Discrete and Digital Signals and Systems - Digital Signal Processing: An Introduction with MATLAB and Applications - page 119

2.6.9 Applications of IIR Digital Filters

2.6.9.1 The Digital Integrator

If the sampling interval T s is small, then a digital summer provides a good
approximation to an integrator. That is:

Z

T

x ð t Þ dt X x ð n Þ T s ¼ T s X

N N

x ð n Þ where N ¼ T = T s Þ: ð 2 : 34 Þ

n ¼ 0 n ¼ 0

0

Now

y ð k Þ¼ X x ð n Þ¼ x ð k Þþ X

k k 1

x ð n Þ¼ x ð k Þþ y ð k 1 Þ: ð 2 : 35 Þ

n ¼ 0 n ¼ 0

Hence,

Y ð z Þ¼ X ð z Þþ z 1 Y ð z Þ

ð 2 : 36 Þ

1
1 z 1 :

) H ð z Þ¼

Therefore, the integrator transfer function is H i (z) = T s H(z). The transfer function
magnitude and phase is plotted in Fig. ( 2.36 ) and it is apparent from the magnitude
plot that the digital integrator is low-pass in nature. It is the inverse of the digital
differentiator, which is a high-pass filter.
MATLAB the pole-zero diagram can be plotted in MATLAB using
zplane(A,B) , where A is the vector of numerator coefficients and B is the
vector of denominator coefficients. In the above example A = [1 0] and B=[1 -
1] since H(z) = z/(z - 1).

x ( n ) y ( n )

T s

T s /2

z −1

Ω =

− π π

ω T s

0

π /2

P − Z
Diagram

Ω

X Re ( z )

0

− π /2

− π π

Fig. 2.36 The digital integrator with its frequency response and p-z diagram