bg physiological signal processing 06 discrete filters2020mk 9652

Design of FIR Filtersa The Filter Design Problem b FIR Filters with Linear Phase c Design of FIR Filter by Windowing d Design of FIR Filter by Frequency Sampling e Chebyshev Polynomials

Nguyễn Công Phương

PHYSIOLOGICAL SIGNAL PROCESSING

Discrete Filters

I Introduction

II Introduction to Electrophysiology

III Signals and Systems

IV Fourier Analysis

V Signal Sampling and Reconstruction

VI The z-Transform

Discrete Filters

1 Types of Filters

2 Transfer Function and Frequency Response

3 Group Delay and Inverse System

4 Minimum and Linear Phase Filters

5 Design of FIR Filters

6 Design of IIR Filters

7 Structure of Realization of Discrete Filters

Types of Filters (1)

http://reactivex.io/documentation/operators/filter.html

Types of Filters (2)

FILTER

3 Group Delay and Inverse System

4 Minimum and Linear Phase Filters

5 Design of FIR Filters

6 Design of IIR Filters

7 Structure of Realization of Discrete Filters

=

( ) ( )

Transfer Function (2)

System

( ) ( )

( )

M

k k k N

k k k

Consider a system function

Find its corresponding difference equation?

1

2

1 2

( ) [ ] ( ) [ ]

3 Group Delay and Inverse System

4 Minimum and Linear Phase Filters

5 Design of FIR Filters

6 Design of IIR Filters

7 Structure of Realization of Discrete Filters

=

( ) ( )

H j

H j

ω ω



∠



Frequency Response (1)

System

( ) ( )

H j

H j

ω ω

ω

=

( ) 2

cos

Frequency Response (4)

10 o

20 log ( ) 0.3 ( ) 207

dB

H j

ω ω

Frequency Response (5)

0.015 o

/ 2 10 0.9661

0 207

Y Y

Y Y

Frequency Response (6)

10 o

20 log ( ) 5.9 ( ) 345

dB

H j

ω ω

Frequency Response (7)

0.295 o

/ 2 10 0.5070

0 345

Y Y

Y Y

ω

=

Discrete Filters

1 Types of Filters

2 Transfer Function and Frequency Response

3 Group Delay and Inverse System

4 Minimum and Linear Phase Filters

5 Design of FIR Filters

6 Design of IIR Filters

7 Structure of Realization of Discrete Filters

Group Delay and Inverse System

j phase delay

dx

H e d

Group Delay and Inverse System

Group Delay and Inverse System

(3)

An LTI system H(z) with input x[n] and output y[n] is said to be invertible

if we can uniquely determine x[n] from y[n]

[ ] * inv [ ] [ ]

h n h n = δ n

1 ( ) inv ( )

( )

M

k k k

N

k k k

Discrete Filters

1 Types of Filters

2 Transfer Function and Frequency Response

3 Group Delay and Inverse System

4 Minimum and Linear Phase Filters

5 Design of FIR Filters

6 Design of IIR Filters

7 Structure of Realization of Discrete Filters

Minimum and Linear Phase

Filters (1)

• A causal and stable system H(z) with a causal and stable inverse H inv (z) is known as a

• H (z) is minimum – phase if both its poles &

zeros are inside the unit circle.

Minimum and Linear Phase

We can obtain the minimum – phase system by replacing each factor (1 + az–1)

(|a| > 1), by a factor of the form a(1 + a–1z–1)

Minimum and Linear Phase

mix mix max

- 0.5 0 0.5 1 1.5

0 5 1

Minimum and Linear Phase

mix mix max

Minimum and Linear Phase

Minimum and Linear Phase

Minimum and Linear Phase

Discrete Filters

1 Types of Filters

2 Transfer Function and Frequency Response

3 Group Delay and Inverse System

4 Minimum and Linear Phase Filters

5 Design of FIR Filters

6 Design of IIR Filters

7 Structure of Realization of Discrete Filters

Design of FIR Filters

00

( )

1

k k

k k

Design of FIR Filters

a) The Filter Design Problem

b) FIR Filters with Linear Phase

c) Design of FIR Filter by Windowing

d) Design of FIR Filter by Frequency Sampling e) Chebyshev Polynomials & Minimax

The Filter Design Problem (1)

1

Filter Spec trum of input s ignal #2

0.2 0.2002 0.2004 0.2006 0.2008 0.201 0.2012 0.2014 0.2016 0.2018 0.202

Time

-2 -1 0 1

Spec trum of output s ignal #2

The Filter Design Problem (2)

1

Filter #2 Spec trum of input s ignal

0.2 0.2002 0.2004 0.2006 0.2008 0.201 0.2012 0.2014 0.2016 0.2018 0.202

Time

-2 -1 0 1

Spec trum of output s ignal #2

The Filter Design Problem (3)

1

Filter #2 Spec trum of input s ignal

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

/

0 0.5

Output s ignal #2

The Filter Design Problem (4)

1

Filter #2 Spec trum of input s ignal

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

/

0 0.5

Output s ignal #2

The Filter Design Problem (5)

The Filter Design Problem (6)

1 2

The Filter Design Problem (7)

The Filter Design Problem (8)

2 2

The Filter Design Problem (9)

Design of FIR Filters

a) The Filter Design Problem

b) FIR Filters with Linear Phase

c) Design of FIR Filter by Windowing

d) Design of FIR Filter by Frequency Sampling e) Chebyshev Polynomials & Minimax

FIR Filters with Linear Phase (1)

[ ]

c lp

= 6.5

-0.1 0 0.1 0.2 0.3 0.4

FIR Filters with Linear Phase (2)

FIR Filters with Linear Phase

FIR Filters with Linear Phase

FIR Filters with Linear Phase

FIR Filters with Linear Phase

FIR Filters with Linear Phase

FIR Filters with Linear Phase

FIR Filters with Linear Phase

2

M j

2 1

FIR Filters with Linear Phase

FIR Filters with Linear Phase

FIR Filters with Linear Phase

FIR Filters with Linear Phase

M j

2 1

FIR Filters with Linear Phase

FIR Filters with Linear Phase

FIR Filters with Linear Phase

( ) sin [ ]cos

2

M j

2 1

FIR Filters with Linear Phase

FIR Filters with Linear Phase

FIR Filters with Linear Phase (9)

FIR Filters with Linear Phase

(10)

( ) 0

1 [ ]cos

1 [ ]sin

FIR Filters with Linear Phase

(11)

( ) 0

FIR Filters with Linear Phase

ω

sin ω

sin 2

ω

Uses

LP, HP, BP, BS multiband filters

LP, BP

Differentiators, Hilbert transformers

Differentiators, Hilbert transformers

FIR Filters with Linear Phase

FIR Filters with Linear Phase

Design of FIR Filters

a) The Filter Design Problem

b) FIR Filters with Linear Phase

c) Design of FIR Filter by Windowing

i Direct Truncation of an Ideal Impulse Response

ii Smoothing the Frequency Response Using Fixed

Windows iii Filter Design Using the Adjustable Kaiser Window

d) Design of FIR Filter by Frequency Sampling

e) Chebyshev Polynomials & Minimax Approximation f) Equiripple Optimum Chebyshev FIR Filter Design g) Design of Some Special FIR Filters

sin( / 2) 1

j M M

j n

Direct Truncation of

an Ideal Impulse Response (4)

ω π

Design of FIR Filters

a) The Filter Design Problem

b) FIR Filters with Linear Phase

c) Design of FIR Filter by Windowing

i Direct Truncation of an Ideal Impulse Response

ii Smoothing the Frequency Response Using Fixed

Windows

iii Filter Design Using the Adjustable Kaiser Window

d) Design of FIR Filter by Frequency Sampling

e) Chebyshev Polynomials & Minimax Approximation f) Equiripple Optimum Chebyshev FIR Filter Design g) Design of Some Special FIR Filters

Smoothing the Frequency Response

Using Fixed Windows (1)

2 / , 0 / 2, [ ] 2 2 / , / 2

0, 5 0,5 cos(2 / ), 0 [ ]

Smoothing the Frequency Response

Using Fixed Windows (2)

Smoothing the Frequency Response

Using Fixed Windows (3)

Window Side lobe

Smoothing the Frequency Response

Using Fixed Windows (4)

Smoothing the Frequency Response

Using Fixed Windows (5)

1 Given the design specification {ω p , ω s , A p , A s }, determine the

ripples δ p , δ s , and set δ = min{δ p , δ s }.

2 Determine the cutoff frequency of the ideal lowpass prototype by

ω

c = (ω p + ω s )/2.

3 Determine the design parameters A = –20log 10 δ & Δω = ω s – ω p

4 From the table choose the window function that provides the

smallest stopband attenuation greater than A.

5 Determine M = L – 1 from Δω in the table If M is odd, we may

increase it by one.

6 Determine the impulse response of the ideal lowpass filter by

7 Compute the impulse response h[n] = h d [n]w[n].

8 Check whether the designed filter satisfies the prescribed

specifications; if not, increase the order M and go back to step 6.

c d

n M

h n

n M

ω π

−

=

−

Smoothing the Frequency Response

Using Fixed Windows (6)

Design a lowpass linear-phase FIR to satisfy the following specifications:

Δω (exact)

δ p ≈ δ s A p

(dB)

A s (dB)

Rect –13 4π/L 1.8π/L 0.09 0.75 21Bartlett –25 8π/L 6.1π/L 0.05 0.45 26Hann –31 8π/L 6.2π/L 0.0063 0.055 44Hamming –41 8π/L 6.6π/L 0.0022 0.019 53Blackman –57 12π/L 11π/L 0.0002 0.0017 74

Smoothing the Frequency Response

Using Fixed Windows (7)

Design a lowpass linear-phase FIR to satisfy the following specifications:

-3 Approximation error

Design of FIR Filters

a) The Filter Design Problem

b) FIR Filters with Linear Phase

c) Design of FIR Filter by Windowing

i Direct Truncation of an Ideal Impulse Response

ii Smoothing the Frequency Response Using Fixed

Windows

iii Filter Design Using the Adjustable Kaiser Window

d) Design of FIR Filter by Frequency Sampling

e) Chebyshev Polynomials & Minimax Approximation f) Equiripple Optimum Chebyshev FIR Filter Design g) Design of Some Special FIR Filters

Filter Design Using the

Adjustable Kaiser Window (1)

2 0

A M

ω

−

=

∆

Filter Design Using the Adjustable Kaiser Window (2)

20

0.2 0.4 0.6 0.8 1

Kaiser, M = 20

= 0 = 5 = 8

-100 -80 -60 -40 -20

Filter Design Using the Adjustable Kaiser Window (3)

1 Given the design specification {ω p , ω s , A p , A s }, determine the

ripples δ p , δ s , and set δ = max{δ p , δ s }.

2 Determine the cutoff frequency of the ideal lowpass prototype by

ω

c = (ω p + ω s )/2.

3 Determine the design parameters A = –20log 10 δ & Δω = ω s – ω p

4 Determine the required values of & M from formulae If M is odd,

we may increase it by one.

5 Determine the impulse response of the ideal lowpass filter by

6 Compute the impulse response h[n] = h d [n]w[n].

7 Check whether the designed filter satisfies the prescribed

specifications; if not, increase the order M and go back to step 6.

c d

n M

h n

n M

ω π

−

=

−

Filter Design Using the Adjustable Kaiser Window (4)

Design a lowpass linear-phase FIR to satisfy the following specifications:

Filter Design Using the Adjustable Kaiser Window (5)

Design a lowpass linear-phase FIR to satisfy the following specifications:

10-3 Approximation error

Filter Design Using the Adjustable Kaiser Window (6)

Ex 2

Design a bandpass filter using a Kaiser window:

( ) 0.01, 0.2 0.99 ( ) 1.01, 0.3 0.7

Filter Design Using the Adjustable Kaiser Window (7)

−

=

−

1 1

40 8

55.7 2.285(0.08 )

Filter Design Using the Adjustable Kaiser Window (8)

10-3 Approxim ation error

Design of FIR Filters

a) The Filter Design Problem

b) FIR Filters with Linear Phase

c) Design of FIR Filter by Windowing

d) Design of FIR Filter by Frequency Sampling

e) Chebyshev Polynomials & Minimax

Design of FIR Filter by Frequency Sampling (1)

Design of FIR Filter by

L

Q  − 

=  

Design of FIR Filter by Frequency Sampling (3)

ω π

Design of FIR Filter by Frequency Sampling (4)

/ 0

/ 0

Design of FIR Filter by Frequency Sampling (5)

/ 0

Design of FIR Filter by Frequency Sampling (6)

1 Choose the order of the filter M by placing at least

two samples in the transition band.

2 For a window design approach obtain samples of the

desired frequency response H d [k] For a smooth

transition band approach, use a straight-line or a

raised-cosine.

3 Compute the (M + 1)-point IDFT of H d [k] to obtain

h [n] For a window design approach multiply h[n] by

the appropriate window function.

4 Compute response H d (e jω ) and verify the design over

passband & stopband.

5 If the specifications are not met, increase M & go back

to step 1.

Design of FIR Filter by Frequency Sampling (7)

Design a lowpass linear-phase FIR to satisfy the following specifications:

Design of FIR Filter by Frequency Sampling (8)

Ex 1

Design of FIR Filters

a) The Filter Design Problem

b) FIR Filters with Linear Phase

c) Design of FIR Filter by Windowing

d) Design of FIR Filter by Frequency Sampling

e) Chebyshev Polynomials & Minimax

Chebyshev Polynomials &

Chebyshev Polynomials &

Chebyshev Polynomials &

Chebyshev Polynomials &

Minimax Approximation (4)

• Chebyshev’s theorem : Off all polynomials of

degree m with coefficient of x m equal to 1, the

Chebyshev polynomial T m (x) multiplied by

2 –(m – 1) has the least maximum amplitude on the interval [–1, 1].

• Alternation theorem : Suppose that f(x) is a

continuous function Then P m (x) is the best

minimax approximating polynomial to f(x) if and only if the error e(x) = f(x) – P m (x) has an (m+2)-point equiripple property.

Chebyshev Polynomials &

( ) ( ) ( ) 0.125

e x = x − P x = x − + x

Design of FIR Filters

a) The Filter Design Problem

b) FIR Filters with Linear Phase

c) Design of FIR Filter by Windowing

d) Design of FIR Filter by Frequency Sampling e) Chebyshev Polynomials & Minimax

Equiripple Optimum Chebyshev

FIR Filter Design (1)

sin , sin( / 2),

odd even

Equiripple Optimum Chebyshev

FIR Filter Design (2)

Equiripple Optimum Chebyshev

FIR Filter Design (3)

R j

then a necessary and sufficient condition that P(ejω) be the unique solution of

is that the weighted error function E(ω) exhibit at least R + 2 alternation in B.

That is, there must exist R + 2 extremal frequencies ω1 < ω2 < …< ωR+2

such that for every k = 1, 2, …, R + 2:

Equiripple Optimum Chebyshev

FIR Filter Design (4)

Equiripple Optimum Chebyshev

FIR Filter Design (5)

Calculate error E(ω)

& find local maxima

Equiripple Optimum Chebyshev

FIR Filter Design (6)

Design a lowpass linear-phase FIR to satisfy the following specifications:

Equiripple Optimum Chebyshev

FIR Filter Design (7)

Design a lowpass linear-phase FIR to satisfy the following specifications:

ωp = 0.25π; ωs = 0.35π; Ap = 0.1dB; As = 50dB.

Ex 1

Equiripple Optimum Chebyshev

FIR Filter Design (8)

Equiripple Optimum Chebyshev

FIR Filter Design (9)

Ex 2

Design a bandpass filter using a Kaiser window:

( ) 0.01, 0.2 0.99 ( ) 1.01, 0.3 0.7

( ) 0.01, 0.78

j j j

H e

H e

H e

ωωω

Equiripple Optimum Chebyshev

FIR Filter Design (10)

Ex 2

Design of FIR Filters

a) The Filter Design Problem

b) FIR Filters with Linear Phase

c) Design of FIR Filter by Windowing

d) Design of FIR Filter by Frequency Sampling

e) Chebyshev Polynomials & Minimax Approximation f) Equiripple Optimum Chebyshev FIR Filter Design

g) Design of Some Special FIR Filters

i Discrete-Time Differentiators

ii Discrete-Time Hilbert Transformers

iii Ideal Raised-Cosine Pulse-Shaping Lowpass Filter

Discrete-Time Differentiators (2)

Design of FIR Filters

a) The Filter Design Problem

b) FIR Filters with Linear Phase

c) Design of FIR Filter by Windowing

d) Design of FIR Filter by Frequency Sampling

e) Chebyshev Polynomials & Minimax Approximation f) Equiripple Optimum Chebyshev FIR Filter Design

g) Design of Some Special FIR Filters

i Discrete-Time Differentiators

ii Discrete-Time Hilbert Transformers

iii Ideal Raised-Cosine Pulse-Shaping Lowpass Filter

Hamming window method

/

0 0.2 0.4 0.6 0.8 1

Frequency sampling method

Design of FIR Filters

a) The Filter Design Problem

b) FIR Filters with Linear Phase

c) Design of FIR Filter by Windowing

d) Design of FIR Filter by Frequency Sampling

e) Chebyshev Polynomials & Minimax Approximation f) Equiripple Optimum Chebyshev FIR Filter Design

g) Design of Some Special FIR Filters

i Discrete-Time Differentiators

ii Discrete-Time Hilbert Transformers

iii Ideal Raised-Cosine Pulse-Shaping Lowpass Filter

Ideal Raised-Cosine Pulse-Shaping Lowpass Filters

n

0 0.1 0.2 0.3

Discrete Filters

1 Types of Filters

2 Transfer Function and Frequency Response

3 Group Delay and Inverse System

4 Minimum and Linear Phase Filters

5 Design of FIR Filters

6 Design of IIR Filters

7 Structure of Realization of Discrete Filters