1
Stability Condition in Terms of
the Pole Locations
? A causal LTI digital filter is BIBO stable if
and only if its impulse response h[n] is
absolutely summable,i.e.,
? We now develop a stability condition in
terms of the pole locations of the transfer
function H(z)
??? ?
?
???n
nh ][S
2
Stability Condition in Terms of
the Pole Locations
? The ROC of the z-transform H(z) of the
impulse response sequence h[n] is defined
by values of |z| = r for which is
absolutely summable
? Thus,if the ROC includes the unit circle |z|
= 1,then the digital filter is stable,and vice
versa
nrnh ?][
3
Stability Condition in Terms of
the Pole Locations
? In addition,for a stable and causal digital
filter for which h[n] is a right-sided
sequence,the ROC will include the unit
circle and entire z-plane including the point
? An FIR digital filter with bounded impulse
response is always stable
??z
4
Stability Condition in Terms of
the Pole Locations
? On the other hand,an IIR filter may be
unstable if not designed properly
? In addition,an originally stable IIR filter
characterized by infinite precision
coefficients may become unstable when
coefficients get quantized due to
implementation
5
Stability Condition in Terms of
the Pole Locations
? Example - Consider the causal IIR transfer
function
? The plot of the impulse response coefficients
is shown on the next slide
21 8 5 0 5 8 608 4 511
1
?? ??? zzzH,.)(
6
Stability Condition in Terms of
the Pole Locations
? As can be seen from the above plot,the
impulse response coefficient h[n] decays
rapidly to zero value as n increases
0 10 20 30 40 50 60 70
0
2
4
6
T i m e i nde x n
A
m
pl
i
t
ude
h[n]
7
Stability Condition in Terms of
the Pole Locations
? The absolute summability condition of h[n]
is satisfied
? Hence,H(z) is a stable transfer function
? Now,consider the case when the transfer
function coefficients are rounded to values
with 2 digits after the decimal point,
21 8508511
1
?? ??? zzzH,.)(
^
8
Stability Condition in Terms of
the Pole Locations
? A plot of the impulse response of is
shown below
][nh^
0 10 20 30 40 50 60 70
0
2
4
6
T i m e i nde x n
A
m
pl
i
t
ude
][nh^
9
Stability Condition in Terms of
the Pole Locations
? In this case,the impulse response coefficient
increases rapidly to a constant value as
n increases
? Hence,the absolute summability condition of
is violated
? Thus,is an unstable transfer function
][nh^
)(zH^
10
Stability Condition in Terms of
the Pole Locations
? The stability testing of a IIR transfer
function is therefore an important problem
? In most cases it is difficult to compute the
infinite sum
? For a causal IIR transfer function,the sum S
can be computed approximately as
??? ? ? ???n nh ][S
? ??? 10Kn nh ][S K
11
Stability Condition in Terms of
the Pole Locations
? The partial sum is computed for increasing
values of K until the difference between a
series of consecutive values of is
smaller than some arbitrarily chosen small
number,which is typically
? For a transfer function of very high order
this approach may not be satisfactory
? An alternate,easy-to-test,stability condition
is developed next
S K
610?
12
Stability Condition in Terms of
the Pole Locations
? Consider the causal IIR digital filter with a
rational transfer function H(z) given by
? Its impulse response {h[n]} is a right-sided
sequence
? The ROC of H(z) is exterior to a circle
going through the pole farthest from z = 0
?
?
?
?
?
?
? N
k
k
k
M
k
k
k
zd
zp
zH
0
0)(
13
Stability Condition in Terms of
the Pole Locations
? But stability requires that {h[n]} be
absolutely summable
? This in turn implies that the DTFT
of {h[n]} exists
? Now,if the ROC of the z-transform H(z)
includes the unit circle,then
??? ? jezj zHeH )()(
)( ?jeH
14
Stability Condition in Terms of
the Pole Locations
? Conclusion,All poles of a causal stable
transfer function H(z) must be strictly inside
the unit circle
? The stability region (shown shaded) in the
z-plane is shown below
1 Re z
j Im z
1?
j?
j
unit circle
stability region
15
Stability Condition in Terms of
the Pole Locations
? Example - The factored form of
is
which has a real pole at z = 0.902 and a real
pole at z = 0.943
? Since both poles are inside the unit circle,
H(z) is BIBO stable
21 8 5 0 5 8 6.0845.01
1)(
?? ??? zzzH
)94 3.01)(90 2.01(
1)(
11 ?? ??? zzzH
16
Stability Condition in Terms of
the Pole Locations
? Example - The factored form of
is
which has a real pole on the unit circle at z =
1 and the other pole inside the unit circle
? Since one pole is not inside but on the unit
circle,H(z) is unstable
21 85.085.11
1)(
?? ??? zzzH
^
)85.01)(1(
1)(
11 ?? ??? zzzH
^
17
Types of Transfer Functions
? The time-domain classification of an LTI
digital transfer function sequence is based
on the length of its impulse response,
- Finite impulse response (FIR) transfer
function
- Infinite impulse response (IIR) transfer
function
18
Types of Transfer Functions
? Several other classifications are also used
? In the case of digital transfer functions with
frequency-selective frequency responses,
one classification is based on the shape of
the magnitude function or the form
of the phase function q(?)
? Based on this four types of ideal filters are
usually defined
|)(| ?jeH
19
Ideal Filters
? A digital filter designed to pass signal
components of certain frequencies without
distortion should have a frequency response
equal to one at these frequencies,and
should have a frequency response equal to
zero at all other frequencies
20
Ideal Filters
? The range of frequencies where the
frequency response takes the value of one is
called the passband
? The range of frequencies where the
frequency response takes the value of zero
is called the stopband
21
Ideal Filters
? Frequency responses of the four popular types
of ideal digital filters with real impulse
response coefficients are shown below,
Lowpass Highpass
Bandpass Bandstop
22
Ideal Filters
? Lowpass filter,Passband -
Stopband -
? Highpass filter,Passband -
Stopband -
? Bandpass filter,Passband -
Stopband -
? Bandstop filter,Stopband -
Passband -
c????0
????? c
????? c
c????0
21 cc ?????
10 c???? ????2cand
21 cc ?????
10 c???? ????? 2cand
23
Ideal Filters
? The frequencies,,and are called
the cutoff frequencies
? An ideal filter has a magnitude response
equal to one in the passband and zero in the
stopband,and has a zero phase everywhere
c? 1c? 2c?
24
Ideal Filters
? Earlier in the course we derived the inverse
DTFT of the frequency response
of the ideal lowpass filter,
? We have also shown that the above impulse
response is not absolutely summable,and
hence,the corresponding transfer function
is not BIBO stable
)( ?jLP eH
sin[ ] sin c,c c c
LP
nnh n n
n
? ? ?
? ? ?
??? ? ? ? ? ? ?
????
25
Ideal Filters
? Also,is not causal and is of doubly
infinite length
? The remaining three ideal filters are also
characterized by doubly infinite,noncausal
impulse responses and are not absolutely
summable
? Thus,the ideal filters with the ideal,brick
wall” frequency responses cannot be
realized with finite dimensional LTI filter
][nhLP
26
Ideal Filters
? To develop stable and realizable transfer
functions,the ideal frequency response
specifications are relaxed by including a
transition band between the passband and
the stopband
? This permits the magnitude response to
decay slowly from its maximum value in
the passband to the zero value in the
stopband
27
Ideal Filters
? Moreover,the magnitude response is allowed
to vary by a small amount (ripple) both in the
passband and the stopband
? Typical magnitude response specifications of
a lowpass filter are shown below
28
Zero-Phase and Linear-Phase
Transfer Functions
? A second classification of a transfer
function is with respect to its phase
characteristics
? In many applications,it is necessary that the
digital filter designed does not distort the
phase of the input signal components with
frequencies in the passband
29
Zero-Phase and Linear-Phase
Transfer Functions
? One way to avoid any phase distortion is to
make the frequency response of the filter
real and nonnegative,i.e.,to design the
filter with a zero phase characteristic
? However,it is possible to design a causal
digital filter with a zero phase
30
Zero-Phase and Linear-Phase
Transfer Functions
? For non-real-time processing of real-valued
input signals of finite length,zero-phase
filtering can be very simply implemented
by relaxing the causality requirement
? One zero-phase filtering scheme is sketched
below
x[n] v[n] u[n] w[n] H(z) H(z)
][][],[][ nwnynvnu ????
31
Zero-Phase and Linear-Phase
Transfer Functions
? It is easy to verify the above scheme in the
frequency domain
? Let,,,,and
denote the DTFTs of x[n],v[n],
u[n],w[n],and y[n],respectively
? From the figure shown earlier and making
use of the symmetry relations we arrive at
the relations between various DTFTs as
given on the next slide
)( ?jeX )( ?jeV )( ?jeU )( ?jeW
)( ?jeY
32
Zero-Phase and Linear-Phase
Transfer Functions
? Combining the above equations we get
x[n] v[n] u[n] w[n] H(z) H(z)
][][],[][ nwnynvnu ????
( ) ( ) ( ),j j jV e H e X e? ? ?? )()()( ??? ? jjj eUeHeW
,)(*)( ?? ? jj eVeU )(*)( ?? ? jj eWeY
)(*)(*)(*)( ???? ?? jjjj eUeHeWeY
)()()(*)()(* ????? ?? jjjjj eXeHeHeVeH )()( 2 ??? jj eXeH
33
Zero-Phase and Linear-Phase
Transfer Functions
? The function filtfilt implements the above
zero-phase filtering scheme
? In the case of a causal transfer function with
a nonzero phase response,the phase
distortion can be avoided by ensuring that
the transfer function has a unity magnitude
and a linear-phase characteristic in the
frequency band of interest
34
Zero-Phase and Linear-Phase
Transfer Functions
? The most general type of a filter with a
linear phase has a frequency response given
by
which has a linear phase from ? = 0 to ? =
2?
? Note also
Djj eeH ??? ?)(
1)( ??jeH
D??? )(
35
Zero-Phase and Linear-Phase
Transfer Functions
? The output y[n] of this filter to an input
is then given by
? If and represent the continuous-
time signals whose sampled versions,
sampled at t = nT,are x[n] and y[n] given
above,then the delay between and
is precisely the group delay of amount D
njAenx ??][
)(][ DnjnjDj AeeAeny ????? ??
)(txa
)(txa
)(tya
)(tya
36
Zero-Phase and Linear-Phase
Transfer Functions
? If D is an integer,then y[n] is identical to
x[n],but delayed by D samples
? If D is not an integer,y[n],being delayed by
a fractional part,is not identical to x[n]
? In the latter case,the waveform of the
underlying continuous-time output is
identical to the waveform of the underlying
continuous-time input and delayed D units
of time
37
Zero-Phase and Linear-Phase
Transfer Functions
? If it is desired to pass input signal
components in a certain frequency range
undistorted in both magnitude and phase,
then the transfer function should exhibit a
unity magnitude response and a linear-phase
response in the band of interest
38
Zero-Phase and Linear-Phase
Transfer Functions
? Figure below shows the frequency response
if a lowpass filter with a linear-phase
characteristic in the passband
39
Zero-Phase and Linear-Phase
Transfer Functions
? Since the signal components in the stopband
are blocked,the phase response in the
stopband can be of any shape
? Example - Determine the impulse response
of an ideal lowpass filter with a linear phase
response,
??
??? )( j
LP eH ?????
??????
c
c
nj oe
,0
0,
40
Zero-Phase and Linear-Phase
Transfer Functions
? Applying the frequency-shifting property of
the DTFT to the impulse response of an ideal
zero-phase lowpass filter we arrive at
? As before,the above filter is noncausal and
of doubly infinite length,and hence,
unrealizable
()[ ] sin c,c c o
LP
nnh n n??
??
???? ? ? ? ? ?
????
41
Zero-Phase and Linear-Phase
Transfer Functions
? By truncating the impulse response to a
finite number of terms,a realizable FIR
approximation to the ideal lowpass filter
can be developed
? The truncated approximation may or may
not exhibit linear phase,depending on the
value of chosen on
42
Zero-Phase and Linear-Phase
Transfer Functions
? If we choose = N/2 with N a positive
integer,the truncated and shifted
approximation
will be a length N+1 causal linear-phase
FIR filter
on
NnNn Nnnh cLP ???? ??? 0,)2/( )2/(si n][^
43
Zero-Phase and Linear-Phase
Transfer Functions
? Figure below shows the filter coefficients
obtained using the function sinc for two
different values of N
0 2 4 6 8 10 12
- 0.2
0
0.2
0.4
0.6
T i m e i nde x n
A
m
pl
i
t
ude
N = 12
0 2 4 6 8 10 12
- 0.2
0
0.2
0.4
0.6
T i m e i nde x n
A
m
pl
i
t
ude
N = 13
44
Zero-Phase and Linear-Phase
Transfer Functions
? Because of the symmetry of the impulse
response coefficients as indicated in the two
figures,the frequency response of the
truncated approximation can be expressed as,
where,called the zero-phase
response or amplitude response,is a real
function of ?
)(][)( 2/
0
??? ??
?
??? ?
LP
Nj
N
n
nj
LP
j
LP HeenheH ^ ^
~
)(?LPH~