What is the relation between block-pulse function and Heaviside Step function?

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Block-pulse function:

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Heaviside-Step function

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Are they the same functions? Is the block pulse function just the discrete form of the Heaviside step func. ?

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No. Your block-pulse functions are supported (i.e. nonzero) on a bounded set, while the Heaviside function is nonzero for all large inputs.