1. EEE4176 Applications of Digital Signal Processing
Fall 2011
Assistant Prof. Yangmo Yoo
Dept. of Electronic Engineering
Sogang University

2. Lecture 2: Realization of Digital Filters

3. Direct Realization of IIR Filters
Causal, rational, stable rational function
Direct realization
where 3

4. Direct Realization of IIR Filters
Canonical form
Transposed form

5. Direct Realization of IIR Filters
Direct form or direct realization (Canonical form)
use minimal possible number of delays
2N+1 multipliers and 2N adders
recursive realization. Transient response -> steady-state response
represents delays, i.e., memory devices like latches or registers
Initialization of filter : memories are initialized either to zero or to
other proper values.
Transposed direct realization
Reverse signal flow directions in all lines
Interchanging the input and the output

6. Parallel Realization of IIR Filters
Rational transfer function
: FIR + IIR
※ Note: If is complex, then is also a pole and
※ Order = (real poles) (complex poles)
where real numbers
depend on

7. Parallel Realization of IIR Filters
Ex:

8. Parallel Realization of IIR Filters
: direct and parallel forms require the same multipliers and adders
Direct form requires multiplications and additions
Parallel from requires multiplications and additions
# of delays are same.
Better sensitivity of the frequency response to finite word length

9. Cascade Realization of IIR Filters
Can be used to realize any IIR filters. Cascade realization
Most widely used. less sensitive to quantization error
The realization is not unique
exist multiple ways of paring poles and zeros to make each biquads.
multiple ways of ordering the biquads in the cascade connection
Filtering performance can differ for different realization.
Biquad 9

10. Cascade Realization of IIR Filters
Pairing in cascade realization
1. Select the pair of complex poles nearest to the unit circles and pairs them with nearest complex zeros
To make the mag respose as flat as possible
Before the quantization errors accumulate too much, use poles nearest to the unit circle.
2. Continue to pair the poles and zeros according to the same rule.
After complex zeros are exhausted, use real zeros
3. Pair the real poles. 10

11. Coupled Cascade Realization
When the word length is short and filter has poles near z= + - 1
When quantization error can make the filter unstable
: complex pole 11

12. Direct Realization of FIR Filters
Standard form
: N+1 Mul. N adders
Transposed form
Cascaded form is seldom used for FIR 12

13. Direct Realization of FIR FIlters
Structure for linear phase FIR filters
can reduce the # of multiplications by half. 13

14. State-space Representation
State-space Concept 14

15. State-space Representation
State equation
Output equation
: state vector
where
• State matrix and input matrix
• output matrix
• direct transmission matrix 15

16. State-space Representation
For transposed direct transformation
Two representations are dual or transpose to each other
Note:
A rational transfer function has an infinite number of state-space representations
You can assign the state variables in many different ways.
# of state vectors = # of memories

17. State-space Representation
Application of state space
Impulse response for and
can be obtained without using z-transform and partial fraction expansion 17

18. State-space Representation
General system response when the system is relaxed.
Z-transform solution and Transfer function 18

19. State-space Representation
Application example: Step Initialization for IIR MTI Filtering
MTI(Moving Target Indicator) Filter is used for
Moving target detection in RADAR, SONAR, etc.
Blood flow detection in Ultrasound Imaging
MTI(Moving Target Indicator) involves
the detection of Doppler frequency shift from a moving object
the elimination of clutter signals, which are generated by slow motions of sensors and non-target objects and
MTI(Moving Target Indicator) Filtering
High-pass filtering
Length of an input data sequence is small
Transient effect is severe. 19

20. State-space Representation
Application example: Step Initialization for IIR MTI Filtering
Transient problem
Input signal can be modeled
as a sum of
ac signal + step function
(i.e., dc offset)
The response will contain
the step response 20

21. State-space Representation
Application example: Step Initialization for IIR MTI Filtering
Step response 21

22. State-space Representation
Application example: Step Initialization for IIR MTI Filtering
Modified filter
Conventional filter
Modified filter
Step response of the modified filter 22

23. Finite Word Length Problem
Possible problems in practical filter implementation
Input data has quantization error
Coefficients of the difference equation are quantized
Alter the pole and zero locations
Intermediate data involving multiplications and additions are quantized before moving on to next step.
Because of the finite word length, saturation or overflow can occur
Coefficient quantization effects in digital filters
Coefficient quantization affects filter response and possibly stability
# of bits to represent the coefficients :
For stability and 23

24. Finite Word Length Problem
Quantized pole location in direct realization
pole:
High coefficient accuracy is required for a narrow band filter and high sampling rate because of the problem with the poles near 24

25. Finite Word Length Problem
Quantized pole location in coupled realization
Real and imaginary parts of the poles are independently quantized
Thus the pole locations are distributed uniformly in the unit circle.
Quantization of the numerator coefficients
1. Zeros of 4 common analog filters are on the imaginary axis
2. Digital filters obtained by the bilinear transformation have poles on the unit circle
which are not sensitive to quantization error
Sensitivity of poles
Parallel or cascade realization is better than direct realization
※ Pair the poles which are not close to each other. 25

26. Finite Word Length Problem
Quantization effect on the frequency response
Assuming the perturbations in the parameters are small, the approximate perturbation in the mag response by a first-order Taylor series is given by
for a set of real parameters,
to be quantized. 26

27. Finite Word Length Problem27

28. Scaling in Fixed-point arithmetic
Analog filter
Saturation
Polarity preserved
Digital filter
Overflow
Reserved polarity
In IIR case
This causes too much problem
Either or is scaled to prevent the overflow 28

29. Scaling in Fixed-point arithmetic
Time-domain scaling
where
If , then 29

30. Scaling in Fixed-point arithmetic
Frequency-domain scaling
1-norm bound
Infinite-norm bound 
Define
: Infinite norm of
: 1-norm of
Note: Time-domain scaling is more conservative
than freq-domain scaling  30

31. Scaling in Fixed-point arithmetic
Scaling of Inner signals
2's complement arithmetic is preferred.
Addition of all numbers does not overflow even if overflows occur in partial sums
Transposed direct realization has an advantage over the direct realization
Use of saturation mode is recommended
Scaling in parallel and cascade realization
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Quantization noise
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