Design of FIR Filters

120.07.2015 OVGU 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

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

Mn01nw where nwnhnh d

else0

Mn0nhnh d

n

njd

jd enheH

deeH21

nh njjdd


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

Mn0M

n2cos46.054.0nw


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


QUESTIONS ?


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