Infinite Impulse Response Filters - Digital Signal Processing in Power System Protection and Control - page 61

h ð n Þ¼ h a ðÞ t ¼ nT s j ; ð 5 : 21 Þ

where h a (t) is an impulse response of analog filter with transfer function H a (s).
Frequency response of digital filter having impulse response as above is Fourier
transform of equation:

h ð t Þ¼ X

1

h a ð nT S Þ d ð t nT S Þ:

ð 5 : 22 Þ

n ¼ 0

and is equal to:

:

X

1

H ð jx Þ¼ 1
T S

H a j x k 2p
T S

ð 5 : 23 Þ

k ¼1

Though the impulse response of analog filter is in a way ''stored'' in digital
system, it may happen that frequency responses of analog and digital filters differ
substantially. The reason of that is aliasing known from sampling theorem. It is
very important then to match parameters of analog origin and the sampling fre-
quency so that the replicas of analog filter frequency responses do not (or just a
little) overlap.
Described method is rarely applied in above-presented way. The most fre-
quently the difference equation of sought digital filter is obtained from analog
origin transfer function presented in the form of partial expansion (transfer
function poles must be known):

Lh a ðf ¼ H a ðÞ¼ X

N

A k
s s k :

ð 5 : 24 Þ

k ¼ 1

Then:

h a ðÞ¼ X

N

A k exp s k ðÞ 5 : 25 Þ

k ¼ 1

and

h ð n Þ¼ X

N

A k exp ð s k nT S Þ; ð 5 : 26 Þ

k ¼ 1

which further yields:

H ðÞ¼ X

N

A k
1 exp s k T s

Þ z 1 :

ð 5 : 27 Þ

ð

k ¼ 1

The resulting discrete transfer function ( 5.27 ) allows getting the sought

difference equation needed for digital filter realization.