Analog Signals and Systems - Digital Signal Processing: An Introduction with MATLAB and Applications - page 33

2. For non-periodic signals: E ¼ R 1

1 j X ð f Þj 2 df , where X(f) is the FT of the signal.
A plot of |X k | 2 versus f (defined at discrete frequencies f = kf o , f o being the
fundamental frequency) is known as the power spectrum, or the power spec-
tral density (PSD) of the signal. The function |X(f)| 2 versus f is called the
energy spectrum or energy spectral density (ESD) of the signal.

1.2.5.3 The Wiener-Kinchin Theorem

The Wiener-Kinchin Theorem states that for a periodic signal x(t), its PSD is the
FT of its autocorrelation function:

Z

T o

F s $ f X

1

R x ð s Þ¼ 1
T o

j X k j 2 d ð f k = T o Þ
|{z}
PSD

x ð k Þ x ð s þ k Þ dk ! :

k ¼1

0

For an energy signal x(t), the ESD is the FT of its autocorrelation:
R x ð s Þ¼ Z

1

F s $ f j X ð f Þj 2
| {z }
ESD

x ð k Þ x ð s þ k Þ dk ! :

1

A similar relation for random signals will be presented later.

1.2.5.4 Examples

Example 1: A periodic current signal x(t) flowing through a 1X resistor has the

form:

x ð t Þ¼ 10 cos ð 2t Þþ 4sin ð 6t Þ A ;

1. Determine whether x(t) is a power or energy signal.
2. Find the complex Fourier series of x(t).
3. Find the fundamental period T o and the fundamental radian frequency, x o .
4. Find the Fourier transform of the signal.
5. Plot the magnitude spectrum of the signal.
6. Plot the power spectrum of the signal.
7. Find the signal power from the time domain and from the frequency domain.

Solution:

1. Since x(t) is periodic, it is a power signal.
2. Since x(t) is already in the form of a trigonometric FS, one can just use Euler's
formula (Tables, Formula 1) to obtain the complex FS: