1. Workshop on Digital Signal and Image processing
Filters
Speaker:
Dr. Rubaiyat Yasmin
Dept. of Information and Communication Engineering, R.U.
25 May, 2012

2. Introduction: What is filter? Why needed? Electronic circuits
Perform signal processing functions
Why needed?
To remove unwanted frequency components from the signal,
To enhance wanted ones.

3. Objective: Basic understanding Analog Filters Digital Filters
Adaptive Filters

4. Basic Types of Filters:
Four Main Filter Types:
Low-pass
High-pass
Band-pass
Band-reject

5. Basic Types of Filters (Contd)
Pass band, H(f)=1
Stop band, H(f)=0
Cut-off frequency

6. Frequency Response Curve
Low pass filter:
Frequency Response Curve
power = amp2; amp = sqrt(power)
1/2 power = sqrt(2)/2 amp = ~71% amp

7. Frequency Response Curve
High pass filter:
Passes high frequencies, attenuates lows.
Frequency Response Curve

8. Frequency Response Curve
Band pass filter:
Frequency Response Curve
Passes band of frequencies, attenuates those above and below band.
Most common in implementations of discrete Fourier transform to separate out harmonics.

9. Frequency Response Curve
Band reject filter:
Stops band of frequencies, passes those above and below band.
Most common in removing electric hum (50 Hertz A/C).
Frequency Response Curve

10. Analogue designs Common Approach :
Frequencies are specified in the Ω domain (in rad/s))
Exist for all the standard filter types
(lowpass, highpass, bandpass, bandreject).
Common Approach :
To define a standard lowpass filter, and
To use standard analogue-analogue
transformations from lowpass to the other
types, prior to performing the bilinear transform.
Important families:
Butterworth
Chebyshev
Elliptic

11. Butterworth Filter:
Maximally flat

12. Chebyshev Filter: Equiripple response in pass-band (up to ωc),
Monotonically decreasing in stop-band

13. Elliptic Filter:
Equiripple in both pass-band and stop-band

14. Other Types of Analogue Filter
Bessel filters, which are almost linear phase.
• Involve different degrees of flexibility and trade-offs in specifying transition bandwidth, ripple amplitude in pass-band/stop-band and phase linearity.
For a given band edge frequency, ripple specification, and filter order, narrower transition bandwidth can be traded off against worse phase linearity

15. Why Digital:

17. Basic concept :

18. What is a Digital Filter?
Numerical procedure or algorithm that transforms a given sequence of numbers into a second sequence that has some more desirable properties.
Input sequence
Output sequence
Digital Filter xn yn

19. Desired features Depend on the application, for example Input Signal
Output
generated by sensing device (microphone)
having less noise or interferences
speech
with reduced
redundancy for more efficient transmission

20. Examples of filtering operations
Noise suppression:
•  received radio signals
•  signals received by image sensors (TV, infrared imaging devices)
•  electrical signals measured from human body (brain heart, neurological signals)
•  signals recorded on analog media such as analog magnetic tapes

21. Enhancement of selected frequency ranges:
•  treble and bass control or graphic
equalizers
increase sound level and high and low level frequencies to compensate for the lower sensitivity of the ear
•  enhancement of edges in images
improve recognition of object (by human or computer)

22. Specific operations: • differentiation • integration
•  Hilbert transform
These operations can be
approximated by digital filters operating on the sampled input signal

23. The Basic of Digital Filters
Filters work by using one or both of the following methods:
Delay a copy of the input signal (by x number of samples), and combine the delayed input signal with the new input signal.
(Finite Impulse Response, FIR, or feedforward filter)
Delay a copy of the output signal (by x number of samples), and combine it with the new input signal.
(Infinite Impulse Response, IIR, feedback filter)
FIR filters are “finite” there is a specific limit to the number of times that any delayed sample is added to a new input sample (one!).
IIR filters, as with any feedback device, create a loop, hence the term infinite. The feedback, or loop, is scaled so that it rarely is infinite. (think high school PA system)
Both types of filters are essentially delay units.
FIR Filters
IIR Filters

24. FIR Filters
FIR filter of length M (order N=M-1, order - number of delays)
The Order of the filter is equal to the number of samples you look back

26. Frequency-domain Equivalent:
The impulse response is of finite length M
FIR filters have only zeros (no poles), hence known also as all-zero filters
FIR filters also known as feedforward or non-recursive, or transversal
Frequency-domain Equivalent:
Now N-point DFT (Y(k)) and then N-point IDFT (y(n)) can be used to compute standard convolution product and thus to perform linear filtering (given how efficient FFT is)

27. FIR Design Methods • Impulse response truncation
the simplest design method, has undesirable
frequency domain-characteristics, not very useful
•  Windowing design method
simple and convenient but not optimal, i.e.
order achieved is not minimum possible
•  Optimal filter design methods

28. Approximated filters obtained by truncation

29. FIR Filter Design Consider the Window method:
Determine ideal response function
If length of ideal function is too long, multiply ideal response by a finite length window function
Note that multiplication by window in time domain means convolution (and smearing) in the frequency domain

30. FIR Window Design Concept
Lowpass filter: cutoff at 0.2 fs .
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.6
0.8 1
1.2
Frequency (fraction of fs)
Amplitude (linear scale)

31. FIR Design Concept (cont.)
Time domain response (Inverse DTFT)
0.4
0.35
0.3
0.25
0.2
Amplitude
0.15
0.1
0.05
-0.05
-0.1
-60
-40
-20 20 40 60
Sample Index

32. FIR Design Concept Window function to limit response length
1.2 1
0.8
Hamming window
0.6
Amplitude
0.4
0.2
-0.2
-60
-40
-20 20 40 60
Sample Index

33. FIR Design Concept (cont.)
Windowed and shifted (causal) result
0.4
0.35
0.3
0.25
0.2
Amplitude
0.15
0.1
0.05
-0.05
-0.1 5 10 15 20 25 30 35 40
Sample Index

34. FIR Design Concept Resulting frequency response of filter 10 -10
-10
Magnitude (dB)
-20
-30
-40
-50
-60
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
Frequency (fraction of fs)

35. Window Design Method
Rectangular Window Frequency Response

36. Truncated Filter Increasing the dimension of the window M:
The width of the main lobe decreases
The area under side lobes remain constant

37. Solution to Sharp Discontinuity of Rectangular Window
Use windows with no abrupt discontinuity in their time- domain response and consequently
low side-lobes in their frequency response.
In this case, the reduced ripple comes at the expense of a wider transition region
However, this can be compensated for by increasing the length of the filter.

38. Alternative Windows -Time Domain
• Hanning
• Hamming
• Blackman
Window Characteristics:
A wider transition region (wider main-lobe) is compensated by much lower side-lobes and thus less ripples.

39. Filter realised with rectangular/Hanning window
M = 16
M = 40
There are much less ripples for the Hanning window but that the transition width has increased
Transition width can be improved by increasing the size of the Hanning window to M = 40

40. Windows characteristics
•  Fundamental trade-off between main-lobe width and side-lobe amplitude
•  As window smoother, peak side-lobe
decreases, but the main-lobe width increases.
•  Need to increase window length to achieve same transition bandwidth.
Specification necessary for Window Design Method

41. Window Design Procedure
•  Ideal frequency response has infinite impulse response
•  To be implemented in practice it has to be
-  truncated
-  shifted to the right (to make is causal)
•  Truncation is just pre-multiplication by a rectangular window
-  the filter of a large order has a narrow transition band
-  however, sharp discontinuity results in side-lobe interference independent of the filter’s order and shape Gibbs phenomenon
•  Windows with no abrupt discontinuity can be used to reduce
Gibbs oscillations (e.g. Hanning, Hamming, Blackman)

42. Windowed FIR filter design procedure
1.  Select a suitable window function
2.  Specify an ideal response Hd(ω)
3.  Compute the coefficients of the ideal filter hd(n)
4.  Multiply the ideal coefficients by the window function to give the filter coefficients
5.  Evaluate the frequency response of the resulting filter
and iterate if necessary (typically, it means increase M if the constraints you have been given have not been
satisfied)

43. FIR Filter Design Using Windows
FIR filter design based on windows is simple and robust, however, it is not optimal:
•  The resulting pass-band and stop-band parameters are equal even though often the specification is more strict in the stop band than in the pass band
unnecessary high accuracy in the pass band
•  The ripple of the window is not uniform (decays as we move away from discontinuity points according to side-lobe pattern of the window)
by allowing more freedom in the ripple behaviour we may be able to reduce filter’s order and hence its complexity
FIR Design by Optimisation
Least-Square Method

44. Design of IIR Filters IIR as a class of LTI Filters
Difference equation:
Transfer function:
To give an Infinite Impulse Response (IIR), a filter must be recursive, that is, incorporate feedback N ≠ 0, M ≠ 0

45. IIR Filters Design from an Analogue Prototype
Almost all methods rely on converting an analogue filter to a digital one
Analogue to Digital Conversion:
Analogue
Digital

46. Methods: Stable analogue filters become stable digital filters
Impulse Invariant method
Matched z-transform method
Backward Difference Method
Z- 0.5 = 0.5
- 0.5
= 0.5
Stable analogue filters become stable digital filters
No high-pass filter possible!
Poles are conned to a relatively small set of frequencies

47. Imaginary axis in the s domain are not mapped to the unit circle
The frequency response will be considerably distorted
Analogue high-pass filter cannot be mapped to digital high-pass
Poles of the digital filter cannot lie in the correct region
Rarely used
Correction :
Bilinear transform
Most popular method
frequency response same characteristics impulse response may be quite different

48. Properties of the Bilinear Transform
Two important cases:
1. With σ=0
Imaginary (frequency) axis in the s-plane maps to the unit circle in the z-plane
Left half s-plane maps onto the interior of the unit circle
2. With σ <0,
s-plane
z-plane
Suitable frequency response, stability for digital filter

49. Design using the bilinear transform
The steps of the bilinear transform method are as follows: 1.
“Warp” the digital critical (e.g. band-edge or "corner") frequencies ωi , in
other words compute the
corresponding analogue critical frequencies Ωi= tan(ωi/2). 2.
Design an analogue filter which satisfies the resulting filter response
specification.
3. Apply the bilinear transform

50. Example: Magnitude Frequency Response

51. Designing high-pass, band-pass and band-stop filters
Concentrated on IIR filters with low-pass
characteristics.
• Various techniques available to transform
a low-pass filter into a high-pass/band
pass/band-stop filters.
Frequency transformation in the analogue domain

52. Transformation table:

53. Adaptive Digital Filters
Top
Adaptive Digital Filters
9.6
• Adaptive digital filters are self learning filters, whereby an FIR (or IIR) is designed based on the characteristics of input signals. No other frequency response information or specification information is available.
• An adaptive digital filter is often represented by a signal flow graph with adaptive nature of weights shown:
x(k-1) x(k-2)
Adaptive
x(k)
Weights
Input w0
w1 w 2
d(k)
- +
y(k)
e(k)
Output
Error
August 2007, Version 3.8/21/07 For Academic Use Only. All Rights Reserved

54. Adaptive Signal Processing:
Top
Adaptive Signal Processing:
9.7
“The aim is to adapt the digital filter such that the input signal x(k) is filtered to produce y(k) which when subtracted from desired signal d(k), will minimise the power of the error signal e(k).”
desired
d(k)
signal +
input
y(k)
e(k)
signal
Adaptive FIR Σ
x(k)
Digital Filter Output-
error signal
signal
Adaptive Algorithm
e(k) = d(k) - y(k) y(k) = Filter(x(k))
August 2007, Version 3.8/21/07 For Academic Use Only. All Rights Reserved

55. Adaptive Filter Nomenclature
If the digital filter is FIR or all-zero, the adaptive system can also be called Moving Average or MA.
If the digital filter is all-pole, the adaptive system can also be called Autoregressive or AR.
If the digital filter is an IIR with zeros, the adaptive system can also be called ARMA.
This presentation addresses FIR or MA filters only.

56. Coefficient Adaptation
The principal behind determining the coefficients of the filter model is to maximize the statistical correlation between the desired signal and the coefficients.
Typically, this is done by minimizing the correlation between the error signal and the filter state as is relevant to the coefficients.
If the adaptive filter is working, the error signal decreases in magnitude, which slows down the movement of the coefficients. The filter is therefore converging to a solution.

57. Applications of Adaptive Filters
System identification: adaptive equalization
The adaptive filter attempts to model an unknown external system.
Interference cancellation
The adaptive filter attempts to isolate the component of a primary signal that is not part of a reference signal.
Linear prediction
This is like interference cancellation, but the adaptive filter uses a delayed version of the primary signal as the reference.

58. Architectures 9.9 Delay Unknown x(k) Adaptive Σ Σ s(k) System Filter -
Top
Architectures
9.9
Delay
s(k) + n)k)
d(k) y(k) +e(k)
d(k)
Unknown x(k)
Adaptive
x(k) Adaptive
y(k) +e(k) Σ
n’(k)
Σ s(k)
System Filter -
Filter -
Noise Cancellation
Inverse System Identification
Unknown System
d(k) x(k) y(k) +e(k)
d(k) +
Adaptive
x(k)
Adaptive Filter
y(k)
e(k) Delay Σ Σ -
s(k)
Filter -
Prediction
System Identification
August 2007, Version 3.8/21/07 For Academic Use Only. All Rights Reserved

59. Application Examples • System Identification:
Top
Application Examples
9.10
• System Identification:
• Channel identification; Echo Cancellation
• Inverse System Identification:
• Digital communications equalisation.
• Noise Cancellation:
• Active Noise Cancellation; Interference cancellation for CDMA
• Prediction:
• Periodic noise suppression; Periodic signal extraction; Speech coders; CMDA interference suppression.
August 2007, Version 3.8/21/07 For Academic Use Only. All Rights Reserved

60. Least-Mean-Square (LMS) Algorithm
Linear adaptive filtering algorithm
Differs from steepest descent
Widely used for its simplicity
Consists of:
1) A filtering process
(mainly FIR model)
2) An adaptive process

61. Parameters: M = # of taps (length of filter)
μ = step-size parameter
Filter output is: y(n) = ŵH(n)u(n)
Error signal is: e(n) = d(n) – y(n)
Tap-weight vector: ŵ(n+1) = ŵ(n) + μu(n)e*(n)

62. The Recursive Least-Squares (RLS) Algorithm
1. Let n=1
2. Compute the gain vector
k(n)= p(n-1)u(n)/1+uT(n) p(n-1)u(n)
3. Compute the true estimation error (n)
4. Update the estimate of the coefficient vector
ŵ(n) = ŵ(n-1) + k(n) (n)
5. Update the error correlation matrix
p(n)= p(n-1)-k(n)uT(n)p(n-1)
6. Increment n by 1, go back to step 2
Convergence is better but computationally expensive

63. Thank You