Digital filters

Lecturer: Dr. Bui Ha Duc Email: ducbh@hcmute.edu.vn

Department of Mechatronics
1
Signal Filter
 Signal Filter is a frequency selective LTI system
𝑀−1
1
𝑦 𝑛 = ෍ 𝑥[𝑛 − 𝑘]
𝑀
𝑘=0

Analog filters Digital filters:

• Electronic components are cheap. • Better performance
• Large dynamic range in amplitude and • Programmable.
frequency. • One hardware design can implement
• Real-time. many different filters.
• Low stability of resistors, capacitors • The characteristics of digital filters
and inductors due to temperature. are predictable.
• Difficult to get the components • Can be evaluated in simulation before
accuracy as calculated by the formula implementing in hardware.

2
Digital filters
 Digital filters are a very important part of DSP:
 signal separation: noise, interference, other signals
E.g. measuring the electrical activity of a baby's heart (EKG) while still in
the womb; extracting EOG

 signal restoration
E.g. recovering an audio recording made with poor equipment,
calibration

3
 Remove noise on raw RSSI data

4
 Remove Eye blinks from EEG signal

5
 Remove noise on raw GPS data

6
Ideal filters

Output of a digital filter:

𝒀 𝒆𝒋𝝎 = 𝑯 𝒆𝒋𝝎 𝑿(𝒆𝒋𝝎 )

Frequency Filter Input

frequencies

7
Ideal Filter
 The output of ideal filter is
 distortionless response over one or more frequency bands
 zero response at the remaining frequencies

−𝒋𝝎𝒏𝒅 , 𝝎 ≤ 𝝎
𝑯 𝒆𝒋𝝎 =൝ 𝒆 𝒄
𝒐, 𝝎𝒄 ≤ 𝝎 ≤ 𝝅

Ideal lowpass filter

8
Ideal Filter
 Invert Fourier transform of an ideal lowpass filter
𝒋𝝎
𝒔𝒊𝒏(𝝎𝒄 (𝒏 − 𝒏𝒅 ))
𝑯 𝒆 → 𝒉𝒍𝒑 𝒏 =
𝝅(𝒏 − 𝒏𝒅 )

→ Not practical

9
Practical filter
 Practical filter is usually done by minimizing some
approximation error between the non-ideal filter and a
prototype ideal filter

 A good filter should have only a small ripple in the passband,

high attenuation in the stopband, and very narrow transition
bands
10
Distortion of signals passing through LTI
systems
 Distortionless response systems
𝑦 𝑛 = 𝐺𝑥 𝑛 − 𝑛𝑑 , 𝐺 and 𝑛𝑑 𝑎𝑟𝑒 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡𝑠

 Taking Fourier transform on both sides:

𝒀 𝒆𝒋𝝎 = 𝑮𝒆−𝒋𝝎𝒏𝒅 𝑿(𝒆𝒋𝝎 )

 Frequency response:
MATLAB
𝑯 𝒆𝒋𝝎 = 𝑮𝒆−𝒋𝝎𝒏𝒅 = 𝑯 𝒆𝒋𝝎 𝒆𝒋∠𝑯(𝒆^𝒋𝝎 ) freqz(b,a)
𝑯 𝒆𝒋𝝎 =𝑮 abs()

∠𝑯 𝒆𝒋𝝎 = −𝝎𝒏𝒅 angle()

11
Distortion of signals passing through LTI
systems
 A filter introduces magnitude distortion 𝑯 𝒆𝒋𝝎 ≠𝑮
 A filter introduces phase or delay distortion if ∠𝑯 𝒆𝒋𝝎 ≠ −𝝎𝒏𝒅

12
 Distortion of signals passing through LTI
systems
Example:
1 1
Input x[𝑛] = cos 𝜔𝑛 − cos 3𝜔𝑛 + cos(5𝜔𝑛)
3 5

Suppose that the output y[n] given by:

y[𝑛] = 𝑐1 cos 𝜔𝑛 + 𝜙1 − 𝑐2 cos 3𝜔𝑛 + 𝜙2 + 𝑐3 cos(5𝜔𝑛 + 𝜙3 )

13
Distortion of signals passing through LTI
systems

14
Distortion of signals passing through LTI
systems

15
Example
 Determine the magnitude and phase response of this
filter

16
Example

17
Magnitude and phase response functions and input–output signals for the filter

18
Group delay
 Group delay is a convenient way to check the linearity of
phase response
 Group delay is defined as the negative of the slope of the
phase:
𝑑(∠𝑯 𝒆𝒋𝝎
𝜏𝑔𝑑 = −
𝑑𝜔
 Linear-phase response have constant group delay

19
Group delay
 Analyze this frequency response

20
Filter design procedure
1. Specification: Specify the desired frequency response
function characteristics to address the needs of a specific
application.
2. Approximation: Approximate the desired frequency
response function
3. Quantization: Quantize the filter coefficients at the
required fixed-point arithmetic representation.
4. Verification: Check whether the filter satisfies the
performance requirements by simulation or testing with real
data.
5. Implementation: Implement the system obtained in
hardware, software.

21
Filter specifications

Practical filter:
• the passband responses are not perfectly flat
• the stopband responses cannot completely reject (eliminate) bands
of frequencies
• the transition between passband and stopband regions takes place
over a finite transition band.
22
Filter specifications

23
Decibel
 A bel (in honor of Alexander Graham Bell) means that
the power is changed by a factor of ten
 E.g 3 bel of amplification produces an output signal with
10×10×10 = 1000 times the power of the input
 dB =1/10 bel, every ten decibels mean that the power has
changed by a factor of ten
𝑃2
𝑑𝐵 = 10𝑙𝑜𝑔10
𝑃1
 Every twenty decibels mean that the amplitude has
changed by a factor of ten.
𝐴2
𝑑𝐵 = 20𝑙𝑜𝑔10
𝐴1

24
Filter specifications
 Absolute specifications
 In the passband
1 − 𝛿𝑝 ≤ 𝐻 𝑒 𝑗𝜔 ≤ 1 + 𝛿𝑝 , 0 ≤ 𝜔 ≤ 𝜔𝑝
 In the stopband
𝐻 𝑒 𝑗𝜔 ≤ 𝛿𝑠 , 𝜔𝑠 ≤ 𝜔 ≤ 𝜋
 Relative specifications
 passband ripple
1+𝛿𝑝
𝐴𝑝 = 20𝑙𝑜𝑔10 𝑑𝐵
1−𝛿𝑝
 stopband attenuation
1 + 𝛿𝑝
𝐴𝑠 = 20𝑙𝑜𝑔10 ≈ −20𝑙𝑜𝑔10 (𝛿𝑠 ) dB
𝛿𝑠

25
Filter specifications
Then
𝐴𝑝 ≤ 𝐻 𝑒 𝑗𝜔 , 𝑖𝑛 𝑑𝐵 ≤ 0, 0 ≤ 𝜔 ≤ 𝜔𝑝

𝐻 𝑒 𝑗𝜔 , 𝑖𝑛 𝑑𝐵 ≤ −𝐴𝑠 , 𝜔𝑠 ≤ 𝜔 ≤ 𝜋

 In a well designed filter, typically 𝐴𝑝 ≈ 0 and 𝐴𝑠 ≫ 1

 The smaller 𝛿𝑝 and 𝛿𝑠 , and the narrower the transition band,
the more complicated (higher order) is the required filter.

26
Example:
 A filter specified have a magnitude response in the
passband within ±0.001, and the minimum stopband
attenuation is given as 𝛿𝑠 =0.001.
 𝐴𝑝 , 𝐴𝑠 =?

1 + 0.001
𝐴𝑝 = 20𝑙𝑜𝑔10 = 0.0174 𝑑𝐵
1 − 0.001
𝐴𝑠 = −20𝑙𝑜𝑔10 (0.001)=60 dB

27
 Example:
 A lowpass digital filter is specified by the following
relative specifications:
𝜔𝑝 = 0.3π, 𝐴𝑝 = 0.5 dB; 𝜔𝑠 = 0.5π, 𝐴𝑠 = 40 dB.
𝛿𝑝 , 𝛿𝑠 = ?

28
FINITE IMPULSE RESPONSE FILTERS
 FIR system:
𝑀

𝑦 𝑛 = ෍ 𝑏𝑘 𝑥[𝑛 − 𝑘]
𝑘=0
𝑏𝑛 , 𝑛 = 0, … , 𝑀
and ℎ 𝑛 =ቊ
0, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒

31
Characteristics of FIR filters
 FIR filters are always stable
 The filter has finite memory
 The design of linear-phase filters can be guaranteed
 FIR filters can be easily implemented on most DSP
processors, but may result in high computational cost

32
FIR filters with linear phase
 Frequency response
∞

𝑯 𝒆𝒋𝝎 = ෍ 𝒉[𝒌]𝒆−𝒋𝝎𝒌
𝒌=−∞
 An FIR filter has linear phase if its coefficients /impulse
response satisfy the following symmetric condition
ℎ 𝑛 = ±ℎ 𝑀 − 𝑎 , 𝑀 𝑖𝑠 𝑎 𝑖𝑛𝑡𝑒𝑔𝑒𝑟

33
Moving Average Filters
 Moving Average Filters:
𝑀−1
1
𝑦 𝑛 = ෍ 𝑥[𝑛 − 𝑘]
𝑀
𝑘=0

 Advantages:
 Easy to understand and implement
 Less computation time

34
Moving Average Filters
 Noise Reduction vs. Step Response

Increasing the number of points lead to?

35
Moving Average Filters
 Frequency Response of moving average filter
𝜔𝑀
sin( )
𝐻 𝑒 𝑗𝜔 = 2 𝑒 −𝑗𝜔(𝑀−1)/2
𝜔
𝑀𝑠𝑖𝑛( )
2

𝜔𝑀
sin( )
𝐻 𝑒 𝑗𝜔 = 2
𝜔
𝑀𝑠𝑖𝑛( )
2

36
Moving Average Filters

Transitional band?
Stopband attenuation?

The moving average filter is

a good smoothing filter but
a bad low-pass-filter

Frequency Response of moving average filter

37
 Multiple pass

Multiple-pass
moving average
filters involve passing
the input signal
through a moving
average filter two or
more times

38
Example of Moving Average filter
The file djw6576.txt contains the weekly opening value
x[n], 0 ≤ n ≤ N−1, of the Dow Jones Industrial Average
for N = 600 weeks starting in January 1965
 Investigate the effect of changing M

39
Summary of Moving Average filter
 Optimal filter for the following tasks:
 Reducing random noise while retaining the sharp step
response
 Useful for time domain encoded signals
 Worst filter concerning frequency encoded signals (no
frequency separation capabilities !)

40
Windowed-sinc filter
 Windowed-sinc filters are used to separate different
bands of frequencies.
 Advantages:
 stable
 very good overall performance in the frequency domain.
(poor performance in time domain.)
 if done by standard convolution: needs much computation
power.
 less execution time necessary when realized by FFT-programm.

41
Windowed-sinc filter
 Motivation

Ideal lowpass filter ⟶ infinite

impulse response

42
Windows-sinc filter

If we make impulse response

become finite result from the abrupt discontinuity

Need to remove the abrupt end! But how?

43
 Solution: multiplying the modified-sinc with a window to
reduce the abruptness of the truncated ends and thereby
improve the frequency response

44
 X

45
46
Design Windowed-sinc filters
 2 parameters must be determined:
 the cutoff frequency, 𝑓𝑐
 the order of the filter kernel, M.

 After fc and M have been selected, the filter kernel is

calculated from the relation:
𝑠𝑖𝑛(2𝜋𝑓𝑐 (𝑛 − 𝑛𝑑 ))
ℎ 𝑛 =𝐴 x w[n]
𝜋(𝑛 − 𝑛𝑑 )

47
 • Cut-off frequency according to the sampling rate
• Filter length M defines the transition bandwith
4
𝑀≈
𝑏𝑎𝑛𝑑𝑤𝑖𝑑𝑡ℎ
48
49
The order of the filter kernel
𝑏𝑛 , 𝑛 = 0, … , 𝑀
 ℎ 𝑛 =ቊ
0, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
 M is the order of the filter
 A filter with order M has L = M+1 coefficients
 M should be an even integer

50
The order of the filter kernel
 Type-I FIR filter:
 has a symmetric impulse response
 even order M
 Consider the case M = 4

51
The order of the filter kernel
 In general, the frequency response of a type-I FIR filter
can be expressed as:

52
The order of the filter kernel
 Type-II FIR filter:
 has a symmetric impulse response
 odd order M

 Consider the case M = 5

53
The order of the filter kernel
 In general, the frequency response of a type-II FIR filter
can be expressed as:

 At 𝜔 = 𝜋, 𝐴 𝑒 𝑗𝜔 = 0, independent of h[k]

filters with a frequency response that is nonzero

at ω = π, for example highpass filter, band stop
filter, cannot be implemented

54
Design of FIR filters using windows

A window useful for filter design should have a Fourier

transform that has a narrow mainlobe, a small relative peak
sidelobe, and good side lobe roll-off
55
FIR filter design using windows
Procedure:
1. Given the design specifications {𝜔𝑝 , 𝜔𝑠 , 𝐴𝑝 , 𝐴𝑠 }, determine the ripples 𝛿𝑝 and 𝛿𝑠 and
set 𝛿 = min{𝛿𝑝 , 𝛿𝑠 }.
2. Determine the cutoff frequency of the ideal lowpass prototype by 𝜔𝑐 = (𝜔𝑝 +
𝜔𝑠 )/2.
3. Determine the design parameters 𝐴𝑝 , 𝐴𝑠 and ∆𝜔 = 𝜔𝑠 − 𝜔𝑝 .
4. Choose the window function that provides the smallest stopband attenuation
greater than A.
5. Determine the required value of M = L−1 by selecting the corresponding value of ω
from the column labeled “exact ω”. If M is odd, we may increase it by one to have a
flexible type-I filter.
6. Determine the impulse response of the ideal lowpass filter by
𝑠𝑖𝑛(𝜔𝑐 (𝑛 − 𝑀/2))
ℎ𝑑 𝑛 =
𝜋(𝑛 − 𝑀/2)
7. Compute the impulse response ℎ[𝑛] = ℎ𝑑 [𝑛]𝑤[𝑛] using the chosen window.
8. Check whether the designed filter satisfies the prescribed specifications; if not,
increase the order M and go back to step 6.

56
Design of FIR filters using windows

Properties of commonly used windows (L = M + 1)

57
Design of FIR filters by frequency sampling
Example:
Design a lowpass linear-phase FIR filter to satisfy the
following specifications:
𝜔𝑝 = 0.25𝜋, 𝜔𝑠 = 0.35𝜋, 𝐴𝑝 = 0.1 𝑑𝐵, 𝐴𝑠 = 50 𝑑𝐵.

58
Kaiser window
 Kaiser window is an windows with adjustable parameters
designed to optimize the trade-off between main-lobe width
and sidelobe height

𝑛−𝛼 2
𝐼0 𝛽 1 − 𝛼
𝑤𝑛 = , 0≤𝑛≤𝑀
𝐼0 (𝛽)
0, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
Where
∞ (𝑥/2)𝑚
𝛼 = 𝑀/2 and 𝐼0 𝑥 = 1 + σ𝑚=1
𝑚!

python function: signal.kaiserord(L,beta)

59
Kaiser window

60
Kaiser window

61
 Example
Design a lowpass filter to remove the high frequency noise
above 1,600 Hz with the following filter specifications:
passband frequency range: 0- 1,600Hz; passband ripple: 0.02
dB; stopband frequency range: 1,800 - 4,000 Hz; stopband
attenuation: 50 dB.
 Use the designed lowpass filter to filter the noisy speech
(nspeech.mat)

62
Summary of Windowed-sinc filters
 incredible performance
 But trade-off in execution time

63
PARKS–MCCLELLAN METHOD
 One of the most popular optimal design method used in
the industry due to its efficiency and flexibility
 The algorithms minimize the maximum error between
the desired frequency response and actual frequency
response
𝐸 = 𝑚𝑎𝑥 𝐻𝑑 𝑒 𝑗𝜔 − 𝐻 𝑒 𝑗𝜔

64
 Parks–McClellan algorithm
65
 Calculate the weights

 Use Remez algorithm to calculate filter coefficients

f_cutoff = 500
order = cycles*fs // f_cutoff
# Build the filter using the Remez method
# Pass band: [0, 0.75 * f_cutoff] (begin to transition at 75% of our cutoff), gain: 1
# Stop band: [f_cutoff, fs/2] (attenuate everything above the cutoff), gain: 0
hpm = scipy.signal.remez(order,
[0, 0.75 * f_cutoff, f_cutoff, fs/2], # Our frequency bands
[1, 0], # Our target gain for each band
fs=fs)
# Apply the filter to a signal
y = scipy.signal.convolve(x, hpm)

66
INFINITE IMPULSE RESPONSE FILTERS
 IIR system:
𝑀 𝑀

𝑦 𝑛 = ෍ 𝑏𝑘 𝑥[𝑛 − 𝑘] − ෍ 𝑎𝑘 𝑦[𝑛 − 𝑘]
𝑘=0 𝑘=1
 ℎ 𝑛 is a infinite sequence

 The objective in IIR filter design is to determine the coefficients

so that its frequency response H(ejω) approximates an ideal
desired response Hd(ejω)

67
IIR filters
 Step 1 Convert the discrete-time design
specifications into continuous-time specifications.
 Step 2 Design a continuous-time filter, that is, obtain a
system function Hc(s) that satisfies the continuous-time
specifications.
 Step 3 Convert Hc(s) to an appropriate system function
H(z), which meets the specifications, using a continuous-
time to discrete-time transformation.

68
Characteristics of IIR filters
 IIR filters cannot have linear phase

 IIR filter not always stable

 IIR filters involves both the magnitude and phase

responses

 IIR filters require a much lower filter order than FIR

filters

IIR filters are suitable for lower cost system

69
IIR filters
 Consider the following IIR system:
1
𝑦[𝑛] = (𝑥[𝑛] − 𝑥[𝑛 − 2]) + 𝑦[𝑛 − 1]
2
1. What is the order of this system?
2. What are b and a for this system?
3. What is its impulse response?

70
Transformation of continuous-time filters
to discrete-time IIR filters
 Impulse-invariance transformation
ℎ 𝑛 = 𝑇𝑑 ℎ𝑐 𝑛𝑇𝑑
𝑇𝑑 is the sampling period

 This transformation is known as impulse-invariance

because it preserves the shape of the impulse
response
 Frequency response:
𝑗𝜔
𝜔
𝐻 𝑒 = 𝐻𝑐 𝑗 , 𝜔 <𝜋
𝑇𝑑
If
𝐻𝑐 𝑗Ω = 0, Ω ≥ 𝜋/𝑇𝑑
71
Transformation of continuous-time filters
to discrete-time IIR filters
 Bilinear transformation
 invertible nonlinear mapping between the s-plane and the z-
plane
2 1 − 𝑧 −1
𝑠=𝑓 𝑧 =
𝑇𝑑 1 + 𝑧 −1

 Td has no effect on the design process since the bilinear

transformation does not involve any sampling
operation.

72
Transformation of continuous-time filters
to discrete-time IIR filters

73
Common IIR filters

74
Butterworth filter
 Butterworth lowpas filter (LPF) was proposed by Stephen
Butterworth in 1930
 any filter with its frequency response H(Ω) that satisfies
the equation

is a low pass filter with order N and cutoff frequency Ωc.

 Laplace transform H(s) :

75
Butterworth filter
 Design IIR low pass filter with specifications as

76
Butterworth filter

N=2

77
Butterworth filter

Design plots for the eighth-order digital lowpass Butterworth filter

78
Butterworth filter
 We note that Butterworth filters have the maximum
flatness, close to the ideal response.
 Butterworth filters have a very smooth group-delay
response
 Python functions for Butterworth filters:
 b, a = scipy.signal.butter(order, fc, fs)

79
Butterworth filter
 Order of the filter is determined based on pass-band
ripple, stop-band attenuation and transition band

fc = 500 # Cutoff at 500 Hz

fstop = 1000 # End the transition band at 1000 Hz
Ap = 3 # 3dB ripple
As = 60 # 60 dB attenuation of the stop-band

# Get the smallest order for the filter

order = scipy.signal.buttord(fc, fstop, Ap, As , fs=fs)

# Now construct the filter

b, a = scipy.signal.butter(order, fc, fs=fs)

80
Butterworth filter
 Use scipy.signal.butter to construct a high-pass filter
(fs=44100) with the following properties:
1. fc = 50
2. a transition width of 250 Hz,
3. at most 1 dB of pass-band loss, and
4. stop-band attenuation of 60 dB.
What is the order of the resulting filter?

81
Chebyshev filter
 named after the mathematician Pafnuty Chebyshev
 There are two types of Chebyshev filters:
 Type 1: have a steeper transition, higher pass-band ripple than
Butterworth filters
# Get the order and discard the second output (natural frequency)
order, _ = scipy.signal.cheb1ord(fc, fstop, ripple, attenuation, fs=fs)
# Build the filter
b, a = scipy.signal.cheby1(order, ripple, fc, fs=fs)

 Type 2: have a steeper transition, lower stop-band attenuation

than Butterworth filters
order2, _ = scipy.signal.cheb2ord(fc, fstop, ripple, attenuation, fs=fs)
b2, a2 = scipy.signal.cheby2(order2, attenuation, fstop, fs=fs)

82
Chebyshev filter

Design plots for the fifth-order digital lowpass Chebyshev filter

83
Elliptic filters
 Elliptic filters have extremely steep transitions but allow
ripple in both the pass- and stop-bands
 Python code to create Elliptic filters
# Get the order and discard the second output (natural frequency)
order, _ = scipy.signal.ellipord(fc, fstop, ripple, attenuation, fs=fs)

# Build the filter

b, a = scipy.signal.ellip(order, ripple, attenuation, fc, fs=fs)

84
Exercise
Implement a low-pass filter at fs = 44100 with the following
specification:
 Cut-off frequency Hz
 No more than 1.5 dB of ripple in the pass-band (less is acceptable)
 At least 80 dB of attenuation in the stop-band (more is acceptable)
 A transition band of no more than 500 Hz (less is acceptable)

a. Which of the IIR filter(s) should be used to implement a

filter with these constraints?
b. Determine the order of the filter. If there are multiple filter
types that will work, which one gives the smallest order?
c. Implement an FIR filter with the given constraints. What
order does the filter need to be to satisfy the constraints?

85