Hossein Sameti Department of Computer Engineering Sharif University of Technology

CE 40763Digital Signal Processing

Fall 1992

Design of digital FIR filtersusing the Windowing Technique

Hossein SametiDepartment of Computer Engineering

Sharif University of Technology

Design of Digital Filters

LTI Systemsh(n)

FIR IIR

With rational transfer function

)()()(

zQzPzH

No rational transfer function

)()()(

zQzPzH

Determine coefficients of h(n) [or P(z)

and Q(z)]

2Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

Design Stages1. Specifications Application dependent 2. Design h(n) Determine coefficients of h(n)3. Realization Direct form I,II, cascade and parallel4. Implementation Programming in Matlab/C, DSP, ASIC,…

Design of FIR filters◦ Windowing

Design of digital filters


IDTFT of ideal low-pass filter:

Motivation: impulse response of ideal-low-pass filter

c

cX01

)(

c

c

dedeXnx njnj

2

121 )()(

njeee

njnx

njnjnj

ccc

c

2][

21)(

n

nnx c

sin)(


Motivation: impulse response of ideal low-pass filter

Multiply by a rectangular window

• It can be shown that if we have a linear-phase ideal filter and we multiply it by a symmetric window function, we end up with a linear-phase FIR filter.


Incorporation of Generalized Linear Phase

Windows are designed with linear phase in mind◦ Symmetric around M/2

So their Fourier transform are of the form

Will keep symmetry properties of the desired impulse response

Assume symmetric desired response

With symmetric window

◦Periodic convolution of real functions

00

w M n n Mw n

else

/ 2 where is a real and evenj j j M je eW e W e e W e

/ 2j j j Md eH e H e e

12

jj je e eA e H e W e d


The steps in the design of FIR filters using windows are as follows:1. Start with the desired frequency response results in the

sinc function in time domain 2. Compute3. Determine the appropriate window function w(n)4. Calculate

Design of FIR filters using windows

)(dH)()}({ nhHIDTFT dd

)()()( nwnhnh d

A finite-length window function


Two properties should be considered: 1) The amplitude is unity in the pass band and

it is zero in the stop band: 2) The phase is linear:

Desired frequency response

)()()( jmd eHH

)(mH

)(H


Example: Design of a high-pass FIR filter

• First, we have to decide on the type of the filter.

• Assume Type I filter (linear-phase) 2

1,,0

NoddN

jmd eHH )()(

c


Example: Design of a high-pass FIR filter j

md eHH )()(

otherwise0

1)( cc

dH

otherwise0)( cc

j

deH

deeHIDTFTnh njjdd

c

c

21

)}({)(

))(sin()(

)1()(

nn

nh c

n

d IIR filter

c

otherwise0

1)( cc

mH

otherwise0

)( ccdH


Example: Design of a high-pass FIR filter

n0 1 2 3 4

1.0

3.0

)()()( nwnhnh d

3.0

5.0

5 6

1.0

7N

• It is a high-pass FIR filter with 7 taps that approximates the high-pass IIR filter.

• How can we quickly check that the resulting FIR filter has the desired properties that we were looking for? (i.e., it is a high-pass linear-phase filter)?


Reminder: DTFT Pairs

Hossein Sameti, ECE, UBC, Summer 2012 Originally Prepared by: Mehrdad Fatourechi, 12

Windowing in frequency domain)()()( nwnhnh d )(*)()( WHH d

• What condition should we impose on W(ω) so that H (ω) looks like Hd(ω) ?• Impulse function in the frequency domain, means an

infinitely-long constant in the time-domain

• Larger window means more computation


Windowing in Frequency Domain Windowed frequency response

The windowed version is smeared version of desired response

If w[n]=1 for all n, then W(ej) is pulse train with 2 period

12

jj jdH e H e W e d





Rationale for the shape of the filter

(Oppenheim and Schaffer, 2009)

Ideal filter

RectangularWindow function

NM 1


Filter Specifications

p 0Pass-band: sStop-band:

p 1Pass-band ripple:

Stop-band ripple: s 2

Transition width: ps

• What is the ideal situation?(Oppenheim and Schaffer, 2009)


Filter Specifications

)(log20 H


ObservationsWidth of transition is not sharp!

• Ripples in the passband / stopband are proportional to the peaks of side lobes of the window.

• The width of transition depends on the width of the main lobe of the window.


Controlling the width of the main lobe

• Q: How can we control the transition width (size of the main lobe)?

• A1: using the size of the window Uncertainty principle


Controlling the width of the main lobe

• Q: How can we control the size of transition width (size of the main lobe)?• A2: Shape of the window; in other words, windows

with a fixed size that have different shapes can have different main lobe width.• Rectangular window Smallest; and Blackman

largest main lobe width


Controlling the peak of the side lobe

• Q: How can we control the peak of the side lobes so that we can get a good ripple behavior in the FIR filter?• A: using the shape of the window


Controlling the peak of the side lobe

• Q: Can we control the peak of the side lobes by changing the size of the window?

• A: It can be shown that changes are not significant.


Demonstration using Kaiser window


Properties of Windows

Prefer windows that concentrate around DC in frequency◦ Less smearing, closer approximation

Prefer window that has minimal span in time ◦ Less coefficient in designed filter, computationally efficient

So we want concentration in time and in frequency◦ Contradictory requirements

Example: Rectangular window

1

0

/ 2

11

sin 1 / 2sin / 2

j MMj j n

jn

j M

eW e ee

Me


Rectangular Window

Narrowest main lobe◦ 4/(M+1)◦ Sharpest transitions at

discontinuities in frequency

Large side lobes◦ -13 dB◦ Large oscillation around

discontinuities Simplest window possible

1 00

n Mw n

else


Bartlett (Triangular) Window

Medium main lobe◦ 8/M

Side lobes◦ -25 dB

Hamming window performs better

Simple equation

2 / 0 / 2

2 2 / / 20

n M n Mw n n M M n M

else


Hanning Window

Medium main lobe◦ 8/M

Side lobes◦ -31 dB

Hamming window performs better

Same complexity as Hamming

1 21 cos 02

0

n n Mw n M

else


Hamming Window

Medium main lobe◦8/M

Good side lobes◦ -41 dB

Simpler than Blackman

20.54 0.46cos 0

0

n n Mw n M

else


Blackman Window

Large main lobe◦ 12/M

Very good side lobes◦ -57 dB

Complex equation

20.42 0.5cos0

40.08cos

0

nM

n Mw n n

Melse


Frequency response of some popular windows (M=50)

rectangular Bartlett

HanningHamming

Blackman


Peak Approximation Error


Approximation Error, defined in passband and stopband.

Peake Approximation Error is the maximum value of

Comparison of different windows


Good design strategy

Shape of the window

Main lobe

Side lobe

width of the window

Main lobe

Good design strategy: 1) Use shape to control the behavior of the side lobe.

2) Use width to control the behavior of the main lobe.


Kaiser window

Mn

otherwiseI

nInw

0

0)(

))]1(([)(

0

5.02

0

:0I Zeroth order modified Bessel function of the first kind

...)!3(2)!2(2)!1(2

1)( 26

6

24

4

22

2

0 xxxxI

:1M Number of taps M2

: Parameter to control the shape of the Kaiser window and thus the trade-off between the width of the main lobe and the peak of the side lobe.


Demonstration of Kaiser window

M=20


Demonstration of Kaiser window

6


Comparison with popular windows


1. Calculate the transition bandwidth2. Calculate

3. Choose

4. Choose

Design Guidelines using Kaiser window

ps 10log20A

285.2

82 AM


Example: Design of LPF using Kaiser window 4.0pSpecs:

6.0s

01.0p

001.0s


Example: Design of LPF using Kaiser window 4.0pSpecs:

6.0s

01.0p

001.0s001.0 60log20 10 A

563.5 2.0 37M Type II

filterUse Bessel equation to get w(n)


Example: Design of LPF using Kaiser window


Example: Design of LPF using Kaiser window

Q: Does it satisfy the specs?


Windowing method is a fast and efficient solution to design FIR filters.

Using Kaiser windows, the window can be chosen automatically.

Summary


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Hossein Sameti Department of Computer Engineering Sharif University of Technology

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