Compare ideal Interpolation filter and interpolation by LSE FIR filter(2)

Advisor : Dr. Yuan-AN Kao

Student: Bill Chen

Outline

• FIR Filter by Windowing• Comparison (Simulation)

• Conclusion

• Reference

Design of FIR Filter By Windowing(1/2)

-

-

-

( ) [ ]

[ ]

1[ ] ( )

2

j j nd d

n

d

j j nd d

Ideal desired frequency response

H e h n e

h n is the corresponding impulse response sequence

h n H e e d

Design of FIR Filter By Windowing (1/2)

[ ] [ ]

[ ] ,0[ ]

0 ,

[ ]

-

define h n is to define a new system with impulse response h ndh n n Mdh n

otherwise

we can respresent h n as the product of the desired impulse

response and a finite duration

" " [ ]

[ ] [ ] [ ] , 0

1 ( - )( ) ( ) ( )-2

window w n

h n h n w n n Md

j j jH e H e W e dd

Kaiser Window & Simulation

2 1 20

0

0

[ (1 [( ) / ] ) ], 0

[ ] ( )

0,

2

( ) - mod .

0.1102( 8.7)

=

I nn M

w n I

otherwise

where M

I represents the zeroth order ified Bessel function of the first kind

A

0.4

10

,A>50

0.5842(A-21) 0.07886( 21) ,21 50

0 ,A<21

20log

A A

A

Kaiser Window (Simulation)

M+1=55

Alpha=0.5M

Beta

Kaiser Window (Simulation)

Comparison(1/14)Filter coefficient M=55

Interpolation filter by LSE FIR filter

Upsample=5

Cutoff freq=0.2pi

Passband freq=0.15pi

Stopband freq=0.25pi

Ideal interpolation filter with Kaiser Window

Alpha=0.5*(M-1)

Beta

Comparison (2/14)Filter coefficient M=55

Interpolation filter by LSE FIR filter

Upsample=5

Cutoff freq=0.2pi

Passband freq=0.15pi

Stopband freq=0.25pi

Ideal interpolation filter with Kaiser Window

Alpha=0.5*(M-1)

Beta

Comparison (3/14)Filter coefficient M=55

Interpolation filter by LSE FIR filter

Upsample=5

Cutoff freq=0.2pi

Passband freq=0.1pi

Stopband freq=0.3pi

Ideal interpolation filter with Kaiser Window

Alpha=0.5*(M-1)

Beta

Comparison (4/14)

Filter coefficient M=55

Interpolation filter by LSE FIR filter

Upsample=5

Cutoff freq=0.2pi

Passband freq=0.1pi

Stopband freq=0.3pi

Ideal interpolation filter with Kaiser Window

Alpha=0.5*(M-1)

Beta

Comparison (5/14)Filter coefficient M=55

Interpolation filter by LSE FIR filter

Upsample=5

Cutoff freq=0.2pi

Passband freq=0.17pi

Stopband freq=0.23pi

Ideal interpolation filter with Kaiser Window

Alpha=0.5*(M-1)

Beta

Comparison (6/14)Filter coefficient M=55

Interpolation filter by LSE FIR filter

Upsample=5

Cutoff freq=0.2pi

Passband freq=0.17pi

Stopband freq=0.23pi

Ideal interpolation filter with Kaiser Window

Alpha=0.5*(M-1)

Beta

Comparison (7/14)Filter coefficient M=11

Interpolation filter by LSE FIR filter

Upsample=5

Cutoff freq=0.2pi

Passband freq=0.15pi

Stopband freq=0.25pi

Ideal interpolation filter with Kaiser Window

Alpha=0.5*(M-1)

Beta0

Comparison (8/14)Filter coefficient M=11

Interpolation filter by LSE FIR filter

Upsample=5

Cutoff freq=0.2pi

Passband freq=0.15pi

Stopband freq=0.25pi

Ideal interpolation filter with Kaiser Window

Alpha=0.5*(M-1)

Beta0

Comparison (9/14)Filter coefficient M=11

Interpolation filter by LSE FIR filter

Upsample=5

Cutoff freq=0.2pi

Passband freq=0.15pi

Stopband freq=0.25pi

Ideal interpolation filter with Kaiser Window

Alpha=0.5*(M-1)

Beta3

Comparison (10/14)Filter coefficient M=11

Interpolation filter by LSE FIR filter

Upsample=5

Cutoff freq=0.2pi

Passband freq=0.15pi

Stopband freq=0.25pi

Ideal interpolation filter with Kaiser Window

Alpha=0.5*(M-1)

Beta3

Comparison (11/14)Filter coefficient M=11

Interpolation filter by LSE FIR filter

Upsample=5

Cutoff freq=0.2pi

Passband freq=0.15pi

Stopband freq=0.25pi

Ideal interpolation filter with Kaiser Window

Alpha=0.5*(M-1)

Beta6

Comparison (12/14)Filter coefficient M=11

Interpolation filter by LSE FIR filter

Upsample=5

Cutoff freq=0.2pi

Passband freq=0.15pi

Stopband freq=0.25pi

Ideal interpolation filter with Kaiser Window

Alpha=0.5*(M-1)

Beta6

Comparison (13/14)Filter coefficient M=11

Interpolation filter by LSE FIR filter

Upsample=5

Cutoff freq=0.2pi

Passband freq=0.15pi

Stopband freq=0.25pi

Comparison (14/14)Filter coefficient M=11

Interpolation filter by LSE FIR filter

Upsample=5

Cutoff freq=0.2pi

Passband freq=0.15pi

Stopband freq=0.25pi

Conclusion

Reference

• F.M.Gardner, ”Interpolation in digital modems-Part I :Fundamental” IEEE Trans.Commun.,vol.41 pp.502-508,Mar.1993

• J.V.,F.L.,T.S.,andM.R. ”The effects of quantizing the fractional interval in interpolation filters”

• Heinrich Meyr ,Marc Moeneclaey ,Stefan A. Fechtel “Digital Communication Receivers”. New York :Wiley 1997

• C. S. Burrus, A. W. Soewito and R. A. Gopnath, “Least Squared Error FIR Filter Design with Transition Bands,” IEEE Trans. Signal Processing, vol. 40, No. 6, pp.1327-1338, June 1992.

• Heinrich Meyr ,Marc Moeneclaey ,Stefan A. Fechtel “Digital Communication Receivers”. New York :Wiley 1997

• Alan V. Oppenheim ,Ronald W. Schafer with John R. Buck “Discrete-Time Signal Processing”.