Dsp Processors III

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DSP PROCESSORS-III


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Module 3


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Syllabus

Digital Filters

IIR

FIR

Adaptive filters DIT and DIF algorithms


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Discrete-time system

Discrete-t ime systemhas discrete-time input and output signals


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A discrete-time filter is a discrete-time system that passescertain frequency components and stops others.

A Digital filter can be either an IIR or FIRfilter.

An IIRfilter usually requires less cost, i.e., less computationand memory.

However, an FIRfilter usually has better performance,

especially in the phase response( a generalized linear phase isrequired).

Discrete-Time Filters


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IIR filter

IIR filter have infinite-duration impulseresponses, hence they can be matched to analog

filters, all of which generally have infinitely longimpulse responses.


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Analysis of Ideal Filters

Consider an ideal low pass filter.

Over period (-, ), the frequency response of an ideal low passfilter is defined as

.

otherwise,0

||),jexp()(H

c

Let xi(n)=Aiexp (jin) be a frequency component of the input signal.

Then, the corresponding output signal of this filter is

yi(n)=Ai exp (jin) H (i).

If|i| c, thenyi(n)=Aiexp [ji (n-)].

ie, xi(n) is passed with a constant delay .


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.

n,

n,)n(

)n(sin

)n(h

c

c

If|i|>c, thenyi(n)=0.

ie, xi(n) is stopped.

The impulse response of the ideal low pass filter is

Since h(n)0 for n


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Prototype Analog Filters

IIR filter design techniques rely on existing analogfilters to obtain digital filters.

We designate these analog filters asprototype filters.

The prototypes widely used in practice

Butterworth lowpass Chebyshev lowpass (Type I and II)


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Butterworth Lowpass Filter

This filter is characterized by the property that itsmagnitude response is flatin both passband and

stopband.

Moderate phase distortion

N

c

a jH 22

1

1|)(|

The magnitude-squared response of an N-order lowpassfilter


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0 10 20 30 40 50 60 70 80 90 1000

0.2

0.4

0.6

0.8

1

1.2

N=1

N=2

N=200

N=100

Plot of the magnitude-squared response


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Properties of Butterworth Filter

Magnitude Response

|Ha(0)|2 =1

|Ha(jc) |2 =0.5, for all N

|Ha(j)|2monotonically decrease for Approaches to ideal filter when N


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Chebyshev Lowpass Filter

Chebyshev -I filtersHave equirippleresponse in thepassband

Chebyshev -II filtersHave equirippleresponse instopband

Butterworth filters

Have monotonic response in both bands

We note that by choosing a filter that has an equripple ratherthan a monotonic behavior, we can obtain a low-orderfilter.

(The maximum delay, in samples, used in creating eachoutput sample is called the orderof the filter )

Therefore Chebyshev filters provide lower orderthan

Buttworth filters for the same specifications.


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Chebyshev -I filters

Chebyshev -II filters


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The magnitude-squared response of Chebyshev -I filter

xx

xxNxT

T

jH

N

c

N

a

1)),(cosh(cosh

10)),(coscos()(

1

1|)(|

1

1

22

2

N is the order of the filter, Epsilon is the passband ripple factor

(a) For 0


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Observations

At x=0 (or =0);

|Ha(j0)|2 = 1; for N odd;

= 1/(1+2); for N even

At x=1 (or = c);

|Ha(j1)|2 =1/(1+2);for all N.

For 0


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Impulse Invariance Transformation

The digital filter impulse response look similar to that of a frequency-selective analog filter.

basic principle is sampling of impulse response of an analog filter

Sample ha(t) at some sampling interval T to obtain h(n)

h(n)=ha(nT)

Sincez = e jwon the unit circle ands = j on the imaginary axis,we have the following transformation from thes-plane to thez-plane:

z=e sT

also,

and

Tjjw eeorTw

k

ak

T

jsH

T

zH21

)((Frequency-domainaliasing

formula)


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Complex-plane mapping in impulse invariance transformation


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Properties:

Sigma < 0, maps into |z| 0, maps into |Z| >1 (outside of the Unit circle)

Causality and Stability are the same without changing Aliasing occur if filter not exactly band-limited

Let digital lowpass filter specifications be wp ws Rp and As


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Let digital lowpass filterspecifications be wp,ws,Rp and As.

To determine H(z) , first design an equivalent analog filter and then

mapping it into the desired digital filter.

Design Procedure:

1. Choose Tand determine the analog frequencies:

p =wp/T, s = ws/T

2. Design an analogfilter Ha(s) using the specifications with one of theprototypes of the previous section.

3. Using partial fraction expansion, expand Ha(s) into

4. Now transform analog poles {pk} into digital poles {epkT} to obtain the

digital filter

Nk

k

ka

ps

RsH

1)(

N

k Tp

k

ze

RzH

k1 1

1

)(

Ad


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Advantages

It is astable design and the frequencies and w are linearlyrelated.

Disadvantage

We should expect some aliasingof the analog frequencyresponse, and in some cases this aliasing is intolerable.

This design method is useful only when the analog filter isessentially band-limited to a lowpass or band pass filter in

which there are no oscillations in the stopband.

i i f i


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Bilinear Transformation

transforms continuous-time system representationsto discrete-time and vice versa

This mapping is the besttransformation method.

It maps positions on the j axis, in the s-plane to theunit circle in the z-plane

For every feature in the frequency response of the analogfilter, there is a corresponding feature, with identical gain

and phase shift, in the frequency response of the digital

filter basic principle: application of the trapezoidal formula for

numerical integration of differential equation

2/1

2/1

1

121

1

sT

sT

zz

z

Ts


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Complex-plane mapping in bilinear transformation


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Observations

Sigma < 0|z| < 1,

Sigma = 0|z| = 1,

Sigma > 0|z| > 1

The entire left half-plane maps into the inside of theUnit circle. This is a stabletransformation.

The imaginary axis maps onto the Unit circle in aone-to-one fashion. Hence there is no aliasingin the

frequency domain.

Relation of to is nonlinear = 2tan-1(T/2) = 2tan(/2)/T;

Given the digital filter specifications wp ws Rp and As we want to


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Given the digital filter specifications wp,ws,Rp and As, we want to

determine H(z).

The design steps:

1. Choose a value forT. this is arbitrary, and we may set T = 1.

2. Prewarp the cutoff freq. wp and ws;

ie, calculate p and s usingp = 2/T*tan(wp/2), s = 2/T*tan(ws/2)

3. Design an analog filter Ha(s) to meet the specifications.

4. Finally set

And simplify to obtain H(z) as a rational function in z-1

1

1

112)(

zz

THzH a


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Advantage of the bilinear Transformation

It is a stable design There is no aliasing

There is no restriction on the type of filter that canbe transformed.

FIR FILTER


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FIR FILTER

A finite impulse response filter is a type of a digital filter.

The impulse response is 'finite' because it settles to zero ina finite number of sample intervals. The impulse response of an Nth order FIR filter lasts for

N+1 samples, and then dies to zero.

Properties: Inherently stable - all the poles are located at the origin

and thus are located within the unit circle.

Require no feedback- any rounding errors are notcompounded by summed iterations. This also makesimplementation simpler.

can be designed to be linear phase - which means the phasechange is proportional to the frequency.

Disadvantage:


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Disadvantage:

considerably more computation power is required comparedwith a similar IIR filter.

This is especially true when low frequencies are to be affected

by the filter.

Design methods:

Window design method Frequency Sampling method Weighted least squares design

Minimax design Equiripple design


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Window design method

Simplest FIR filter design is window function technique Specific window should be chosen according to desiredconstraints

the impulse response of the causal FIR filter isobtained by windowing the ideal filter.


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windowing hd(n), we can obtain the impulse response of a

causal FIR lowpass filter. ie,

h(n) = hd(n) w(n)

where w(n) is a window function and is equal to 0 for n


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Window functions

The rectangular window is defined as

.otherwise,0

1Nn0,1)n(w

The Bartlett window is defined as

.

otherwise,0

1Nn1)/2(N),1N/(n22

2/)1N(n0),1N/(n2

)n(w


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.otherwise,0

1Nn0,

1N

n2cos5.05.0

)n(w

The Hamming window is defined as

.

otherwise,0

1Nn0,1Nn2cos46.054.0

)n(w

The Blackman window is defined as

.

otherwise,0

1Nn0,1N

n4cos08.0

1N

n2cos5.042.0

)n(w

The Hann window is defined as


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Bartlett WindowRectangular Window

Hann Window


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Blackman Window

Hamming Window


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It can be seen that when the length of the window is fixed, a narrow

transition band corresponds to large ripples

Type of Window

Width of

Transition Band

Peak Amplitude

of Ripples

Rectangular 4/N -21dB

Bartlett 8/(N-1) -25dB

Hann 8/(N-1) -44dBHamming 8/(N-1) -53dB

Blackman 12/(N-1) -74dB

Features of Commonly Used Windows.


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The Kaiser window is defined as

.

otherwise0,

1Nn0,)(I

1

1N

n21I

)n(w

0

2

0

Here, I0() is the zero-order modified Bessel function of the first kindand is a shape parameter.

should be selected to obtain a good tradeoff between the width of

the transition band and the amplitudes of the ripples.


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The windowing method is carried out in the following

steps:

1. Determine the type of the window according to the

specification for the amplitudes of the ripples.

2. Find the length of the window according to the

specification for the width of the transition band.

3. Find

4. Determine the impulse response of the causal FIR lowpass

filter

Ad ti filt


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Adaptive filter An adaptive filter is a filter that self-adjustsits

transfer function according to an optimizingalgorithm.

Because of the complexity of the optimizingalgorithms, most adaptive filters are digital filters

that perform digital signal processing and adapt theirperformance based on the input signal.

In IIR and FIR filters process parameters are

known in advance or variation is assumed to beknown

Adaptive filters are best suited when signalconditions are slowly changing or there is large

uncertainty and filter has to compensate for that

Basic adaptive filter:


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Here the adaptive filter output y is compared with adesired signal d to yield a error signal e which is fed

back to adaptive filterCoefficients of adaptive filter is adjusted or optimized

using a least mean square algorithm based on theerror signal

Adaptive filter

d

x

+

-

e

y

p

Output of adaptive filter:


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p p

y(n)= wk(n) x(n-k)

where wk(n) represent N weights for a specific time n.

The performance measure is based on the error signal.

The error signal, e(n) = d(n)-y(n)

The weights of wk(n) are adjusted such that a mean squared

error function is minimized

Mean square error function is E[e2

(n)],where E represents theexpected value

K=0

N-1

Least mean square algorithm:


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Least mean square algorithm:

The simple and effective means of updating the weightswithout averaging or differentiating

Wk(n+1) = wk(n)+2e(n). x(n-k) , k=0,1,,N-1Where x(n) is the input and is adaptive step size.

For each time n, weight wk(n) is updated or replaced bynew coefficients, unless the error function is zero.

After the output y(n), error signal e(n) and wk(n) areupdated for a specific time n, new sample is taken and theadaptation process is repeated

Applications of adaptive filters


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Applications of adaptive filters

Noise cancellation

Signal prediction

Adaptive feedback cancellation

Echo cancellation

For noise cancellation:


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For noise cancellation:

The basic adaptive structure is modified for a noisecancellation system

Adaptive filter

d+n

n

+

-

e

The desired signal d is corrupted by noise n.

The input to the adaptive filter is noise n that is correlated with noise n (noise

n could come from same source as n but modified by environment) The output y is adapted to noise n

When this happens the error signal approaches to desired signal d.

The overall output is e and not the adaptive filter output y.

y

For system identification:


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y

The same input u is given to the unknown system (plant) and to theadaptive filter.

The error signal e is the difference between response d and the filter

output y. The error signal e is fed to the adaptive filter and is used to update thefilter coefficients, until y = d

When this happens the adaption process is finished and e approaches tozero.

Thus the unknown system is modeled.

Signal Prediction:


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Signal Prediction: Used to provide a prediction of the present value of a random signal

This is like interference cancellation, but the adaptive filter uses adelayed version of the primary signal as the reference.

Parameters

u = input of adaptive filter = delayed version of random signal

y = output of adaptive filter

d = desired response = random signal

e = d - y =estimation error = system output

DFT


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DFT

The discrete Fourier transform (DFT) is one of the

specific forms of Fourier analysis.

It transforms one function into another, which is calledthefrequency domain representation of the original

function (the time domain).

IDFT does the inverse operation

But the DFT requires an input function that is discrete andwhose non-zero values have a limited (finite) duration.

Such inputs are often created by sampling a continuousfunction, like a person's voice


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The FFT is based on the divide-and-conquer paradigm


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The DIF FFT is the transpose of the DIT FFT


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The DIF FFT is the transposeof the DIT FFT

To obtain flowgraph transposes:

Reverse direction of flowgraph arrows Interchange input(s) and output(s)

DIT butterfly: DIF butterfly:

Comparing DIT and DIF structures:


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Comparing DIT and DIF structures:

DIT FFT structure: DIF FFT structure:

Steps involved in DIF FFT algorithm:


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Data shuffling not required N=2p & input is separated into two parts .

The first set consists of first N/2 input samples with n ranging from0 to N/2-1

Second set consists of the remaining N/2 input samples with n

ranging from N/2 to N-1

X(k) is decimated into even & odd numbered points (DIF)decomposing an N point DFT into 2 N/2Point DFTs

The procedure is repeated till we get a 2-point DFT.

The 2-point DFT is then converted to butterfly to get the completestructure.


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DIF or DIT?


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In terms of computational work load, both perform exactlythe same number of butterflies.

Each butterfly requires exactly one complex multiply andtwo complex adds.

The most significant difference between simple DIF

and DIT algorithms is that DIF starts with normalorder input and generates bit reversed order output.

In contrast, DIT starts with bit reversed orderinput and generates normal order output.

So use DIF for the forward transform and DIT forthe inverse transform

Advantages:


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g

(i) Reasonably good saving in terms of computation, i.e. it

is efficient

(ii) Short Program required

(iii) Easy to understand

Disadvantages:

(i) A relatively large number of operations still required,

especially multiplications

(ii) Problem of the generation of WNfor n=0,1,2,N-1.

Applications of FFT


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Applications of FFT

Linear convolution

(1) Append zeros to the two sequences of lengthsNand

M, to make them of lengths an integer power of two

that is larger than or equal toM+N-1.(2) Apply FFT to both zero appended sequences

(3) Multiply the two transformed domain sequences

(4) Apply inverse FFT to the new multiplied sequence


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THE END

Dsp Processors III

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