NOT time invariant, because the
output for x1(t+delta) is in general NOT the same as the output shifted
in time by delta for input x1(t).

You can most certainly find text books that define linearity
differently than I did above. I find the definition above to be a more
useful one, however, and it seems to be the one generally accepted in
practice, even if it's not stated accurately in words.

Cheers,
Tom

FIGHT! FIGHT! FIGHT!
 

K7ITM wrote:
"If you say that in the system (y = mx + b) that b is the response to
zero input (x=0) then you will conclude the system is nonlinear."

Why? the factor (b) is a constant, a value to be added to (mx) to total
a value for (y).

(mx) is a straight line. Every value of (b) produces a straight lline
parallel with lhe line y=mx when b=0. Factor (b) is merely the offset
value of the sloped line in the x direction.

y=mx+b is listed in math books as a defining example of a linear
equation. (When plotted, a linear equation produces a straight line.)
y=mx+b has a special name: "The point slope formula". Perfectly
descriptive, too.

To clarify everything, graph a few values for yourself.

Best regards, Richard Harrison, KB5WZI

FIGHT! FIGHT! FIGHT!
 

Richard Harrison wrote:
(mx) is a straight line. Every value of (b) produces a straight lline
parallel with lhe line y=mx when b=0. Factor (b) is merely the offset
value of the sloped line in the x direction.

Why must superposition preserve 'b' (as someone has
asserted?)
--
73, Cecil http://www.qsl.net/w5dxp

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Richard Harrison wrote:
K7ITM wrote:
"If you say that in the system (y = mx + b) that b is the response to
zero input (x=0) then you will conclude the system is nonlinear."

Why? the factor (b) is a constant, a value to be added to (mx) to total
a value for (y).


Because, from 3.3, 'Network Theorems,' in "The Electrical Engineering
Handbook,"
Ed. Richard C. Dorf, Boca Raton: CRC Press LLC, 2000 (you can find it
online),

"The superposition condition: If the input to the system, e1, causes a
response, r1, and if an input to the system, e2, causes a response, r2,
then a response, r1 + r2, will occur when the input is e1 + e2."

If you take the response to x0=0 to be b, then the response to, say,
x1=1 must be m+b, and to x2=2 must be 2*m+b. Then for superposition,
and therefore for linearity, the response to (x1+x2)=3 must be
m+b+2*m+b = 3*m+2*b, which it is not: it is 3*m+b. I do not suggest a
change in the definition of superposition or of homogeneity (which
seems to be simply a subset of superpostion anyway) or of linearity. I
only suggest that "response" be interpreted as the change in output
which occurs when you go from input a to input b. I've generally not
seen "response" to be very well defined in those texts which define
linearity, though it's quite possible I've missed it. In the PDF file
for all of Chapter 3 of the book quoted above, it is certainly not.

I guess I'd prefer, basically, to say that the following relationship
defines a linear system (MIMO even) -- but it is NOT sufficient to
describe all linear systems:

dx/dt = Ax + Bu
y = Cx + Du
where u is a vector of independent input variables, y is a vector of
dependent output variables, x represents state variables for the
system, and A, B, C and C are matrices of coefficients which are
constant in a time-invariant system, but which may be variable with
time in a system which is time variant.

Cheers,
Tom



(mx) is a straight line. Every value of (b) produces a straight lline
parallel with lhe line y=mx when b=0. Factor (b) is merely the offset
value of the sloped line in the x direction.

y=mx+b is listed in math books as a defining example of a linear
equation. (When plotted, a linear equation produces a straight line.)
y=mx+b has a special name: "The point slope formula". Perfectly
descriptive, too.

To clarify everything, graph a few values for yourself.

Best regards, Richard Harrison, KB5WZI