Solution to ECE Test #12 S09
1.
The device which accepts an impulse excitation, produces a rectangular response and
is used in modeling the action of a DAC is called a zero-order hold.
2.
Digital filters whose impulse response goes to zero at a finite time and stays there
forever after that are called finite-duration impulse response (FIR) filters
3.
Digital filters whose impulse response approaches zero asymptotically as time
approaches infinity are called infinite-duration impulse response (IIR) filters
d
4.
When trying to design an approximation to an ideal differentiator ( y ( t ) = ( x ( t )) )
dt
using the step invariant technique we encounter a major problem.
Y ( s ) = s X ( s ) ⇒ H ( s ) = s ⇒ H −1 ( s ) = s × 1 / s = 1 ⇒ h −1 ( t ) = δ ( t )
The next step would be to sample the CT step response. But, since the step response is a
CT impulse it cannot be sampled.
5.
A lowpass analog filter has unity gain at a frequency of zero and a single real pole at
s = −20 . A digital filter is designed to approximate it using the impulse-invariant technique
with a sampling rate of 200 Hz. Find the following.
h ( t ) = 20 e−20 t u ( t )
z
H ( z ) = 20
z − 0.9048
(
h −1 ( t ) = 20 1 − e
H (s) =
−20 t
) u (t )
h [ n ] = 20 ( 0.9048 ) u [ n ]
20
H −1 ( s ) =
s ( s + 20 )
n
z2
H −1 ( z ) = 20
( z − 1) ( z − 0.9048 )
20
n
⇒ h ( t ) = 20 e−20 t u ( t ) ⇒ h [ n ] = 20 e−20 n / 2000 u [ n ] = 20 ( 0.9048 ) u [ n ]
s + 20
20
H −1 ( s ) =
⇒ h −1 ( t ) = 20 1 − e−20 t u ( t )
s ( s + 20 )
(
)
z
z
z
z2
⇒ H −1 ( z ) = 20
= 20 2
z − 0.9048
z − 1 z − 0.9048
z − 1.9048 z + 0.9048
2
z
z
H −1 ( z ) = 20 2
= 20 z × 2
z − 1.9048 z + 0.9048
z − 1.9048 z + 0.9048
2
20 z 2
20 z
H −1 ( z ) =
⇒ lim h [ n ] = lim ( z − 1)
z →1
( z − 1) ( z − 0.9048 ) n→∞ −1
( z − 1) ( z − 0.9048 )
H ( z ) = 20
20 z 2
= lim
= 210.1
z →1 ( z − 0.9048 )
What is the numerical final value of the digital filter to a unit sequence? 210.1
Solution to ECE Test #12 S09
1.
Digital filters whose impulse response goes to zero at a finite time and stays there
forever after that are called finite-duration impulse response (FIR) filters
2.
Digital filters whose impulse response approaches zero asymptotically as time
approaches infinity are called infinite-duration impulse response (IIR) filters
3.
The device which accepts an impulse excitation, produces a rectangular response and
is used in modeling the action of a DAC is called a zero-order hold.
d
4.
When trying to design an approximation to an ideal differentiator ( y ( t ) = ( x ( t )) )
dt
using the step invariant technique we encounter a major problem.
Y ( s ) = s X ( s ) ⇒ H ( s ) = s ⇒ H −1 ( s ) = s × 1 / s = 1 ⇒ h −1 ( t ) = δ ( t )
The next step would be to sample the CT step response. But, since the step response is a
CT impulse it cannot be sampled.
5.
A lowpass analog filter has unity gain at a frequency of zero and a single real pole at
s = −30 . A digital filter is designed to approximate it using the impulse-invariant technique
with a sampling rate of 200 Hz. Find the following.
h ( t ) = 30 e−30 t u ( t )
z
H ( z ) = 30
z − 0.8607
(
h −1 ( t ) = 30 1 − e
H (s) =
−30 t
) u (t )
h [ n ] = 30 ( 0.8607 ) u [ n ]
30
H −1 ( s ) =
s ( s + 30 )
n
z2
H −1 ( z ) = 30
( z − 1) ( z − 0.8607 )
30
n
⇒ h ( t ) = 30 e−30 t u ( t ) ⇒ h [ n ] = 30 e−30 n / 2000 u [ n ] = 30 ( 0.8607 ) u [ n ]
s + 30
30
H −1 ( s ) =
⇒ h −1 ( t ) = 30 1 − e−30 t u ( t )
s ( s + 30 )
(
)
z
z
z
z2
⇒ H −1 ( z ) = 30
= 30 2
z − 0.8607
z − 1 z − 0.8607
z − 1.8607 z + 0.8607
2
z
z
H −1 ( z ) = 30 2
= 30 z × 2
z − 1.8607 z + 0.8607
z − 1.8607 z + 0.8607
2
30 z 2
30 z
H −1 ( z ) =
⇒ lim h [ n ] = lim ( z − 1)
z →1
( z − 1) ( z − 0.8607 ) n→∞ −1
( z − 1) ( z − 0.8607 )
H ( z ) = 30
30 z 2
= lim
= 215.36
z →1 ( z − 0.8607 )
What is the numerical final value of the digital filter to a unit sequence? 215.36