Page 233 - Schaum's Outline of Theory and Problems of Signals and Systems

FOURIER ANALYSIS OF TIME SIGNALS AND SYSTEMS [CHAP. 5

Equation (5.64) is called Parseual's identity (or Parseual's theorem) for the Fourier
transform. Note that the quantity on the left-hand side of Eq. (5.64) is the normalized
energy content E of x(t) [Eq. (1.14)]. Parseval's identity says that this energy content E
can be computed by integrating I x(w)12 over all frequencies w. For this reason I x(w)l2 is
often referred to as the energy-density spectrum of x(t), and Eq. (5.64) is also known as
the energy theorem.
Table 5-1 contains a summary of the properties of the Fourier transform presented in
this section. Some common signals and their Fourier transforms are given in Table 5-2.

Table 5-1. Properties of the Fourier Transform
Property Signal Fourier transform

Linearity
Time shifting
Frequency shifting

Time scaling
Time reversal
Duality

Time differentiation

Frequency differentiation

Integration

Convolution

Multiplication
Real signal

Even component
Odd component
Parseval's relations