eee507_IIRFiltersImplement

EEE 507 - Lecture 14 Copyright 2004 by Lina J. Karam

IIR Filters Infinite-length impulse response => cannot use convolution or DFT

to realize filter IIR filter generally implemented using difference equations and

recursive implementation General system response defined using z- transform

! Frequency response

= z-Transform evaluated on unit circle in the M-Dcomplex plane

! of IIR filter have poles and zeros => stability issue

jezzHH == )()(

1|| =z)(zH


IIR Filters

Why interest in z-Transform?! DFT computable! z-Transform

" Poles and zeros => can describe filter characteristics" Implementation, how to build a filter

Example:

How to break H(z) into components has a lot to do with hardware implementation (for example, get chips implementing a 2nd order digital filter on and hook these together).

)(1 zH )(2 zH )(3 zH

)().().()( 321 zHzHzHzH =


IIR Filters Issues:

! Polynomials do not factor in M-D case! Difference equations harder to deal with (direction of recursion, ordering

relation) than 1-D! Stability analysis much more complicated than 1-D


IIR Filters 2-D Difference Equation

! Difference equation used to implement IIR filters! Sums have finite limits ! Size of b is related to the filter order ! The system is LSI if it is at initial rest! Order of difference equation = size of region of support of

=1 21 2

),(),(),(),( 221121221121r rl l

rnrnxrralnlnyllb

),( 21 llb


IIR Filters 2-D Difference Equation

Recursive Computability

Note: and were normalized by dividing by To use this as an algorithm, the output samples on the right need to be

computed before the one on the left! We can calculate output given input on a sample-by-sample basis! We have to find out which step to execute first in order to be able to compute all

needed output samples recursively If an ordering exists so that all needed output samples on the right side, are

available when needed, then the system is recursively computable.

)0,0(),(,),(),(

),(),(),(

21

nconvolutio a similar to

221121

22112121

1 2

1 2

=

lllnlnyllb

rnrnxrranny

l l

r r

44444 344444 21

),( 21 llb ),( 21 rra )0,0(b

=1 21 2

),(),(),(),( 221121221121r rl l

rnrnxrralnlnyllb


IIR Filters

r2

Weight&sumn2

n1r1

Input mask

Weight&sum+ n2

n1

l2

l1

Output maskInput array x

output y

-+

)0,0(),( ,),(),(-),(),(),( 21221121221121211 21 2

= lllnlnyllbrnrnxrrannyl lr r

Recursive Computability


IIR Filters Recursive Computability

! Examples of recursively-computable output masks" First-quadrant or causal filter

Boundary conditions for output y needed in L-shaped region and are indicated by open circles

Output mask can be swept upward column by column, or left to right row by row, or over any family of parallel lines with negative slope

}0,0{for 0),( 221121 NlNlllb

N1+1

N2+1

n2

n1

N1

N2


IIR Filters

Recursive Computability! Examples of recursively-computable output masks

" Non-symmetric half-plane (NSHP) filter

Shape of boundary condition band depends on filter order NSHP is a generalization of quadrant filter NSHP filters can be further generalized by either a reflection or a

rotation of the output mask or by a combination of both operations

N1+1

N2+1

n2

n1

N3

{2

N43421

231++ NN



! Not all masks are recursively computable! Examples of non-recursively computable output masks

Position of hole does not lie on a corner

! In general, there are only two types of recursively computable output masks" Quadrant-plane (4)" Non-symmetric half-plane (NSHP) (8)



! Boundary conditions" For a recursive system, how do you choose the boundary conditions?" If the system is LSI, the initial conditions must be zero and they must lie

outside the support of the output y(n1,n2)

! Ordering the computations" Output samples can be computed in any one several possible orderings =>

computation only partially ordered" Partial ordering represented by precedence graph



! Example: Computation Ordering and Precedence Graph

N1+1

N2+1

n2

n1Initial condition region

Output Mask

The first point that can be calculated is the origin

xx

The next point to be computed is (0,1) or (1,0), or we can calculate both of them in parallel if the hardware permits.

xx

x

(0,0)

(1,0)(0,1)

(2,0)(1,1)(0,2)

Precedence Graph



! Not all the possible computation orderings are equivalent" Amount of storage" Degree of parallelism

! Example: " 3x3 output mask" Output evaluated on NxN square region

N-1

n2

n1

N-1

N-1

n2

n1

N-1

Dashed region indicatesthe needed storage.


IIR filters Recursive Computability

! Example: Computation Ordering storage/parallelism (continued)

- Dashed region indicates samples to be buffered.- If output mask is of size L1xL2, column-by-column implementation requires the

storage of (L1-1)N+(L2-1) samples. - Output samples along any diagonal can be computed independently => all output

samples along a diagonal can be computed simultaneously.- Diagonal versus column (or row) implementation

" Diagonal implementation is maximally parallel if parallel hardware available." Diagonal more efficient computationally due to possible parallel

implementation but more storage required.Note: each diagonal has a different length => must allow storage for the longest one.

N-1

n2

n1

N-1

N-1

n2

n1

N-1


Implementation of IIR Filters Direct Form (recursive) Implementation

! Rearrange difference equation to express output sample in terms of input samples and previously computed output samples

! Requires filter to be recursively computable.

where .

-

=

),(

)0,0(

22112122112121

211 21 2

),(),(),(),(),(mm

m ml lmnmnxmmblnlnxllanny

1)0,0( =b

),(),(),(),( 21212121 nnhnnynnnnx ==

=

),(

)0,0(

2211212121

211 2

),(),(),(),(mm

m mmnmnhmmbnnannh

=

),(

)0,0(

2121211

21

12121

211 2

21

1 2

),(),(),(),(mm

m mz

mm

l lz zzHzzmmbzzllazzH

,),(),(),(

21

2121 zzB

zzAzzHz

zz =

1)0,0( ;),(),(

),(),(

1 2

21

1 2

212121

12

112121

==

=

bzzmmbzzB

zzllazzA

m m

mmz

l lzwhere


Implementation of IIR Filters Direct Form (recursive) Implementation

! can be implemented as the cascade of two filters: Direct Form I

" Assume output mask is swept column-by-column: If N2 output points desired per column

- For filter with an input mask, we need to store input samples.

- For filter with an output mask, we need to store computed output samples.

- Alternative Implementation (Direct Form II)

),( 21 zzHz

),( 21 zzAz),( 21 nnx ),( 21 nny

),(1

21 zzBz

21 LL 1)1( 221 + LNL

),( 21 zzBz

),( 21 zzAz

21 MM

),( 21 zzAz),( 21 nnx ),( 21 nn ),( 21 nny

),(1

21 zzBz

FIR Purely IIR

1)1( 221 + MNM


Implementation of IIR Filters

Direct Form (recursive) Implementation

! Aternative Implementation: Direct Form II

" Reduces storage required." Storage shared between the feedback and feedforward stages: only the

samples of need to be stored.! Cascade and parallel implementations can be used to design specific

IIR filters" Contrary to 1-D case, a general 2-D IIR filter cannot be decomposed into a

cascade of simpler filters" Cascading or interconnecting filters in parallel can be used to synthesize

some IIR filters" Example: Bandpass formed by cascading

a lowpass and a highpass filter

),( 21 nn

),( 21 zzAz),( 21 nnx ),( 21 nn ),( 21 nny

),(1

21 zzBz

2

1


Iterative implementation for 2-D IIR filters! Recursive versus Iterative Implementation

! Iterative method is an attempt to design and implement non-recursively computable IIR filters

! Iteratively generates better estimates of output y(n1,n2) given the desired transfer function Hz(z1,z2).

Recursive Implementation

Efficient Requires recursive computability

Requires initial conditionsDoes not require storing all input sample

Iterative Implementation

Not so efficient Recursive computability not required (does

not have to be casual)Can use boundary conditionAll input samples need to be available and stored (such as in image processing)



Iterative implementation for 2-D IIR filters! Basic Iterative implementation

),(1),(

),(),(),(

21

21

21

2121 zzC

zzAzzBzzAzzH

z

z

z

zz

==

43421atordeno

zz zzBzzCmin

2121 ),(1),( where =

),().,(),( 212121 zzXzzHzzY zzz =

),().,(),().,(),( 2121212121 zzYzzCzzXzzAzzY zzzzz +=

(*)),(),(),(),(),( 2121212121 nnynncnnxnnanny +=



Implementation of IIR Filters Iterative implementation for 2-D IIR filters

! Basic Iterative implementation

- Approach:1. Guess2. Substitute in the RHS of to get a better estimate3. Repeat Step 2 until satisfied

- Note: and are FIR filters- Issue: will estimates converge to true output?

Answer: Yes, but infinite number of iterations needed => after some finite number of iterations, we get a very

good estimate (small error)

),( 21 nny

),( 21 nna ),( 21 nnc

)(

),(),(),(),(),( 21121212121 nnynncnnxnnanny ii +=),().,(),().,(),( 21121212121 += izzzi YCXAY

(*)),(),(),(),(),( 2121212121 nnynncnnxnnanny +=



! Basic Iterative implementation

In frequency domain, this becomes:

Convergence Proof:- Let the initial guess be Y-1(1, 2)= 0.

- Convergence condition:

),().,(),( 2121210 XAY =

M

),().,().,(),().,(),( 2121212121211 XACXAY +=

=

=

I

i

iI CXAY

021212121 ),(),().,(),(

1),( 21



! Basic Iterative implementation" Disadvantage:

- Convergence condition is too restrictive since stability only requires

- Not all 2-D stable IIR filters can be implemented with this basic method

! Solution: Multiply denominator and numerator by where=constant parameter

=> Generalized Iterative Implementation

1),( 21 C

),( 21

* B

1),( 21


Iterative implementation for 2-D IIR filters! Generalized Iterative implementation

- Set

- Choose paramter in the range

221

2121

21

2121

),(),(),(

),(),(),(

BAB

BAH

==

4434421atordeno

BCmin

22121 ),(1),( =


221),(),(max

20

21

B



! Generalized Iterative implementation

- Iterative computation:Initial Guess: Y-1(1, 2) = 0.

- Note: True phase of obtained directly:

),(),(lim 2121 YYII =

),(),(),().,().,(),( 2112121212121 += ii YCXABY

I""iteration at ,),(1),(1),(),(),(),(

21

211

212121*

21

CCXABY

I

I

+=+

(),( ),( 121021 YYY =

),(),(

),((1),(),(

),(),(),(

),(),(),(

21

21

21

21212

21

2121

21

2121

XY

CAB

BAB

BAH =

===


Iterative implementation for 2-D IIR filters! Generalized Iterative implementation

- Error:

- Stop iteration when

where is the desired tolerance (usually, a small positive constant), and is a specified band of frequencies.

- Note: Error distribution can be controlled on desired bands by using a weighting function => replace

121

21

2121 ),(1),(

),(),( += II

CYYE

eee507_IIRFiltersImplement

Documents

Transcript of eee507_IIRFiltersImplement