2.4. Non-Periodic Discrete Signal (Fourier Sum Transform)

(1)

X() = x(n) e^-jn,  - .

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

Ex	= |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

X() = |X()| e ^j().

where

() = arg X().

The mapping   -> ()   (and its graph) is called the phase spectrum of the signal x(n) ,   |X()|   is the amplitude spectrum and

S_xx() = |X()|²

the energy spectrum.

If   x(n)   is a real valued signal, then

X^*() = X(-)

or equivalently

|X(-)| = |X()|                   even

and

arg X(-) = -arg X()        odd

Also

S_xx(-) = S_xx()                 even

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

X(z) = x(n) z^-n,  r₁ < | z | < r₂ .

Thus

X() = x(n) e^-jn = X(z) _{z = e ^j}

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) = x1(n) * x2(n) <-> X() = X1() X2()

rxx(l) <-> Sxx()

rxy(l) <-> Sxy() = X() Y(-)

( rxy(l) = x(l) * y(-l) )

nx(n) <-> jX'()