Chapter 14 Finite Impulse Response (FIR) Filters

Chapter 14Chapter 14

Finite Impulse Response (FIR) FiltersFinite Impulse Response (FIR) Filters

Dr. Naim Dahnoun, Bristol University, (c) Texas Instruments 2004

Chapter 14, Slide 2

Learning ObjectivesLearning Objectives

Introduction to the theory behind FIR Introduction to the theory behind FIR filters:filters: Properties (including aliasing).Properties (including aliasing). Coefficient calculation.Coefficient calculation. Structure selection.Structure selection.

Implementation in Matlab, C, assembly Implementation in Matlab, C, assembly and linear assembly.and linear assembly.


Chapter 14, Slide 3

IntroductionIntroduction

Amongst all the obvious advantages that digital filters offer, the FIR filter can Amongst all the obvious advantages that digital filters offer, the FIR filter can guarantee linear phase characteristics.guarantee linear phase characteristics.

Neither analogue or IIR filters can achieve this.Neither analogue or IIR filters can achieve this. There are many commercially available software packages for filter design. However, There are many commercially available software packages for filter design. However,

without basic theoretical knowledge of the FIR filter, it will be difficult to use them.without basic theoretical knowledge of the FIR filter, it will be difficult to use them.


Chapter 14, Slide 4

Properties of an FIR FilterProperties of an FIR Filter

Filter coefficients:Filter coefficients:

1

0

N

kk knxbny

x[n] x[n] represents the filter input,represents the filter input,bbk k represents the filter coefficients,represents the filter coefficients,

y[n] y[n] represents the filter output,represents the filter output,NN is the number of filter coefficients is the number of filter coefficients

(order of the filter).(order of the filter).


Chapter 14, Slide 5



1

0

N

kk knxbny

z-1

+

z-1 z-1

+ +

z-1

y(n)

x(n)

x xxxb0 b1 b2 bN-1

FIR equationFIR equation

Filter structureFilter structure


Chapter 14, Slide 6



1

0

N

kk knxbny

If the signal x[n] is replaced by an impulse If the signal x[n] is replaced by an impulse [n] then: [n] then:

1

0

N

kk knbny

Nbbby k 100 10


Chapter 14, Slide 7



1

0

N

kk knxbny


1

0

N

kk knbny

Nnbnbnbny k 110


Chapter 14, Slide 8



1

0

N

kk knxbny


1

0

N

kk knbny

kn for 0

knfor 1kn


Chapter 14, Slide 9



1

0

N

kk knxbny

Finally: Finally:

khb

hb

hb

k

1

0

1

0


Chapter 14, Slide 10



1

0

N

kk knxbny

With: With: khbk

The coefficients of a filter are the same as The coefficients of a filter are the same as the impulse response samples of the filter.the impulse response samples of the filter.



Frequency Response of an FIR FilterFrequency Response of an FIR Filter

By taking the z-transform of h[n], H(z):By taking the z-transform of h[n], H(z):

Replacing z by eReplacing z by ejj in order to find the in order to find the frequency response leads to:frequency response leads to:

1

0

N

n

nznhzH

1

0

N

n

jnj

ezenheHzH j




Since eSince e-j2-j2kk = 1 then: = 1 then:

Therefore:Therefore:

1

0

1

0

22

N

n

jnN

n

jn

ezenhenhzH

jkj eHeH 2

FIR filters have a periodic frequency FIR filters have a periodic frequency response and the period is 2response and the period is 2..




Frequency Frequency response:response:

FIRFIR y[n]y[n]x[n]x[n]

FFss/2/2FFss/2/2

FreqFreq

FreqFreq

x[n

]x[

n]

y[n

]y[

n]

jkj eHeH 2




Solution: Use an anti-aliasing filter.Solution: Use an anti-aliasing filter.

FIRFIR y[n]y[n]x[n]x[n]ADCADC

Analogue Analogue Anti-AliasingAnti-Aliasing

x(t)x(t)

FFss/2/2FFss/2/2FreqFreqFreqFreq

x(t)

x(t)

y[n

]y[

n]



Phase Linearity of an FIR FilterPhase Linearity of an FIR Filter

A causal FIR filter whose impulse response is symmetrical is A causal FIR filter whose impulse response is symmetrical is guaranteed to have a linear phase response.guaranteed to have a linear phase response.

0n

h(n)

1 n n+1 2n+12n

N = 2n + 2

0n

h(n)

1 n n+1 2n2n-1n-1

N = 2n + 1

Even symmetryEven symmetry Odd symmetryOdd symmetry




A causal FIR filter whose impulse response is symmetrical (ie h[n] = h[N-A causal FIR filter whose impulse response is symmetrical (ie h[n] = h[N-1-n] for n = 0, 1, …, N-1) is guaranteed to have a linear phase response.1-n] for n = 0, 1, …, N-1) is guaranteed to have a linear phase response.

Condition Phase

2

1Nk Phase Property Filter Type

1 nNhnh

Positive Symmetryk Linear phase

Odd Symmetry – Type 1

Even Symmetry – Type 2




Application of 90° linear phase shift:Application of 90° linear phase shift:

Signal separationSignal separation

9090oo

delaydelay

9090oo

delaydelay

++

++

++

--

II

QQ

ReverseReverse

ForwardForward

IHIH

QHQH

tBtAI rf sincos

tBtAQ rf cossin tBtA

tBtAIH

rf

rf

cossin

2sin

2cos

tBQIH rcos2

tBIQH fsin2



Design ProcedureDesign Procedure To fully design and implement a filter five steps are required:To fully design and implement a filter five steps are required:

(1)(1) Filter specification.Filter specification.

(2)(2) Coefficient calculation.Coefficient calculation.

(3)(3) Structure selection.Structure selection.

(4)(4) Simulation (optional).Simulation (optional).

(5)(5) Implementation.Implementation.



Filter Specification - Step 1Filter Specification - Step 1

(a)

1

f(norm)fc : cut-off frequency

pass-band stop-band

pass-band stop-bandtransition band

1

s

pass-bandripple

stop-bandripple

fpb : pass-band frequency

fsb : stop-band frequency

f(norm)

(b)

p1

s

p0

-3

p1

fs/2

fc : cut-off frequency

fs/2

|H(f)|(dB)

|H(f)|(linear)

|H(f)|



Coefficient Calculation - Step 2Coefficient Calculation - Step 2 There are several different methods available, the most popular are:There are several different methods available, the most popular are:

Window method.Window method. Frequency sampling.Frequency sampling. Parks-McClellan.Parks-McClellan.

We will just consider the window method.We will just consider the window method.



Window MethodWindow Method

First stage of this method is to calculate the coefficients of the First stage of this method is to calculate the coefficients of the ideal filterideal filter.. This is calculated as follows:This is calculated as follows:

0nfor

0nfor

2

sin2

12

1

2

1

c

c

cc

nj

njd

fn

nf

de

deHnh

c

c



Window MethodWindow Method Second stage of this method is to select a window function based on the passband or Second stage of this method is to select a window function based on the passband or

attenuation specifications, then determine the filter length based on the required width of the attenuation specifications, then determine the filter length based on the required width of the transition band.transition band.

W i n d o w T y p e N o r m a l i s e d T r a n s i t i o nW i d t h ( f ( H z ) )

P a s s b a n d R i p p l e ( d B ) S t o p b a n d A t t e n u a t i o n( d B )

R e c t a n g u l a rN

9.00 . 7 4 1 6 2 1

H a n n i n gN

1.30 . 0 5 4 6 4 4

H a m m i n gN

3.30 . 0 1 9 4 5 3

B l a c k m a nN

5.50 . 0 0 1 7 7 4

K a i s e r

54.493.2

N

96.871.5

N

0 . 0 2 7 4

0 . 0 0 0 2 7 5

5 0

9 0

13284.12.1

3.33.3

kHz

kHzfNUsing the Hamming Using the Hamming

Window:Window:




The third stage is to calculate the set of truncated or windowed The third stage is to calculate the set of truncated or windowed impulse response coefficients, h[n]:impulse response coefficients, h[n]:

nWnhnh d even Nfor

odd Nfor

22

2

1

2

1

Nn

N

Nn

N

133

2cos46.054.0

2

cos46.054.0

n

N

nnW

forfor

Where:Where:6666 nforfor




Matlab code for calculating coefficients:Matlab code for calculating coefficients:close all;clear all;

fc = 8000/44100; % cut-off frequencyN = 133; % number of tapsn = -((N-1)/2):((N-1)/2);n = n+(n==0)*eps; % avoiding division by zero

[h] = sin(n*2*pi*fc)./(n*pi); % generate sequence of ideal coefficients[w] = 0.54 + 0.46*cos(2*pi*n/N); % generate window functiond = h.*w; % window the ideal coefficients

[g,f] = freqz(d,1,512,44100); % transform into frequency domain for plotting

figure(1)plot(f,20*log10(abs(g))); % plot transfer functionaxis([0 2*10^4 -70 10]);

figure(2);stem(d); % plot coefficient valuesxlabel('Coefficient number');ylabel ('Value');title('Truncated Impulse Response');

figure(3)freqz(d,1,512,44100); % use freqz to plot magnitude and phase responseaxis([0 2*10^4 -70 10]);




0 0.5 1 1.5 2

x 104

-6000

-4000

-2000

0

Frequency (Hz)

Pha

se (

degr

ees)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

x 104

-60

-40

-20

0

Frequency (Hz)

Mag

nitu

de (

dB)

0 20 40 60 80 100 120 140-0.1

0

0.1

0.2

0.3

0.4

Coefficient number, n

h(n

)

Truncated Impulse Response



Realisation Structure Selection - Step 3Realisation Structure Selection - Step 3

1

0

N

k

kk zbzH

zXzHzY 1....1 110 Nnxbnxbnxbny N

z-1

z-1

z-1

+ + +

b0

b2

bN-1

y(n)

x(n)

b1

Direct form structure for an FIR filter:Direct form structure for an FIR filter:




1

0

N

k

kk zbzH

Linear phase structures:Linear phase structures:

N even:N even:

N Odd:N Odd:

1

2

0

1

N

k

kNkk zzbzH

2

1

0

2

1

2

11

N

k

N

NkNk

k zbzzbzH





(a) N even.(a) N even.

(b) N odd.(b) N odd.+

b0

+

+

+

+b

1

+b

2

+b

N/2-1

y(n)

(a)

z-1

z-1

z-1

z-1

z-1

+

b0

+

+

+

+b

1

+b

2

+b (N-3)/2

y(n)

x(n)

b(N-1)/2

+

(b)

z-1

z-1

z-1

z-1

z-1

z-1




1

0

N

k

kk zbzH

Cascade structures:Cascade structures:

M

kkk

NN

NN

N

k

kk

zbzbb

zb

bz

b

bz

b

bb

zbzbzbbzbzH

1

22,

11,0

1

0

12

0

21

0

10

11

22

110

1

0

1

...1

...





1

0

N

k

kk zbzH

Cascade structures:Cascade structures:

z -1

+b

1,1

x(n)

z -1

+

b1,2

z -1

+b

2,1

z -1

+

b2,2

z -1

+b

M,1

z -1

+

bM,2

y(n)b0




Implementation - Step 5Implementation - Step 5 Implementation procedure in ‘C’ with fixed-point:Implementation procedure in ‘C’ with fixed-point:

Set up the codec (Set up the codec (\Links\\Links\CodecSetup.pdfCodecSetup.pdf).).

Transform: to ‘C’ code. Transform: to ‘C’ code.

((\Links\\Links\FIRFixed.pdfFIRFixed.pdf)) Configure timer 1 to generate an interrupt at 8000Hz (Configure timer 1 to generate an interrupt at 8000Hz (\Links\\Links\TimerSetup.pdfTimerSetup.pdf).). Set the interrupt generator to generate an interrupt to invoke the Interrupt Service Routine (ISR) (Set the interrupt generator to generate an interrupt to invoke the Interrupt Service Routine (ISR) (\Links\\Links\InterruptSetup.pdfInterruptSetup.pdf).).

1

0

N

kk knxbny



Implementation - Step 5Implementation - Step 5 Implementation procedure in ‘C’ with floating-point:Implementation procedure in ‘C’ with floating-point:

Same set up as fixed-point plus:Same set up as fixed-point plus: Convert the input signal to floating-point format.Convert the input signal to floating-point format. Convert the coefficients to floating-point format.Convert the coefficients to floating-point format. With floating-point multiplications there is no need for the shift required when using Q15 format.With floating-point multiplications there is no need for the shift required when using Q15 format.

See See \Links\\Links\FIRFloat.pdfFIRFloat.pdf



Implementation - Step 5Implementation - Step 5 Implementation procedure in assembly:Implementation procedure in assembly:

Same set up as fixed-point, however:Same set up as fixed-point, however: is written in assembly.is written in assembly.

((\Links\\Links\FIRFixedAsm.pdfFIRFixedAsm.pdf))

The ISR is now declared as external.The ISR is now declared as external.

1

0

N

kk knxbny



Implementation - Step 5Implementation - Step 5 Implementation procedure in assembly:Implementation procedure in assembly:

The filter implementation in assembly is now using The filter implementation in assembly is now using circular addressingcircular addressing and therefore: and therefore: The circular pointers and block size register are selected and initialised by setting the appropriate values of the AMR bit fields.The circular pointers and block size register are selected and initialised by setting the appropriate values of the AMR bit fields. The data is now aligned using:The data is now aligned using:

Set the initial value of the circular pointers, see Set the initial value of the circular pointers, see \Links\\Links\FIRFixedAsm.pdfFIRFixedAsm.pdf..

#pragma DATA_ALIGN (symbol, constant (bytes))#pragma DATA_ALIGN (symbol, constant (bytes))



Implementation - Step 5Implementation - Step 5

y0 = b0*x0y0 = b0*x0 + b1*x1+ b1*x1 + b2*x2+ b2*x2 + b3*x3+ b3*x3

Circular addressing link slide.Circular addressing link slide.

timetime

y[n

]y[

n]

00 11 22

bb00

bb11

bb22

bb33

xx00

xx11

xx22

xx33



xx44

xx11

xx22

xx33



y1 = b0*x4y1 = b0*x4 + b1*x1+ b1*x1 + b3*x3+ b3*x3+ b2*x2+ b2*x2


timetime

y[n

]y[

n]

00 11 22

bb00

bb11

bb22

bb33



xx44

xx55

xx22

xx33






timetime

y[n

]y[

n]

00 11 22

bb00

bb11

bb22

bb33



FIR CodeFIR Code

Code location:Code location: Code\Chapter 14 - Finite Impulse Response FiltersCode\Chapter 14 - Finite Impulse Response Filters

Projects:Projects: Fixed Point in C:Fixed Point in C: \FIR_C_Fixed\\FIR_C_Fixed\ Floating Point in C:Floating Point in C: \FIR_C_Float\\FIR_C_Float\ Fixed Point in Assembly:Fixed Point in Assembly: \FIR_Asm_Fixed\\FIR_Asm_Fixed\ Floating Point in Assembly:Floating Point in Assembly: \FIR_Asm_Float\\FIR_Asm_Float\

Chapter 14Chapter 14

Finite Impulse Response (FIR) FiltersFinite Impulse Response (FIR) Filters

- End -- End -

Chapter 14 Finite Impulse Response (FIR) Filters

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