Linear Phase Finite Impulse Response (FIR) Digital Filters, Summaries of Microelectronic Circuits

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Finite Impulse Response (FIR)

Digital Filters (I)

Types of linear phase FIR filters

Yogananda Isukapalli

Key characteristic features of FIR filters

1. The basic FIR filter is characterized by the following two

equations:

å

=

1

0

N

k

y n h k x n k

å

=

1

0

N

k

k

H z h k z

where h(k), k=0,1,…,N-1, are the impulse response

coefficients of the filter, H(z) is the transfer function

and N the length of the filter.

2. FIR filters can have an exactly linear phase response.

3. FIR filters are simple to implement with all DSP processors

available having architectures that are suited to FIR filtering.

a

0 w c

p w

k

0 w c

p w

Phase

Magnitude

Example) Ideal Lowpass filter

• A filter is said to have a linear phase response if,

J w b aw

J w aw

= -

= -

( )

( )

where a and b are constants.

H(e

jw

k e

• jwa

passband

0 otherwise

Magnitude response = |H(e

jw

)| = k

Phase response (q(w)) = < H(e

jw

) = - wa

Follows: y[n] = kx[n-a] : Linear phase implies that the

output is a replica of x[n] {LPF} with a time shift of a

• p - w u - w l

0 w l

w u

p w

• p - w u - w l

0 w l

w u

p w

h[n] = h[N- 1 - n] n = 0,1,….(N-1)/

h[0] = h[10]

h[1] = h[9]

h[2] = h[8]

h[3] = h[7]

h[4] = h[6]

h[5] = h[5]

Example) For positive symmetry and N = 11 odd length

8

0 1 2 3 4 5 6 7 8 9 10

N = 11

0 1 2 3 4 5 6 7 8 9

N = 10

h[n] = h[10- 1 - n] = h[9-n]

h[0] = h[9]

h[1] = h[8]

h[2] = h[7]

h[3] = h[6]

h[4] = h[5]

Consider Frequency Response :

= + + +

= =

---

=

jw j w jw jw j w

z e

jw

H e h e h e h e e

H e H z jw

2 2

( ) [ 2 ] [ [ 1 ] [ 3 ] ]

( ) ( ) (T 1 )

j w j w j w

h e h e e

2 2 2

[ [ 0 ] [ 4 ] ]

[ ]

[ [ 2 ] 2 [ ]cos( ( 2 ))

[ [ 2 ] 2 [ 1 ]cos 2 [ 0 ]cos 2

2

1

0

2

2

j w j w

n

j w

j w

e H e

e h h n w n

e h h w h w

• q

=

ú

û

ù

ê

ë

é

å

Phase = - 2 w

( Linear Phase form)

Group Delay :

( phase )

dw

d

T g

=

passband

2 w

0 w p

p w

H

0 w p

p w

Group delay is constant over the passband for linear phase

filters.

g

T

2. Type II FIR linear phase system

The impulse response is positive symmetric and

N is an even integer

h [ n ]= h [ N - 1 - n ],

0 £ n £( N / 2 )- 1

The frequency response is

å

=

=

( / 2 ) 1

0

( ) [ ]

N

n

jw jwn

H e hne

)] ,

2

1

( ) [ ]cos[ (

/ 2

1

( 1 )/ 2

ï þ

ï

ý

ü

ï î

ï

í

ì

= - å

=

N

n

jw jw N

H e e bn wn

b [ n ] 2 h [ N / 2 n ],

where

= -

n = 1 , 2 ,... N / 2.

3. Type III FIR linear phase system

The impulse response is negative-symmetric and

N an odd integer.

h [ n ]= - h [ N - 1 - n ],^0 £^ n £( N -^1 )/^2

The frequency response is

å

=

=

( 1 )/ 2

0

( ) [ ]

N

n

jw jwn

H e hne

( ) [ ]sin( ) ,

( 1 )/ 2

1

( 1 )/ 2

ï þ

ï

ý

ü

ï î

ï

í

ì

= å

=

N

n

jw jw N

H e je a n wn

a [ n ] 2 h [(( N 1 )/ 2 ) n ],

where

= - -

n = 1 , 2 ,...( N - 1 )/ 2.

Fig: A summary of four types of linear phase FIR filters

1

Fig: A comparison of the impulse of the four types of linear phase FIR filters

1

• The phase delay (for type 1 and 2 filters) or group delay (for

all four types) is expressible in terms of the number of

coefficients of the filter.

• Thus they can be corrected to give a zero phase or group delay

response.

• For example,

T

N

T p

÷

ø

ö

ç

è

æ -

=

2

1 Phase delay for

types 1 and 2

T

N

T g

÷

ø

ö

ç

è

æ - -

=

2

1 p

Group delay for

types 3 and 4

where T is the sampling period.

References

1. “Digital Signal Processing – A Practical Approach” -

Emmanuel C. Ifeachor and Barrie W. Jervis

Second Edition