5: filters - faculty of engineering | imperial college · pdf file•minimum phase +...

5: Filters

5: Filters

• Difference Equations

• FIR Filters

• FIR Symmetries +

• IIR Frequency Response

• Negating z +

• Cubing z +

• Scaling z +

• Low-pass filter +

• Allpass filters +

• Group Delay +

• Minimum Phase +

• Linear Phase Filters

• Summary

• MATLAB routines

DSP and Digital Filters (2017-10159) Filters: 5 – 1 / 15

Difference Equations

5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines


Most useful LTI systems can be described bya difference equation:

y[n] =∑M

r=0 b[r]x[n− r]−∑Nr=1 a[r]y[n− r]


5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines



y[n] =∑M

r=0 b[r]x[n− r]−∑Nr=1 a[r]y[n− r]

(1) Always causal.


5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines



y[n] =∑M

r=0 b[r]x[n− r]−∑Nr=1 a[r]y[n− r]

(1) Always causal.(2) Order of system is max(M,N), the highest r with a[r] 6= 0 orb[r] 6= 0.


5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines



y[n] =∑M

r=0 b[r]x[n− r]−∑Nr=1 a[r]y[n− r]

⇔ ∑Nr=0 a[r]y[n− r] =

∑Mr=0 b[r]x[n− r] with a[0] = 1

(1) Always causal.(2) Order of system is max(M,N), the highest r with a[r] 6= 0 orb[r] 6= 0.


5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines



y[n] =∑M

r=0 b[r]x[n− r]−∑Nr=1 a[r]y[n− r]

⇔ ∑Nr=0 a[r]y[n− r] =

∑Mr=0 b[r]x[n− r] with a[0] = 1

(1) Always causal.(2) Order of system is max(M,N), the highest r with a[r] 6= 0 orb[r] 6= 0.(3) We assume that a[0] = 1


5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines



y[n] =∑M

r=0 b[r]x[n− r]−∑Nr=1 a[r]y[n− r]

⇔ ∑Nr=0 a[r]y[n− r] =

∑Mr=0 b[r]x[n− r] with a[0] = 1

⇔ a[n] ∗ y[n] = b[n] ∗ x[n]



5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines



y[n] =∑M

r=0 b[r]x[n− r]−∑Nr=1 a[r]y[n− r]

⇔ ∑Nr=0 a[r]y[n− r] =

∑Mr=0 b[r]x[n− r] with a[0] = 1

⇔ a[n] ∗ y[n] = b[n] ∗ x[n]

⇔ Y (z) = B(z)A(z)X(z)



5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines



y[n] =∑M

r=0 b[r]x[n− r]−∑Nr=1 a[r]y[n− r]

⇔ ∑Nr=0 a[r]y[n− r] =

∑Mr=0 b[r]x[n− r] with a[0] = 1

⇔ a[n] ∗ y[n] = b[n] ∗ x[n]

⇔ Y (z) = B(z)A(z)X(z)

(1) Always causal.(2) Order of system is max(M,N), the highest r with a[r] 6= 0 orb[r] 6= 0.(3) We assume that a[0] = 1; if not, divide A(z) and B(z) by a[0].


5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines



y[n] =∑M

r=0 b[r]x[n− r]−∑Nr=1 a[r]y[n− r]

⇔ ∑Nr=0 a[r]y[n− r] =

∑Mr=0 b[r]x[n− r] with a[0] = 1

⇔ a[n] ∗ y[n] = b[n] ∗ x[n]

⇔ Y (z) = B(z)A(z)X(z)

(1) Always causal.(2) Order of system is max(M,N), the highest r with a[r] 6= 0 orb[r] 6= 0.(3) We assume that a[0] = 1; if not, divide A(z) and B(z) by a[0].(4) Filter is BIBO stable iff roots of A(z) all lie within the unit circle.


5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines



y[n] =∑M

r=0 b[r]x[n− r]−∑Nr=1 a[r]y[n− r]

⇔ ∑Nr=0 a[r]y[n− r] =

∑Mr=0 b[r]x[n− r] with a[0] = 1

⇔ a[n] ∗ y[n] = b[n] ∗ x[n]

⇔ Y (z) = B(z)A(z)X(z)

⇔ Y (ejω) = B(ejω)A(ejω)X(ejω)



5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines



y[n] =∑M

r=0 b[r]x[n− r]−∑Nr=1 a[r]y[n− r]

⇔ ∑Nr=0 a[r]y[n− r] =

∑Mr=0 b[r]x[n− r] with a[0] = 1

⇔ a[n] ∗ y[n] = b[n] ∗ x[n]

⇔ Y (z) = B(z)A(z)X(z)

⇔ Y (ejω) = B(ejω)A(ejω)X(ejω)


Note negative sign in first equation.Authors in some SP fields reverse the sign of the a[n]: BAD IDEA.

FIR Filters

5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines


A(z) = 1: Finite Impulse Response (FIR) filter: Y (z) = B(z)X(z).

FIR Filters

5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines


A(z) = 1: Finite Impulse Response (FIR) filter: Y (z) = B(z)X(z).Impulse response is b[n] and is of length M + 1.

FIR Filters

5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines



Frequency response is B(ejω) and is the DTFT of b[n].

FIR Filters

5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines



Frequency response is B(ejω) and is the DTFT of b[n].Comprises M complex sinusoids + const:

b[0] + b[1]e−jω + · · ·+ b[M ]e−jMω

FIR Filters

5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines




b[0] + b[1]e−jω + · · ·+ b[M ]e−jMω

Small M⇒response contains only low “quefrencies”

M=4

0 1 2 30

0.5

1

ω

FIR Filters

5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines




b[0] + b[1]e−jω + · · ·+ b[M ]e−jMω


Symmetrical b[n]⇒H(ejω)ejMω

2 consists of M2 cosine waves [+ const]

M=4

0 1 2 30

0.5

1

ω

FIR Filters

5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines




b[0] + b[1]e−jω + · · ·+ b[M ]e−jMω




M=4 M=14

0 1 2 30

0.5

1

ω0 1 2 3

0

0.5

1

ω

FIR Filters

5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines




b[0] + b[1]e−jω + · · ·+ b[M ]e−jMω




M=4 M=14 M=24

0 1 2 30

0.5

1

ω0 1 2 3

0

0.5

1

ω0 1 2 3

0

0.5

1

ω

FIR Filters

5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines




b[0] + b[1]e−jω + · · ·+ b[M ]e−jMω




M=4 M=14 M=24

0 1 2 30

0.5

1

ω0 1 2 3

0

0.5

1

ω0 1 2 3

0

0.5

1

ω

Rule of thumb: Fastest possible transition ∆ω ≥ 2πM

(marked line)

FIR Symmetries +

5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines


B(ejω) is determined by the zeros of zMB(z) =∑M

r=0 b[M − r]zr

FIR Symmetries +

5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines



r=0 b[M − r]zr

Real b[n] ⇒ conjugate zero pairs: z ⇒ z∗

Real:[1, −1.28, 0.64]

-1 0 1

-1

-0.5

0

0.5

1

z

-2 0 20

1

2

ω (rad/sample)

FIR Symmetries +

5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines



r=0 b[M − r]zr


Symmetric: b[n] = b[M − n] ⇒ reciprocal zero pairs: z ⇒ z−1

Real: Symmetric:[1, −1.28, 0.64] [1, −1.64 + 0.27j, 1]

-1 0 1

-1

-0.5

0

0.5

1

z-1 0 1

-1

-0.5

0

0.5

1

z

-2 0 20

1

2

ω (rad/sample)-2 0 2

0

1

2

3

ω (rad/sample)

FIR Symmetries +

5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines



r=0 b[M − r]zr


Symmetric: b[n] = b[M − n] ⇒ reciprocal zero pairs: z ⇒ z−1

Real + Symmetric b[n] ⇒ conjugate+reciprocal groups of fouror else pairs on the real axis

Real: Symmetric: Real + Symmetric:[1, −1.28, 0.64] [1, −1.64 + 0.27j, 1] [1, −3.28, 4.7625, −3.28, 1]

-1 0 1

-1

-0.5

0

0.5

1

z-1 0 1

-1

-0.5

0

0.5

1

z-1 0 1

-1

-0.5

0

0.5

1

z

-2 0 20

1

2


0

1

2

3


0

5

10

ω (rad/sample)

IIR Frequency Response

5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines


Factorize H(z) = B(z)A(z)=

b[0]∏

Mi=1(1−qiz

−1)∏

Ni=1(1−piz−1)


5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines



b[0]∏

Mi=1(1−qiz

−1)∏

Ni=1(1−piz−1)

Roots of A(z) and B(z) are the “poles” {pi} and “zeros” {qi} of H(z)


5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines



b[0]∏

Mi=1(1−qiz

−1)∏

Ni=1(1−piz−1)

Roots of A(z) and B(z) are the “poles” {pi} and “zeros” {qi} of H(z)Also an additional N −M zeros at the origin (affect phase only)


5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines



b[0]∏

Mi=1(1−qiz

−1)∏

Ni=1(1−piz−1)


∣

∣H(ejω)∣

∣ =|b[0]||z−M |∏M

i=1|z−qi|

|z−N |∏

Ni=1|z−pi|

for z = ejω


5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines



b[0]∏

Mi=1(1−qiz

−1)∏

Ni=1(1−piz−1)


∣

∣H(ejω)∣

∣ =|b[0]||z−M |∏M

i=1|z−qi|

|z−N |∏

Ni=1|z−pi|

for z = ejω

Example:

H(z) = 2+2.4z−1

1−0.96z−1+0.64z−2

0 1 2 30

5

10

ω


5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines



b[0]∏

Mi=1(1−qiz

−1)∏

Ni=1(1−piz−1)


∣

∣H(ejω)∣

∣ =|b[0]||z−M |∏M

i=1|z−qi|

|z−N |∏

Ni=1|z−pi|

for z = ejω

Example:

H(z) = 2+2.4z−1

1−0.96z−1+0.64z−2=2(1+1.2z−1)

(1−(0.48−0.64j)z−1)(1−(0.48+0.64j)z−1)

0 1 2 30

5

10

ω-1 0 1

-1

-0.5

0

0.5

1

ℜ(z)

ℑ


5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines



b[0]∏

Mi=1(1−qiz

−1)∏

Ni=1(1−piz−1)


∣

∣H(ejω)∣

∣ =|b[0]||z−M |∏M

i=1|z−qi|

|z−N |∏

Ni=1|z−pi|

for z = ejω

Example:

H(z) = 2+2.4z−1

1−0.96z−1+0.64z−2=2(1+1.2z−1)

(1−(0.48−0.64j)z−1)(1−(0.48+0.64j)z−1)

At ω = 1.3:∣

∣H(ejω)∣

∣ = 2×1.761.62×0.39

0 1 2 30

5

10

ω-1 0 1

-1

-0.5

0

0.5

1

1.76 1.62

0.39

ℜ(z)

ℑ


5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines



b[0]∏

Mi=1(1−qiz

−1)∏

Ni=1(1−piz−1)


∣

∣H(ejω)∣

∣ =|b[0]||z−M |∏M

i=1|z−qi|

|z−N |∏

Ni=1|z−pi|

for z = ejω

Example:

H(z) = 2+2.4z−1

1−0.96z−1+0.64z−2=2(1+1.2z−1)

(1−(0.48−0.64j)z−1)(1−(0.48+0.64j)z−1)

At ω = 1.3:∣

∣H(ejω)∣

∣ = 2×1.761.62×0.39= 5.6

0 1 2 30

5

10

ω-1 0 1

-1

-0.5

0

0.5

1

1.76 1.62

0.39

ℜ(z)

ℑ


5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines



b[0]∏

Mi=1(1−qiz

−1)∏

Ni=1(1−piz−1)


∣

∣H(ejω)∣

∣ =|b[0]||z−M |∏M

i=1|z−qi|

|z−N |∏

Ni=1|z−pi|

for z = ejω

Example:

H(z) = 2+2.4z−1

1−0.96z−1+0.64z−2=2(1+1.2z−1)

(1−(0.48−0.64j)z−1)(1−(0.48+0.64j)z−1)

At ω = 1.3:∣

∣H(ejω)∣

∣ = 2×1.761.62×0.39= 5.6

∠H(ejω) = (0.6 + 1.3)− (1.7 + 2.2) = −1.97

0 1 2 30

5

10

ω-1 0 1

-1

-0.5

0

0.5

1

1.76 1.62

0.39

ℜ(z)

ℑ

-1 0 1

-1

-0.5

0

0.5

1

0.6

1.7

2.2

1.3

ℜ(z)

ℑ

Negating z +

5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines


Given a filter H(z) we can form a new one HR(z) = H(−z)

Negating z +

5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines


Given a filter H(z) we can form a new one HR(z) = H(−z)Negate all odd powers of z, i.e. negate alternate a[n] and b[n]

Negating z +

5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines



Example: H(z) = 2+2.4z−1

1−0.96z−1+0.64z−2

Negating z +

5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines




1−0.96z−1+0.64z−2

-1 0 1

-1

-0.5

0

0.5

1

ℜ(z)

ℑ

Negating z +

5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines




1−0.96z−1+0.64z−2

-1 0 1

-1

-0.5

0

0.5

1

ℜ(z)

ℑ

0 1 2 30

5

10

ω

Negating z +

5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines




1−0.96z−1+0.64z−2

-1 0 1

-1

-0.5

0

0.5

1

ℜ(z)

ℑ

0 1 2 30

5

10

ω

Negate z: HR(z) =2−2.4z−1

1+0.96z−1+0.64z−2 Negate odd coefficients

Negating z +

5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines




1−0.96z−1+0.64z−2

-1 0 1

-1

-0.5

0

0.5

1

ℜ(z)

ℑ

0 1 2 30

5

10

ω



-1 0 1

-1

-0.5

0

0.5

1

ℜ(z)

ℑ

Pole and zero positions are negated

Negating z +

5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines




1−0.96z−1+0.64z−2

-1 0 1

-1

-0.5

0

0.5

1

ℜ(z)

ℑ

0 1 2 30

5

10

ω



-1 0 1

-1

-0.5

0

0.5

1

ℜ(z)

ℑ

0 1 2 30

5

10

ω

Pole and zero positions are negated, response is flipped and conjugated.

Cubing z +

5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines


Given a filter H(z) we can form a new one HC(z) = H(z3)

Cubing z +

5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines


Given a filter H(z) we can form a new one HC(z) = H(z3)Insert two zeros between each a[n] and b[n] term

Cubing z +

5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines




1−0.96z−1+0.64z−2

Cubing z +

5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines




1−0.96z−1+0.64z−2

-1 0 1

-1

-0.5

0

0.5

1

ℜ(z)

ℑ

Cubing z +

5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines




1−0.96z−1+0.64z−2

-1 0 1

-1

-0.5

0

0.5

1

ℜ(z)

ℑ

-2 0 20

5

10

ω (rad/sample)

Cubing z +

5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines




1−0.96z−1+0.64z−2

-1 0 1

-1

-0.5

0

0.5

1

ℜ(z)

ℑ

-2 0 20

5

10

ω (rad/sample)

Cube z: HC(z) =2+2.4z−3

1−0.96z−3+0.64z−6 Insert 2 zeros between coefs

Cubing z +

5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines




1−0.96z−1+0.64z−2

-1 0 1

-1

-0.5

0

0.5

1

ℜ(z)

ℑ

-2 0 20

5

10

ω (rad/sample)

Cube z: HC(z) =2+2.4z−3


-1 0 1

-1

-0.5

0

0.5

1

z

Pole and zero positions are replicated

Cubing z +

5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines




1−0.96z−1+0.64z−2

-1 0 1

-1

-0.5

0

0.5

1

ℜ(z)

ℑ

-2 0 20

5

10

ω (rad/sample)

Cube z: HC(z) =2+2.4z−3


-1 0 1

-1

-0.5

0

0.5

1

z-2 0 2

0

5

10

ω (rad/sample)

C

Pole and zero positions are replicated, magnitude response replicated.

Scaling z +

5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines


Given a filter H(z) we can form a new one HS(z) = H( zα)

Scaling z +

5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines



Multiply a[n] and b[n] by αn

Scaling z +

5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines





1−0.96z−1+0.64z−2

Scaling z +

5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines





1−0.96z−1+0.64z−2

-1 0 1

-1

-0.5

0

0.5

1

ℜ(z)

ℑ

Scaling z +

5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines





1−0.96z−1+0.64z−2

-1 0 1

-1

-0.5

0

0.5

1

ℜ(z)

ℑ

0 1 2 30

5

10

ω

Scaling z +

5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines





1−0.96z−1+0.64z−2

-1 0 1

-1

-0.5

0

0.5

1

ℜ(z)

ℑ

0 1 2 30

5

10

ω

Scale z: HS(z) = H( z1.1 ) =

2+2.64z−1

1−1.056z−1+0.7744z−2

Scaling z +

5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines





1−0.96z−1+0.64z−2

-1 0 1

-1

-0.5

0

0.5

1

ℜ(z)

ℑ

0 1 2 30

5

10

ω

Scale z: HS(z) = H( z1.1 ) =

2+2.64z−1

1−1.056z−1+0.7744z−2

-1 0 1

-1

-0.5

0

0.5

1

z

Pole and zero positions are multiplied by α

Scaling z +

5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines





1−0.96z−1+0.64z−2

-1 0 1

-1

-0.5

0

0.5

1

ℜ(z)

ℑ

0 1 2 30

5

10

ω

Scale z: HS(z) = H( z1.1 ) =

2+2.64z−1

1−1.056z−1+0.7744z−2

-1 0 1

-1

-0.5

0

0.5

1

z0 1 2 3

0

5

10

15

20

ω (rad/s)

Pole and zero positions are multiplied by α, α > 1 ⇒peaks sharpened.

Scaling z +

5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines





1−0.96z−1+0.64z−2

-1 0 1

-1

-0.5

0

0.5

1

ℜ(z)

ℑ

0 1 2 30

5

10

ω

Scale z: HS(z) = H( z1.1 ) =

2+2.64z−1

1−1.056z−1+0.7744z−2

-1 0 1

-1

-0.5

0

0.5

1

z0 1 2 3

0

5

10

15

20

ω (rad/s)

Pole and zero positions are multiplied by α, α > 1 ⇒peaks sharpened.Pole at z = p gives peak bandwidth ≈ 2 |log |p|| ≈ 2 (1− |p|)

Scaling z +

5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines





1−0.96z−1+0.64z−2

-1 0 1

-1

-0.5

0

0.5

1

ℜ(z)

ℑ

0 1 2 30

5

10

ω

Scale z: HS(z) = H( z1.1 ) =

2+2.64z−1

1−1.056z−1+0.7744z−2

-1 0 1

-1

-0.5

0

0.5

1

z0 1 2 3

0

5

10

15

20

ω (rad/s)

Pole and zero positions are multiplied by α, α > 1 ⇒peaks sharpened.Pole at z = p gives peak bandwidth ≈ 2 |log |p|| ≈ 2 (1− |p|)For pole near unit circle, decrease bandwidth by ≈ 2 logα

Low-pass filter +

5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines


1st order low pass filter: extremely commony[n] = (1− p)x[n] + py[n− 1]

Low-pass filter +

5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines


1st order low pass filter: extremely commony[n] = (1− p)x[n] + py[n− 1]⇒ H(z) = 1−p

1−pz−1

-1 0 1

-1

-0.5

0

0.5

1

ℜ(z)

ℑ

Low-pass filter +

5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines



1−pz−1

Impulse response:h[n] = (1− p)pn

-1 0 1

-1

-0.5

0

0.5

1

ℜ(z)

ℑ

p=0.80

Low-pass filter +

5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines



1−pz−1

Impulse response:h[n] = (1− p)pn = (1− p)e−

nτ

where τ = 1− ln p

is the time constant in samples.

-1 0 1

-1

-0.5

0

0.5

1

ℜ(z)

ℑ

p=0.80

Low-pass filter +

5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines



1−pz−1


nτ



Magnitude response:∣

∣H(ejω)∣

∣ = 1−p√1−2p cosω+p2

-1 0 1

-1

-0.5

0

0.5

1

ℜ(z)

ℑ

p=0.80

0 1 2 30

0.5

1

ω (rad/sample)

Low-pass filter +

5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines



1−pz−1


nτ




∣H(ejω)∣

∣ = 1−p√1−2p cosω+p2

-1 0 1

-1

-0.5

0

0.5

1

ℜ(z)

ℑ

p=0.80

0.1 1

-20

-10

0

ω (rad/sample)

Low-pass filter +

5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines



1−pz−1


nτ




∣H(ejω)∣

∣ = 1−p√1−2p cosω+p2

Low-pass filter with DC gain of unity.

-1 0 1

-1

-0.5

0

0.5

1

ℜ(z)

ℑ

p=0.80

0.1 1

-20

-10

0

ω (rad/sample)

Low-pass filter +

5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines



1−pz−1


nτ




∣H(ejω)∣

∣ = 1−p√1−2p cosω+p2


3 dB frequency is ω3dB = cos−1(

1− (1−p)2

2p

)

-1 0 1

-1

-0.5

0

0.5

1

ℜ(z)

ℑ

p=0.80

0.1 1

-20

-10

0

ω (rad/sample)

Low-pass filter +

5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines



1−pz−1


nτ




∣H(ejω)∣

∣ = 1−p√1−2p cosω+p2



1− (1−p)2

2p

)

≈ 2 1−p1+p

-1 0 1

-1

-0.5

0

0.5

1

ℜ(z)

ℑ

p=0.80

0.1 1

-20

-10

0

ω (rad/sample)

Low-pass filter +

5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines



1−pz−1


nτ




∣H(ejω)∣

∣ = 1−p√1−2p cosω+p2



1− (1−p)2

2p

)

≈ 2 1−p1+p

≈ 1τ

-1 0 1

-1

-0.5

0

0.5

1

ℜ(z)

ℑ

p=0.80

0.1 1

-20

-10

0

1/τω (rad/sample)

Low-pass filter +

5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines



1−pz−1


nτ




∣H(ejω)∣

∣ = 1−p√1−2p cosω+p2



1− (1−p)2

2p

)

≈ 2 1−p1+p

≈ 1τ

Compare continuous time: HC(jω) =1

1+jωτ

-1 0 1

-1

-0.5

0

0.5

1

ℜ(z)

ℑ

p=0.80

0.1 1

-20

-10

0

1/τ

HC(jω)

ω (rad/sample)

Low-pass filter +

5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines



1−pz−1


nτ




∣H(ejω)∣

∣ = 1−p√1−2p cosω+p2



1− (1−p)2

2p

)

≈ 2 1−p1+p

≈ 1τ


1+jωτ

Indistinguishable for low ω but H(ejω) is periodic, HC(jω) is not

-1 0 1

-1

-0.5

0

0.5

1

ℜ(z)

ℑ

p=0.80

0.1 1

-20

-10

0

1/τ

HC(jω)

ω (rad/sample)

Low-pass filter +

5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines



1−pz−1


nτ




∣H(ejω)∣

∣ = 1−p√1−2p cosω+p2



1− (1−p)2

2p

)

≈ 2 1−p1+p

≈ 1τ


1+jωτ

Indistinguishable for low ω but H(ejω) is periodic, HC(jω) is not

-1 0 1

-1

-0.5

0

0.5

1

ℜ(z)

ℑ

p=0.80

0.01 0.1

-30

-20

-10

0

H(jω)

HC(jω)

1/τ 2πω (rad/sample)

Allpass filters +

5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines


If H(z) = B(z)A(z) with b[n] = a∗[M − n] then we have an allpass filter:

⇒ H(ejω) =∑

Mr=0 a∗[M−r]e−jωr

∑Mr=0 a[r]e−jωr

= e−jωM∑

Ms=0 a∗[s]ejωs


[s = M − r]

Allpass filters +

5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines



⇒ H(ejω) =∑



= e−jωM∑

Ms=0 a∗[s]ejωs


[s = M − r]

The two sums are complex conjugates

Allpass filters +

5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines



⇒ H(ejω) =∑



= e−jωM∑

Ms=0 a∗[s]ejωs


[s = M − r]

The two sums are complex conjugates ⇒ they have the same magnitude

Allpass filters +

5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines



⇒ H(ejω) =∑



= e−jωM∑

Ms=0 a∗[s]ejωs


[s = M − r]

The two sums are complex conjugates ⇒ they have the same magnitudeHence

∣

∣H(ejω)∣

∣ = 1∀ω

Allpass filters +

5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines



⇒ H(ejω) =∑



= e−jωM∑

Ms=0 a∗[s]ejωs


[s = M − r]


∣

∣H(ejω)∣

∣ = 1∀ω ⇔ “allpass”

Allpass filters +

5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines



⇒ H(ejω) =∑



= e−jωM∑

Ms=0 a∗[s]ejωs


[s = M − r]


∣

∣H(ejω)∣


However phase is not constant: ∠H(ejω) = −ωM − 2∠A(ejω)

Allpass filters +

5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines



⇒ H(ejω) =∑



= e−jωM∑

Ms=0 a∗[s]ejωs


[s = M − r]


∣

∣H(ejω)∣



1st order allpass: H(z) = −p+z−1

1−pz−1

0 1 2 30

0.2

0.4

0.6

0.8

1

ω0 1 2 3

-3

-2

-1

0

ω

∠

Allpass filters +

5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines



⇒ H(ejω) =∑



= e−jωM∑

Ms=0 a∗[s]ejωs


[s = M − r]


∣

∣H(ejω)∣




1−pz−1 = −p 1−p−1z−1

1−pz−1

Pole at p and zero at p−1

0 1 2 30

0.2

0.4

0.6

0.8

1

ω0 1 2 3

-3

-2

-1

0

ω

∠

-1 0 1

-1

-0.5

0

0.5

1

ℜ(z)

ℑ

Allpass filters +

5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines



⇒ H(ejω) =∑



= e−jωM∑

Ms=0 a∗[s]ejωs


[s = M − r]


∣

∣H(ejω)∣




1−pz−1 = −p 1−p−1z−1

1−pz−1

Pole at p and zero at p−1: “reflected in unit circle”

0 1 2 30

0.2

0.4

0.6

0.8

1

ω0 1 2 3

-3

-2

-1

0

ω

∠

-1 0 1

-1

-0.5

0

0.5

1

ℜ(z)

ℑ

Allpass filters +

5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines



⇒ H(ejω) =∑



= e−jωM∑

Ms=0 a∗[s]ejωs


[s = M − r]


∣

∣H(ejω)∣




1−pz−1 = −p 1−p−1z−1

1−pz−1


Constant distance ratio:∣

∣ejω − p∣

∣ = |p|∣

∣

∣ejω − 1

p

∣

∣

∣∀ω

0 1 2 30

0.2

0.4

0.6

0.8

1

ω0 1 2 3

-3

-2

-1

0

ω

∠

-1 0 1

-1

-0.5

0

0.5

1

ℜ(z)

ℑ

Allpass filters +

5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines



⇒ H(ejω) =∑



= e−jωM∑

Ms=0 a∗[s]ejωs


[s = M − r]


∣

∣H(ejω)∣




1−pz−1 = −p 1−p−1z−1

1−pz−1


Constant distance ratio:∣

∣ejω − p∣

∣ = |p|∣

∣

∣ejω − 1

p

∣

∣

∣∀ω

0 1 2 30

0.2

0.4

0.6

0.8

1

ω0 1 2 3

-3

-2

-1

0

ω

∠

-1 0 1

-1

-0.5

0

0.5

1

ℜ(z)

ℑ

In an allpass filter, the zeros are the poles reflected in the unit circle.

Group Delay +

5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines


Group delay: τH(ejω) = −d∠H(ejω)dω

= delay of the modulation envelope.

Group Delay +

5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines




Trick to get at phase: lnH(ejω) = ln∣

∣H(ejω)∣

∣+ j∠H(ejω)

Group Delay +

5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines





∣H(ejω)∣

∣+ j∠H(ejω)

τH =−d(ℑ(lnH(ejω)))

dω

Group Delay +

5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines





∣H(ejω)∣

∣+ j∠H(ejω)


dω= ℑ

(

−1H(ejω)

dH(ejω)dω

)

Group Delay +

5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines





∣H(ejω)∣

∣+ j∠H(ejω)


dω= ℑ

(

−1H(ejω)

dH(ejω)dω

)

= ℜ(

−zH(z)

dHdz

)∣

∣

∣

z=ejω

Group Delay +

5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines





∣H(ejω)∣

∣+ j∠H(ejω)


dω= ℑ

(

−1H(ejω)

dH(ejω)dω

)

= ℜ(

−zH(z)

dHdz

)∣

∣

∣

z=ejω

H(ejω) =∑∞

n=0 h[n]e−jnω

Group Delay +

5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines





∣H(ejω)∣

∣+ j∠H(ejω)


dω= ℑ

(

−1H(ejω)

dH(ejω)dω

)

= ℜ(

−zH(z)

dHdz

)∣

∣

∣

z=ejω

H(ejω) =∑∞

n=0 h[n]e−jnω

dH(ejω)dω

=∑∞

n=0 −jnh[n]e−jnω

Group Delay +

5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines





∣H(ejω)∣

∣+ j∠H(ejω)


dω= ℑ

(

−1H(ejω)

dH(ejω)dω

)

= ℜ(

−zH(z)

dHdz

)∣

∣

∣

z=ejω

H(ejω) =∑∞

n=0 h[n]e−jnω= F (h[n]) [F = DTFT]

dH(ejω)dω

=∑∞

n=0 −jnh[n]e−jnω

Group Delay +

5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines





∣H(ejω)∣

∣+ j∠H(ejω)


dω= ℑ

(

−1H(ejω)

dH(ejω)dω

)

= ℜ(

−zH(z)

dHdz

)∣

∣

∣

z=ejω

H(ejω) =∑∞


dH(ejω)dω

=∑∞

n=0 −jnh[n]e−jnω= −jF (nh[n])

Group Delay +

5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines





∣H(ejω)∣

∣+ j∠H(ejω)


dω= ℑ

(

−1H(ejω)

dH(ejω)dω

)

= ℜ(

−zH(z)

dHdz

)∣

∣

∣

z=ejω

H(ejω) =∑∞


dH(ejω)dω

=∑∞


τH = ℑ(

−1H(ejω)

dH(ejω)dω

)

Group Delay +

5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines





∣H(ejω)∣

∣+ j∠H(ejω)


dω= ℑ

(

−1H(ejω)

dH(ejω)dω

)

= ℜ(

−zH(z)

dHdz

)∣

∣

∣

z=ejω

H(ejω) =∑∞


dH(ejω)dω

=∑∞


τH = ℑ(

−1H(ejω)

dH(ejω)dω

)

= ℑ(

jF(nh[n])F(h[n])

)

Group Delay +

5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines





∣H(ejω)∣

∣+ j∠H(ejω)


dω= ℑ

(

−1H(ejω)

dH(ejω)dω

)

= ℜ(

−zH(z)

dHdz

)∣

∣

∣

z=ejω

H(ejω) =∑∞


dH(ejω)dω

=∑∞


τH = ℑ(

−1H(ejω)

dH(ejω)dω

)

= ℑ(

jF(nh[n])F(h[n])

)

= ℜ(

F(nh[n])F(h[n])

)

Group Delay +

5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines





∣H(ejω)∣

∣+ j∠H(ejω)


dω= ℑ

(

−1H(ejω)

dH(ejω)dω

)

= ℜ(

−zH(z)

dHdz

)∣

∣

∣

z=ejω

H(ejω) =∑∞


dH(ejω)dω

=∑∞


τH = ℑ(

−1H(ejω)

dH(ejω)dω

)

= ℑ(

jF(nh[n])F(h[n])

)

= ℜ(

F(nh[n])F(h[n])

)

Example: H(z) = 11−pz−1

Group Delay +

5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines





∣H(ejω)∣

∣+ j∠H(ejω)


dω= ℑ

(

−1H(ejω)

dH(ejω)dω

)

= ℜ(

−zH(z)

dHdz

)∣

∣

∣

z=ejω

H(ejω) =∑∞


dH(ejω)dω

=∑∞


τH = ℑ(

−1H(ejω)

dH(ejω)dω

)

= ℑ(

jF(nh[n])F(h[n])

)

= ℜ(

F(nh[n])F(h[n])

)

Example: H(z) = 11−pz−1

-1 0 1

-1

-0.5

0

0.5

1

ℜ(z)

ℑ

0 1 2 3

-0.8

-0.6

-0.4

-0.2

0

p=0.80

ω

∠

Group Delay +

5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines





∣H(ejω)∣

∣+ j∠H(ejω)


dω= ℑ

(

−1H(ejω)

dH(ejω)dω

)

= ℜ(

−zH(z)

dHdz

)∣

∣

∣

z=ejω

H(ejω) =∑∞


dH(ejω)dω

=∑∞


τH = ℑ(

−1H(ejω)

dH(ejω)dω

)

= ℑ(

jF(nh[n])F(h[n])

)

= ℜ(

F(nh[n])F(h[n])

)

Example: H(z) = 11−pz−1⇒ τH = −τ[1 −p]

-1 0 1

-1

-0.5

0

0.5

1

ℜ(z)

ℑ

0 1 2 3

-0.8

-0.6

-0.4

-0.2

0

p=0.80

ω

∠

Group Delay +

5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines





∣H(ejω)∣

∣+ j∠H(ejω)


dω= ℑ

(

−1H(ejω)

dH(ejω)dω

)

= ℜ(

−zH(z)

dHdz

)∣

∣

∣

z=ejω

H(ejω) =∑∞


dH(ejω)dω

=∑∞


τH = ℑ(

−1H(ejω)

dH(ejω)dω

)

= ℑ(

jF(nh[n])F(h[n])

)

= ℜ(

F(nh[n])F(h[n])

)

Example: H(z) = 11−pz−1⇒ τH = −τ[1 −p]= −ℜ

(

−pe−jω

1−pe−jω

)

-1 0 1

-1

-0.5

0

0.5

1

ℜ(z)

ℑ

0 1 2 3

-0.8

-0.6

-0.4

-0.2

0

p=0.80

ω

∠

Group Delay +

5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines





∣H(ejω)∣

∣+ j∠H(ejω)


dω= ℑ

(

−1H(ejω)

dH(ejω)dω

)

= ℜ(

−zH(z)

dHdz

)∣

∣

∣

z=ejω

H(ejω) =∑∞


dH(ejω)dω

=∑∞


τH = ℑ(

−1H(ejω)

dH(ejω)dω

)

= ℑ(

jF(nh[n])F(h[n])

)

= ℜ(

F(nh[n])F(h[n])

)


(

−pe−jω

1−pe−jω

)

-1 0 1

-1

-0.5

0

0.5

1

ℜ(z)

ℑ

0 1 2 3

-0.8

-0.6

-0.4

-0.2

0

p=0.80

ω

∠

0 1 2 3

0

1

2

3 p=0.80

ω

τ H

Group Delay +

5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines





∣H(ejω)∣

∣+ j∠H(ejω)


dω= ℑ

(

−1H(ejω)

dH(ejω)dω

)

= ℜ(

−zH(z)

dHdz

)∣

∣

∣

z=ejω

H(ejω) =∑∞


dH(ejω)dω

=∑∞


τH = ℑ(

−1H(ejω)

dH(ejω)dω

)

= ℑ(

jF(nh[n])F(h[n])

)

= ℜ(

F(nh[n])F(h[n])

)


(

−pe−jω

1−pe−jω

)

-1 0 1

-1

-0.5

0

0.5

1

ℜ(z)

ℑ

0 1 2 3

-0.8

-0.6

-0.4

-0.2

0

p=0.80

ω

∠

0 1 2 3

0

1

2

3 p=0.80

ω

τ HAverage group delay (over ω) = (# poles – # zeros) within the unit circle

Group Delay +

5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines





∣H(ejω)∣

∣+ j∠H(ejω)


dω= ℑ

(

−1H(ejω)

dH(ejω)dω

)

= ℜ(

−zH(z)

dHdz

)∣

∣

∣

z=ejω

H(ejω) =∑∞


dH(ejω)dω

=∑∞


τH = ℑ(

−1H(ejω)

dH(ejω)dω

)

= ℑ(

jF(nh[n])F(h[n])

)

= ℜ(

F(nh[n])F(h[n])

)


(

−pe−jω

1−pe−jω

)

-1 0 1

-1

-0.5

0

0.5

1

ℜ(z)

ℑ

0 1 2 3

-0.8

-0.6

-0.4

-0.2

0

p=0.80

ω

∠

0 1 2 3

0

1

2

3 p=0.80

ω

τ HAverage group delay (over ω) = (# poles – # zeros) within the unit circle

Zeros on the unit circle count –½

Minimum Phase +

5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines


Average group delay (over ω) = (# poles – # zeros) within the unit circle

-1 0 1

-1

-0.5

0

0.5

1

ℜ(z)

ℑ

Minimum Phase +

5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines



• zeros on the unit circle count –½

-1 0 1

-1

-0.5

0

0.5

1

ℜ(z)

ℑ

Minimum Phase +

5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines




0 1 2 30

1

2

3

4

ω

-1 0 1

-1

-0.5

0

0.5

1

ℜ(z)

ℑ

Minimum Phase +

5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines




0 1 2 30

1

2

3

4

ω

-1 0 1

-1

-0.5

0

0.5

1

ℜ(z)

ℑ

0 1 2 3

-10

-5

0

ω

∠

Minimum Phase +

5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines




0 1 2 30

1

2

3

4

ω

-1 0 1

-1

-0.5

0

0.5

1

ℜ(z)

ℑ

0 1 2 3

-10

-5

0

ω

∠

0 1 2 3-10

0

10

20

30

ω

τ H

Minimum Phase +

5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines




0 1 2 30

1

2

3

4

ω

-1 0 1

-1

-0.5

0

0.5

1

ℜ(z)

ℑ

0 1 2 3

-10

-5

0

ω

∠

-1 0 1

-1

-0.5

0

0.5

1

ℜ(z)

ℑ

0 1 2 3-10

0

10

20

30

ω

τ H

Minimum Phase +

5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines




Reflecting an interior zero to the exteriormultiplies

∣

∣H(ejω)∣

∣ by a constant butincreases average group delay by 1 sample. 0 1 2 3

0

1

2

3

4

ω

-1 0 1

-1

-0.5

0

0.5

1

ℜ(z)

ℑ

0 1 2 3

-10

-5

0

ω

∠

-1 0 1

-1

-0.5

0

0.5

1

ℜ(z)

ℑ

0 1 2 3-10

0

10

20

30

ω

τ H

Minimum Phase +

5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines





∣

∣H(ejω)∣


0

1

2

3

4

ω

-1 0 1

-1

-0.5

0

0.5

1

ℜ(z)

ℑ

0 1 2 3

-10

-5

0

ω

∠

-1 0 1

-1

-0.5

0

0.5

1

ℜ(z)

ℑ

0 1 2 3-10

0

10

20

30

ω

τ H

A filter with all zeros inside the unit circle is a minimum phase filter:

Minimum Phase +

5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines





∣

∣H(ejω)∣


0

1

2

3

4

ω

-1 0 1

-1

-0.5

0

0.5

1

ℜ(z)

ℑ

0 1 2 3

-10

-5

0

ω

∠

-1 0 1

-1

-0.5

0

0.5

1

ℜ(z)

ℑ

0 1 2 3-10

0

10

20

30

ω

τ H

A filter with all zeros inside the unit circle is a minimum phase filter:• Lowest possible group delay for a given magnitude response

Minimum Phase +

5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines





∣

∣H(ejω)∣


0

1

2

3

4

ω

-1 0 1

-1

-0.5

0

0.5

1

ℜ(z)

ℑ

0 1 2 3

-10

-5

0

ω

∠

-1 0 1

-1

-0.5

0

0.5

1

ℜ(z)

ℑ

0 1 2 3-10

0

10

20

30

ω

τ H

A filter with all zeros inside the unit circle is a minimum phase filter:• Lowest possible group delay for a given magnitude response• Energy in h[n] is concentrated towards n = 0

Linear Phase Filters

5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines


The phase of a linear phase filter is: ∠H(ejω) = θ0 − αω


5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines



Equivalently constant group delay: τH = −d∠H(ejω)dω

= α


5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines




= α

A filter has linear phase iff h[n] is symmetric or antisymmetric:h[n] = h[M − n] ∀n or else h[n] = −h[M − n] ∀n


5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines




= α


M can be even (⇒ ∃ mid point) or odd (⇒ ∄ mid point)


5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines




= α



Proof ⇐:2H(ejω) =

∑M0 h[n]e−jωn +

∑M0 h[M − n]e−jω(M−n)


5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines




= α






= e−jωM2

∑M0 h[n]e−jω(n−M

2 ) + h[M − n]ejω(n−M2 )


5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines




= α






= e−jωM2


2 ) + h[M − n]ejω(n−M2 )

h[n] symmetric:

2H(ejω) = 2e−jωM2

∑M0 h[n] cos

(

n− M2

)

ω


5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines




= α






= e−jωM2


2 ) + h[M − n]ejω(n−M2 )

h[n] symmetric:


∑M0 h[n] cos

(

n− M2

)

ω

h[n] anti-symmetric:

2H(ejω) = −2je−jωM2

∑M0 h[n] sin

(

n− M2

)

ω


5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines




= α






= e−jωM2


2 ) + h[M − n]ejω(n−M2 )

h[n] symmetric:


∑M0 h[n] cos

(

n− M2

)

ω

h[n] anti-symmetric:

2H(ejω) = −2je−jωM2

∑M0 h[n] sin

(

n− M2

)

ω

= 2e−j(π2 +ωM

2 )∑M

0 h[n] sin(

n− M2

)

ω

Summary

5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines


• Useful filters have difference equations:

Summary

5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines



◦ Freq response determined by pole/zero positions

Summary

5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines



◦ Freq response determined by pole/zero positions◦ N −M zeros at origin (or M −N poles)

Summary

5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines



◦ Freq response determined by pole/zero positions◦ N −M zeros at origin (or M −N poles)◦ Geometric construction of |H(ejω)|

⊲ Pole bandwidth ≈ 2 |log |p|| ≈ 2 (1− |p|)

Summary

5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines




⊲ Pole bandwidth ≈ 2 |log |p|| ≈ 2 (1− |p|)◦ Stable if poles have |p| < 1

Summary

5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines





• Allpass filter: a[n] = b[M − n]

Summary

5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines





• Allpass filter: a[n] = b[M − n]◦ Reflecting a zero in unit circle leaves |H(ejω)| unchanged

Summary

5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines






• Group delay: τH(

ejω)

= −d∠H(ejω)dω

samples

Summary

5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines







ejω)

= −d∠H(ejω)dω

samples

◦ Symmetrical h[n] ⇔ τH(

ejω)

= M2 ∀ω

Summary

5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines







ejω)

= −d∠H(ejω)dω

samples


ejω)

= M2 ∀ω

◦ Average τH over ω = (# poles – # zeros) within the unit circle

Summary

5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines







ejω)

= −d∠H(ejω)dω

samples


ejω)

= M2 ∀ω


• Minimum phase if zeros have |q| ≤ 1

Summary

5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines







ejω)

= −d∠H(ejω)dω

samples


ejω)

= M2 ∀ω


• Minimum phase if zeros have |q| ≤ 1◦ Lowest possible group delay for given |H(ejω)|

Summary

5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines







ejω)

= −d∠H(ejω)dω

samples


ejω)

= M2 ∀ω



• Linear phase = Constant group Delay = symmetric/antisymmetric h[n]

Summary

5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines







ejω)

= −d∠H(ejω)dω

samples


ejω)

= M2 ∀ω



• Linear phase = Constant group Delay = symmetric/antisymmetric h[n]

For further details see Mitra: 6, 7.

MATLAB routines

5: Filters


• FIR Filters



• Negating z +

• Cubing z +

• Scaling z +



• Group Delay +

• Minimum Phase +


• Summary

• MATLAB routines


filter filter a signalimpz Impulse response

residuez partial fraction expansiongrpdelay Group Delay

freqz Calculate filter frequency response

5: filters - faculty of engineering | imperial college · pdf file•minimum phase +...

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