Discrete and Digital Signals and Systems - Digital Signal Processing: An Introduction with MATLAB and Applications - page 88

2.5.1.6 The Wiener-Kinchin Theorem

For periodic signals: R x ð k Þ !j X k j 2
|{z}
PSD

:

:

For non-periodic signals: R x ð k Þ ! T s j X s ð f Þj 2
|{z}
ESD

;

2.6 Digital Filters and Their Applications

The advancement of digital computers during the 1960s paved the way for many
analog electronic circuits to be emulated in digital computers. The advantages of
this kind of digital emulation are many—greater accuracy, greater flexibility,
lower cost, lower power requirements, greater reproducibility, etc. This section of
the topic focuses on a particular type of digital emulation-namely the emulation of
conventional filters with digital computer hardware. This type of emulation is
commonly referred to as digital filtering.

2.6.1 Ideal Digital Filters

Ideal digital LP, HP, BP, and BS filters can be defined by simple analogy with
analog filters. In exploiting the similarities between digital and analog filters,
however, it is also important to keep in mind that there are some key differences.
One of the key differences is that there is a periodicity in the frequency response of
digital filters.

2.6.1.1 Mathematical Formulation

The Digital LPF

The transfer function of a digital LPF over the principal frequency domain
(- f s /2, f s /2) is given by

H L ð f Þ¼ H L ð e j2pfT s Þ¼ 1 ;

0 j f j f c

0 ; f c j f j f s = 2 :

[see Fig. ( 2.15 )].