Session 3 DSP Intro

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M.S. Ramaiah School of Advanced Studies, Bengaluru

Session 3:Time-Domain Filtering

Session delivered by:

Chandan N.

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Session Objectives • To discuss several noise reduction examples and experiments to

introduce important time-domain techniques for processing digital signals and analyzing simple DSP systems.

• To understand the fundamental time-domain DSP concepts which is helpful and more interesting to examine real-world DSP applications with the help of interactive MATLAB tools

• To understand the concepts of FIR and IIR filter

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• Introduction to Digital systems • Moving-Average Filters • Digital Filters • Realization of FIR Filters • Nonlinear Filters • Examples • Introduction to FIR and IIR • Structures for FIR Systems • Structures for IIR Systems

Session Topics

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Introduction

• A DSP system (or algorithm) performs operations on digital signals to achieve predetermined objectives.

• In some applications, the single-input, single-output DSP system processes an input signal x(n) to produce an output signal y(n).

• The general relationship between x(n) and y(n) is described as y(n) = F [x(n)],

where F denotes the function of the digital system.

Figure : General block diagram of digital system

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Introduction

• DSP system consists of the interconnection of adders, multipliers, and delay units.

• A digital filter alters the spectral content of input signals in a specified manner.

• Common filtering objectives include removing noises, improving signal quality, extracting information from signals, and separating signal components that have been previously combined.

• A digital filter is a mathematical algorithm that can be implemented in digital hardware and software and operates on a digital input signal for achieving filtering objectives.

• A digital filter can be classified as being linear or nonlinear, time invariant or time varying.

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• To process signals, we have to design and implement systems called filters.

• The filter design issue is influenced by such factors as

• The type of the filter: IIR or FIR

• The form of its implementation: structures

• Different filter structures dictate different design strategies.

Introduction

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• Since our filters are LTI systems, we need the following three elements to describe digital filter structures. – Adder – Multiplier (Gain) – Delay element (shift or memory)

x1(n)

x2(n)

x1(n)+x2(n) x(n) ax(n)a

x(n) x(n-1)1/z

Introduction

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1;1)(

)()( 011

110

0

0 =++++++

=== −−

−−

=

−

=

−

∑

∑a

zazazbzbb

za

zb

zAzBzH N

N

MM

N

n

nn

M

n

nn

The system function of an IIR filter is given by

The difference equation representation of an IIR filter is expressed as

∑ ∑= =

−−−=M

m

N

mmm mnyamnxbny

0 1)()()(

Structures for IIR Systems

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Three different structures can be used to implement an IIR filter: Direct Form:

In this form, there are two parts to this filter, the moving average part and the recursive part (or the numerator and denominator parts)

Two version: direct form I and direct form II Cascade Form:

The system function H(z) is factored into smaller second-order sections, called biquads. H(z) is then represented as a product of these biquads.

Each biquad is implemented in a direct form, and the entire system function is implemented as a cascade of biquad sections.

Parallel Form: H(z) is represented as a sum of smaller second-order sections. Each section is again implemented in a direct form. The entire system function is implemented as a parallel network of sections.


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Direct Form I Structure:

Consists of the zeros of H(z) Consists of the poles of H(z)

∑ ∑ = =

− − − = M

k

N

k k k k n y a k n x b n y

0 1 ) ( ) ( ) (


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)2()1()2()1()()( 21210 −−−−−+−+= nyanyanxbnxbnxbny


As the name suggests, the difference equation is implemented as given using delays, multipliers, and adders.

For the purpose of illustration, Let M = N = 2,

x( n) y( n)b0

b1

b2

- a1

- a2

1/ z

1/ z

1/ z

1/ z

H1( z) H2( z)

Direct Form I structure


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)1()()1()( 101 −++−−= nxbnxbnyany

Given a LTI system with a rational transfer function H(z)

11

110

1)( −

−

−+

=zazbbzH

Direct Form I structure


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We can implement the system with the following pair of coupled difference equations: )1()()(

)()1()(

10

1

−+=+−−=

nwbnwbnynxnwanw

The commutative law of the convolution


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Direct Form II Structure:


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Direct Form II Structure: x( n) y( n)

H1( z)

- a1

- a2

1/ z

1/ z

H2( z)

b0

b1

b2

1/ z

1/ z

x( n) y( n)

H( z)

- a1

- a2

1/ z

1/ z

b0

b1

b2

The commutative law of the convolution


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Cascade Form:

• In this form the system function H(z) is written as a product of second-order section with real coefficients.

• This is done by factoring the numerator and denominator polynomials into their respective roots and then combining either a complex conjugate root pair or any two real roots into second-order polynomials.


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We assume that N is an even integer. Then Cascade Form:

22,

11,

22,

11,

10

11

1

0

11

110

11

11

1)(

00

1

−−

−−

=

−−

−−

−−

−−

++++

Π=

+++

+++=

++++++

=

zAzAzBzB

b

zazazz

b

zazazbzbbzH

kk

kkK

k

NN

Nbb

bb

NN

NN

N

Where, K is equal to N/2, and Bk,1, Bk,2, Ak,1,

Ak,2 are real numbers representing the coefficients of second-order section.


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)()();()(

,,2,1,11

)()()(

101

22,

11,

22,

11,1

zYzYzXbzYwith

KkzAzAzBzB

zYzYzH

K

kk

kk

k

kk

==

=++++

==

+

−−

−−+

Biquad Section:

Is called the k-th biquad section. The input to the k-th biquad section is the output from the (k-1)-th section, while the output from the k-th biquad is the input to the (k+1)-th biquad. Each biquad section can be implemented in direct form II.

Yk(n)=XK+1(n) Yk+1(n)

-Ak,1

-Ak,2

1/z

1/z

Bk,1

Bk,2


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Example:

x(n) y(n)

-A1,1

-A1,2

1/z

1/z

B1,1

B1,2

-A2,1

-A2,2

1/z

1/z

B2,1

B2,2

b0

Cascade form structure for N=4


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Example:

Determine the cascade structure the following transfer function:


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( ) ( )( )( )( )

( )( )

( )( )1

1

1

1

11

11

21

21

z25.01z1

z5.01z1

z25.01z5.01z1z1

z125.0z75.01zz21

zH

−

−

−

−

−−

−−

−−

−−

−+

−+

=

−−++

=+−++

=

Example:

Cascade of Direct Form I

Cascade of Direct Form II

Determine the cascade structure the following transfer function:


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Parallel Form: In this form the system function H(z) is written as a sum of second-

order section using partial fraction expansion.

NMifonly

NM

k

kk

K

k kk

kk

NMifonly

NM

k

kkN

N

NM

NN

MM

zCzAzA

zBB

zCzazazbzbb

zazazbzbb

zAzBzH

≥

−

=

−

=−−

−

≥

−

=

−−−

−−

−−

−−

∑∑

∑

+++

+=

+++++++

=

++++++

==

012

2,1

1,

11,0,

01

1

110

11

110

1

1

ˆˆˆ1)(

)()(

K=N/2, and

B,A are real numbers


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The second-order Section:

∑ <==

=++

+== −−

−+

NMzYzYzXzHzYwith

KkzAzA

zBBzYzYzH

kkk

kk

kk

k

kk

),()(),()()(

,,2,1,1)(

)()( 22,

11,

11,0,1

Is the k-th proper rational biquad section.

The filter input is available to all biquad section as well as to the polynomial section if M>=N (which is an FIR part)

The output from these sections is summed to form the filter output.

Each biquad section can be implemented in direct form II.


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Parallel form structure for N=4 (M=N=4)

C0 B1,0

-A1,1

-A1,2

B1,1

B2,0

-A2,1

-A2,2

B2,1

x(n) y(n) 1/z 1/z

1/z

1/z 25

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Example: Determine the parallel structure the following transfer function:

H(z) must be expanded in partial fractions:


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Example: (Cont.)

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Three different structures can be used to implement an FIR filter:

Direct Form.

Cascade Form.

Linear Phase Form.

Frequency Sampling Form.

Lattice Form.

Structures for FIR Systems

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Direct Form :

The difference equation is implemented as a tapped delay line since there are no feedback paths. Note that since the denominator is equal to unity, there is

only one direct form structure.


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Example : Determine the Direct form structure for the following FIR Filter:


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2/);1(

1

)(

22,

11,10

1

0

11

0

10

11

110

MKzBzBb

zb

bzbbb

zbzbbzH

kk

K

k

MM

MM

=++Π=

+++=

+++=

−−

=

−−−

−−

−

Cascade Form :

Cascade form FIR structure for 6-order FIR filter


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Linear Phase System :

A LTI system is said to have Linear Phase if the frequency response has the form

where α real is a real number

A system is said to have Generalized Linear Phase if the frequency response has the form where A(ejw) is a

real-valued function of ω, and β is a constant

Often the term linear phase is used to denote a system that has either linear or generalized linear phase.

20,,)( πβππαβ ±=≤<−−=∠ orwweH jw


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For an FIR filter with a real-valued impulse response of length M, a sufficient condition for this filter to have generalized linear phase is that the h(n) be whether symmetric or anti symmetric.

For a causal FIR filter with M Length and impulse over [0,M-1] interval, the linear-phase conditions

1

0( ) ( )

Mj j k

kH e h k eω ω

−−

=

= ∑1

0( ) ( ) ( )

M

ky n h k x n k

−

=

= −∑

10,2/);1()(10,0);1()(

−≤≤±=−−−=−≤≤=−−=

MnnMhnhMnnMhnh

πββ

The length of the impulse response of the FIR filter (M) can be even or odd.



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Linear phase filters may be classified into 4 types, depending upon whether h(n) is symmetric or anti symmetric and whether M is even or odd.


1. Type I Linear Phase: Symmetrical and M even 2. Type II Linear Phase: Symmetrical and M odd 3. Type III Linear Phase: Anti-symmetrical and M even 4. Type IV Linear Phase: Anti-symmetrical and M odd


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16M =

h(1)=h(14)

( )h n

n

h(0)=h(15)

h(2)=h(13)

h(7)=h(8)

Type I Linear Phase:

Symmetric and M even 11 2

2

0

1( ) 2 ( )cos2

MMjj

k

MH e e h k kωω ω

−−−

=

− = −

∑

Symmetrical Impulse Response, M: Even


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Symmetrical Impulse Response, M: Odd ( )h n

n

15M =

h(0)=h(14)

“h(7)=h(7)”

h(1)=h(13)

h(6)=h(8)

Type II Linear Phase:

Symmetric and M odd

H(ejw)


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Antisymmetrical Impulse Response, M: Even 16M =( )h n

n

h(0)=-h(15)

h(7)=-h(8)

h(1)=-h(14)

Type III Linear Phase:

Anti Symmetric and M even

H(ejw)


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Antisymmetrical Impulse Response, M: Odd

n

( )h n 17M =

h(0)=-h(16)

h(1)=-h(15)

h(8)=-h(8)=0

h(7)=-h(9)

!

Type IV Linear Phase:

Anti Symmetric and M odd

H(ejw)


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++−+−++−+=+−++−++−+=

)]2()1([)]1()([)1()2()1()()(

10

0110

MnxnxbMnxnxbMnxbMnxbnxbnxbny

Linear Phase Form : Consider the difference equation with a symmetric impulse response.

The linear-phase structure is essentially a direct form draw differently to save on multiplications.

For M odd


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Example:

Determine the Linear phase structure for the following FIR Filter:

The length of the filter is 7


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Example:

7654321 )0()1()2()3()3()2()1()0()( −−−−−−− +++++++= zhzhzhzhzhzhzhhzH

Determine the Linear phase structure for the following FIR Filter:

The length of the filter is 8


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Moving-Average Filters: Structures And Equations

• As shown in Figure below, the effect of noise causes signal samples to fluctuate from the original values; thus the noise may be removed by averaging several adjacent samples.

• The moving (running)-average filter is a simple example of a digital filter.

• An L-point moving-average filter is defined by the following input/output (I/O) equation

where each output signal y(n) is the average of L consecutive

input signal samples.

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Example

Figure : Original (open circles) and corrupted (x) signals

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Moving-Average Filters

• Implementation of above Equation requires L − 1 additions and L memory locations for storing signal sequence x(n), x(n − 1), . . . , x(n − L + 1) in a memory buffer.

• As illustrated in Figure below, the signal samples used to compute the output signal y(n) at time n are L samples included in the window at time n. These samples are almost the same as the samples used in the previous window at time n − 1 to compute y(n − 1), except that the oldest sample x(n − L) in the window at time n − 1 is replaced by the newest sample x(n) of the window at time n.

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Moving-Average Filters • Thus previous equation can be computed as

• Therefore, the averaged signal, y(n), can be computed recursively based on the previous result y(n − 1). This recursive equation can be realized by using only two additions.

• However, we need L + 1 memory locations for keeping L + 1 signal samples {x(n), x(n − 1) . . . x(n − L)} and another memory location for storing y(n − 1).

• The recursive equation given in above equation is often used in DSP algorithms, which involves the output feedback term y(n − 1) for computing current output signal y(n).

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Example • To remove the noise that corrupts the sine wave as shown in Figure we

implement the moving-average filter with L = 5, 10, and 20, using the MATLAB code example.m.

• The original sine wave, noisy sine wave, and filter output for L = 5 are shown in Figure in next slide

• This figure shows that the moving-average filter with L = 5 is able to reduce the noise, but the output signal is different from the original sine wave in terms of amplitude and phase.

• In addition, the output waveform is not as smooth as the original sine wave, which indicates that the moving-average filter has failed to completely remove all the noise components.

• In addition, we plot the filter outputs for L = 5, 10, and 20 in Figure for comparison. This figure shows that the filter with L = 10 can remove more noise than the filter with L = 5 because the filter output is smoother for L = 10; however, the amplitude of the filter output is further attenuated. When the filter length L is increased to 20, the sine wave component is attenuated completely.

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Figure : Performance of moving-average filter, L = 5 48

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Figure : Moving-average filter outputs, L = 5, 10, and 20 49

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Digital Filters

• The I/O equation given below can be generalized as a difference equation with L parameters, expressed as

where bl are the filter coefficients. • The moving-average filter coefficients are all equal as bl = 1/L. • We can use filter design techniques to determine different sets of

coefficients for a given specification to achieve better performance. • Define a unit impulse function as

FIR Filter Equation

Substituting x(n) = δ(n), the output is called the impulse response of the filter, h(n), and can be expressed as

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Example

• The moving-average filter defined above is an FIR filter of length L (order L − 1) with the same coefficients bl = 1/L.

• Consider an FIR Hanning filter of length L = 5 with the coefficient set {0.1 0.2 0.4 0.2 0.1}.

• Similar to Example , we can implement this filter with MATLAB script example2_4.m and compare the performance with the moving-average filter of the same length.

• The outputs of both filters are shown in Figure . The results show that the five-point Hanning filter has less attenuation than the moving-average filter. Therefore, we show that better performance can be achieved by using different filter coefficients derived from filter design techniques.

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Example

Figure :Comparison of moving-average and Hanning filters, L = 5

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Nonlinear Filters

• An efficient technique for reducing impulse noise is the use of a nonlinear filter, median filter.

• An L-point running median filter can be implemented as a first-in first-out buffer of length L to store input signals x(n), x(n − 1), . . . , x(n − L + 1).

• These samples are moved to a new sorting buffer, where the elements are ordered by magnitude.

• The output of the running median filter y(n) is simply the median of the L numbers in the sorting buffer.

• Medians will not smear out discontinuities in the signal if the signal has no other discontinuities within L/2 samples and will approximately follow low-order trends in the signal.

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Session Summary

• Signals represent a message or information and are represented mathematically as a function of one or more independent variables, e.g. time.

• A system is an operation that transforms input signal x into output signal y. • The signals encountered in the real world such as speech, music, and noise

are random signals. • The moving (running)-average filter is a simple example of a digital filter • An efficient technique for reducing impulse noise is the use of a nonlinear

filter, median filter. • Realisation of a digital filter amounts to constructing a block diagram of

the internal structure of the filter. • Realisation of LTI filters need only three blocks.

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Session 3 DSP Intro