Fundamentals of System Analysis and Synthesis - Digital Signal Processing in Power System Protection and Control - page 51

Fig. 4.6 Frequency
responses of the systems from
Example 4.15

(a)

1
)

H ∗

( j ω

0.5

0

0 0.2 0.4 0.6 0.8 1

(b)

1
)

H ∗ ( j ω

0.5

0

0 0.2 0.4 0.6 0.8 1

(c)

10
) H ∗ ( j ω

5

0

0 0.2 0.4 0.6 0.8 1

f f S

cos

2

H ð jx Þ¼ 0 : 5 ½ 1 þ exp ð jxT S Þ ¼ exp j xT S
2

xT S

(a)

sin

2

H ð jx Þ¼ 0 : 5 ½ 1 exp ð jxT S Þ ¼ j exp j xT S
2

xT S

(b)

H ð jx Þ¼ P k ¼ 0 ð 0 : 9 Þ k exp ð jxT S Þ¼ 1

(c)

1 0 : 9 exp ð jxT S Þ
One can observe that the first system has a low-pass filtering characteristic (in
the frequency range up to half of the sampling rate), while the second one is a
high-pass filter. The absolute value of the latter system is described by:

1
½ cos ð xT S Þ 0 : 9 2 þ sin 2 ð xT S Þ

H ð jx Þ j j ¼ q :

The absolute values of the calculated frequency responses are shown in

Fig. 4.6 .

References

1. Brigham E (1988) The fast Fourier transform and its applications. Prentice Hall Inc.,
Englewood Cliffs
2. Dyke PPG (1999) An introduction to Laplace transforms and Fourier series. Springer, London
3. Graf U (2004) Applied Laplace transforms and Z-transforms for scientists and engineers:
a computational approach using a mathematica package. Birkhäuser, Basel
4. Hamming RW (1986) Numerical methods for scientists and engineers. McGraw-Hill,

New York
5. James JF (2004) A student's guide to Fourier transforms: with applications in physics and
engineering. Cambridge University Press, Cambridge