文档介绍：Fundamentals of Measurement Technology(3)
Prof. Wang Boxiong
Unit impulse function
Assuming a rectangular pulse pΔ(t) of a width Δ and an amplitude 1/Δ, its area is equal to 1. As Δ→0, the limit of pΔ(t) is called the unit impulse function or delta function denoted by δ(t).
Fourier transforms of power signals
Fig. Rectangular pulse function and delta function δ(t)
δ(t) is a pulse with unbounded amplitude and zero time duration. This impulse function must be treated as a so-called generalized function.
Properties:

The two properties for the impulse function can be conveniently summarized into one defining equation for δ(t).
provided x(t) is continuous at t=t0.
Fourier transforms of power signals
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The Fourier transform of the impulse function δ(t):
Fourier transform pair:
Fourier transforms of power signals
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Fig. δ(t) and its Fourier transform
Fourier transforms of power signals
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Fig. δ(t-t0) and its Fourier transform
Using the symmetry property, we can derive the transform pairs:
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Fourier transforms of power signals
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Fig. The unity and its Fourier transform
Furthermore, we have the following relation:
Fourier transforms of power signals
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Fig. Convolution of an arbitrary function with a unit impulse
Sinusoidal functions

Using the transform pair
we see
Similarly,
Fourier transforms of power signals
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Fourier transforms of power signals
Fig. Sinusoidal functions and their spectra
The Signum Function
The signum function, denoted by sgn(t), is defined as
If
then
Suppose we differentiate the signum function. Its derivative is 2δ(t):
Fourier transforms of power signals
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