(1)
The frequency may also be given on the interval [0,2], because X() is 2 periodic:
X( + 2k) | = x(n) e^{-j(+2k)n} |
= x(n) e^{-jn} e^{-j2kn} | |
= x(n) e^{-jn} = X(). | |
As a periodic function X() has a Fourier series expansion which is (1). The x(n) are the Fourier coefficients,
(2)
x(n) = | 1 | X() e^{ jn} d. |
2 | ||
The energy of the signal x(n) is
E_{x } | = |x(n)|^{2} |
= x(n) x^{*}(n) | |
= x(n) (1/2) X^{*}() e^{-jn} d | |
= (1/2) X^{*}() x(n) e^{-jn} d | |
= (1/2) |X()|^{2} d. | |
This is the Parseval identity for a non-periodic discrete signal. The Fourier transform X() is, in general, complex valued whence it can be written in the form
where
The mapping -> () (and its graph) is called the phase spectrum of the signal x(n) , |X()| is the amplitude spectrum and
the energy spectrum.
If x(n) is a real valued signal, then
or equivalently
and
Also
From the above it follows that it is enough to know X() for 0 .
Example 2.4.1: Energy spectrum
Example 2.4.2: Reconstruction of a signal
Example 2.4.3: Fourier transform, amplitude spectrum and phase spectrum
The Z-transform (bilateral) of a signal x(n) is defined by the equation
Thus
i.e. the Fourier transform is the Z-transform evaluated on the unit circle.
Example 2.4.4: Fourier transform and Z-transform
Example 2.4.5: Z- and Fourier transforms of some signals
Remark A real even signal x(n) has a real and even Fourier transform. A real odd signal x(n) has a purely imaginary Fourier transform.
Example 2.4.6: Real even signal, Fourier transform, amplitude- and phase spectrum
The Fourier sum transform has the corresponding properties as the Fourier integral transform, see A6.1 and 4.2. For example
x(n-k) <-> X() e^{ jk} |
x(-n) <-> X(-) |
x(n) = x_{1}(n) * x_{2}(n) <-> X() = X_{1}() X_{2}() |
r_{xx}(l) <-> S_{xx}() |
r_{xy}(l) <-> S_{xy}() = X() Y(-) |
( r_{xy}(l) = x(l) * y(-l) ) |
nx(n) <-> jX'() |