Design of FIR Filters

120.07.2015OVGU Präsentation
Design of FIR Filters
Aranya Sarkar
M.Sc- Electrical Engineering and Information Technology
20.07.2015

 Introduction- digital filters
 FIR filters, advantages and disadvantages
 Frequency response of FIR filters
 Design methods
 Windowing techniques
 Optimum filter designing and various techniques
 Alternation Theorem
 Parks- Mcclellan Algorithm
 Conclusion
 References
Contents

 performs mathematical operation on
a sampled discrete time signal
to reduce or enhance certain aspects
Advantages
 Software programmable
 Requires only arithmetic functions
 Do not drift with temperature or humidity
 Superior performance-to-cost ratio
 Do not suffer from manufacturing defects or aging
Digital Filter-Introduction

 A filter whose response has finite duration
 Non recursive since unlike IIR filters, the feedback is not there
Fig. FIR Filter of order n
FIR Filters

Advantages:
 Unconditionally stable
 Simple to implement
 Linear
 Non Causal
Disadvantages:
 Expensive due to large order
 Requires more memory
 Time consuming process
Advantages and Disadvantages of FIR Filters

 Let’s consider the desired impulse response of the FIR is hd[n].
 DTFT of hd[n] is 𝐻 𝑑 𝜔 = ℎ=−∞
∞
ℎ 𝑑(𝑛)𝑒−𝑗𝜔𝑛
hd[n] should be finite. So we need to truncate it from 0 to M to have an
order of M+1.
Considering an ideal low pass filter:
Frequency response of FIR filter

 Windowing Technique
 Frequency Sampling
 Equiripple Design
Basic Design Methods

 Simplest way of designing FIR filters
Method is all discrete-time no continuous-time involved
Start with ideal frequency response
 The easiest way to obtain a causal FIR filter from ideal is
 More generally
Filter Design by Windowing
       


 

else0
Mn01
nwwherenwnhnh d
   


 

else0
Mn0nh
nh d
   




n
nj
d
j
d enheH     

 



deeH
2
1
nh njj
dd

 Narrowest main lob
-4/(M+1)
-Sharpest transitions
at discontinuites in frequency
 Large side lobs
-13 dB
-Large oscillation
around discontinuities
• Simplest window possible
Rectangular Window

 Medium main lob
-8/M
 Side lobs
-25 dB
 Hamming window performs better
 Simple equation
Bartlett (Triangular) Window
 








else0
Mn2/MM/n22
2/Mn0M/n2
nw

 Medium main lob
- 8/M
 Side lobs
- 41 dB
 Simpler than Blackman
Hamming Window
 









 


else0
Mn0
M
n2
cos46.054.0
nw

Comparison of Frequency Response of The Windows

 Though ´windowing method is simple, it is not the most effective
 Rectangular windowing is optimum In one sense since they
minimise the mean squared approximation error to desired
response, but causes errors around discontinuities
Most popular alternative method: Parks-McClellan Algorithm
 Uses minimax error method for function approximation
Optimum Filter Design

 Weighted-least-squares method
Chebyshev method
WLS-Chebyshev method
 Parks-Mcclellan algorithm
Different Methods of Optimum Filter Designing by
Approximation

 Often called the Remez exchange method
 This method designs an optimal linear phase filter
 This is the standard method for FIR filter design
 This methodology for designing symmetric filters that minimize
filter length for a particular set of design constraints {ωp, ωs, δ p, δ
s}
 The computational effort is linearly proportional to the length of
the filter
 In Matlab, this method is available as remez().
Parks- McClellan

 The resulting filters minimize the maximum error between the
desired frequency response and the actual frequency response by
spreading the approximation error uniformly over each band
 Such filters that exhibit equiripple behavior in both the passband
and the stopband, and are sometimes called equiripple filters
Parks- McClellan Method

 The polynomial of degree L that minimizes the maximum error will
have at least L+2 extrema.
 The optimal frequency response will just touch
the maximum ripple bounds.
 Extrema must occur at the pass and stop band edges and
at either ω=0 or π or both.
 The derivative of a polynomial of degree L is a polynomial of
degree L-1, which can be zero in at most L-1 places. So the
maximum number of local extrema is the L-1 local extrema plus
the 4 band edges. That is L+3.
 The alternation theorem doesn’t directly suggest a method for
computing the optimal filter
Alternation Theorem

Parks-McClellan Algorithm

 FIR filters allow the design of linear phase filters, which eliminate
the possibility of signal phase distortion
 Two methods of linear phase FIR design were discussed:
-The ideal window method
-The optimal Parks-McClellan method
 FIR is advantageous due to linearity and stability
 The disadvantages of FIR include expensiveness and that the
process is time consuming
Conclusion

 Digital Signal Processing, Alan V.Oppenheim/Ronald W. Schafer,
ISBN-10 0132146355, Prentice Hall, June 1974
 Digital Signal Processing: A computer based approach, Sanjit
K.Mehta, ISBN 9780072513783, Mcgraw Hill, 1997
 Digital Signal Processing, P.Ramesh Babu, ISBN 8187328525,
Scitech Publications, 2003
 Parks-McClellan FIR Filter Design, Eman R.El-
Taweel/MaysoonA.Abu Shamla, Islamic University-Gaza, 2nd May,
2007
References

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Design of FIR Filters

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