Convolution properteis and the imaginary unit

[atrus_ovis]

finding the FT of x(t)=sin(πt) sin(50πt) :

( '*' is the convolution operator)

its FT X(Ω)=(1/2π) F(sin(πt)) * F(sin(50πt))

= (1/2π) jπ(δ(Ω+π)-δ(Ω-π)) * (jπ (δ(Ω+π)-δ(Ω-π)) ) (a)

from my professor's solution it next goes:

= (π/2) (-δ(Ω+π)+δ(Ω-π)) * ((-δ(Ω+π)+δ(Ω-π)) ) (b)

1)How did the signs change?
2)Where did the j's go? ( i suspect 1,2 are related =P )

also,
3) doesn't the 1/π cancel out the π coefficients in each quantity to be convoluted?
from wiki:
a (f * g) = (a f) * g = f * (a g)

why is that? i mean, why isn't it:
a (f * g) = (a f) *(a g)

?

 

[LCKurtz]

finding the FT of x(t)=sin(πt) sin(50πt) :

( '*' is the convolution operator)

its FT X(Ω)=(1/2π) F(sin(πt)) * F(sin(50πt))

= (1/2π) jπ(δ(Ω+π)-δ(Ω-π)) * (jπ (δ(Ω+π)-δ(Ω-π)) ) (a)

from my professor's solution it next goes:

= (π/2) (-δ(Ω+π)+δ(Ω-π)) * ((-δ(Ω+π)+δ(Ω-π)) ) (b)

1)How did the signs change?
2)Where did the j's go? ( i suspect 1,2 are related =P )

I think he is just using j²=-1 and inadvertently applied it to both factors instead of just the first.

also,
3) doesn't the 1/π cancel out the π coefficients in each quantity to be convoluted?
from wiki:
a (f * g) = (a f) * g = f * (a g)

why is that? i mean, why isn't it:
a (f * g) = (a f) *(a g)

?

Because constants come outside the integrals. a multiplies only one factor at a time.

 

[atrus_ovis]

I think he is just using j2=-1 and inadvertently applied it to both factors instead of just the first.

But where does the 2nd j come from?

 

[LCKurtz]

= (1/2π) jπ(δ(Ω+π)-δ(Ω-π)) * (jπ (δ(Ω+π)-δ(Ω-π)) ) (a)

But where does the 2nd j come from?

Right there. I see two of them.

 

[atrus_ovis]

Just a sec,i'm confused.

Because constants come outside the integrals. a multiplies only one factor at a time.

What integrals?

Also, the sign is inverted at both the operands of the convolution.
We had Aj*jB , we should have Aj*j*B = -A*B or A*-B , but it says -A*-B

 

[LCKurtz]

I think he is just using j²=-1 and inadvertently applied it to both factors instead of just the first.

Just a sec,i'm confused.


Also, the sign is inverted at both the operands of the convolution.
We had Aj*jB , we should have Aj*j*B = -A*B or A*-B , but it says -A*-B

I agree; that's what I said in the first place above -- I think he made a typo.

 

[atrus_ovis]

Oh, sorry.
Okay,thanks.