Chapter 7 LTI Discrete-Time Systems in the Transform Domain.
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Transcript of Chapter 7 LTI Discrete-Time Systems in the Transform Domain.
Chapter 7
LTI Discrete-Time SysteLTI Discrete-Time Systems in the Transform Doms in the Transform Do
mainmain
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
Types of Transfer Functions
In the case of digital transfer functions with frequency-selective frequency responses, there are two types of classifications
(1) Classification based on the shape of the magnitude function |H (e jω) |
(2) Classification based on the the form of the phase function θ(ω)
§7.1 Classification Based onMagnitude Characteristics
One common classification is based on an ideal magnitude response
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
§7.1.1 Digital Filters with Ideal Magnitude Responses
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
§7.1.1 Digital Filters with Ideal Magnitude Responses
Frequency responses of the four popular types of ideal digital filters with real impulse response coefficients are shown below:
§7.1.1 Digital Filters with Ideal Magnitude Responses
Lowpass filter: Passband -0≤ω≤ωc
Stopband -ωc≤ω≤π Highpass filter: Passband -ωc≤ω≤π
Stopband -0≤ω≤ωc
Bandpass filter: Passband -ωc1≤ω≤ωc2
Stopband -0≤ω≤ωc1 and ωc2≤ω≤π Bandstop filter: Stopband -ωc1≤ω≤ωc2
Passband -0≤ω≤ωc1 and ωc2≤ω≤π
§7.1.1 Digital Filters with Ideal Magnitude Responses
The frequenciesωc ,ωc1 ,andωc2 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
§7.1.1 Digital Filters with Ideal Magnitude Responses
Earlier in the course we derived the inverse DTFT of the frequency response HLP(e jω) of the ideal lowpass filter:
nn
nnh c
LP ,sin
][
We have also shown that the above impulse response is not absolutely summable, and hence, the corresponding transfer function is not BIBO stable
§7.1.1 Digital Filters with Ideal Magnitude Responses
Also, HLP[n] 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
§7.1.1 Digital Filters with Ideal Magnitude Responses
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
§7.1.1 Digital Filters with Ideal Magnitude Responses
Moreover, the magnitude response is allowed to vary by a small amount both in the passband and the stopband
Typical magnitude response specifications of a lowpass filter are shown the right
§7.1.2 Bounded Real TransferFunctions
A causal stable real-coefficient transfer function H(z) is defined as a bounded real
(BR) transfer function if
|HLP(e jω)|≤1 for all values of ω Let x[n] and y[n] denote, respectively, the in
put and output of a digital filter characterized by a BR transfer function H(z)
with X(e jω) and Y(e jω) denoting their DTFTs
§7.1.2 Bounded Real TransferFunctions
Then the condition |H(e jω)|≤1 implies that22
)()( jj eXeY
nn
nxny22
][][
Integrating the above from –π to π, and applying Parseval’s relation we get
§7.1.2 Bounded Real TransferFunctions
Thus, for all finite-energy inputs, the output energy is less than or equal to the input energy implying that a digital filter characterized by a BR transfer function can be viewed as a passive structure
If |H(e jω)|=1, then the output energy is equal to the input energy, and such a digital filter is therefore a lossless system
§7.1.2 Bounded Real TransferFunctions
A causal stable real-coefficient transfer function H(z) with |H(e jω)|=1 is thus called a lossless bounded real (LBR) transfer function
The BR and LBR transfer functions are the keys to the realization of digital filters with low coefficient sensitivity
§7.1.2 Bounded Real TransferFunctions
Example – Consider the causal stable IIR
transfer function
10,1
)(1
z
KzH
cos2)1()()()(
2
212
K
zHzHeH jez
j
where K is a real constant Its square-magnitude function is given by
§7.1.2 Bounded Real TransferFunctions
The maximum value of |H(e jω)|2 is obtained when 2αcosω in the denominator is maximum and the minimum value is a obtained when 2αcosω is a minimum
For α > 0, maximum value of 2αcosωis equal to 2α at ω=0, and minimum value is -2α at ω=π
§7.1.2 Bounded Real TransferFunctions
Thus, for α>0, the maximum value of |H(ej
ω)|2 is equal to K2/(1-α)2 at ω=0 and the minimum value is equal to K2/(1+α)2 at ω=π
On the other hand, for α<0, the maximum value of 2αcosω is equal to -2α at ω=πand the minimum value is equal to 2α at ω= 0
§7.1.2 Bounded Real TransferFunctions
Here, the maximum value of |H(ejω)|2 is equal to K2/(1-α)2 at ω=π and the minimum value is equal to K2/(1-α)2 at ω=0
Hence, the maximum value can be made equal to 1 by choosing K=±(1-α), in which case the minimum value becomes (1-α)2/(1+α)2
§7.1.2 Bounded Real TransferFunctions
is a BR function for K=±(1-α) Plots of the magnitude function for α=±0.5 w
ith values of K chosen to make H(z) a BR
function are shown on the next slide
10,1
)(1
z
KzH
Hence,
§7.1.2 Bounded Real TransferFunctions
§7.1.3 Allpass Transfer Function
Definition An IIR transfer function A(z) with unity magni
tude response for all frequencies, i.e., |A(ejω)|2= 1, for all ω
is called an allpass transfer function An M-th order causal real-coefficient allpass
transfer function is of the form
MM
MM
MMMM
zdzdzd
zzdzddzAM
1
11
1
11
11
1)(
§7.1.3 Allpass Transfer Function If we denote the denominator polynomials of
AM(z) as DM(z):
DM(z)=1+d1z-1+···+dM-1z-M+1+dMz-M
then it follows that AM(z) can be written as:
Note from the above that if z=re jФ is a pole of a real coefficient allpass transfer function, then it has a zero at
)(
)()(
1
zD
zDzzA
M
MM
M
jer
z 1
§7.1.3 Allpass Transfer Function The numerator of a real-coefficient allpass t
ransfer function is said to be the mirror- image polynomial of the denominator, and vice versa
)()(~ 1 zDzzD M
MM
We shall use the notation to denote the mirror-image polynomial of a degree-M polynomial DM
(z), i.e.,
)(~
zDM
§7.1.3 Allpass Transfer Function
implies that the poles and zeros of a real- coefficient allpass function exhibit mirror- image symmetry in the z-plane
)(
)()(
1
zD
zDzzA
M
MM
M
321
321
3 2.018.04.01
4.018.02.0)(
zzz
zzzzA
The expression
§7.1.3 Allpass Transfer Function
To show that we observe that
)(
)()(
11
zD
zDzzA
M
MM
M
)(
)(
)(
)()()(
1
11
zD
zDz
zD
zDzzAzA
M
MM
M
MM
MM
1)()()( 12
jezMMj
M zAzAeA
Therefore
Hence
§7.1.3 Allpass Transfer Function
Now, the poles of a causal stable transfer function must lie inside the unit circle in the z-plane
Hence, all zeros of a causal stable allpass transfer function must lie outside the unit circle in a mirror-image symmetry with its poles situated inside the unit circle
§7.1.3 Allpass Transfer Function Figure below shows the principal value of th
e phase of the 3rd-order allpass function
321
321
3 2.018.04.01
4.018.02.0)(
zzz
zzzzA
Note the discontinuity by the amount of 2π
in the phase θ(ω)
§7.1.3 Allpass Transfer Function
If we unwrap the phase by removing the discontinuity, we arrive at the unwrapped phase function θc(ω) indicated below
Note: The unwrapped phase function is a continuous function of ω
§7.1.3 Allpass Transfer Function The unwrapped phase function of any arbitr
ary causal stable allpass function is a continuous function of ω
Properties (1) A causal stable real-coefficient allpass tr
ansfer function is a lossless bounded real
(LBR) function or, equivalently, a causal stable allpass filter is a lossless structure
§7.1.3 Allpass Transfer Function (2) The magnitude function of a stable allpa
ss function A(z) satisfies:
1,1
1,1
1,1
)(
zfor
zfor
zfor
zA
)]([)(
cd
d
(3) Let τ(ω) denote the group delay function of an allpass filter A(z),i.e.,
§7.1.3 Allpass Transfer Function
The unwrapped phase function θc(ω) of a stable allpass function is a monotonically decreasing function of ω so that τ(ω) is everywhere positive in the range 0<ω<π
The group delay of an M-th order stable real-coefficient allpass transfer function satisfies:
Md 0
)(
§7.1.3 Allpass Transfer FunctionA Simple Application
A simple but often used application of an allpass filter is as a delay equalizer
Let G(z) be the transfer function of a digital filter designed to meet a prescribed magnitude response
The nonlinear phase response of G(z) can be corrected by cascading it with an allpass filter A(z) so that the overall cascade has a constant group delay in the band of interest
§7.1.3 Allpass Transfer Function
Since |A(ejω)|=1, we have
|G(ejω)A(ejω)|=|G(ejω)| Overall group delay is the given by the sum
of the group delays of G(z) and A(z)
G(z) A(z)
§7.1.3 Allpass Transfer Function
Example – Figure below shows the group delay of a 4th order elliptic filter with the following specifications: ωp=0.3π,δp=1dB, δs=35dB
§7.1.3 Allpass Transfer Function
Figure below shows the group delay of the original elliptic filter cascaded with an 8th order allpass section designed to equalize the group delay in the passband
§7.2 Classification Based on PhaseCharacteristics
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
§7.2.1 Zero-Phase Transfer Function
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 not possible to design a causal digital filter with a zero phase
§7.2.1 Zero-Phase Transfer Function
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
H(z)x[n] v[n]
u[n]=v[-n]
H(z)u[n] w[n]
y[n]=w[-n]
§7.2.1 Zero-Phase Transfer Function It is easy to verify the above scheme in the f
requency domain Let X(ejω), V(ejω), U(ejω), W(ejω) and Y(ejω) den
ote 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
§7.2.1 Zero-Phase Transfer Function
V(ejω)=H(ejω)X(ejω), W(ejω)=H(ejω)U(ejω)
U(ejω)=V*(ejω), Y(ejω)=W*(ejω) Combining the above equations we get
Y(ejω)=W*(ejω)=H*(ejω)U*(ejω)= H*(ejω)V(ejω)=H*(ejω)H(ejω)X(ejω)= |H(ejω)|2X(ejω)
H(z)x[n] v[n]
u[n]=v[-n]
H(z)u[n] w[n]
y[n]=w[-n]
§7.2.1 Zero-Phase Transfer Function
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
§7.2.1 Zero-Phase Transfer Function
The most general type of a filter with a linear phase has a frequency response given by
H(ejω)=e-jωD
which has a linear phase from ω=0 to ω=2π
Note also |H(ejω)|=1 τ(ω)=D
§7.2.2 Linear-Phase TransferFunction
The output y[n] of this filter to an input x[n]=Aejωn is then given by
y[n]=Ae-jωDejωn=Aejω(n-D)
If xa(t) and ya(t) represent the continuous-time signals whose sampled versions, sampled at t = nT ,are x[n] and y[n] given above, then the delay between xa(t) and ya(t) isprecisely the group delay of amount D
§7.2.2 Linear-Phase TransferFunction
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
§7.2.2 Linear-Phase TransferFunction
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
§7.2.2 Linear-Phase TransferFunction
Figure below shows the frequency response if a lowpass filter with a linear-phase characteristic in the passband
§7.2.2 Linear-Phase TransferFunction
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:
c
cnj
jLP
eeH
,0
0,)(
0
§7.2.2 Linear-Phase TransferFunction
Applying the frequency-shifting property of the DTFT to the impulse response of an ideal zero-phase lowpass filter we arrive at
n
nn
nnnh c
LP ,)(
)(sin][
0
0
As before, the above filter is noncausal and of doubly infinite length, and hence, unrealizable
§7.2.2 Linear-Phase TransferFunction
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 n0 chosen
§7.2.2 Linear-Phase TransferFunction
If we choose n0=N/2 with N a positive integer, the truncated and shifted approximation
NnNn
Nnnh c
LP
0,
)2/(
)2/(sin][ˆ
will be a length N+1 causal linear-phase FIR filter
§7.2.2 Linear-Phase TransferFunction
Figure below shows the filter coefficients obtained using the function sinc for two different values of N
§7.2.2 Linear-Phase Transfer Function
Zero-Phase Response Because of the symmetry of the impulse res
ponse coefficients as indicated in the two figures, the frequency response of the truncated approximation can be expresse as:
)(~
][ˆ)(ˆ 2/
0
LP
NjnjN
nLP
jLP HeenheH
where , called the zero-phase response or amplitude response, is a real function of ω
)(~ LPH
§7.2.3 Minimum-Phase and Maximum- Phase Transfer Functions
Consider the two 1st-order transfer function:
1,1,1
)(,)( 21
baaz
bzzH
az
bzzH
Both transfer functions have a pole inside the unit circle at the same location z=-a and are stable
But the zero of H1(z) is inside the unit circle at z=-b, whereas, the zero of H2(z) is at z=-1/b situated in a mirror-image symmetry
§7.2.3 Minimum-Phase and Maximum- Phase Transfer Functions
Figure below shows the pole-zero plots of the two transfer functions
§7.2.3 Minimum-Phase and Maximum- Phase Transfer Functions
However, both transfer functions have an identical magnitude function as
)()()()( 122
111
zHzHzHzH
cos
sintan
cos
sintan)](arg[ 11
1
abeH j
cossintan
cos1sintan)](arg[ 11
2
abbeH j
The corresponding phase functions are
§7.2.3 Minimum-Phase and Maximum- Phase Transfer Functions
Figure below shows the unwrapped phase responses of the two transfer functions for a=0.8 and b=-0.5
§7.2.3 Minimum-Phase and Maximum- Phase Transfer Functions
From this figure it follows that H2(z) has an excess phase lag with respect to H1(z)
The excess phase lag property of H2(z) with respect to H1(z) can also be explained by observing that we can write
)()(
2
11)(
1 zAzH
bz
bz
az
bz
az
bzzH
§7.2.3 Minimum-Phase and Maximum- Phase Transfer Functions
where A(z)=(bz+1)/(z+b) is a stable allpass function
The phase function of H1(z) and H2(z) are thus related through
)](arg[)](arg[)](arg[ 12 jjj eAeHeH
As the unwrapped phase function of a stable first-order allpass function is a negative function of ω, it follows from the above that H2(z) has indeed an excess phase lag with respect to H1(z)
§7.2.3 Minimum-Phase and Maximum- Phase Transfer Functions
Generalizing the above result, let Hm(z) be
a causal stable transfer function with all zeros inside the unit circle and let H(z) be another causal stable transfer function satifying |H(ejω)|=|Hm(ejω)|
These two transfer functions are then related through H(z) =Hm(z) A (z) where A (z) is a causal stable allpass function
§7.2.3 Minimum-Phase and Maximum- Phase Transfer Functions
The unwrapped phase functions of Hm(z) and H(z) are thus related through
)](arg[)](arg[)](arg[ jjm
j eAeHeH H(z) has an excess phase lag with respect t
o Hm(z) A causal stable transfer function with all zer
os inside the unit circle is called a minimum-phase transfer function
§7.2.3 Minimum-Phase and Maximum- Phase Transfer Functions
A causal stable transfer function with all zeros outside the unit circle is called a maximum-phase transfer function
A causal stable transfer function with zeros inside and outside the unit circle is called a mixed-phase transfer function
§7.2.3 Minimum-Phase and Maximum- Phase Transfer Functions
Example – Consider the mixed-phase transfer function
)5.01)(2.01(
)4.0)(3.01(2)(
11
11
zz
zzzH
functionAllpass
1
1
functionphaseMinimum
11
11
4.014.0
)5.01)(2.01()4.01)(3.01(2
)(
zz
zzzz
zH
We can rewrite H(z) as
§7.3 Types of Linear-Phase FIR Transfer Functions
It is impossible to design an IIR transfer function with an exact linear-phase
It is always possible to design an FIR transfer function with an exact linear-phase response
We now develop the forms of the linear- phase FIR transfer function H(z) with real impluse response h[n]
called the amplitude response, also called the zero-phase response, is a real function of ω
§7.3 Types of Linear-Phase FIR Transfer Functions
If H(z) is to have a linear-phase, its frequency response must be of the form
where c and β are constants, and ,)(H
nN
n
znhzH
0
][)(
)()( )( HeeH cjj
Let
§7.3 Types of Linear-Phase FIR Transfer Functions
For a real impulse response, the magnitude response |H(ejω)| is an even function of ω, i.e.,
)()( jj eHeH Since ,the amplitude)()( HeH j
)()( HH
response is then either an even function or an odd function of ω, i.e.
§7.3 Types of Linear-Phase FIR Transfer Functions
The frequency response satisfies the relation
)()( * jj eHeH
)()( )()( HeHe cjcj
If is an even function, then the above relation leads to
)(H
jj ee
or, equivalently, the relation
implying that either β=0 or β=π
§7.3 Types of Linear-Phase FIR Transfer Functions
Substituting the value of β in the above we get
)()( )( HeeH cjj
)()( )( jcj eHeH
)(
0
][)()( ncjN
n
jjc enheHeH
From
we have
§7.3 Types of Linear-Phase FIR Transfer Functions
Replacing ω with -ω in the previous equation we get
)(
0
][)( lcjN
l
elhH
)(
0
][)( nNcjN
n
enNhH
Making a change of variable l=N-n, we rewrite the above equation as
§7.3 Types of Linear-Phase FIR Transfer Functions
The above leads to the condition
As , we have )()( HH
)()( ][][ nNcjncj enNhenh
NnnNhnh 0,][][
Thus the FIR filter with an even amplitude ewsponse will have linear phase if it has a symmetric impulse response
with c=-N/2
§7.3 Types of Linear-Phase FIR Transfer Functions
The above is satisfied if β=π/2 or β=-π/2 Then
If is an odd function of ω, then from)(H
)()( )()( HeHe cjcj
we get as jj ee )()( HH
)()( )( HeeH cjj
)()( HjeeH jcj
reduces to
§7.3 Types of Linear-Phase FIR Transfer Functions
The last equation can be rewritten as
)(
0
][)()( ncjN
n
jjc enhjeHjeH
As , from the above we get)()( HH
)(
0
][)(
cj
N
ehjH
§7.3 Types of Linear-Phase FIR Transfer Functions
Making a change of variable l=N-n we rewrite the last equation as
)(
0
][)(
cj
N
ehjH
)(
0
][)( ncjN
n
enhjH
Equation the above with
we arrive at the condition for linear phase as
§7.3 Types of Linear-Phase FIR Transfer Functions
h[n]=h[n-N] , 0≤n≤N
with c=-N/2 Therefore, a FIR filter with an odd amplitude
response will have linear-phase response if it has an antisymmetric impulse response
§7.3 Types of Linear-Phase FIR Transfer Functions
Since the length of the impulse response can be either even or odd, we can define four types of linear-phase FIR transfer functions
For an antisymmetric FIR filter of odd length, i.e., N even
h[N/2] = 0 We examine next the each of the 4 cases
§7.3 Types of Linear-Phase FIR Transfer Functions
Type 1: N=8 Type 2: N=7
Type 3: N=8 Type 4: N=7
§7.3 Types of Linear-Phase FIR Transfer Functions
Type 1: Symmetric Impulse Response with Odd Length
In this case, the degree N is even Assume N=8 for simplicity The transfer function H(z) is given by
87654
321
]8[]7[]6[]5[]4[]3[]2[]1[]0[)(
zhzhzhzhzhzhzhzhhzH
§7.3 Types of Linear-Phase FIR Transfer Functions
Because of symmetry, we have h[0]=h[8], h[1]=h[7], h[2]=h[6], h[3]=h[5]
Thus we can write
H[z]=h[0](1+z-8)+h[1](z-1+z-7)+ h[2](z-2+z-6)
+h[3](z-3+z-5)+ h[4]z-4
=z-4{h[0](z4+z-4)+h[1](z3+z-3)+h[2](z2+z-2)
+h[3](z+z-1)+h[4]}
§7.3 Types of Linear-Phase FIR Transfer Functions
The corresponding frequency response is then given by
]}4[)cos(]3[2)2cos(]2[2
)3cos(]1[2)4cos(]0[2{)( 4
hhh
hheeH jj
The quantity inside the braces is a real function of w, and can assume positive or negative values in the range 0||
§7.3 Types of Linear-Phase FIR Transfer Functions
The phase here is given by
θ(ω)=-4ω+β
where β is either 0 or π,and hence, it is a linear function of ω
The group delay is given by
4)(
)(
d
d
indicating a constant group delay of 4 samples
§7.3 Types of Linear-Phase FIR Transfer Functions
In the general case for Type 1 FIR filters, the frequency response is of the form
)()( 2/ HeeH jNj
)cos(]2
[2]2
[)(2/
1
nnN
hN
hHN
n
where the amplitude response , also called the zero-phase response, is of the form
)(H
§7.3 Types of Linear-Phase FIR Transfer Functions
which is seen to be a slightly modified version of a length-7 moving-average FIR filter
The above transfer function has a symmetric impulse response and therefore a linear phase response
]2
1
2
1[
6
1)( 654321
0 zzzzzzzH
Example – Consider
§7.3 Types of Linear-Phase FIR Transfer Functions
A plot of the magnitude response of H0(z) along with that of the 7-point moving-average filter is shown below
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
/
Mag
nitu
de
modified filtermoving-average
§7.3 Types of Linear-Phase FIR Transfer Functions
Note the improved magnitude response obtained by simply changing the first and the last impulse response coefficients of a moving-average (MA) filter
It can be shown that we can express
)1(6
1)1(
2
1)( 543211
0 zzzzzzzH
which is seen to be a cascade of a 2-point MA filter with a 6-point MA filter
Thus, H0(z) has a double zero at z=-1, i.e., (ω=π)
§7.3 Types of Linear-Phase FIR Transfer Functions
Type 2: Symmetric Impulse Response with Even Length
In this case, the degree N is odd Assume N=7 for simplicity The transfer function if of the form
H(z)=h[0]+h[1]z-1 +h[2]z-2 +h[3]z-3 +h[4]z-4
+h[5]z-5 +h[6]z-6 +h[7]z-7
§7.3 Types of Linear-Phase FIR Transfer Functions
Making use of the symmetry of the impulse response coefficients, the transfer function can be written as
H(z)=h[0](1+z-7)+h[1](z-1+z-6)
+h[2](z-2+z-5)+h[3](z-3+z-4)
=z-7/2{h[0](z7/2+z-7/2 )+h[1](z5/2+z-5/2 )
+h[2](z3/2+z-3/2 )+h[3](z1/2+z-1/2 )}
§7.3 Types of Linear-Phase FIR Transfer Functions
The corresponding frequency response is given by
)}2
cos(]3[2)2
3cos(]2[2
)2
5cos(]1[2)
2
7cos(]0[2{)( 2/7
hh
hheeH jj
As before, the quantity inside the braces is a real function of ω, and can assume positive or negative values in the range 0≤|ω|≤π
§7.3 Types of Linear-Phase FIR Transfer Functions
Here the phase function is given by θ(ω)=-7/2ω+βwhere again is either 0 or π
As a result, the phase is also a linear function of ω
The corresponding group delay is τ(ω)=7/2indicating a group delay of 7/2 samples
§7.3 Types of Linear-Phase FIR Transfer Functions
The expression for the frequency response in the general case for Type 2 FIR filters is of the form
)(~
)( 2/ HeeH jNj
))2
1(cos(]
2
1[2)(
~ 2/)1(
1
nnN
hHN
n
where the amplitude response is given by
§7.3 Types of Linear-Phase FIR Transfer Functions
Type 3:Antiymmetric Impulse Response with Odd Length
In this case, the degree N is even Assume N=8 for simplicity Applying the symmetry condition we get
H(z)=z-4{h[0](z4-z-4)+h[1](z3-z-3)
+h[2](z2-z-2)+h[3](z-z-1)}
§7.3 Types of Linear-Phase FIR Transfer Functions
It also exhibits a linear phase response given by
2
4)(
)}sin(]3[2)2sin(]2[2
)3sin(]1[2)4sin(]0[2{)( 2/4
hh
hheeeH jjj
The corresponding frequency response is given by
where β is either 0 or π
§7.3 Types of Linear-Phase FIR Transfer Functions
The group delay here is τ(ω)=4
indicating a constant group delay of 4 samples In the general case
)(~
)( 2/ HjeeH jNj
)sin(]2
[2)(~ 2/
1
nnN
hHN
n
where the amplitude response is of the form
§7.3 Types of Linear-Phase FIR Transfer Functions
Type 4:Antiymmetric Impulse Response with Even Length
In this case, the degree N is even Assume N=7 for simplicity Applying the symmetry condition we get
H(z)=z-7/2{h[0](z7/2-z-7/2)+h[1](z5/2-z-5/2)
+h[2](z3/2-z-3/2)+h[3](z1/2-z-1/2)}
§7.3 Types of Linear-Phase FIR Transfer Functions
The corresponding frequency response is given by
It again exhibits a linear phase response given by
θ(ω)=-7/2ω+π/2+β
where β is either 0 or π
)]}2
sin(]3[2)]2
3sin(]2[2
)]2
5sin(]1[2)]
2
7sin(]0[2{)( 2/2/7
hh
hheeeH jjj
§7.3 Types of Linear-Phase FIR Transfer Functions
The group delay is constant andis given byτ(ω)=7/2
)(~
)( 2/ HjeeH jNj
))2
1(sin(]
2
1[2)(
~ 2/)1(
1
nnN
hHN
n
In the general case we have
where now the amplitude response is of the form
§7.3 Types of Linear-Phase FIR Transfer Functions
General Form of Frequency Response In each of the four types of linear-phase FIR
filters, the frequency response is of the form
)(~
)( 2/ HejeeH jjNj The amplitude response for each of th
e four types of linear-phase FIR filters can become negative over certain frequency ranges, typically in the stopband
)(~ H
§7.3 Types of Linear-Phase FIR Transfer Functions
The magnitude and phase responses of the linear-phase FIR are given by
0)(~
,2
0)(~
,2)(
)(~
)(
HforN
HforN
HeH j
The group delay in each case isτ(ω)=N/2
§7.3 Types of Linear-Phase FIR Transfer Functions
Note that, even though the group delay is constant, since in general |H(ejω)| is not aconstant, the output waveform is not a replica of the input waveform
An FIR filter with a frequency response that is a real function of ω is often called a zero- phase filter
Such a filter must have a noncausal impulse response
§7.3.1 Zero Location of Linear-Phase FIR Transfer Functions
Consider first an FIR filter with a symmetric impulse response: h[n]=h[N-n]
Its transfer function can be written as
nN
n
nN
n
znNhznhzH
00
][][)(
mN
m
NmNN
m
nN
n
zmhzzmhznNh
000
][][][
By making a change of variable m=N-n, we can write
§7.3.1 Zero Location of Linear-Phase FIR Transfer Functions
Hence for an FIR filter with a symmetric impulse response of length N+1 we have
)(][ 1
0
zHzmh mN
m
)()( 1 zHzzH N
But,
A real-coefficient polynomial H(z) satisfying the above condition is called a mirror-image polynomial (MIP)
§7.3.1 Zero Location of Linear-Phase FIR Transfer Functions
Now consider first an FIR filter with an antisymmetric impulse response:
nN
n
nN
n
znNhznhzH
00
][][)(
)(][][ 1
00
zHzzmhznNh NmNN
m
nN
n
][][ nNhnh Its transfer function can be written as
By making a change of variable m=N-n, we get
§7.3.1 Zero Location of Linear-Phase FIR Transfer Functions
Hence, the transfer function H(z) of an FIR filter with an antisymmetric impulse response satisfies the condition
)()( 1 zHzzH N
A real-coefficient polynomial H(z) satisfying the above condition is called a antimirror-image polynomial (AIP)
Hence, a zero at z=ξo is associated with a zero at
0z
§7.3.1 Zero Location of Linear-Phase FIR Transfer Functions
It follows from the relation H(z)=±z-NH(z-1)
that if z=ξo is a zero of H(z), so is z=1/ξo Moreover, for an FIR filter with a real impuls
e response, the zeros of H(z) occur in complex conjugate pairs
§7.3.1 Zero Location of Linear-Phase FIR Transfer Functions
Thus, a complex zero that is not on the unit circle is associated with a set of 4 zeros given by
jj er
zrez 1
,
jez
A zero on the unit circle appear as a pair
as its reciprocal is also its complex conjugate
§7.3.1 Zero Location of Linear-Phase FIR Transfer Functions
Since a zero at z =±1 is its own reciprocal, it can appear only singly
Now a Type 2 FIR filter satisfies H(z)=z-NH(z-1)
with degree N odd Hence, H(-1)=(-1)-NH(-1)=-H(-1)
implying H(-1)=0 , i.e., H(z) must have a zero at z=-1
§7.3.1 Zero Location of Linear-Phase FIR Transfer Functions
Likewise, a Type 3 or 4 FIR filter satisfies
H(z)=-z-NH(z-1) Thus H(-1)=-(1)-NH(1)=-H(1)
implying that H(z) must have a zero at z = 1 On the other hand, only the Type 3 FIR filter
is restricted to have a zero at since here the degree N is even and hence,
H(-1)=-(-1)-NH(-1)=-H(-1)
§7.3.1 Zero Location of Linear-Phase FIR Transfer Functions
Typical zero locations shown below
§7.3.1 Zero Location of Linear-Phase FIR Transfer Functions
Summarizing
(1) Type 1 FIR filter: Either an even number or no zeros at z =1 and z =-1
(2) Type 2 FIR filter: Either an even number or no zeros at z =1, and an odd number of zeros at z =-1
(3) Type 3 FIR filter: An odd number of zeros at z =1 and z =-1
§7.3.1 Zero Location of Linear-Phase FIR Transfer Functions
(4) Type 4 FIR filter: An odd number of zeros at z =1, and either an even number of no zeros at z =-1
The presence of zeros at z =±1 leads to the following limitations on the use of these linear-phase transfer functions for designing frequency-selective filters
§7.3.1 Zero Location of Linear-Phase FIR Transfer Functions
A Type 2 FIR filter cannot be used to design a highpass filter since it always has a zero z =-1
A Type 3 FIR filter has zeros at both z =1 and z =-1, and hence cannot be used to design either a lowpass or a highpass or a bandstop filter
§7.3.1 Zero Location of Linear-Phase FIR Transfer Functions
A Type 4 FIR filter is not appropriate to design lowpass and bandstop filters due to the presence of a zero at z =1
Type 1 FIR filter has no such restrictions and can be used to design almost any type of filter
§7.4 Simple Digital Filters
Later in the course we shall review various methods of designing frequency-selective filters satisfying prescribed specifications
We now describe several low-order FIR and IIR digital filters with reasonable selective frequency responses that often are satisfactory in a number of applications
§7.4.1 Simple FIR Digital Filters
FIR digital filters considered here have integer-valued impulse response coefficients
These filters are employed in a number of practical applications, primarily because of their simplicity, which makes them amenable to inexpensive hardware implementations
§7.4.1 Simple FIR Digital Filters
Lowpass FIR Digital Filters The simplest lowpass FIR digital filter is the
2-point moving-average filter given by
z
zzzH
2
1)1(
2
1)( 1
0
The above transfer function has a zero at z=-1 and a pole at z=0
Note that here the pole vector has a unity magnitude for all values of ω
§7.4.1 Simple FIR Digital Filters
On the other hand, as ω increases from 0 to π, the magnitude of the zero vector decreases from a value of 2, the diameter of the unit circle, to 0
Hence, the magnitude response |H0(ejω)| is a monotonically decreasing function of ωfrom ω=0 to ω=π
§7.4.1 Simple FIR Digital Filters
The maximum value of the magnitude function is 1 at ω = 0, and the minimum value is 0 at ω = π, i.e.,
0)(,1)( 00
0 jj eHeH
)2/cos()( 2/0 jj eeH
The frequency response of the above filter is given by
§7.4.1 Simple FIR Digital Filters
can be seen to be a monotonically decreasing function of ω
)2/cos()(0 jeH
The magnitude response
§7.4.1 Simple FIR Digital Filters
The frequency ω=ωc at which
)(2
1)( 0
00jj eHeH c
since the dc gain G(0)=20log10|H(ej0)|=0
is of practical interest since here the gain G(ωc ) in dB is given by
dBe
eG
j
jc
c
32log20log20
log20)(
100
10
10
§7.4.1 Simple FIR Digital Filters Thus, the gain G(ω) at ω=ωc is approximately
3 dB less than the gain atω=0
As a result, ωc is called the 3-dB cutoff frequency
To determine the value of ωc we set
2
1)2/(cos)( 22
0 cj ceH
which yields ωc =π/2
§7.4.1 Simple FIR Digital Filters
The 3-dB cutoff frequency ωc can be consid
ered as the passband edge frequency As a result, for the filter H0(z) the passband
width is approximately π/2 Note: H0(z) has a zero at z=-1 or ω=π, which
is in the stopband of the filter
§7.4.1 Simple FIR Digital Filters
A cascade of the simple FIR filter
)1(2
1)( 1
0 zzH
results in an improved lowpass frequency response as illustrated below for a cascade of 3 sections
§7.4.1 Simple FIR Digital Filters
The 3-dB cutoff frequency of a cascade of M sections is given by
)2(cos2 2/11 Mc
For M = 3, the above yields ωc=0.302π
Thus, the cascade of first-order sections yields a sharper magnitude response but at the expense of a decrease in the width of the passband
§7.4.1 Simple FIR Digital Filters A better approximation to the ideal lowpass f
ilter is given by a higher-order moving- average filter
Signals with rapid fluctuations in sample values are generally associated with high- frequency components
These high-frequency components are essentially removed by an moving-average filter resulting in a smoother output
§7.4.1 Simple FIR Digital Filters
Highpass FIR Digital Filters The simplest highpass FIR filter is obtained f
rom the simplest lowpass FIR filter by replacing z with -z
This results in
)1(2
1)( 1
1 zzH
§7.4.1 Simple FIR Digital Filters Corresponding frequency response is given
by)2/sin()( 2/
1 jj jeeH whose magnitude response is plotted below
§7.4.1 Simple FIR Digital Filters
The monotonically increasing behavior of the magnitude function can again be demonstrated by examining the pole-zero pattern of the transfer function H1(z)
The highpass transfer function H1(z) has a zero at z =1 or ω=0 which is in the
§7.4.1 Simple FIR Digital Filters
Improved highpass magnitude response can again be obtained by cascading several sections of the first-order highpass filter
Alternately, a higher-order highpass filter of the form
1
01 )1(1
)(M
n
nn zM
zH
is obtained by replacing z with -z in the transfer in function of a moving average filter
§7.4.1 Simple FIR Digital Filters
An application of the FIR highpass filters is in moving-target-indicator (MTI) radars
In these radars, interfering signals, called clutters, are generated from fixed objects in the path of the radar beam
The clutter, generated mainly from ground echoes and weather returns, has frequency components near zero frequency (dc)
§7.4.1 Simple FIR Digital Filters
The clutter can be removed by filtering the radar return signal through a two-pulse canceler, which is the first-order FIR highpass filter H1(z)=1/2(1-z-1)
For a more effective removal it may be necessary to use a three-pulse canceler obtained by cascading two two-pulse cancelers
§7.4.2 Simple IIR Digital Filters
Lowpass IIR Digital Filters We have shown earlier that the first-order ca
usal IIR transfer function
10,1
)(1
z
KH z
has a lowpass magnitude response for α>0
§7.4.2 Simple IIR Digital Filters
An improved lowpass magnitude response is obtained by adding a factor (1+z-1) to the numerator of transfer function
10,1
)1()(
1
1
z
zKH z
This forces the magnitude response to have a zero at ω=π in the stopband of the
§7.4.2 Simple IIR Digital Filters
On the other hand, the first-order causal IIR transfer function
01,1
)(1
z
KH z
has a highpass magnitude response for α< 0
§7.4.2 Simple IIR Digital Filters
However, the modified transfer function obtained with the addition of a factor (1+z-1) to the numerator
01,1
)1()(
1
1
z
zKH z
exhibits a lowpass magnitude response
§7.4.2 Simple IIR Digital Filters
The modified first-order lowpass transfer function for both positive and negative values of α is then given by
10,1
)1()(
1
1
z
zKH LP z
As ω increases from 0 to π, the magnitude of the zero vector decreases from a value of 2 to 0
§7.4.2 Simple IIR Digital Filters
The maximum values of the magnitude function is 2K/(1-α) at ω=0 and the minimum value is 0 at ω=π, i.e.,
0)(,1
2)( 0
j
LPj
LP eHK
eH
Therefore, |HLP(ejω)| is a monotonically decreasing function of ω from ω=0 to ω=π
§7.4.2 Simple IIR Digital Filters For most applications, it is usual to have a d
c gain of 0 dB, that is to have |HLP(ej0)| =1 To this end, we choose K=(1-α)/2 resulting i
n the first-order IIR lowpass transfer function
10,1
1
2
1)(
1
1
z
zzH LP
The above transfer function has a zero at i.e., at ω=π which is in the stopband
§7.4.2 Simple IIR Digital Filters
Lowpass IIR Digital Filters A first-order causal lowpass IIR digital filter h
as a transfer function given by
1
1
1
1
2
1)(
z
zzH LP
where |α|<1 for stability The above transfer function has a zero at
z =-1 i.e., at ω=π which is in the stopband
§7.4.2 Simple IIR Digital Filters
HLP(z) has a real pole at z =α As ω increases from 0 to π, the magnitude of
the zero vector decreases from a value of 2 to 0, whereas, for a positive value of α, the magnitude of the pole vector increases from a value of 1-α to 1+α
The maximum value of the magnitude function is 1 at ω= 0, and the minimum value is 0 at ω=π
§7.4.2 Simple IIR Digital Filters i.e., |HLP(ej0)| =1, |HLP(ejπ)| =0
Therefore, |HLP(ejω)| is a monotonically decreasing function of ω from ω=0 to ω=π as indicated below
§7.4.2 Simple IIR Digital Filters
The squared magnitude function is given by
)cos21(2
)cos1()1()(
2
22
jLP eH
22
222
)cos21(2
sin)21()1()(
d
eHd jLP
The derivative of |HLP(ejω)|2 with respect to ω is given by
§7.4.2 Simple IIR Digital Filters
d|HLP(ejω)|2/dω≤0 in the range 0≤ω≤π
verifying again the monotonically decreasing behavior of the magnitude function
To determine the 3-dB cutoff frequency we set
2
1)(
2cj
LP eH
in the expression for the square magnitude function resulting in
§7.4.2 Simple IIR Digital Filters
The above quadratic equation can be solved for α yielding two solutions
cc cos21)cos1()1( 22
2
1
)cos21(2
)cos1()1(2
2
c
c
21
2cos
c
or
which when solved yields
§7.4.2 Simple IIR Digital Filters The solution resulting in a stable transfer fun
ction HLP(z) is given by
c
c
cos
sin1
)cos21(2
)cos1()1()(
2
22
jLP eH
It follows from
that HLP(z) is a BR function for |α|<1
§7.4.2 Simple IIR Digital Filters
Highpass IIR Digital Filters A first-order causal highpass IIR digital filter
has a transfer function given by
1
1
1
1
2
1)(
z
zzH HP
where |α|<1 for stability The above transfer function has a zero at z=
1 i.e., at ω=0 which is in the stopband
§7.4.2 Simple IIR Digital Filters Its 3-dB cutoff frequency ωc is given by
cc cos/)sin1( which is the same as that of HLP(z)
Magnitude and gain responses of HHP(z) are shown below
§7.4.2 Simple IIR Digital Filters
HHP(z) is a BR function for |α|<1
Example – Design a first-order highpass digital filter with a 3-dB cutoff frequency of 0.8π
Now, sin(ωc)=sin(0.8π)=0.587785 and cos(0.8
π)=-0.80902 Therefore
α=(1-sinωc)/cosωc=-0.5095245
§7.4.2 Simple IIR Digital Filters
Therefore
1
1
1
1
5095245.01
124538.0
1
1
2
1)(
z
z
z
zzH LP
§7.4.2 Simple IIR Digital Filters
Bandpass IIR Digital Filters A 2nd-order bandpass digital transfer functio
n is given by
]2cos2cos)1(2)1(1[2
)2cos1()1(
)(
2222
2
2
jBP eH
21
2
)1(1
1
2
1)(
zz
zzH BP
Its squared magnitude function is
§7.4.2 Simple IIR Digital Filters
|HBP(ejω)|2 goes to zero at ω=0 and ω=π
It assumes a maximum value of 1 at ω=ω0, called the center frequency of the bandpass filter, where
)(cos 10
The frequencies ωc1 and ωc2 where |HBP(ejω)|2 becomes 1/2 are called the 3-dB cutoff frequencies
§7.4.2 Simple IIR Digital Filters
The difference between the two cutoff frequencies, assumingωc2 >ωc1 is called the 3-dB bandwidth and is given by
21
21 1
2cos
ccwB
The transfer function HBP(z) is a BR function if |α|<1 and |β|<1
§7.4.2 Simple IIR Digital Filters
Plots of |HBP(ejω)| are shown below
§7.4.2 Simple IIR Digital Filters
Example – Design a 2nd order bandpass digital filter with center frequency at 0.4π and a 3-dB bandwidth of 0.1π
Here β=cos(ω0)=cos(0.4π)=0.309017
and9510565.0)1.0cos()cos(
1
22
wB
The solution of the above equation yields:
α=1.376382 and α=0.72654253
§7.4.2 Simple IIR Digital Filters The corresponding transfer functions are
21
2
37638.17343424.01
118819.0)(
zz
zzH BP
21
2
72654253.0533531.01
113673.0)(
zz
zzH BP
and
The poles of H’BP(z) are at z=0.3671712±
j1.11425636 and have a magnitude>1
§7.4.2 Simple IIR Digital Filters
Thus, the poles of H’BP(z) are outside the unit circle making the transfer function unstable
On the other hand, the poles of H”BP(z) are at z= 0.2667655±j0.85095546 and have a magnitude of 0.8523746
Hence H”BP(z) is BIBOstable Later we outline a simpler stability test
§7.4.2 Simple IIR Digital Filters Figures below show the plots of the magnitu
de function and the group delay of H”BP(z)
§7.4.2 Simple IIR Digital Filters
Bandstop IIR Digital Filters A 2nd-order bandstop digital filter has a tran
sfer function given by
21
21
)1(1
21
2
1)(
zz
zzzH BS
The transfer function HBS(z) is a BR function if |α|<1 and |β|<1
§7.4.2 Simple IIR Digital Filters
Its magnitude response is plotted below
§7.4.2 Simple IIR Digital Filters
Here, the magnitude function takes the maximum value of 1 at ω=0 and ω=π
It goes to 0 at ω=ω0 , whereω0, called the notch frequency, is given by
ω0=cos-1(β)
The digital transfer function HBS(z) is more commonly called a notch filter
§7.4.2 Simple IIR Digital Filters
The frequenciesωc2 andωc1 where |HBS(ejω)|2 becomes 1/2 are called the 3-dB cutoff frequencies
The difference between the two cutoff frequencies, assuming ωc2 >ωc1 is called the 3-dB notch bandwidth and is given by
21
21 1
2cos
ccwB
§7.4.2 Simple IIR Digital Filters
Higher-Order IIR Digital Filters By cascading the simple digital filters discus
sed so far, we can implement digital filters with sharper magnitude responses
Consider a cascade of K first-order lowpass sections characterized by the transfer
1
1
1
1
2
1)(
z
zzH LP
§7.4.2 Simple IIR Digital Filters The overall structure has a transfer function
given by K
LP z
zzG
1
1
1
1
2
1)(
K
LP jG
)cos21(2
)cos1()1()(
2
22
The corresponding squared-magnitude function is given by
§7.4.2 Simple IIR Digital Filters To determine the relation between its 3-dB c
utoff frequencyωc and the parameter α, we set
2
1
)cos21(2
)cos1()1(2
2
K
c
c
c
cc
C
CCC
cos1
2sincos)1(1 2
which when solved for α, yields for a stable
GLP(z):
§7.4.2 Simple IIR Digital Filters
for K=1
KKC /)1(2
c
c
cos
sin1
where
It should be noted that the expression for αgiven earilier reduces to
§7.4.2 Simple IIR Digital Filters Example – Design a lowpass filter with a 3-d
B cutoff frequency at ωc=0.4π using a single first-order section and a cascade of 4 first-order sections, and compare their gain responses
For the single first-order lowpass filter we have
1584.0)4.0cos(
)4.0sin(1
cos
sin1
c
c
§7.4.2 Simple IIR Digital Filters For the cascade of 4 first-order sections, we
substitute K=4 and get
6818.122 4/)14(/)1( KKC
251.0)4.0cos(6818.11
)6818.1()6818.1(2)4.0sin()4.0cos(6818.11(1
cos1
2sincos)1(1
2
2
c
cc
C
CCC
Next we compute
§7.4.2 Simple IIR Digital Filters The gain responses of the two filters are sho
wn below As can be seen, cascading has resulted in a
sharper roll-off in the gain response
§7.4.3 Comb Filters
The simple filters discussed so far are characterized either by a single passband and/or a single stopband
There are applications where filters with multiple passbands and stopbands are required
The comb filter is an example of such filters
§7.4.3 Comb Filters In its most general form, a comb filter has a f
requency response that is a periodic function of ω with a period 2π/L, where L is a positive integer
If H(z) is a filter with a single passband and/or a single stopband, a comb filter can be easily generated from it by replacing each delay in its realization with L delays resulting in a structure with a transfer function given by G(z)=H(zL)
§7.4.3 Comb Filters
If |H(ejω)|exhibits a peak at ωp,then |G(ejω)| will exhibit L peaks at ωpk/L, 0≤k≤L-1 in the frequency range 0≤ω<2π
Likewise, if |H(ejω)| has a notch atω0, then |G(ejω)| will have L notches at ω0k/L, 0≤k≤L-1 in the frequency range 0≤ω<2π
A comb filter can be generated from either an FIR or an IIR prototype filter
§7.4.3 Comb Filters For example, the comb filter generated from th
e prototype lowpass FIR filter H0(z)=1/2(1+z-1) has a transfer function
)1(2
1)()( 00
LL zzHzG
|G0(ejω)| has L notchesat ω=(2k+1)π/L and L peaks at ω=2πk/L, 0≤k≤L-1, in the frequency range 0≤ω<2π
§7.4.3 Comb Filters For example, the comb filter generated from t
he prototype highpass FIR filter
H1(z)=1/2(1-z-1) has a transfer function
)1(
1)(
1
zM
zzH
M
|G1(ejω)| has L peaksatω=(2k+1)π/L and L notches at ω=2πk/L, 0≤k≤L-1, in the frequency range 0≤ω<2π
§7.4.3 Comb Filters Depending on applications, comb filters wit
h other types of periodic magnitude responses can be easily generated by appropriately choosing the prototype filter
For example, the M-point moving average filter
)1(
1)(
1
zM
zzH
M
has been used as a prototype
§7.4.3 Comb Filters This filter has a peak magnitude at ω=0, an
d M-1 notches at ω=2πl/M, 1≤l≤M-1 The corresponding comb filter has a transfer
function
)1(
1)(
L
LM
zM
zzG
whose magnitude has L peaks at ω=2πk/L, 0≤k≤L-1 and L(M-1) notches at ω=2πk/LM, 1≤k≤L(M-1)
§7.5 Complementary TransferFunctions
A set of digital transfer functions with complementary characteristics often finds useful applications in practice
Four useful complementary relations are described next along with some applications
§7.5.1 Delay-Complementary Transfer Functions
A set of L transfer functions, {Hi(z)}, 0≤i≤L-1, is defined to be delay-complementary of each other if the sum of their transfer functions is equal to some integer multiple of unit delays, i.e.,
0,)(1
0
0
L
i
ni zzH
where n0 is a nonnegative integer
§7.5.1 Delay-Complementary Transfer Functions
A delay-complementary pair {H0(z), {H1(z)}, can be readily designed if one of the pairs is a known Type 1 FIR transfer function of odd length
Let H0(z) be a Type 1 FIR transfer function of length M=2K+1
Then its delay-complementary transfer function is given by
)()( 01 zHzzH K
§7.5.1 Delay-Complementary Transfer Functions
Let the magnitude response of H0(z) be equal to and 1±δp in the passband and less than or equal to δs in the stopband where δp and δs are very small numbers
Now the frequency response of H0(z) can be expressed as
)()( 00 HeeH jKj
where is the amplitude response)(0 H
§7.5.1 Delay-Complementary Transfer Functions
Its delay-complementary transfer function H1
(z) has a frequency response given by)](1[)()( 011 HeHeeH jKjKj
Now, in the passband, ,1)(1 0 pp H
and in the stopband, ss H )(0
It follows from the above equation that in the stopband, and in the passband,
pp H )(1
ss H 1)(1 1
§7.5.1 Delay-Complementary Transfer Functions
As a result, H1(z) has a complementary magnitude response characteristic to that of H0(z) with a stopband exactly identical to the passband of H0(z) , and a passband that is exactly identical to the stopband of H0(z)
Thus, if H0(z) is a lowpass filter, H1(z) will be a highpass filter, and vice versa
§7.5.1 Delay-Complementary Transfer Functions
The frequency ω0 at which
5.0)()( 10 HH
the gain responses of both filters are 6 dB below their maximum values
The frequency ω0 is thus called the 6 dB crossover frequency
§7.5.1 Delay-Complementary Transfer Functions
Example – Consider the Type 1 bandstop transfer function
)45541()1(64
1)( 121084242 zzzzzzzH BS
)45541()1(64
1)()(
121084242
10
zzzzzz
zHzzH BSBP
Its delay-complementary Type 1 bandpass transfer function is given by
§7.5.1 Delay-Complementary Transfer Functions
Plots of the magnitude responses of HBS(z) and HBP(z) are shown below
§7.5.2 Allpass Complementary Transfer Functions
A set of M digital transfer functions, {Hi(z)}, 0≤i≤M-1, is defined to be allpass-complementary of each other, if the sum of their transfer functions is equal to an allpass function, i.e.,
1
0
)()(M
ii zAzH
§7.5.3 Power-Complementary Transfer Functions
A set of M digital transfer functions, {Hi(z)}, 0≤i≤M-1, is defined to be power-complementary of each other, if the sum of their square-magnitude responses is equal to a constant K for all values of ω, i.e.,
1
0
2allfor,)(
M
i
ji KeH
§7.5.3 Power-Complementary Transfer Functions
By analytic continuation, the above property is equal to
1
0
1 allfor,)()(M
iii KzHzH
for real coefficient Hi(z) Usually, by scaling the transfer functions, the
power-complementary property is defined for K=1
§7.5.3 Power-Complementary Transfer Functions
For a pair of power-complementary transfer functions, H0(z) and H1(z), the frequency ω0 where |H0(ejω0)|2= |H1(ejω0)|2 =0.5, is called the cross-over frequency
At this frequency the gain responses of both filters are 3-dB below their maximum values
As a result, ω0 is called the 3-dB cross-over frequency
§7.5.3 Power-Complementary Transfer Functions
Example – Consider the two transfer functions H0(z) and H1(z) given by
)]()([2
1)(
)]()([2
1)(
101
100
zAzAzH
zAzAzH
where A0(z) ana A1(z) are stable allpass transfer functions
Note that H0(z)+H1(z)=A0(z) Hence, H0(z) and H1(z) are allpass complement
ary
§7.5.3 Power-Complementary Transfer Functions
It can be shown that H0(z) and H1(z) are also power-complementary
Moreover, H0(z) and H1(z) are bounded-real transfer functions
§7.5.4 Double-Complementary Transfer Functions
A set of M transfer functions satisfying both the allpass complementary and the power- complementary properties is known as a doubly-complementary set
A pair of doubly-complementary IIR transfer functions, H0(z) and H1(z), with a sum of allpass decomposition can be simply realized as indicated below
§7.5.4 Double-Complementary Transfer Functions
)(
)()(
)(
)()( 1
10
0 zX
zYzH
zX
zYzH
A0(z)
A1(z)
●
●
●
X(z)Y0(z)
Y1(z)
1/2
-1
§7.5.4 Double-Complementary Transfer Functions
Example – The first-order lowpass transfer function
1
1
1
1
2
1)(
z
zzH LP
)]()([2
1
11
2
1)( 101
1
zAzAz
zzH LP
1
1
10 1)(,1)(
z
zzAzA
can be expressed as
where
§7.5.4 Double-Complementary Transfer Functions
Its power-complementary highpass transfer function is thus given by
1
1
1
1
10
11
21
11
21)]()([
21)(
zz
zzzAzAzHLP
The above expression is precisely the first- order highpass transfer function described earlier
Complementary TransferFunctions
Figure below demonstrates the allpass complementary property and the power complementary property of HLP(z) and HHP(z)
§7.5.5 Power-Symmetric Filtersand Conjugate Quadrature Filters
A real-coefficient causal digital filter with a transfer function H(z) is said to be a power- symmetric filter if it satisfies the condition
KzHzHzHzH )()()()( 11
where K>0 is aconstant
§7.5.5 Power-Symmetric Filtersand Conjugate Quadrature Filters
It can be shown that the gain function G(ω) of a power-symmetric transfer function at ω=π is given by
dBK 3log10 10
constanta)()()()( 11 zGzGzHzH
If we defined G(z)= H(-z), then it follows from the definition of the power-symmetric filter that H(z) and G(z) are power- complementary as
§7.5.5 Power-Symmetric Filtersand Conjugate Quadrature Filters
Conjugate Quadratic Filter If a power-symmetric filter has an FIR transf
er function H(z) of order N, then the FIR digital filter with a transfer function
)()( 1 zHzzG N
is called a conjugate quadratic filter of H(z) and vice-versa
§7.5.5 Power-Symmetric Filtersand Conjugate Quadrature Filters
It follows from the definition that G(z) is also a power-symmetric causal filter
It also can be seen that a pair of conjugate quadratic filters H(z) and G(z) are also power-complementary
§7.5.5 Power-Symmetric Filtersand Conjugate Quadrature Filters
Example – Let H(z)=1-2z-1+6z-2+3z-3 We form
100)345043()345043(
)3621)(3621()3621)(3621(
)()()()(
313
313
32321
32321
11
zzzzzzzz
zzzzzzzzzzzz
zHzHzHzH
H(z) is a power-symmetric transfer function
§7.8 Digital Two-Pairs
The LTI discrete-time systems considered so far are single-input, single-output structures characterized by a transfer function
Often, such a system can be efficiently realized by interconnecting two-input, two- output structures, more commonly called two-pairs
§7.8 Digital Two-Pairs
Figures below show two commonly used block diagram representations of a two-pair
Here Y1 and Y2 denote the two outputs X1 and X2 denote the two inputs,where the dependencies on the variable z has been omitted for simplicity
X1
Y1
Y2
X2
X1
X2
Y1
Y2
§7.8.1 Characterization The input-output relation of a digital two- pai
r is given by
2
1
2221
1211
2
1
X
X
tt
tt
Y
Y
2221
1211
tt
tt
In the above relation the matrix τ given by
is called the transfer matrix of the two-pair
§7.8.1 Characterization
It follows from the input-output relation that the transfer parameters can be found as follows:
02
2220
1
221
02
1120
1
111
12
12
,
,
XX
XX
XY
tXY
t
XY
tXY
t
§7.8.1 Characterization
An alternate characterization of the two-pair is in terms of its chain parameters
2
2
1
1
X
Y
DC
BA
Y
X
DC
BA
where the matrix Γ given by
is called the chain matrix of the two-pair
§7.8.1 Characterization
The relation between the transfer parameters and the chain parameters are given by
21
22112112
21
11
21
22
21
22211211
,,,1
,1,,
ttttt
Dtt
Ctt
Bt
A
ACt
At
ABCADt
ACt
§7.8.2 Two-Pair InterconnectionSchemes
Cascade Connection – Γ-cascade
2
2
1
1
2
2
1
1
X
Y
DC
BA
Y
X
X
Y
DC
BA
Y
X
DC
BA
DC
BA1X
1Y 2Y
2X
2Y 1X
1Y 2X Here
§7.8.2 Two-Pair InterconnectionSchemes
But from figure, X”1=Y’
2 and Y”1=X’
2 Substituting the above relations in the first e
quation on the previous slide and combining the two equations we get
2
2
1
1
X
Y
DC
BA
DC
BA
Y
X
DC
BA
DC
BA
DC
BA
Hence
§7.8.2 Two-Pair InterconnectionSchemes
Cascade Connection –τ-cascade
2
1
2221
1211
2
1
2
1
2221
1211
2
1
X
X
tt
tt
Y
Y
X
X
tt
tt
Y
Y Here
1X 1Y
2Y 2X
2Y
1X 1Y
2X
2221
1211
tt
tt
2221
1211
tt
tt
§7.8.2 Two-Pair InterconnectionSchemes
But from figure, X”1=Y’
1 and X’2 = Y”
2
Substituting the above relations in the first equation on the previous slide and combining the two equations we get
2
1
2221
1211
2221
1211
2
1
X
X
tt
tt
tt
tt
Y
Y
2221
1211
2221
1211
2221
1211
tt
tt
tt
tt
tt
tt
Hence,
§7.8.2 Two-Pair InterconnectionSchemes
Constrained Two-Pair
)(1)(
)()(
)(
22
211211
1
1
zGtzGtt
t
zGBAzGDC
XY
zH
It can be shown that
Y2
G(z)X1
X2Y1H(z)
§7.9 Algebraic Stability Test
We have shown that the BIBO stability of a causal rational transfer function requires that all its poles be inside the unit circle
For very high-order transfer functions, it is very difficult to determine the pole locations analytically
Root locations can of course be determined on a computer by some type of root finding algorithms
§7.9.1 The Stability Triangle
We now outline a simple algebraic test that does not require the determination of pole locations
The Stability Triangle For a 2nd-order transfer function the stability
can be easily checked by examining its denominator coefficients
§7.9.1 The Stability Triangle
denote the denominator of the transfer function
In terms of its poles, D(z) can be expressed as
22
111)( zdzdzD
221
121
12
11 )(1)1)(1()( zzzzzD
212211 ,)( dd
Let
Comparing the last two equations we get
§7.9.1 The Stability Triangle
Now the coefficient d2 is given by the product of the poles
Hence we must have
1,1 21
12 d
21 1 dd
The poles are inside the unit circle if
It can be shown that the second coefficient condition is given by
§7.9.1 The Stability Triangle
The region in the (d1, d2)-plane where the two coefficient condition are satisfied, called the stability triangle, is shown below
§7.9.1 The Stability Triangle
Example – Consider the two 2nd-order bandpass transfer functions designed earlier:
21
2
37638.17343424.01118819.0)(
zzzzHBP
21
2
72654253.0533531.01113673.0)(
zzzzHBP
§7.9.1 The Stability Triangle
In the case of H’BP(z), we observe that
d1=-0.7343424, d2=1.3763819 Since here |d2|>1, H’
BP(z) is unstable On the other hand, in the case of H”
BP(z), we observe that
d1=-0.53353098, d2=0.726542528 Here, |d2|<1 and |d1|<1+ d2, and hence H”
BP(z) is BIBO stable
§7.9.2 A Stability Test Procedure
Let DM(z) denote the denominator of an M-th order causal IIR transfer function H(z):
iM
i iM zdzD
0)(
)()(
)(1
zDzDz
zAM
MM
M
where we assume d2=1 for simplicity Defined an M-th order allpass transfer functi
on:
§7.9.2 A Stability Test Procedure
then it follows that
MM
MM
MMMMM
M zdzdzdzdzzdzdzdd
zA
1
12
21
1
12
21
1
1)(
M
i iM zzD1
1)1()(
M
i iM
Md1
)1(
Or, equivalently
If we express
§7.9.2 A Stability Test Procedure
Now for stability we must have, |λi|<1, which implies the condition |dM|<1
Define
kM= AM(∞)<1= dM
Then a necessary condition for stability of AM
(z), and hence,the transfer function H(z) is given by
12 Mk
§7.9.2 A Stability Test Procedure
Assume the above condition holds We now form a new function
)(1
)()(1
)()(1 zAd
dzAz
zAkkzA
zzAMM
MM
MM
MMM
)1(1
)2(2
11
)1()2(1
12
1 1)(
MM
MM
MMMM
M zdzdzdzzdzdd
zA
Substituting the rational form of AM(z) in the above equation we get
§7.9.2 A Stability Test Procedure
Hence, AM-1(z) is an allpass function of order M-1
Now the poles λo of AM-1(z) are given by the roots of the equation
11,1 2
Mid
dddid
M
iMMi
MoM k
A 1)(
where
§7.9.2 A Stability Test Procedure
If AM(z) is a stable allpass function, then
Hence 1)( oMA By assumption 12 Mk
Thus, if AM(z) is a stable allpass function, then the condition holds only if |λo | <1
1for,1
1for,1
1for,1
)(
z
z
z
zAM
§7.9.2 A Stability Test Procedure
Thus, if AM(z) is a stable allpass function and , then AM-1(z) is also a stable allpass function of one order lower
12 Mk
We now prove the converse, i.e., if AM-1(z) is a stable allpass function and , then AM
(z) is also a stable allpass function
12 Mk
Or, in other words AM-1(z) is a stable allpass function
§7.9.2 A Stability Test Procedure
To this end, we express AM(z) in terms of A
M-1(z) arriving at
)(1)(
)(1
11
1
zAzkzAzk
zAMM
MMM
MM k
A 1)( 011
0
By assumption holds12 Mk
If ζ0 is a pole of AM(z) , then
§7.9.2 A Stability Test Procedure
|AM-1(ζ0)|>|ζ0|
The above condition implies |AM-1(ζ0)|>1 if |ζ0|≥1
Assume AM-1(z) is a stable allpass function
Then AM-1(z) ≤1 for |z|≥1
Thus, for |ζ0|≥1, we should have
|AM-1(ζ0)| ≤1
Therefore, i.e., 1)( 011
0 MA
§7.9.2 A Stability Test Procedure
Thus there is a contradiction On the other hand, if |ζ0|<1 then from
AM-1(z) >1 for |z|<1
we have |AM-1(ζ0)|>1 The above condition does not violate the co
ndition |AM-1(ζ0)|>|ζ0|
§7.9.2 A Stability Test Procedure
Summarizing, a necessary and sufficient set of conditions for the causal allpass function AM-1(z) to be stable is therefore:
Thus, if and if AM-1(z) is a stable allpass function, then AM(z) is also a stable allpass function
12 Mk
(1) , and
(2) The allpass function AM-1(z) is stable
12 Mk
§7.9.2 A Stability Test Procedure Thus, once we have checked the condition
, we test next for the stability of the lower-order allpass function AM-1(z)
12 Mk
The process is then repeated, generating a set of coefficients:
and a set of allpass functions of decreasing order:
121 ,,...,, kkkk MM
1)(),(),(...,),(),( 0121 zAzAzAzAzA MM
§7.9.2 A Stability Test Procedure The allpass function AM(z) is atable if and onl
y if for i12 ik
12341)( 234
zzzz
zH
432
23
14
12
23
34
4
234
234
4
1
41
41
21
43
143
21
41
41
)(dzdzdzdz
zdzdzdzd
zzzz
zzzzzA
141)( 444 dAk Note:
Example – Test the stability of
From H(z) we generate a 4-th order allpass function
§7.9.2 A Stability Test Procedure
we determine the coefficients {d’i} of the thir
d-order allpass function A3(z) from the dcoefficients { d’
i} of A4(z):
31,1 2
4
44
iddddi
d ii
151
52
1511
11511
52
151
11
)(23
23
32
23
1
11
22
33
3
zzz
zzz
zdzdzdzdzdzd
zA
Using
§7.9.2 A Stability Test Procedure
Following the above procedure, we derive the next two lower-order allpass functions:
1151)( 333 dAk
10153
110153
)(
22479
224159
1224159
22479
)( 12
2
2
z
zzA
zz
zzzA
Note:
§7.9.2 A Stability Test Procedure
Since all of the stability conditions are satisfied, A4(z) and hence H(z) are stable
Note: It is not necessary to derive A1(z) since A2(z) can be tested for stability using the coefficient conditions
110153)(1
122479)(
12
22
Ak
Ak
Note: