1. Chapter 6. Digital Filter Structures and Designs Section 6. 0-6
Chapter 6. Digital Filter Structures and Designs Section and
6.1 Block Diagram and Signal Flow Graph
6.2 Structures of IIR Filters
6.3 FIR Filter Design: Basic
6.4 FIR Filter Design: Windowing
6.5 IIR Filter Design: Impulse Invariance
6.6 IIR Filter Design Examples

2. 6.1.1 Filter Transfer Function
Linear constant-coefficient difference equations (LCCDEs).
Taking the two sided Z-transform we have
H(z) is a filter transfer function, and can be used to describe any digital filter.
Expressed as a diagram or signal flow graph
Implemented as a digital circuit

3. 6.1.2 Block Diagram and Signal Flow Graph
A signal flow graph representation of LCCDE is same as a block diagram, and it is a network of directed branches that connect at nodes which are variables.
Block diagram representation of a first-
order digital filter.
Structure of the signal flow graph.
Structure of the signal flow graph
with the delay branch indicated by z-1.

4. 6.1.3 Block Diagram: Direct Form I Realization
This structure (non-canonic, direct form I) can be too sensitive to finite word-length errors (quantization errors) – errors are summed,
fed back and re-amplified over and over.

5. 6.1.4 Block Diagram: Direct Form II Realization
This structure (canonic direct from, direct form II) requires less delay elements. The minimum number of delays is max(N, M).

6. 6.1.5 Example of LTI Implementation
Consider the LTI system with system (transfer) function
we have two implementation as follows

7. 6.1.6 Signal Flow Graph: Direct Forms
Given the LCCDE
Signal Flow Chart of Direct From I
Signal Flow Chart of Direct From II

8. 6.2.1 Structure of IIR: Cascade Form
By factoring the numerator and denominator we can write
which can be drawn as a cascade of smaller sections:
Advantages: Smaller sections – less feedback error.
Disadvantages: Errors fed from section-to-section.

9. 6.2.2 Parallel Realization
By performing a partial fraction expansion we can write
which can be drawn as a parallel sum of smaller sections
Advantages: smaller sections- less feedback error, and error confined to each section. 

10. 6.2.3 Structure of IIR: Example (Cascade)
Given a two-order system
Cascade Structure (Not unique)

11. 6.2.4 Structure of IIR: Example (Parallel)
Given a two-order system
Parallel Structure (Not unique)
Parallel-form structure using second-
order system (form I)
Parallel-form structure using
first-order system

12. 6.3.1 Digital Filter Design
Given a set specifications or stated constraints on
magnitude spectrum
phase spectrum
Find where,
The constraints may include
zero, small, or linear phase
specific bandlimit within a passband
amount of ripple within a passband
amount of ripple within a stopband
sharpness of transitions between passband/stopband
filter order K, M.

13. 6.3.2 Finite Impulse Response (FIR) Filter Design
Here it is assumed that
Hence
And so the unit pulse response of the filter is clearly:
Problem: Given specifications on and ,
find
FIR filters are often called non-recursive for obvious reasons.

14. 6.3.3 FIR Filter: Advantages and Disadvantages
Always stable (assume non-recursive implementation).
Quantization noise is not much of a problem.
Can be designed to have exact linear phase even when causal, while meeting a prescribed phase to arbitrary accuracy.
Design complexity generally linear.
Transients have a finite duration.
Disadvantages:
A high-order filter is generally needed to satisfy the stated specification – so more coefficients are needed with more storage and computation.

15. 6.3.4 FIR Filter Design: Linear Phase Condition
Definition: The digital filter
is linear phase if
for some real number C. If C=0, then the filter has zero- phase, which is only possible when the filter is non-causal.
Achieving linear phase is quite important in applications where is desirable not to distort the signal phase much –i.e., where the frequency locations are critical, such as speech signals.
Many applications benefit be the linear phase thought as
shaping frequencies according to the magnitude spectrum.
Time-shifting the response by an amount -C

16. 6.3.5 Linear Phase Condition
Theorem: a causal FIR filter with unit pulse response
is linear phase if h(n) is even symmetric:
Proof: Suppose M is odd. Then
which finishes the proof, why?
Consider case of M even to be an exercise.

17. 6.4.1 FIR Filter Design: Windowing
Goal: Design an FIR digital filter with M+1 coefficients that approximates a desired frequency response
with
Usually d(n) cannot be realized for some reasons.
d(n) has infinite duration if contains discontinuities;
If d(n) is non-causal and we want it causal;
If d(n) is longer than can be computed efficiently;
It’s generally desirable to have few coefficients;
Windowing is the simplest approach to FIR filter design. One can proceed naively, and thus obtain poor results. But with little care (basic windowing strategies), windowing can be very effective.

18. 6.4.2 General Windowing Approach
Define
where
Then designed filter then has frequency response
Observations: We desire conflicting goals
be time-limited to
be spectrally localized – impulse-like, if

19. 6.4.3 Truncation Windowing Rectangular window:
The designed filter has frequency response
where
Is the frequency response a good approximation to the desired frequency response ?
Actually, for M give is optimal in the mean square sense (MSE).
However, while rectangular windowing does the best global MSE job, it suffers dramatically at frequencies.

21. 6.4.4 Triangle (Bartlett) Windowing
Suppose that
which
Note that (ignoring the shift) is a positive function, hence
must rise monotonically at a jump discontinuity (why?).
In the prior example, using the triangular window gives an approximation with smooth, but wider transition.

22. Ripples vs. Transition Width
6.4.5 Windowing: Trade-off
Ripples vs. Transition Width
Rectangular window has a sharp transition but severe ripple.
Triangular window has no ripple but a very wide transition.

23. 6.4.6 Other Windows
Other windows attempt to optimize this trade-off. Widely used windows that give intermediate results are:
Hamming Window:
Hanning Window:
Blackman Window:

24. 6.4.7 Windowing Comparisons
Rectangular: transition width is optimized.
Blackman: Ripple is minimized.

25. 6.4.8 Kaiser Window Design Here
Where and represents the zeroth-order modified Bessel function of the first kind, and there are two important parameters: M, .
For M held constant, increasing  reduces sidelobe but increase mainlobe width.
For  held constant, increasing M reduces mainlobe width but does not affect sidelobes much.
Kaiser developed an empirical but careful design procedure for windowing a filter having sharp discontinuity (e.g. an ideal LPF).

27. 6.5.1 Infinite Impulse Response (IIR) Filter Design
Here it is assumed that
Problem: Given a desired response , Find:
and
IIR filters are often called recursive for obvious reason.
IIR filter advantages: A lower-order filter is generally sufficient to satisfy the stated specification – so fewer coefficients are needed (less storage and less computation).
IIR filter disadvantages:
Not necessarily stable.
Quantization noise can be a problem.
Cannot be designed to have exact linear phase when causal.

28. 6.5.2 Design of Discrete-Time IIR from Continuous-Time Filters
The art of continuous-time IIR filter design is highly advanced.
Many useful continuous-time IIR filter design methods have relatively simple closed form design formulas.
The standard approximation methods that work well for continuous-time IIR filters do not lead to simple closed-form design formulas when these methods are applied directly to the discrete-time IIR case.

29. 6.5.3 Recall Impulse Invariance
Impulse invariance: a discrete-time system is defined by sampling the impulse response of a continuous-time system by the sampling rate Td, i.e.,
Relationship between the frequency response of the discrete-time and continuous-time filters
If the continuous-time filter is band-limited, so that
then if

30. 6.5.4 IIR Filter Design by Impulse Response
Goal: Approximate a desired discrete-time frequency response by time-sampling an analog system with impulse response
If were strictly or effectively band-limited to the low-pass frequency interval
then can be regarded as equivalent to an ideal digital filter:

31. Then
or using the ideal relationship between analog and digital frequency
the digital filter that provides the equivalent system is
Thus, in an ideal situation, the desired digital filter could be obtained by
by appropriate selection of
Since the analog prototypes cannot be strictly band-limited, the above can only be approximated.
Impulse invariance is a procedure for obtaining the coefficients of an IIR filter approximating the above relationship.

32. 6.5.5 Sampling the Impulse Response
Suppose we (scale) and sample the impulse response of a (causal) analog prototype :
Then

33. 6.5.6 Aliasing Effect
It is very difficult to compensate for aliasing effects in the impulse invariance approach. Therefore, either sufficiently high sampling rate must be used or another design procedure must be used.
Assume a sufficiently high sampling rate such that is effectively band-limited to
Of course the scaled analog prototype must agree satisfactorily with the digital specification by selection of, e.g.,
Aliasing is an important consideration with impulse invariance design. One common strategy is that no more than 10%, or more stringently, 1% of the energy be aliased.

34. 6.5.7 Implementation of Impulse Invariance Design
Let have L distinct poles (typical for our prototypes): so
The sampled filter is then
The designed digital transfer function is then
Which is ideal for a parallel realization of first-order section or even better combined into second-order sections.

35. 6.5.8 Comments on Stability
If a pole pm of the analog filter lies in the left-hand plane (LHP), then also
So the corresponding pole of will lie inside the unit circle. Then the causal filter will be stable.
Hence the important property

36. 6.6.1 IIR Filter Design Example-1
Let us consider the design of a low-pass discrete-time filter by applying impulse invariance to an appropriate Butterworth continuous-time filter. The specifications for the discrete filter are
For simplicity, in the impulse invariance design, we will take Ts=1, hence we have
Thus, the specifications on the analog filter are
Since the magnitude response of an analog Butterworth filter is a monotonic function of frequency, the specifications are simplified into

37. 6.6.2 Filter Design Example-2
Specifically, the magnitude function of a Butterworth filter is
so we need to determine N and  to meet the desired specifications
which lead to the equations with equality
The solutions are and If we use N=6,
then

38. 6.6.3 Filter Design Example-3
Given N=6, the 12 poles of the magnitude-squared function
are uniformly distributed in angle of a circle of Consequently, the poles of are the three pole pairs in the left half of the s-plane as

39. 6.6.4 Filter Design Example-4
From the 6 poles of , we have
We use a partial fraction expansion, and , we have a parallel form of the digital filter.
The frequency-response functions of the discrete-time filter is