Page 216 - Mechanical Engineers' Handbook (Volume 2)

3 Basic Digital Filter 205

Then map the zeros from the prototype using Eqs. (40) and (41):

→ 1 (54)

1
1

→ 1 (55)
2
1

→ 1
3
1 (56)
Step 4. Expand the numerator and denominator polynomials:
(z
) (z
)
1
6
(z pz ) (z pz )
6
1
1 3z 2 3z 4 z 6
(57)
(1 1.091z 1 3.632z 2 2.616z 3 4.642z 4 1.982z 5 2.407z 6
Step 5. Normalize the transfer function so that it will have unity gain in the passband. For
this, we estimate
j
M max
H(e )] (58)
0
Then compute

1
H normalized (z) H(z) (59)
M
For this example, M 14.45.
Step 6. Realize the difference equation from inverse z-transformation of the derived transfer
function:

y 1.091y 3.632y 2.616y 4.642y 1.982y
n n 1 n 2 n 3 n 4 n 5
2.407y 0.0692 * x 0.2076x 0.2076x 0.0692x (60)
n 6 n n 2 n 4 n 6

3.5 Frequency-Domain Filtering
It is possible to filter the data in the frequency domain. The method involves the use of the
Fourier transform. We Fourier transform the data, multiply by the desired frequency response,
then inverse Fourier transform the data. This is similar to the FIR filters discussed earlier.
Deriving the FIR coefficients by performing a discrete cosine transform (DCT) of the desired
frequency response and then convolving the coefficients with the data is equivalent to filtering
the data in the frequency domain. One difference, however, is that the frequency domain
filtering is generally done on blocks of data and not on streaming data, as is done in the
time domain, which can be of concern when processing highly nonstationary data with abrupt
transients. The inverse Fourier transform is

1
F {F( )} ƒ( )e j t d (61)
1
2