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FUNDER - Theoretical Background |
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DIRAC'S DELTA DEFINITION A Dirac's delta,
In all cases,
For instance, the sequence of rectangle functions is
Conventionally, a Dirac's delta can be plotted as an arrow of height one pointing upwards. In this way, the product of a delta by a constant can be easily represented as an arrow whose height is the value of the constant. This bypasses the fact that delta sequences diverge at the origin.
FUNDAMENTAL PROPERTY Dirac's delta can alternatively be defined as the mathematical object with the fundamental property
This property can be written in non-integral form as
The area under a delta follows immediately:
In particular, the area under any function in the above delta sequences, eqs. (1)-(4) is one. SIFTING PROPERTY - ANY DETERMINISTIC SIGNAL IS A SUM OF DELTAS Upon a shift in time of a Dirac's delta, the fundamental property leads to the important sifting property:
This equation states that any function of time,
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,
is a limit of any delta sequence. Examples are:
,
can be reconstructed as a sum (integral) of scaled deltas. In other
words, any deterministic signal can be represented as a superposition
of deltas. Hence, impulses become fundamental signals.
