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120.07.2015 OVGU Präsentation
Design of FIR Filters
Aranya Sarkar M.Sc- Electrical Engineering and Information Technology
20.07.2015
220.07.2015 OVGU Präsentation
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
320.07.2015 OVGU Präsentation
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
420.07.2015 OVGU Präsentation
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
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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
620.07.2015 OVGU Präsentation
Let’s consider the desired impulse response of the FIR is hd[n].
DTFT of hd[n] is
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
720.07.2015 OVGU Präsentation
Windowing Technique
Frequency Sampling
Equiripple Design
Basic Design Methods
820.07.2015 OVGU Präsentation
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
Mn01nw where nwnhnh d
else0
Mn0nhnh d
n
njd
jd enheH
deeH21
nh njjdd
920.07.2015 OVGU Präsentation
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
1020.07.2015 OVGU Präsentation
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
1120.07.2015 OVGU Präsentation
Medium main lob
- 8/M
Side lobs
- 41 dB
Simpler than Blackman
Hamming Window
else0
Mn0M
n2cos46.054.0nw
1220.07.2015 OVGU Präsentation
Comparison of Frequency Response of The Windows
1320.07.2015 OVGU Präsentation
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
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Weighted-least-squares method
Chebyshev method
WLS-Chebyshev method
Parks-Mcclellan algorithm
Different Methods of Optimum Filter Designing by Approximation
1520.07.2015 OVGU Präsentation
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
1620.07.2015 OVGU Präsentation
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
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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
1820.07.2015 OVGU Präsentation
Parks-McClellan Algorithm
1920.07.2015 OVGU Präsentation
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
2020.07.2015 OVGU Präsentation
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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QUESTIONS ?
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Thank You For Your Attention