The IIR FILTERsThese are highly sensitive to coefficients

which may affect stability. The magnitude-frequency response of these

filters is established while the phase-frequency response is poor. These filters have short time delay

and better time response.

The FIR filters FIR have truncated time-response. Thus their frequency response is poor. These can be designed for time-limited

as well as frequency-limited response.

FIR filters are inherently stable. Can be designed for Linear phase

performance. …..contd

Advantages of FIR filters

Any arbitrary magnitude response can be designed using frequency sampling technique. They are inherently stable.It is the first choice of the designer if the time delay is not important, even though components required are many more times.

…..contd

Properties of FIR filters….It is simple to implementation.

• The finite word length effect is far less severe on frequency performance.

• Non-causal filters can be designed for the use of mathematical manipulations.• To reduce the computation time of

convolution the long discrete sequences, FFT algorithms are used.

Properties of FIR filters…. FFT can be implemented on hardware as well as on soft-ware. FIR filter are implemented non-recursively. But these can be mathematically expressed recursively. It has no support of analog filters.Computer Aided Designs are used to design such filters.

Comparison between IIR and FIR

by example 10.01.The following transfer functions, one is recursive and other is non-recursive. Both yield identical

magnitude-frequency response. We Compare their computational and storage

requirements. Recursive Transfer function

H1(z) = (bo+ b1 z-1 + b2 z-2) / (1+a1 z-1 + a 2 z-2)where

[bo b1 b2] = [ 0.4981819 0.9274777 0.4981819] [a1 a2] =[ -0.6744878 - 0.3633482]

and

Example 10.01 contd…Non recursive transfer function:

whereh(0) = h(11) = 0.54603280 x 10-2

h(1) = h(10) = -0.45068750 x 10-1

h(2) = h(9) = 0.69169420 x 10-1

h(3) = h(8) = -0.55384370 x 10-1

h(4) = h(7) = -0.63428410 x 10-1

h(5) = h(6) = 0.57892400 x 100

H2 z( )

0

11

k

h k( ) z k

=

Summary:Computational and storage requirements

Thus we see that IIR filter requires far less components and storage space. But since FIR

filter coefficients are symmetrical, the later results in efficient implementation.

item IIR filter H1 (z)

FIR filter H2 (z)

Number of multiplication 5 12Storage elements 2 11Storage locations, coefficients

7 23

Linear Phase response FIR:Tptime-phase delay and Tg time group delay.Phase Delay:

A signal consists of several frequency components.

The phase delay is the amount of time delay each individual frequency components of the

signal suffer while transmitted through a system. Non linear phase characteristics of a system

results in phase distortion at the output due to alteration in the phase relationship of frequency

components of the signal during processing.

Phase delay contd. Non linear phase delay is undesirable in

hi-fi systems such as video-,bio-, data- transmission etc.

Mathematical model of phase delay is:Tp = - ()/

For linear phase response, the following conditions should be satisfied:

where and are constants.

In the expression For a filter of length N,

If the symmetry is positive, = 0 and = (N -1)/2;

and if the symmetry is negative, = /2 and = (N -2)/2).

[Ifeachor,”Digital Signal Procesing”2/e, PH,pp344-348]

Group Delay It is the average time delay of the frequency components of the

composite signal. Mathematically it is defined as:

Tg = -d()/d =

“the derivative of phase wrt frequency”. For no phase distortion, the should

be a constant.

Phase delay and Group Delay

• Phase • Phase delay Tp = - ()/• Group delay Tg = -d()/d= • The phase delay = group delay

if / is a constant.

Phase and Group Delays Phase and Group Delays displayeddisplayed• Figure below shows the waveform of an

amplitude-modulated input and the output generated by an LTI system

phase

Phase and Group DelaysPhase and Group Delays• Note: • The carrier component at the output is delayed

by the phase delay.• the envelope of the output is delayed by the

group delay.• It is relative to the waveform of the underlying

continuous-time input signal• The waveform at the output shows distortion if

the group delay is not constant.

Phase and Group DelaysPhase and Group Delays

• If the distortion is unacceptable then a delay equalizer is cascaded to enable the overall group delay nearly linear over the frequency band of interest

• To keep the magnitude response of the parent system unchanged, the magnitude characteristics of delay equalizer need to be constant over the frequency band of interest.

Necessary and sufficient condition for a

linear phase response filter is: The transfer function of the filter

should be symmetrical. This symmetry can be positive or, negative.

The word-length N, can be even or, odd.

It returns four cases:

Two cases for odd word-lengthOdd coefficients: ao= h[(N-1)/2]; a(n) = 2h[(N-1)/2 - n]

Case Wordlength

Symmetry response

I odd EvenOr,Positive

II odd Odd, Or,Negative

e

j N 1( )2

0

N 12

n

a n( )

=

cos n( )

ej

N 1( )2

2

0

N 12

n

a n( ) sin

=

n( )

Two cases for even word-lengthEven coefficients: b(n) = 2h(N/2 – n)

case Wordlength

symmetry response

III Even Even,Or,Positive

IV Even Odd,Or,Negative

ej

N 1( )2

2

1

N2

n

b n( ) sin

=

n 12

e

j N 1( )2

1

N2

n

b n( ) cos

=

n 12

Even image symmetry with odd and even word length.

e

j N 1( )2

0

N 12

n

a n( )

=

cos n( ) e

j N 1( )2

1

N2

n

b n( ) cos

=

n 12

1st 3rd

FD=1/2

Conclusion…1 Even length filter (IV) always exhibit zero response at FD= 0.5. FD= 0.5 corresponds to half the sampling frequency.

Hence it is not suitable for high pass filters.It has zero response at DC too.

Negative symmetry filters (II & IV) introduces a phase shift of in the phase response. It makes output zero at DC or, zero frequency. not suitable for low pass filters. These are useful in design of differentiator and Hilbert transformers as they require radians phase shift.

Conclusion….2

Type I is

the most versatile

filter.

Conclusion….3• Further note that the phase delay for positive

symmetry (I and III) or group delay in all the four filters is expressible in terms of the coefficients of the word length of the filter.

• And hence can be corrected to give a zero phase or, group delay response.

• Denoting T to be the sampling period, phase delay Tp For filter I and III, = (N-1)T/2;For filter II and IV, = (N – 1 - )T/2.

[Ifeachor,”Digital Signal Processing” PH, 2/e, pp.344-348.

Summery: configuration & standard filters

I II III IV

LP

BP

HP

BS

STEPS IN FIR FILTER DESIGN 1. Filter Specifications:

• Filter transfer function H(z), • Required amplitude and phase responses,

• acceptable tolerances, • sampling frequency and

• the word length of the input data. 2. Coefficient Calculations:

to determine the coefficients of H(z) so as to satisfy the filter specifications.

STEPS IN FIR FILTER DESIGN….3. Realization:

Conversion of the transfer function into suitable structure.

4. Analysis of finite word length effects: Error effect of quantization of input signal,

Effect of coefficient quantization. Optimization of word-length.

5. Implementation: Producing software codes and/or hardware

and performing the actual filtering.

Design specifications1. Pass / Stop band specifications:Magnitude deviation (includes ripple)

Pass/Stop band edge frequency (or frequencies in case of band

pass/stop filter).2. Sampling Frequency.

3. Word length of the filter

Methods of Calculation of FIR Coefficients

1. The Window Method,2. Frequency Sampling Method,

3. Optimal or, Min-max design method. Each method can lead to design of a linear phase FIR filter.

The common mathematical model is:

H z( )

0

n

k

h k( ) z k

=

The window methodA suitable window function w[n] is

selected, required word length is calculated.

Then it is multiplied with the impulse response of a (ideal) LPF. Thus

hw [n] = h[n] w[n]

Or, hw [n] = H[F] W[F].

The window method….• The spectrum of ideal low pass filter

have a jump discontinuity at F = Fc. • But the windowed spectrum shows

over-shoot, ripple and

a finite transition width but

no abrupt jump.

Window method contd… It’s normalized signal magnitude at

F = Fc is 0.5. It corresponds to attenuation of -6 dB. The ripple in pass band and over-shoot is

attributed to Gibb’s phenomena; 9% minimum.

The side-lobs produces the ripple in pass band and stop band.

The ripples in pass band and stop band have odd symmetry.

Window method contd…The transition width is due to main lob. Wider the main lob, wider is the transit band. Wider is the window width, smaller is the width of main-lob.Number of minima and maxima in the pass band and stop band are decided by N.Unlike in Tchebyshev Filters, the peaks here have different heights, maximum near band edges, decaying thereafter.

Note that number of samples equal maxima and minima of a rectangular window in pass- and stop band.

The peak occurs near band edges.Maxima-Minima

Rectangular WindowThis window has two properties: maximum number of alternating maxima and minima and their peaks follow the attenuation at the

rate of –6.02dB per octave or, equivalent -20dB/dec.

Mathematical model of different type windows follows.

Mathematic Models Of Different Type Of Windows

Window Representation ExpressionRectangular wR[n] 1Bartlett wT[n] 1 – {2|n| / (N-1)}Von Hann whn [n] 0.50 + 0.50 cos{2n/(N-1)}Hamming whm [n] 0.54 + 0.46 cos{2n/(N-1)}Blackman wb [n] 0.42 +0.50 cos{2n/(N-1)}

+0.08 cos{4n/(N-1)}Kaiser wK[n,] Io(x1)/Io(x2);

ratio of modified bessel function of order zero; where

x1=( {1 – 4[n/(N-1)]2}); and x2= ()

Characteristics of Windows

We now examine the characteristics of various other type of windows and compare their performances for N=21and N=51.

Before that note various nomeclatures.

Mathematical representation:Nomenclatures

1. GP / GS = Peak Gain of main-lob / side-lobe dB

2. ASL = Side-lobe attenuation = (GP /GS) dB. 3. WM = Half-width of main-lobe

4. W6 / W3 = - 6 dB / -3dB half-width

5. DS = stop-band attenuation dB/dec.

6. FWS = C/N where C= constant of filter.7. WS = Half width in main-lobe to reach the peak

level of first side lob.8. Aws= Peak side-lobe attenuation in dB

9. AWP = Pass band attenuation in dB

Window Gp GS/Gp ASLdB WM WS W 6 W3 DS AWS FWS AWP

Rectangular 1 0.2172 13.3 1 0.81 0.6 0.44 20 21.7 0.92 1.562

BartlettTriangular

0.5 0.0472 26.5 2 1.62 0.88 0.63 40 25

Von HannHanning

0.5 0.0267 31.5 2 1.87 1.0 0.72 60 44 3.21 0.1103

Hamming 0.54 0.0073 42.7 2 1.91 0.9 0.65 20 53 3.47 0.0384

Blackman 0.42 0.0012 58.1 3 2.82 1.14 0.82 60 75.3 5.71 0.003

Kaiser = 0.26

0.4314 0.0010 60 2.98 2.72 1.11 0.80 20

Note: The widths; WM, WS, W6, W3; must be normalized by the window length N.Empirical Values for Kaiser Window depends on the value of defined as:

GP = |sinc(j)| / Io(); ASL = Sinh()/0.22; WM = (1+2); W 6 = (0.661+2)

PROCEDURE OF CALCULATING FILTER COEFFICIENTS USING WINDOW

Specify the desired frequency response of the filter Hd().

Obtain the impulse response hD [n] of the desired filter by inverse Fourier transform. Select a window which satisfies the pass-band attenuation specification.

PROCEDURE OF CALCULATING FILTER COEFFICIENTS USING WINDOW…Determine the number of coefficients

using the appropriate relationship between the filter length and the transition width f expressed as a fraction of the sampling frequency. Obtain the values of w[n] for the chosen window function and that of the actual FIR coefficients h[n] and multiplying them. Plot the response and verify the compliance of specifications.

Summery of ideal impulse response of standard frequency selective

filtersFilter type Ideal impulse response

hD[n]HD [0]

Low Pass 2fcsinc(nc) 2fc

High Pass 1-2fcsinc(nc) 1-2fc

Band Pass 2f2sinc(n2)- 2f1sinc(n1) 2(f2-f1)

Band Stop 2f1sinc(n1)- 2f2 sinc(n2) 1-2(f2-f1)

Note: fc, f1 and f2 are the normalized edge frequencies. N is the length of the filter [Ifeachor: p.353]

Remarks:

• The TF of a filter is an even symmetric function.• It is an ideal transfer function.• It has a linear phase response.• Theoretical value of n . But for an FIR filter, n should be finite.• With finite n, the response will have ripples.• The response will also have at least 9% overs-

hoots near critical frequencies, Gibbs Phenomena.

Remarks: If the n in truncated range is increased, ripple

is reduced so also the overshoot, upto 9%. Increased n means increase in number of

coefficients. Ideal truncation is equivalent to convolving

an ideal filter hD having frequency response sinc() with rectangular frequency window, W().

It is equivalent to multiplication in time domain.

convolution of an ideal filter with a sinc window function.

Peak side lob attenuation

Window Gp GS/Gp ASLdB WM WS W 6 W3 DS AWS FWS AWP

Rectangular 1 0.2172 13.3 1 0.81 0.6 0.44 20 21.7 0.92 1.562

BartlettTriangular

0.5 0.0472 26.5 2 1.62 0.88 0.63 40 25

Von HannHanning

0.5 0.0267 31.5 2 1.87 1.0 0.72 60 44 3.21 0.1103

Hamming 0.54 0.0073 42.7 2 1.91 0.9 0.65 20 53 3.47 0.0384

Blackman 0.42 0.0012 58.1 3 2.82 1.14 0.82 60 75.3 5.71 0.003

Kaiser = 0.26

0.4314 0.0010 60 2.98 2.72 1.11 0.80 20

Note: All widths; WM, WS, W6, W3; must be normalized by the window length N.Empirical Values for Kaiser Window depends on the value of defined as:

GP = |sinc(j)| / Io(); ASL = Sinh()/0.22; WM = (1+2); W 6 = (0.661+2)

Example:Design a low-pass FIR filter to meet the following specs:Pass band edge frequency: 1500 HzTransition width: 500 Hz.Stop-band attenuation AWS= > 50 dBSampling frequency fs = 8000 Hz.

Soln:1. Meaning of given specifications are:

Sampling frequency fs = 8000 Hz.

Pass band edge frequency: fc =1500/8000

Transition width f = 500/8000.Stop-band attenuation AWS= > 50 dB

Design considerations contd…

2. The filter function is HD ()= 2fc sinc(nc).

3. Because of stop-band attenuation characteristics, either of the Hamming,

Blackman or,Kaiser windows

can be used. We use Hamming window:

whm[n] =0.54 + 0.46 cos{2n/(N-1)}

Design considerations contd…4 f = transition band width/sampling frequency

= 0.5/8 =0.0625 = 3.3/N.Thus N = 52.8 53 i.e. for symmetrical window

–26 n 26.

fc’ = fc + f/2 = (1500+ 250)/8000 = 0.21875.5. Calculate values of hD [n] and whm[n] for

–26 n 26 Add 26 to each index so that the indices range

from 0 to 52.6. Plot the response of the design and verify the

specifications.

Calculations:c= 2fc = 1.37452fc=1.3745/ = 0.4375 • hD(n) = 2fc [sin(nc)/ nc]

wn = [0.54 + 0.46cos(2n/N) The input signal to the filter function is a series of

pulses of known width but of different heights manipulated as per the window function.

• The overall is the multiplication of two.

Calculations…

h(n) = hD [n] w D[n] = 0.4375 {[sin(nc)/ nc]}

x {[0.54 + 0.46cos(2n/N)}at n=0, sincesin(nc)/nc = 1, so also cos(0) = 1; h(0) = 0.4375 x[0.54 + 0.46] = 0.4375.Again since 2fc / c = 1/h(n)= [sin(1.3745n)/n] [0.54 +0.46cos(2n/53)]

Coefficient Calculationsn 1 26 hnsin 1.3745n( )

n wn 0.54 0.46( )cos 2

n53

hn

0.3120.061-0.088-0.0560.0350.049

-3-8.888·10-0.04

-3-6.883·100.0290.016-0.019-0.02

-38.716·100.021

-5-1.694·10-0.018

-3-6.752·100.0140.011

-3-8.435·10-0.013

-32.717·100.013

-32.467·10-0.011

wn

0.9930.9720.9370.89

0.8290.7580.6750.5830.4830.3760.2640.1480.03

-0.089-0.206-0.32-0.43

-0.534-0.63

-0.718-0.795-0.861-0.915-0.956-0.984-0.998

hn wn

0.310.059-0.083-0.050.0290.037

-3-5.999·10-0.023

-3-3.323·100.011

-34.234·10-3-2.771·10-4-6.03·10-4-7.74·10-3-4.286·10-65.425·10-37.899·10-33.604·10-3-8.783·10-3-8.066·10-36.705·10

0.012-3-2.486·10

-0.013-3-2.428·10

0.011

n

123456789

1011121314151617181920212223242526

Example:Design a filter for the specifications:Pass band: 150-250 Hz.Transition width: 50 HzPass band ripple: 0.1 dB max.Stop-band attenuation: > 60 dBSampling frequency: 1000 Hz.

Soln: The above is a band pass filter.

1. Interpretations of specifications are:Sampling frequency fs = 1000 Hz.Pass band edge frequency: fc = 150-

250/1000Pass band ripple: p= 0.1 dB max.Transition width f = 50/1000.Stop-band attenuation AWS= > 60 dB

Design considerations contd…2. The filter function is

HD()= 2f2 sinc(n2) -2f1 sinc(n1)

3. Because of stop-band attenuation characteristics, either of the Blackman or, Kaiser windows can be used.

4. From the specifications of pass band and stop-band:20 log(1+p) = 0.1 or, p = 0.0115;

-20 log (s) = 60 dB, or, s = 0.001.

therefore = min(p, s) = 0.001.

Window Gp GS/Gp ASLdB WM WS W 6 W3 DS AWS FWS AWP

Rectangular 1 0.2172 13.3 1 0.81 0.6 0.44 20 21.7 0.92 1.562

BartlettTriangular

0.5 0.0472 26.5 2 1.62 0.88 0.63 40 25

Von HannHanning

0.5 0.0267 31.5 2 1.87 1.0 0.72 60 44 3.21 0.1103

Hamming 0.54 0.0073 42.7 2 1.91 0.9 0.65 20 53 3.47 0.0384

Blackman 0.42 0.0012 58.1 3 2.82 1.14 0.82 60 75.3 5.71 0.003

Kaiser = 0.26

0.4314 0.0010 60 2.98 2.72 1.11 0.80 20

Note: All widths; WM, WS, W6, W3; must be normalized by the window length N.Empirical Values for Kaiser Window depends on the value of defined as:

GP = |sinc(j)| / Io(); ASL = Sinh()/0.22; WM = (1+2); W 6 = (0.661+2)

Design considerations contd…5. We use Blackman window:

wb[n] = 0.42 +0.50 cos{2n/(N-1)}

+ 0.08 cos{4n/(N-1)}f = transition band width/sampling frequency = 50/1000 =0.05 = 5.5/N.hence N 110. i.e. for symmetrical even window

–55 n 55, but for n=0., being an even window. We can choose N = 111.

Design considerations contd…

6. For N =111,Plot the response of the design and verify the specifications.

7. Calculate values of hD [n] and whm [n] for –55 n 55

8. Add 55 to each index so that the indices range from 0 to111.

Comparison of commonly used windows with Kaiser window:

Window type   

Peak normalized side lob amplitude 

Approximate. Width of main-lob

Appx. peak error 20 log

Equivalent Kaiser Window Transition Width

Rectangular -13 4/(M+1) -21 0 1.81/M

Bartlett  -25  8/M -25 1.33 2/37/M

Hanning -31  8/M -44 3.86 5.01/M

Hamming -41  8/M -53 4.86 6/27/M

Blackman -57 12/M -74 7.04 9.19/M

The comparison shows that the Kaiser window is more efficient than any other window in question.

Example: take-up above problem and solve it using Kaiser window.

Soln.specifications are repeated here:Sampling frequency fs = 1000 Hz.Pass band edge frequency: fc = 150-

250/1000Pass band ripple: p= 0.1 dB max.Transition width f = 50/1000.Stop-band attenuation AWS= > 60 dB

Window Gp GS/Gp ASLdB WM WS W 6 W3 DS AWS FWS AWP

Rectangular 1 0.2172 13.3 1 0.81 0.6 0.44 20 21.7 0.92 1.562

BartlettTriangular

0.5 0.0472 26.5 2 1.62 0.88 0.63 40

Von HannHanning

0.5 0.0267 31.5 2 1.87 1.0 0.72 60 44 3.21 0.1103

Hamming 0.54 0.0073 42.7 2 1.91 0.9 0.65 20 53 3.47 0.0384

Blackman 0.42 0.0012 58.1 3 2.82 1.14 0.82 60 75.3 5.71 0.003

Kaiser = 0.26

0.4314 0.0010 60 2.98 2.72 1.11 0.80 20

Note: All widths; WM, WS, W6, W3; must be normalized by the window length N.Empirical Values for Kaiser Window depends on the value of defined as:

GP = |sinc(j)| / Io(); ASL = Sinh()/0.22; WM = (1+2); W 6 = (0.661+2)

Design using Kaiser WindowThe filter function is

HD()= 2f2 sinc(n2) -2f1sinc(n1)Because of stop-band attenuation characteristics, either of the Blackman or, Kaiser windows can be used.From the specifications of pass band and stop-band: 20 log(1+p) = 0.1 dB or, p = 0.0115;-20 log (s) = 60 dB, or, s = 0.001.therefore = min(p, s) = 0.001.f = transition band width/sampling frequency = 50/1

=0.05= (AWS-7.95)/ 14.36N = (60-7.95)/14.36N or, N =72.49 73.

..contd

Design using Kaiser Window…

Calculation of by empirical formulae. = 0 if A 21 dB; = 0.5842(AWS -21)0.4 + 0.07886(A-21)

if A <21<50 dB = 0.1102(AWS -8.7) if A 50

Hence = 5.65Evaluate the coefficients.

Evaluate the performance. plot the graph and verify the performance of

designed filter.

Advantages and disadvantages of the window method.

It is simple to apply and simple to understand. It involves minimum computation.

Lacks flexibility. Both peak pass band and stop-band ripples are nearly equal, limits the choice of

designer.Because of convolution of the spectrum of the

window function and the desired response, pass band and stop-band edge frequencies can not be

precisely specified. Maximum ripple magnitudes in pass-band and stop-band in the filter response is fixed regardless of N

(except in Kaiser Window).

Frequency Sampling Method• Arbitrary frequency response is possible to design. • Since coefficients need not symmetrical, design of

recursive filters possible.• It is possible to compute coefficients as integers.• Unless optimized, the band pass ripple, like in window

method, is not equi height.• Frequency sampling is at equi-angle on unit circle.• Odd coeff. filters will have zeroes at either z = 1 or -1.• Even coefficient filters will simultaneously either have

zeroes or none at z = 1.

Mathematics for Linear Phase response.

• Linear Phase response can be obtained by either even symmetric or, odd symmetric impulse response coefficients.

h n( )1N

0

N 1

k

H k( ) ej2

N

nk

Inverse DFT is expressed as

h n( )1N

0

N 1

k

H k( ) cos2 n k

N

j sin2 n k

N

h n( )1N

0

N 1

k

H k( ) ej2

N

k

ej2

N

nk

h n( )1N

0

N 1

k

H k( ) ej2

N

n ( )k

where = (N-1)/2

h n( )1N

0

N 1

k

H k( ) cos2 n k

N

For all the real coefficients of h(n)

h n( )1N

1

N

21

k

2 H k( ) cos2 n k

N

H 0( )

And if the coefficients are symmetrical too:

For even coefficients, H(0) will be zero.

Example:(Ifeachor: p.382)

Prob.: Design an FIR linear phase filter having pass band 0-5 kHz,

Sampling frequency 18 kHz, filter length 9Soln: N =9 hence N/2 -1 = 4. frequency interval is fs/N = 18/9 = 2 kHz.

|H(k)| = 1 at k =0, 1, 2 = 0 at k= 3, 4

More read Ifeachor PP 382 to 401: matlab: 450 to 453.