5: Filters - Faculty of Engineering | Imperial College · PDF file•Minimum Phase +...

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”

0 1 2 30

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]

0 1 2 30

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

ω0 1 2 3

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

ω0 1 2 3

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

ω0 1 2 3

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

-2 0 20

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

z-1 0 1

-2 0 20

ω (rad/sample)-2 0 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 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

z-1 0 1

-2 0 20

ω (rad/sample)-2 0 2

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: 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

ω-1 0 1

ℜ(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)

∣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

ω-1 0 1

1.76 1.62

ℜ(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)

∣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

ω-1 0 1

1.76 1.62

ℜ(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)

∣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

ω-1 0 1

1.76 1.62

ℜ(z)

-1 0 1

ℜ(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)

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

ℜ(z)

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

ℜ(z)

0 1 2 30

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

ℜ(z)

0 1 2 30

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

ℜ(z)

0 1 2 30

-1 0 1

ℜ(z)

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

ℜ(z)

0 1 2 30

-1 0 1

ℜ(z)

0 1 2 30

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

ℜ(z)

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

ℜ(z)

-2 0 20

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

ℜ(z)

-2 0 20

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

ℜ(z)

-2 0 20

ω (rad/sample)

Cube z: HC(z) =2+2.4z−3

-1 0 1

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

ℜ(z)

-2 0 20

ω (rad/sample)

Cube z: HC(z) =2+2.4z−3

-1 0 1

z-2 0 2

ω (rad/sample)

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

ℜ(z)

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

ℜ(z)

0 1 2 30

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

ℜ(z)

0 1 2 30

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

ℜ(z)

0 1 2 30

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

2+2.64z−1

1−1.056z−1+0.7744z−2

-1 0 1

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

ℜ(z)

0 1 2 30

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

2+2.64z−1

1−1.056z−1+0.7744z−2

-1 0 1

z0 1 2 3

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

ℜ(z)

0 1 2 30

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

2+2.64z−1

1−1.056z−1+0.7744z−2

-1 0 1

z0 1 2 3

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

ℜ(z)

0 1 2 30

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

2+2.64z−1

1−1.056z−1+0.7744z−2

-1 0 1

z0 1 2 3

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

ℜ(z)

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

ℜ(z)

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−

where τ = 1− ln p

is the time constant in samples.

-1 0 1

ℜ(z)

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

Magnitude response:∣

∣H(ejω)∣

∣ = 1−p√1−2p cosω+p2

-1 0 1

ℜ(z)

p=0.80

0 1 2 30

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

∣H(ejω)∣

∣ = 1−p√1−2p cosω+p2

-1 0 1

ℜ(z)

p=0.80

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

∣H(ejω)∣

∣ = 1−p√1−2p cosω+p2

Low-pass filter with DC gain of unity.

-1 0 1

ℜ(z)

p=0.80

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

∣H(ejω)∣

∣ = 1−p√1−2p cosω+p2

3 dB frequency is ω3dB = cos−1(

1− (1−p)2

-1 0 1

ℜ(z)

p=0.80

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

∣H(ejω)∣

∣ = 1−p√1−2p cosω+p2

1− (1−p)2

≈ 2 1−p1+p

-1 0 1

ℜ(z)

p=0.80

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

∣H(ejω)∣

∣ = 1−p√1−2p cosω+p2

1− (1−p)2

≈ 2 1−p1+p

≈ 1τ

-1 0 1

ℜ(z)

p=0.80

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

∣H(ejω)∣

∣ = 1−p√1−2p cosω+p2

1− (1−p)2

≈ 2 1−p1+p

≈ 1τ

Compare continuous time: HC(jω) =1

1+jωτ

-1 0 1

ℜ(z)

p=0.80

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

∣H(ejω)∣

∣ = 1−p√1−2p cosω+p2

1− (1−p)2

≈ 2 1−p1+p

≈ 1τ

1+jωτ

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

-1 0 1

ℜ(z)

p=0.80

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

∣H(ejω)∣

∣ = 1−p√1−2p cosω+p2

1− (1−p)2

≈ 2 1−p1+p

≈ 1τ

1+jωτ

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

-1 0 1

ℜ(z)

p=0.80

0.01 0.1

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

ω0 1 2 3

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

ω0 1 2 3

-1 0 1

ℜ(z)

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

ω0 1 2 3

-1 0 1

ℜ(z)

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

∣∀ω

0 1 2 30

ω0 1 2 3

-1 0 1

ℜ(z)

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

∣∀ω

0 1 2 30

ω0 1 2 3

-1 0 1

ℜ(z)

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)))

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)

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)

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)

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)

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)

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)

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)

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)

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)

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)

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

ℜ(z)

0 1 2 3

p=0.80

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)

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

ℜ(z)

0 1 2 3

p=0.80

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)

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

ℜ(z)

0 1 2 3

p=0.80

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)

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

ℜ(z)

0 1 2 3

p=0.80

0 1 2 3

3 p=0.80

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)

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

ℜ(z)

0 1 2 3

p=0.80

0 1 2 3

3 p=0.80

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)

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

ℜ(z)

0 1 2 3

p=0.80

0 1 2 3

3 p=0.80

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

ℜ(z)

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

ℜ(z)

Minimum Phase +

5: Filters

• FIR Filters

• Negating z +

• Cubing z +

• Scaling z +

• Group Delay +

• Minimum Phase +

• Summary

• MATLAB routines

0 1 2 30

-1 0 1

ℜ(z)

Minimum Phase +

5: Filters

• FIR Filters

• Negating z +

• Cubing z +

• Scaling z +

• Group Delay +

• Minimum Phase +

• Summary

• MATLAB routines

0 1 2 30

-1 0 1

ℜ(z)

0 1 2 3

Minimum Phase +

5: Filters

• FIR Filters

• Negating z +

• Cubing z +

• Scaling z +

• Group Delay +

• Minimum Phase +

• Summary

• MATLAB routines

0 1 2 30

-1 0 1

ℜ(z)

0 1 2 3

0 1 2 3-10

Minimum Phase +

5: Filters

• FIR Filters

• Negating z +

• Cubing z +

• Scaling z +

• Group Delay +

• Minimum Phase +

• Summary

• MATLAB routines

0 1 2 30

-1 0 1

ℜ(z)

0 1 2 3

-1 0 1

ℜ(z)

0 1 2 3-10

Minimum Phase +

5: Filters

• FIR Filters

• Negating z +

• Cubing z +

• Scaling z +

• Group Delay +

• Minimum Phase +

• Summary

• MATLAB routines

0 1 2 30

-1 0 1

ℜ(z)

0 1 2 3

-1 0 1

ℜ(z)

0 1 2 3-10

Minimum Phase +

5: Filters

• FIR Filters

• Negating z +

• Cubing z +

• Scaling z +

• Group Delay +

• Minimum Phase +

• Summary

• MATLAB routines

0 1 2 30

-1 0 1

ℜ(z)

0 1 2 3

-1 0 1

ℜ(z)

0 1 2 3-10

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

-1 0 1

ℜ(z)

0 1 2 3

-1 0 1

ℜ(z)

0 1 2 3-10

Minimum Phase +

5: Filters

• FIR Filters

• Negating z +

• Cubing z +

• Scaling z +

• Group Delay +

• Minimum Phase +

• Summary

• MATLAB routines

∣H(ejω)∣

-1 0 1

ℜ(z)

0 1 2 3

-1 0 1

ℜ(z)

0 1 2 3-10

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)∣

-1 0 1

ℜ(z)

0 1 2 3

-1 0 1

ℜ(z)

0 1 2 3-10

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)∣

-1 0 1

ℜ(z)

0 1 2 3

-1 0 1

ℜ(z)

0 1 2 3-10

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

Minimum Phase +

5: Filters

• FIR Filters

• Negating z +

• Cubing z +

• Scaling z +

• Group Delay +

• Minimum Phase +

• Summary

• MATLAB routines

∣H(ejω)∣

-1 0 1

ℜ(z)

0 1 2 3

-1 0 1

ℜ(z)

0 1 2 3-10

Minimum Phase +

5: Filters

• FIR Filters

• Negating z +

• Cubing z +

• Scaling z +

• Group Delay +

• Minimum Phase +

• Summary

• MATLAB routines

∣H(ejω)∣

-1 0 1

ℜ(z)

0 1 2 3

-1 0 1

ℜ(z)

0 1 2 3-10

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

5: Filters

• FIR Filters

• Negating z +

• Cubing z +

• Scaling z +

• Group Delay +

• Minimum Phase +

• Summary

• MATLAB routines

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

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

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

= 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

= e−jωM2

2 ) + h[M − n]ejω(n−M2 )

h[n] symmetric:

2H(ejω) = 2e−jωM2

∑M0 h[n] cos

n− M2

5: Filters

• FIR Filters

• Negating z +

• Cubing z +

• Scaling z +

• Group Delay +

• Minimum Phase +

• Summary

• MATLAB routines

= e−jωM2

2 ) + h[M − n]ejω(n−M2 )

h[n] symmetric:

∑M0 h[n] cos

n− M2

h[n] anti-symmetric:

2H(ejω) = −2je−jωM2

∑M0 h[n] sin

n− M2

5: Filters

• FIR Filters

• Negating z +

• Cubing z +

• Scaling z +

• Group Delay +

• Minimum Phase +

• Summary

• MATLAB routines

= e−jωM2

2 ) + h[M − n]ejω(n−M2 )

h[n] symmetric:

∑M0 h[n] cos

n− M2

h[n] anti-symmetric:

2H(ejω) = −2je−jωM2

∑M0 h[n] sin

n− M2

= 2e−j(π2 +ωM

2 )∑M

0 h[n] sin(

n− M2

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(

= −d∠H(ejω)dω

samples

Summary

5: Filters

• FIR Filters

• Negating z +

• Cubing z +

• Scaling z +

• Group Delay +

• Minimum Phase +

• Summary

• MATLAB routines

= −d∠H(ejω)dω

samples

◦ Symmetrical h[n] ⇔ τH(

= M2 ∀ω

Summary

5: Filters

• FIR Filters

• Negating z +

• Cubing z +

• Scaling z +

• Group Delay +

• Minimum Phase +

• Summary

• MATLAB routines

= −d∠H(ejω)dω

samples

= 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

= −d∠H(ejω)dω

samples

= 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

= −d∠H(ejω)dω

samples

= 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

= −d∠H(ejω)dω

samples

= 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

= −d∠H(ejω)dω

samples

= 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 +...

Documents

FMAB NEO - 1-Phase AC Filters / DC Filters...1 1-Phase AC Filters / DC Filters FMAB NEO /PG84 EMC Products AC Filter, High Symmetrical Attenuation Bolt and nut M4 Case PK Quick connect

FMCC SOL - 3-Phase AC Filters / DC Filters · 3-Phase AC Filters / DC Filters FMCC SOL ... -25 °C to 100 °C Climatic Category 25/100/21 acc. to IEC 60068-1 ... 64 M12 (2x) 20 (350)

EE247 - Lecture 2 Filtersee247/fa10/files07/lectures/...EE247 - Lecture 2 Filters •Filters: –Nomenclature –Specifications •Quality factor •Magnitude/phase response versus

DiscreteDiscrete--Time Signal Time Signal Processing · 1. more phase control--can design filters that are exactly linear phase 2. guaranteed stability--FIR filters have only zeros,

LINEAR-PHASE FIR FILTERS 1.The amplitude response …eeweb.poly.edu/iselesni/EL713/zoom/linphase.pdf · LINEAR-PHASE FIR FILTERS 1.The amplitude response 2.Why linear-phase? 3.The

Photovoltaicsn-coretech.com/company/Brochure_Photovoltaics_690-889A_e.pdfSingle-phase AC EMI filters DC EMI filters Application Single-phase AC EMI filters are an essential part of

An Analysis and Comparison of Linear Phase Bandpass Filters

FMER SOL - 3-Phase AC Filters / DC Filters · 3-Phase AC Filters / DC Filters FMER SOL /PG80 EMC Products DC filter for photovoltaic converters See below: Approvals and Compliances

LINEAR-PHASE FIR FILTERS DESIGNfaculty.citadel.edu/potisuk/elec423/ppt/fir_lp.pdf · · 2017-10-26Negative of the slope of the phase response of a linear system, i.e., filters 1

A Computer Program for Designing Optimum FIR Linear Phase Digital Filters-q9K

Spatial Filters in Image Processing - Imperial College Londontania/teaching/DIP 2014/Image filters.pdf · Spatial Filters in Image Processing DR TANIA STATHAKI READER (ASSOCIATE PROFFESOR)

Minimum-Phase & All-Pass Filters - McGill University...Minimum-Phase & All-Pass Filters Peter Kabal Department of Electrical & Computer Engineering McGill University Montreal, Canada

Heavy-Duty Imperial XL Filters - Luber-finerluberfiner.com/documents/brochure/en/LF-Imperial-Filter-Broc...Imperial XL Filter. CAPACITY. Luber-finer IMPERIAL XL filters have 1½ times

Phase Relations in Active Filters - Analog Devices

Application of Linear-Phase Digital Crossover Filters to ... (2007-09 AES... · Application of Linear-Phase Digital Crossover Filters to Pair-Wise Symmetric Multi-Way Loudspeakers

FMER SOL - 3-Phase AC Filters / DC Filters · FMER SOL 3-Phase AC Filters / DC Filters /PG80 EMC Products Application standards Application standards where the product can be used

AC 3-Phase 3-Line Filters

Oscillators - ElectronicsThis circuit uses the property of RC filters to cause a phase shift, and by using multiple filters, a feedback circuit with exactly 180° phase shift can be

Three-phase filters - Schiele-Vollmar · PDF filewire output connections ... Three-phase power supplies and UPS ... (690V filters) High potential test voltage:

AC 3-Phase 3-Line Filters FT320-FT3200 Series › wp-content › uploads › 2019 › 08 › ... · AC 3-Phase 3-Line Filters FT320-FT3200 Series • Rated current: 0.5A~2000A •