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Math Help - Sampling and reconstruction

  1. #1
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    Sampling and reconstruction

    From a previous exam that I've been trying to solve for a while there is one section with a series of yes/no questions that I just can't figure out for certain what they should be.

    1) The spectrum of a box filter (in time domain) is also a box function (in frequency domain).

    I've been told this one is false, but why is it false? Also I'm confused by the terminology, what is the difference (if any) between a box filter and a box function?

    I know what a box function is -- any function that is zero over the entire real line except for a small interval where it is equal to the constant A, i.e. the "Boxcar function".

    If I assume as premise for the following reasoning that a box filter is just a type of box function -- used as a type of filter -- then:

    A box filter given a signal in time domain will parse this signal giving only values of 0 and A -- depending on the signal given and the nature of the box function.

    Now, the frequency domain is a frequency graph where the quantified distribution of a specific signal's frequency spectrum is shown.

    I.e. the frequency spectrum resulting from the former function is simply 0 and A -- why is this not a box function in frequency domain?
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by posix_memalign View Post
    From a previous exam that I've been trying to solve for a while there is one section with a series of yes/no questions that I just can't figure out for certain what they should be.

    1) The spectrum of a box filter (in time domain) is also a box function (in frequency domain).

    I've been told this one is false, but why is it false? Also I'm confused by the terminology, what is the difference (if any) between a box filter and a box function?

    I know what a box function is -- any function that is zero over the entire real line except for a small interval where it is equal to the constant A, i.e. the "Boxcar function".

    If I assume as premise for the following reasoning that a box filter is just a type of box function -- used as a type of filter -- then:

    A box filter given a signal in time domain will parse this signal giving only values of 0 and A -- depending on the signal given and the nature of the box function.

    Now, the frequency domain is a frequency graph where the quantified distribution of a specific signal's frequency spectrum is shown.

    I.e. the frequency spectrum resulting from the former function is simply 0 and A -- why is this not a box function in frequency domain?
    I think you may have to be more explicit about what you mean by a box filter and its spectrum.

    If a box filter corresponding to convolution of the input with a top hat function and by the spectrum you mean the FT of the top hat function, then the spectrum is a sinc.

    CB
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  3. #3
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    Quote Originally Posted by CaptainBlack View Post
    I think you may have to be more explicit about what you mean by a box filter and its spectrum.

    If a box filter corresponding to convolution of the input with a top hat function and by the spectrum you mean the FT of the top hat function, then the spectrum is a sinc.

    CB
    Thank you for your reply however that is the whole question from an exam; and I'm still struggling with it.

    The only context is that it is related to "sampling and reconstruction" and "antialiasing" (just some lecture topics), there is no other context to give; the exam is also from a course in computer graphics and is not very mathematical oriented (although it is to some degree, it is just not purely mathematical) -- I think this question is supposed to be a bit trivial, although it isn't trivial to me, what am I missing?

    The second question is:

    "The spectrum of the sinc function (in time domain) is the box function (in frequency domain).", i.e. is this true or false? I'm told it is true, but I can't see why.
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  4. #4
    Grand Panjandrum
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    Quote Originally Posted by posix_memalign View Post
    Thank you for your reply however that is the whole question from an exam; and I'm still struggling with it.

    The only context is that it is related to "sampling and reconstruction" and "antialiasing" (just some lecture topics), there is no other context to give; the exam is also from a course in computer graphics and is not very mathematical oriented (although it is to some degree, it is just not purely mathematical) -- I think this question is supposed to be a bit trivial, although it isn't trivial to me, what am I missing?

    The second question is:

    "The spectrum of the sinc function (in time domain) is the box function (in frequency domain).", i.e. is this true or false? I'm told it is true, but I can't see why.
    Use the inversion formula on the frequency domain representation.

    CB
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