AM Modulation

[Rich Grise]

Question for you analytical types: what would the spectrum look like
for a sine wave that is ever, linearly increasing in amplitude?

v = t * sin(w*t)

John

I don't know about that, but if you modulate a TWT with a sawtooth,
it's called "serrodyne modulation", and it makes a little baby spectrum
(which in jamming transmitters is called the "package") split off
and march up the band, confusing Doppler radar.

Cheers!
Rich

 

[Phil Hobbs]

Question for you analytical types: what would the spectrum look like
for a sine wave that is ever, linearly increasing in amplitude?

v = t * sin(w*t)

Of course the integral doesn't converge, but by analytic continuation
you can do it--you just chuck in a factor of exp(-epsilon*t) and let
epsilon go to 0 after taking the integral--the net result is to chop off
all the contributions from the t=infinity limit. I assume you want it
to turn on at t=0, so the amplitude is t*H(t), where H(t) is the unit
step function.

g'(t) = H(t) has transform G'(f) = i/(2*pi*f), so
g(t) = t*H(t) has transform G(f) = -1/(2*pi*f)**2,

and sin(2*pi*F*t) has transform 0.5*(delta(f+F)-delta(f+F)),

so theoretically (by the convolution theorem)

v(t) = t sin(wt) has transform 1 ( 1 1 )
-------- ( ------- - -------- )
8*pi**2 ( (f-F)**2 (f+F)**2 )

(Modulo a possible sign.)

Cheers,

Phil Hobbs

 

[Jim Thompson]

Of course the integral doesn't converge, but by analytic continuation
you can do it--you just chuck in a factor of exp(-epsilon*t) and let
epsilon go to 0 after taking the integral--the net result is to chop off
all the contributions from the t=infinity limit. I assume you want it
to turn on at t=0, so the amplitude is t*H(t), where H(t) is the unit
step function.

g'(t) = H(t) has transform G'(f) = i/(2*pi*f), so
g(t) = t*H(t) has transform G(f) = -1/(2*pi*f)**2,

and sin(2*pi*F*t) has transform 0.5*(delta(f+F)-delta(f+F)),

so theoretically (by the convolution theorem)

v(t) = t sin(wt) has transform 1 ( 1 1 )
-------- ( ------- - -------- )
8*pi**2 ( (f-F)**2 (f+F)**2 )

(Modulo a possible sign.)

Cheers,

Phil Hobbs

I did it graphically (the MIT method), and I agree with your result...
"circus tent" skirts ;-)

...Jim Thompson

 

[Meindert Sprang]

g'(t) = H(t) has transform G'(f) = i/(2*pi*f), so
g(t) = t*H(t) has transform G(f) = -1/(2*pi*f)**2,

and sin(2*pi*F*t) has transform 0.5*(delta(f+F)-delta(f+F)),

so theoretically (by the convolution theorem)

v(t) = t sin(wt) has transform 1 ( 1 1 )
-------- ( ------- - -------- )
8*pi**2 ( (f-F)**2 (f+F)**2 )

(Modulo a possible sign.)

Has you mother never taught you not to swear or to use foul language?

Meindert

 

[Phil Allison]

"Jim Thompson"


** Troll.





.......... Phil

 

[Phil Hobbs]

Has you mother never taught you not to swear or to use foul language?

Yeah, well, what can I tell you...I took Fourier from Bracewell himself,
and he made it such fun that I can't resist playing with it sometimes.
My bad.

Cheers,

Phil Hobbs

 

[John Larkin]

Yeah, well, what can I tell you...I took Fourier from Bracewell himself,
and he made it such fun that I can't resist playing with it sometimes.
My bad.

Cheers,

Phil Hobbs

Neat! Bracewell is a very cool dude; I wish I'd met him.

http://en.wikipedia.org/wiki/Ronald_N._Bracewell


The HP/Agilent sampling scopes used (maybe still use) the Bracewell
Transform for TDR deconvolution, and he has interesting ideas on
extraterrestial life and stuff.

John

ps - ever get involved in deconvolution? I accidentally invented a
very simple and fast deconvolution algorithm for TDR equalization, and
I wish somebody would explain to me how it works.

 

[Meindert Sprang]

Yeah, well, what can I tell you...I took Fourier from Bracewell himself,
and he made it such fun that I can't resist playing with it sometimes.
My bad.

Well, my problem is, I got stuck in mathematics at that level. I've been
programming FFTs on DPSs and I can perfectly see what the transform does,
algorithm-wise, but these mathematical representations are just plain
chinese to me ;-)

My bad too...... (heading for the pubs instead of going to math classes, but
that was 20 odd years ago. I'm wiser now, I now drink at home ;-))

Meindert

 

[Jonathan Kirwan]

Well, my problem is, I got stuck in mathematics at that level. I've been
programming FFTs on DPSs and I can perfectly see what the transform does,
algorithm-wise, but these mathematical representations are just plain
chinese to me ;-)

My bad too...... (heading for the pubs instead of going to math classes, but
that was 20 odd years ago. I'm wiser now, I now drink at home ;-))

Have you tried reading E. Oran Brigham's edition(s)?

I instantly saw sin() component effect the moment I'd read John's
question. The (1/2j) * (e^jwt - e^-jwt) popped immediately to mind as
equivalent and the last part of that multiplied by the e^2j pi ft
immediately yields the f-f0 f+f0 splitting.) But I didn't bother with
the 't' spectrum to see what shape would be split, at the time. I
followed Phil's response quickly, though.

Anyway, that's where I learned to vaguely read such things, despite
the fact that it was just a passing curiosity and I don't normally
need to apply FTs. His writing worked for me better than other places
I'd looked. Might help you, if you haven't already tried him.

Jon

 

[Phil Hobbs]

The HP/Agilent sampling scopes used (maybe still use) the Bracewell
Transform for TDR deconvolution, and he has interesting ideas on
extraterrestial life and stuff.

John

ps - ever get involved in deconvolution? I accidentally invented a
very simple and fast deconvolution algorithm for TDR equalization, and
I wish somebody would explain to me how it works.

I did some of that for my thesis research, but the most interesting was
for my $10 thermal IR cameras--I needed to take the response of each
pixel and deconvolve it with a filter about 25 or 30 samples long, on a
micro with a capacious 902 bytes of RAM--there was no way I could use
that many frame buffers.

It turned out that the deconvolver could be factored pretty accurately
into a 3-sample FIR filter followed by a 1-sample IIR integrator, which
made it much more feasible. It still wound up in the PC code and not in
the micro, but oh, well.

Cheers,

Phil Hobbs