Discrete and Digital Signals and Systems - Digital Signal Processing: An Introduction with MATLAB and Applications - page 105

þ c ;

/ ¼ X

2 þ p

2

where c = p is to compensate for the change of sign in sin ð 2 X Þ:
The magnitude and phase responses are plotted in Fig. 2.28 . It is seen that the
digital differentiator is a high-pass filter.

Frequency Domain Approach

From Fourier transform Tables one can find the transfer function of the continu-
ous-time differentiator as:

H a ð x Þ¼ jx ð 2 : 25 Þ

which can be transformed into the digital domain as follows:

H ð e jX Þ¼ jX = T s ; p X\p : ð 2 : 26 Þ

Figure ( 2.29 ) shows the magnitude and phase responses of the above transfer
functions (compare with Fig. ( 2.28 )). Note that Fig. ( 2.29 a) represents a theoretical
magnitude response, as practical differentiators are band-limited within a cutoff
frequency x c .
The impulse response of the above filter can be found to be:

Z p

h ð n Þ¼ 1
2p

jX
T s

e jnX dX

p

ð 2 : 27 Þ

cos ð np Þ
n

sin ð np Þ
pn 2

¼ 1
T s

; 1 \n\ 1

For a practical design, the above impulse response should be shifted by 2M ? 1
samples and then truncated. The shifting operation is described by:

h ð n Þ¼ 1
T s

cos ½ð n M Þ p
ð n M = 2 Þ

sin ½ð n M Þ p
p ð n M Þ 2

; 1 \n\ 1; ð 2 : 28 Þ

while the truncation window w(n) is specified by:

h w ð n Þ¼ w ð n Þ h ð n Þ ; 0\n\2M ð 2 : 29 Þ

If a linear phase is required, then the truncation window should be symmetric so
that one can get h(n) =-h(2M - n). The filter in ( 2.29 ) can be implemented
using the approach in Sect. 2.6.4.6 .