Question with sinc function

[kostbill]

sinc function in chinese

Hello.

I have a problem with one exercise, it starts like this:

We sample the signal x( t ) = ( sinc( 4t ) )^2 with the Nyquist frequency.

At the end of the exercise, he gives us that

F[ sinc( x ) ] = rect( f ), F[ x( at ) ] = ( 1/|a| ) X( f/a ), F[ ( x( t ) )^2 ] = F[ x(t) ] * F[ x(t) ].

In order to find the maximum frequency of our signal, I start like that

F[ ( sinc( 4t ) ) ] =
F[ sinc( 4t ) ] * F[ sinc( 4t ) ] =
(1/4) rect( f/4 ) * (1/4) rect( f/4 ) =
(1/16) triangle( f/4 ).

( the convolution of two rectagle functions rect( x ) * rect( x ) = triangle( x ) ).

So, given the definition of the triangle function, my max frequency is 4.

This is my question, the area under triangle( f/4 ) is not equal to 1. Should it be equal to one? I am not sure about that...

Can anyone help?

Thanks a lot.

Added after 11 minutes:

I made a mistake, I forgot to square the funtion, so, the order is

F[ ( sinc( 4t ) )^2 ] =
F[ sinc( 4t ) ] * F[ sinc( 4t ) ] =
(1/4) rect( f/4 ) * (1/4) rect( f/4 ) =
(1/16) triangle( f/4 ).

 

[pixel]

sinc function triangle

a*rect(f/4)*a*rect(f/4)=a^2*triang(f/2)
In the case of 2 rectangle convolution result takes 2 times higher frequency range than rectangle... You can see it when you shift one of the rectangles and evaluate cases for non zero convolution.

****
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in convolution with
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gives:
.....*
...***
.******
********