Linear-Phase FIR Transfer Functions

(1)

Functions 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 impulse response h[n]

Functions Functions

• Let

• If H(z)is to have a linear-phase, its frequency response must be of the form where cand βare constants, and , called the amplitude response, also called the zero-phase response, is a real function of ω

= ∑

= N − n

z n

n h z H

0

] [ ) (

) ( )

(e^ω =e ⁽^ω⁺^β⁾H ω H ^j ^j^c (

(ω) H(

Linear

Linear- -Phase FIR Transfer Phase FIR Transfer Functions

Functions

• For a real impulse response, the magnitude response is an evenfunction of ω, i.e.,

• Since , the amplitude response is then either an evenfunction or an oddfunction of ω, i.e.

| ) (

|H e^j^ω

| ) (

|

| ) (

|H e^j^ω =H e⁻^j^ω

| ) (

|

| ) (

|H e^j^ω = H( ω

) ( )

(−ω =±H ω

H( (

Linear

Linear- -Phase FIR Transfer Phase FIR Transfer Functions

Functions

• The frequency response satisfies the relation or, equivalently, the relation

• If is an even function, then the above relation leads to

implying that eitherβ= 0orβ= π )

*( )

(e^j^ω =H e⁻^j^ω H

) ( )

( ⁽ ⁾

)

(^ω⁺^βH ω =e⁻ ⁻ ^ω⁺^βH −ω e^j^c ( ^j ^c ( (ω)

H(

β

− β= ^j

j e

e

Linear

Linear- -Phase FIR Transfer Phase FIR Transfer Functions

Functions

• From we have

• Substituting the value of βin the above we get

) ( )

(e^ω =e ⁽^ω⁺^β⁾H ω H ^j ^j^c (

) ( )

(ω =e⁻^j⁽^c^ω⁺^β⁾He^j^ω H(

±∑

=

±

=

ω =

+ ω

− ω

ω

− N

n

n c j j

jc H e hne

e H

0

)

] (

[ )

( )

((

Linear

Linear- -Phase FIR Transfer Phase FIR Transfer Functions

Functions

• Replacing ωwith in the previous equation we get

• Making a change of variable , we rewrite the above equation as

±∑

= ω

−

=

+ N ω

c

ej

h H

0

)

] (

[ ) (

l

l l

(

ω

−

n N− l=

∑ −

±

= ω

− =

− + N ω

n

n N c

ej

n N h H

0

)

] (

[ )

((

(2)

Functions Functions

• As , we have

• The above leads to the condition with

• Thus, the FIR filter with an even amplitude responsewill have a linear phase if it has a symmetric impulse response

) ( ) (ω =H −ω H( (

) ( )

( [ ]

]

[ne ^j ^c ⁿ h N ne^j ^c ^N ⁿ h ⁻ ^ω ⁺ = − ^ω ⁺ ⁻

N n n

N h n

h[ ]= [ − ], 0≤ ≤ 2

/ N c=−

Functions Functions

• If is an odd function of ω, then from

we get as

• The above is satisfied if β= π/2orβ

• Then reduces to

2 π/

−

= (ω)

H(

) ( )

( ⁽ ⁾

)

(^ω⁺^βH ω =e⁻ ⁻ ^ω⁺^βH −ω e^j^c ( ^j ^c (

β

− β=− ^j

j e

e

) ( )

(e^ω =e⁽^ω⁺^β⁾H ω H ^j ^j^c (

) ( )

(e^ω =je ^ωH ω H ^j ^jc (

) ( ) (−ω =−H ω

H( (

Linear

Linear- -Phase FIR Transfer Phase FIR Transfer Functions

Functions

• The last equation can be rewritten as

• As , from the above we get

) ( 0

] [ )

( )

( ^N ^j ^c ⁿ

n j

jc H e j h n e

je

H ⁻ ^ω ⁺

= ω ω

− =− ∑

−

= ( ω

) ( 0

] [ )

( ^l

l l

( ω +

∑=

= ω

− j ^Nh e^j ^c H

) ( ) (−ω =−H ω

H( (

Linear

Linear- -Phase FIR Transfer Phase FIR Transfer Functions

Functions

• Making a change of variable we rewrite the last equation as

• Equating the above with

we arrive at the condition for linear phase as

) ( 0

] [ )

( ^l

l l

( ω +

∑=

= ω

− j^Nh e^j ^c H

n N− l=

) ( 0

] [ )

( ^j ^c ⁿ

N n

e n h j

H ⁻ ^ω ⁺

∑=

−

= ( ω

Linear

Linear- -Phase FIR Transfer Phase FIR Transfer Functions

Functions

with

• Therefore, a FIR filter with an odd amplitude response will have linear-phase response if it has an antisymmetric impulse response

N n n

N h n

h[ ]= [ − ], 0≤ ≤ 2

/ N c=−

Linear

Linear- -Phase FIR Transfer Phase FIR Transfer Functions

Functions

• Since the length of the impulse response can be either even or odd, we can define four typesof linear-phase FIR transfer functions

• For an antisymmetric FIR filter of odd length, i.e.,Neven

h[N/2] = 0

• We examine next the each of the 4cases

(3)

Functions Functions

Type 1: N= 8 Type 2: N= 7

Type 3: N= 8 Type 4: N= 7 14

Functions Functions

Type 1: Symmetric Impulse Response with Odd Length

• In this case, the degreeNis even

• AssumeN= 8 for simplicity

• The transfer functionH(z)is given by

3 2

1 2 3

1

0 + ⁻ + ⁻ + ⁻

=h h z h z h z z

H( ) [ ] [] [ ] [ ]

8 7

6 5

4 5 6 7 8

4 ⁻ + ⁻ + ⁻ + ⁻ + ⁻

+h[ ]z h[ ]z h[ ]z h[ ]z h[ ]z

Linear

Linear- -Phase FIR Transfer Phase FIR Transfer Functions

Functions

• Because of symmetry, we haveh[0] = h[8], h[1] = h[7],h[2] = h[6], andh[3] = h[5]

• Thus, we can write

) ](

[ ) ](

[ )

(z =h0 1+z⁻⁸ +h1 z⁻¹+z⁻⁷ H

4 5

3 6

2 3 4

2 ⁻ + ⁻ + ⁻ + ⁻ + ⁻

+h[ ](z z ) h[ ](z z ) h[ ]z ) ](

[ ) ](

[

{ ⁴ ⁴ ³ ³

4 0 ⁻ 1 ⁻

− + + +

=z h z z h z z

]}

[ ) ](

[2 z² z ² h3 z z ¹ h4

h + + + +

+ ⁻ ⁻

Linear

Linear- -Phase FIR Transfer Phase FIR Transfer Functions

Functions

• The corresponding frequency response is then given by

• The quantity inside the braces is a real function ofω, and can assume positive or negative values in the range0≤ω≤π

) 3 cos(

] 1 [ 2 ) 4 cos(

] 0 [ 2 { )

(e^ω =e⁻ ⁴^ω h ω + h ω

H ^j ^j

]}

4 [ ) cos(

] 3 [ 2 ) 2 cos(

] 2 [

2h ω + h ω +h

+

Linear

Linear- -Phase FIR Transfer Phase FIR Transfer Functions

Functions

• The phase function here is given by whereβis either0 orπ, and hence, it is a linear function ofω

• The group delay is given by

indicating a constant group delay of4 samples β

+ ω

−

= ω θ( ) 4

4 )

(ω =− ⁽ ⁾= τ ^θ_ω^ω

d d

Linear

Linear- -Phase FIR Transfer Phase FIR Transfer Functions

Functions

• In the general case for Type 1 FIR filters, the frequency response is of the form where theamplitude response , also called thezero-phase response, is of the form

) ( )

(e^ω =e⁻ ^ω^/²H ω

H ^j ^jN (

(ω) H(

(ω)

H( = + ∑ − ω

= 2 /

1 2

2] 2 [ ]cos( )

[ ^N

n N

N h n n

h

(4)

Linear

Functions Functions

• Example -Consider

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

] [

)

( ⁶

2 5 1 4 3 2 1 2 1 6 0 1

−

− + + + + +

+

= z z z z z z

z H

Linear

Functions Functions

• A plot of the magnitude response of along with that of the 7-point moving- average filter is shown below

) (z H₀

0 0.2 0.4 0.6 0.8 1

ω/π

Magnitude

modified filter moving-average

Linear

Linear- -Phase FIR Transfer Phase FIR Transfer Functions

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 an express

which is seen to be a cascade of a2-point MA filter with a6-point MA filter

• Thus, has a double zero at , i.e., (ω= π)

) (

) ( )

( ¹ ² ³ ⁴ ⁵

6 1 1 2

0 z =11+z⁻ ⋅ 1+z⁻ +z⁻ +z⁻ +z⁻ +z⁻ H

−1

= ) z

H₀(z

Linear

Linear- -Phase FIR Transfer Phase FIR Transfer Functions

Functions

Type 2: Symmetric Impulse Response with Even Length

• In this case, the degreeNis odd

• The transfer function is of the form

3 2

1 2 3

1

0 + ⁻ + ⁻ + ⁻

=h h z h z h z z

H( ) [ ] [] [ ] [ ]

7 6

5

4 5 6 7

4 ⁻ + ⁻ + ⁻ + ⁻

+h[ ]z h[ ]z h[ ]z h[ ]z

Linear

Linear- -Phase FIR Transfer Phase FIR Transfer Functions

Functions

• Making use of the symmetry of the impulse response coefficients, the transfer function can be written as

) ](

[ ) ](

[ )

(z =h0 1+z⁻⁷ +h1 z⁻¹+z⁻⁶ H

) ](

[ ) ](

[2 ⁻²+ ⁻⁵ + 3 ⁻³+ ⁻⁴

+h z z h z z

) ](

[ ) ](

[

{ ^/ ^/ ^/ ^/

/2 7 2 7 2 5 2 5 2

7 0 ⁻ 1 ⁻

− + + +

=z h z z h z z

)}

](

[ ) ](

[2 ³^/²+ ⁻³^/² + 3 ¹^/²+ ⁻¹^/²

+h z z h z z

Linear

Linear- -Phase FIR Transfer Phase FIR Transfer Functions

Functions

• The corresponding frequency response is given by

• As before, the quantity inside the braces is a real function ofω, and can assume positive or negative values in the range

) cos(

] 1 [ 2 ) cos(

] 0 [ 2 { )

(e ^ω =e⁻ ⁷^ω^/² h ⁷₂^ω + h ⁵₂^ω

H ^j ^j

)}

cos(

] 3 [ 2 ) cos(

] 2 [

2 ³₂^ω + ^ω₂

+ h h

π

≤ ω

≤ 0

(5)

Functions Functions

• Here the phase function is given by where againβis either0 orπ

• As a result, the phase is also a linear function ofω

• The corresponding group delay is indicating a group delay of samples

β + ω

−

= ω θ( ) ⁷₂

2 7 2

) 7

(ω = τ

Functions Functions

• The expression for the frequency response in the general case for Type 2 FIR filters is of the form

where the amplitude response is given by (ω)

H( = ⁺∑ − ω −

= 2 + / ) 1 (

1 2

1 ]cos( ( )) [

2

N n

N n n

h

) ( )

(e^ω =e⁻ ^ω^/²H ω

H ^j ^jN (

Linear

Linear- -Phase FIR Transfer Phase FIR Transfer Functions

Functions

Type 3: Antiymmetric Impulse Response with Odd Length

• Applying the symmetry condition we get ) ](

[ ) ](

[ { )

(z =z⁻⁴ h0 z⁴−z⁻⁴ +h1 z³−z⁻³ H

)}

](

[ ) ](

[2 ²− ⁻² + 3 − ⁻¹ +h z z h z z

Linear

Linear- -Phase FIR Transfer Phase FIR Transfer Functions

Functions

• It also exhibits a linear phase response given by

whereβis either0 orπ

) 3 sin(

] 1 [ 2 ) 4 sin(

] 0 [ 2 { )

(e^ω =e⁻ ⁴^ωe^π^/² h ω + h ω

H ^j ^j ^j

)}

sin(

] 3 [ 2 ) 2 sin(

] 2 [

2 ω + ω

+ h h

β + + ω

−

= ω

θ( ) 4 ^π₂

Linear

Linear- -Phase FIR Transfer Phase FIR Transfer Functions

Functions

• The group delay here is

indicating a constant group delay of4 samples

• In the general case

where the amplitude response is of the form 4

) (ω = τ

) ( )

(e^ω = je⁻ ^ω^/²H ω

H ^j ^jN (

(ω)

H( = ∑ − ω

= 2 /

1 [2 ]sin( ) 2

N n

N n n

h

Linear

Linear- -Phase FIR Transfer Phase FIR Transfer Functions

Functions

Type 4: Antiymmetric Impulse Response with Even Length

• Applying the symmetry condition we get ) ](

[ ) ](

[ { )

(z =z⁻⁷^/² h0 z⁷^/²−z⁻⁷^/² +h1 z⁵^/²−z⁻⁵^/² H

)}

](

[ ) ](

[2 ³^/²− ⁻³^/² + 3 ¹^/²− ⁻¹^/²

+h z z h z z

(6)

Linear

Functions Functions

• It again exhibits a linear phase response given by

whereβis either0 orπ

) sin(

] 1 [ 2 ) sin(

] 0 [ 2 { )

(e^ω =e⁻ ⁷^ω^/²e^π^/² h ⁷₂^ω + h ⁵₂^ω

H ^j ^j ^j

)}

sin(

] 3 [ 2 ) sin(

] 2 [

2 ³₂^ω + ^ω₂

+ h h

β + + ω

−

= ω

θ( ) ⁷₂ ^π₂

Linear

Functions Functions

• The group delay is constant and is given by

• In the general case we have

where now the amplitude response is of the form

2

) 7

(ω = τ

) ( )

(e^ω =je⁻ ^ω^/²H ω

H ^j ^jN (

(ω)

H( = ∑⁺ − ω −

= 2 + / ) 1 (

1 2

1 ]sin( ( )) [

2

N n

N n n

h

Linear

Linear- -Phase FIR Transfer Phase FIR Transfer Functions

Functions

General Form of Frequency Response

• In each of the four types of linear-phase FIR filters, the frequency response is of the form

• The amplitude response for each of the four types of linear-phase FIR filters can become negative over certain frequency ranges, typically in the stopband

) ( )

(e^ω =e⁻ ^ω^/²e^βH ω

H ^j ^jN ^j (

(ω) H(

Linear

Linear- -Phase FIR Transfer Phase FIR Transfer Functions

Functions

• Example–Consider the causal Type 1 FIR transfer function

• Its amplitude and phase responses are given by

6 5 4 3 2

1(z)=−1+2z⁻1−3z⁻ +6z⁻ −3z⁻ +2z⁻ −z⁻ H

) 3 cos(

2 ) 2 cos(

4 ) cos(

6 6 )

1(ω = − ω + ω − ω

H(

ω

−

= ω θ₁( ) 3

Linear

Linear- -Phase FIR Transfer Phase FIR Transfer Functions

Functions

• Next, consider the causal Type 1 FIR transfer function

6 5 4 3 2

2(z)=1−2z⁻1+3z⁻ −6z⁻ +3z⁻ −2z⁻ +z⁻ H

) ( )

( ₁

2 ω =−H ω

H( (

π + ω

−

= ω θ₂( ) 3

Linear

Linear- -Phase FIR Transfer Phase FIR Transfer Functions

Functions

• Note:

6 5 4 3 2 1

2(z)=1−2z⁻ +3z⁻ −6z⁻ +3z⁻ −2z⁻ +z⁻ H

) ( )

( ₁

2 ω =−H ω

H( (

π + ω

−

= ω θ₂( ) 3

| ) (

|

| ) (

|H₁e^j^ω = H₂ e^j^ω

(7)

Functions Functions

• Hence, and have identical magnitude responses but phase responses differing by πas shown below

)

1(z

H H₂(z)

0 0.2 0.4 0.6 0.8 1

0 5 10 15 20

ω/π

Amplitude

Amplitude response of H₁(z)

0 0.2 0.4 0.6 0.8 1

-10 -5 0 5

ω/π

Phase, radians

Phase responses of H₁(z) and H₂(z)

H₁(z) H2(z)

Functions Functions

• Example–Consider the causal Type 1 FIR transfer function

6 5 4 2

3(z)=1−2z⁻1+3z⁻ −3z⁻ +2z⁻ −z⁻ H

) 3 sin(

2 ) 2 sin(

4 ) sin(

6 )

3(ω =− ω + ω + ω

H(

3(ω)=−3ω+2^π

θ

Linear

Linear- -Phase FIR Transfer Phase FIR Transfer Functions

Functions

• Note:

6 5 4 2

4(z)=−1+2z⁻1−3z⁻ +3z⁻ −2z⁻ +z⁻ H

) ( )

( ₃

4 ω =−H ω

H( (

4(ω)=−3ω−^π2

θ

| ) (

|

| ) (

|H₃ e^j^ω =H₄ e^j^ω

Linear

Linear- -Phase FIR Transfer Phase FIR Transfer Functions

Functions

• Hence, and have identical magnitude responses but phase responses differing by πas shown below

)

3(z

H H₄(z)

0 0.2 0.4 0.6 0.8 1

-15 -10 -5 0 5

ω/π

Phase, radians

Phase responses of H 3(z) and H

4(z)

H₃(z)

H₄(z)

0 0.2 0.4 0.6 0.8 1

0 2 4 6 8 10

ω/π

Amplitude

Amplitude response of H 3(z)

Linear

Linear- -Phase FIR Transfer Phase FIR Transfer Functions

Functions

• The magnitude and phase responses of the linear-phase FIR are given by

) (

| ) (

|H e^j^ω =H( ω

⎪⎩

⎪⎨

⎧

= ω θ( )

0 ) ( for ,

2 2

<

ω π

− β +

−

≥ ω β

+

−

ω ω

H H

N N

( (

Linear

Linear- -Phase FIR Transfer Phase FIR Transfer Functions

Functions

• Thegroup delayin each case is

• Note that, even though the group delay is constant, since in general is not a constant, the output waveform is not a replica of the input waveform

) 2

(ω =^N τ

| ) (

|H e^j^ω

(8)

Linear

Linear- -Phase FIR Transfer Phase FIR Transfer Functions

Functions

• Note that, even though the group delay is constant, since in general is not a constant, the output waveform is not a replica of the input waveform

• An FIR filter with a frequency response that is a real function of ωis often called a zero- phase filter

• Such a filter must have a noncausal impulse response

Zero Locations of Linear Phase FIR Transfer Functions Phase FIR Transfer Functions

• Consider first an FIR filter with a symmetric impulse response:

• Its transfer function can be written as

• By making a change of variable , we can write

∑

=

−

=

− = −

= ^N

n N n

n

n h N n z

z n h z H

0 0

] [ ]

[ ) (

] [ ]

[n h N n

h = −

n N m= −

∑

∑ =

−

=

+

−

=

− = =

− ^N

m m N N

m

m N N

n

n hmz z hmz

z n N h

0 0

0

] [ ]

[ ]

[

Zero Locations of Linear Zero Locations of Linear- - Phase FIR Transfer Functions Phase FIR Transfer Functions

• But,

• Hence for an FIR filter with a symmetric impulse response of length N+1we have

• A real-coefficient polynomialH(z) satisfying the above condition is called a mirror-image polynomial(MIP)

) ( )

(z =z⁻ H z⁻¹

H ^N

) ( ]

[ ¹

0

= = −

∑

^Nm ^h^m^z ^H ^z m

Zero Locations of Linear Zero Locations of Linear- - Phase FIR Transfer Functions Phase FIR Transfer Functions

• Now consider first an FIR filter with an antisymmetric impulse response:

• Its transfer function can be written as

• By making a change of variable , we get

] [ ]

[n h N n

h =− −

∑

=

−

=

− =− −

= ^N

n N n

n

n h N n z

z n h z H

0 0

] [ ]

[ ) (

) ( ]

[ ]

[ ¹

0 0

−

=

+

−

=

− =− =−

−

−∑^h^N ⁿ^z ∑^N^h^m^z ^z ^N^H ^z

m

m N N

n

n N m= −

Zero Locations of Linear Zero Locations of Linear- - Phase FIR Transfer Functions Phase FIR Transfer Functions

• Hence, the transfer functionH(z) of an FIR filter with an antisymmetric impulse response satisfies the condition

• A real-coefficient polynomialH(z) satisfying the above condition is called a antimirror-image polynomial(AIP)

) ( )

(z =−z⁻ H z⁻¹

H ^N

Zero Locations of Linear Zero Locations of Linear- - Phase FIR Transfer Functions Phase FIR Transfer Functions

• It follows from the relation that if is a zero ofH(z), so is

• Moreover, for an FIR filter with a real impulse response, the zeros ofH(z) occur in complex conjugate pairs

• Hence, a zero at is associated with a zero at

) ( )

(z =±z⁻ H z⁻¹

H ^N

z=ξo

z=1/ξo

*o

z=ξ

(9)

Phase FIR Transfer Functions 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

• A zero on the unit circle appear as a pair as its reciprocal is also its complex conjugate

φ,

=re±^j

z z=¹_re^±^j^φ

φ

=e±^j

z

Phase FIR Transfer Functions Phase FIR Transfer Functions

• Since a zero at is its own reciprocal, it can appear only singly

• Now a Type 2FIR filter satisfies with degreeNodd

• Hence

implying , i.e.,H(z) must have a zero at

±1

= z

) ( )

(z =z⁻ H z⁻¹

H ^N

−1

= z

) ( ) ( ) ( )

(−1 = −1⁻ H −1 =−H −1

H ^N

0 1 =

− ) ( H

Zero Locations of Linear Zero Locations of Linear- - Phase FIR Transfer Functions Phase FIR Transfer Functions

• Likewise, a Type 3or 4FIR filter satisfies

• Thus

implying thatH(z) must have a zero atz= 1

• On the other hand, only the Type 3FIR filter is restricted to have a zero at since here the degreeNis even and hence,

) ( )

(z =−z⁻ H z⁻¹

H ^N

) ( ) ( ) ( )

(1 1 H1 H1

H =− ⁻^N =−

−1

= z

) ( ) ( ) ( )

(−1 =− −1⁻ H −1 =−H −1

H ^N

Zero Locations of Linear Zero Locations of Linear- - Phase FIR Transfer Functions Phase FIR Transfer Functions

• Typical zero locations shown below

−1 1

Type 2 Type 1

−1 1

Type 4 Type 3

−1 1

Zero Locations of Linear Zero Locations of Linear- - Phase FIR Transfer Functions Phase FIR Transfer Functions

• Summarizing

(1) Type 1 FIR filter:Either an even number or no zeros atz= 1 and

(2) Type 2 FIR filter:Either an even number or no zeros atz= 1, and an odd number of zeros at

(3) Type 3 FIR filter:An odd number of zeros atz= 1 and

−1

= z

−1

= z

−1

= z

Zero Locations of Linear Zero Locations of Linear- - Phase FIR Transfer Functions Phase FIR Transfer Functions

(4) Type 4 FIR filter:An odd number of zeros atz= 1, and either an even number or no zeros at

• The presence of zeros at leads to the following limitations on the use of these linear-phase transfer functions for designing frequency-selective filters

−1

= z

±1

= z

(10)

Zero Locations of Linear Phase FIR Transfer Functions Phase FIR Transfer Functions

• A Type 2FIR filter cannot be used to design ahighpassfilter since it always has a zero

• A Type 3FIR filter has zeros at both z= 1 and , and hence cannot be used to design either a lowpassor a highpassor a bandstopfilter

−1

= z

−1

= z

Zero Locations of Linear Phase FIR Transfer Functions Phase FIR Transfer Functions

• A Type 4FIR filter is not appropriate to design lowpassand bandstopfilters due to the presence of a zero atz= 1

• Type 1FIR filter has no such restrictions and can be used to design almost any type of filter