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EEE6209
Advanced Signal Processing (ASP)
Topic 01 - Digital Filters :
Part I
Analogue vs. Digital
Filter basics
Time/Frequency domain parameters
Dr Charith Abhayaratne
Tel: 25893
Email: [email protected]
Office: P/C11
Time/Frequency domain parameters
A framework for filter design
Part II
Time/Frequency domain parameters
What makes a good filter?
Realising other filters from a LPF
Moving average filter
Reference
Proakis: Ch8.1
MATLAB
Commands: filter, freqz, pz2tf, tf2pz, fft, plot, help, fir1, fir2, remez, firls
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Digital filters are used for two general purposes:
(1) separation of signals that have been combined,
E.g., foetus heartbeat monitoring
(2) restoration of signals that have been distorted in
some way.
E.g., Denoising an image
Analogue vs. Digital filters
Advantages of Digital filters?
Disadvantages of digital filters?
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Both x(n) and y(n) are digital signals.
Usually a filter's input and output signals are in the time
Digital filter
x(n) y(n)
)]([)( nxTny
Usually a filter's input and output signals are in the time domain. (Due to A to D conversion process sampling at equal intervals of time)
The second most common way of sampling is at equal intervals in space. E.g., imaging
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Difference Equation
(1.1)
Convolution
(1.2)
==
+=M
k
k
N
k
k knxbknyany01
)()()(
knxkhny
=
= )()()(
(1.3)
(1.4)
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N
N
M
M
k
zaza
zbzbbzH
zXzHzY
=
+++
+++=
=
...1
...)(
)()()(
1
1
1
10
4
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The impulse response
The step response
The frequency response.
Each of these responses contains complete information about the filter, but in a different form. form.
If one of the three is specified, the other two are fixed and can be directly calculated.
All three of these representations are important, because they describe how the filter will react under different circumstances.
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The impulse response
The output when the input is an impulse
)0()( when )()(
)()()(
==
=
=
nxnhny
knxkhnyk
The step response
The output when the input is a step (also called an edge, and an edge response)
It is the integral of the impulse response
Two ways to compute.
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Frequency response
Denoted by H(j)
Magnitude and Phase responses
jNN
j
jM
M
j
ez eaea
ebebbzHjH j
= +++
+++==
...1
...)()(
1
10
Magnitude and Phase responses
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( )
( ) ( )
=
=
==
)()(
)()()()(
jHjH
jHjHejHjH j
7
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Exercise: 2.1
Consider the difference equation
y(n) = 0.5y(n-1) + x(n) x(n-2)
Find out Find out
Impulse response
Step response
Frequency response
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Filter Implementation
The most straightforward way to implement a digital filter is by convolving the input signal with the digital filter's impulse response.
All possible linear filters can be made in this manner. manner.
When the impulse response is used in this way, filter designers give it a special name: the filter kernel.
There is also another way to make digital filters, called recursion.
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When a filter is implemented by convolution, each sample in the output is calculated by weighting the samples in the input, and adding them together.
Recursive filters are an extension of this, using previously calculated values from the output, besides points from the input.
Instead of using a filter kernel, recursive filters are defined by a set of recursion coefficients.
Remember, all linear filters have an impulse response, even if you don't use it to implement the filter.
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To find the impulse response of a recursive filter, simply feed in an impulse, and see what comes out.
The impulse responses of recursive filters are composed of sinusoids that exponentially decay in amplitude.
In principle, this makes their impulse responses infinitely long. However, the amplitude eventually drops below the round-off noise of the system, and infinitely long. However, the amplitude eventually drops below the round-off noise of the system, and the remaining samples can be ignored.
Because of this characteristic, recursive filters are also called Infinite Impulse Response or IIR filters.
In comparison, filters carried out by convolution are called Finite Impulse Response or FIR filters.
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IIR Filters
Duration length of h(n) is infinite. That means
h(n)0 when n.
IIR filters are recursive. That means at least one
coefficient ak0.
Causal IIR filters can be represented as Causal IIR filters can be represented as
IIR filters have at least one nonzero finite pole
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==
=
+==M
k
k
N
k
k
k
knxbknyaknxkhny010
)()()()()(
( ) ( )NN
M
M
N
N
M
M
zpzp
zbzbb
zaza
zbzbbzH
+
+++=
+++
+++=
1...1
...
...1
...)(
1
1
1
10
1
1
1
10
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FIR Filters
Duration length of h(n) is finite. Length is M+1 <
. That means h(n)=0 when n.
FIR filters are non-recursive. That means all
coefficients ak=0 when k>0 and a0=1.
Causal FIR filters can be represented as Causal FIR filters can be represented as
FIR filters are all zero filters. That means all poles
are either 0 finite pole
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==
==M
k
k
M
k
knxbknxkhny00
)()()()(
M
M zbzbbzH +++= ...)( 110
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Information in signals There are many ways that information can be contained
in a signal.
information represented in the time domain,
information represented in the frequency domain
Information represented in the time domain describes when something occurs and what the amplitude of the occurrence is.
Each sample contains information that is interpretable Each sample contains information that is interpretable without reference to any other sample. Even if you have only one sample from this signal, you still know something about what you are measuring.
This is the simplest way for information to be contained in a signal.
The step response describes how information represented in the time domain is being modified by the system
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Information in signals
In contrast, information represented in the frequency domain is more indirect.
Many things in our universe show periodic motion. By measuring the frequency, phase, and amplitude of this periodic motion, information can often be obtained about the system producing the motion.
A single sample, in itself, contains no information A single sample, in itself, contains no information about the periodic motion. The information is contained in the relationship between many points in the signal.
The frequency response shows how information represented in the frequency domain is being changed.
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Filter parameters Step response
Fast step response
No overshoot
Linear phase
Frequency response
Fast roll-off Fast roll-off
Flat passband
Good stopband attenuation
Why step response and not the impulse response is considered?
What about phase related parameters of the frequency response?
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Filter Classification
Four main types
Low pass
High pass
Band pass
Band reject
How to design a high pass filter if we know the impulse
response of the low pass FIR filter h(n)?
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There are two methods
Spectral inversion
Spectral reversal
Spectral Inversion
If low pass filter is H(z)
H(z)1
00 0.5 f
G(z)1
00
fc
fcIf low pass filter is H(z)
The high pass filter G(z) = 1 H(z)
in time domain g(n) = (n) h(n)
Parallel filters
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00 0.5 f
h(n)
(n)
+
x(n)y(n)
-
fc
-
Spectral reversal
If low pass filter is H(z)
The high pass filter G(z) = H(z+0.5)
in time domain g(n) = (1)n h(n)
H(z)1
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1
00 0.5 f
G(z)1
00 0.5 f
fc
0.5-fc
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Show how to use cascade and/or parallel LPF and/or HPF to
realise band-pass and band-reject filters.
LPF H(Z) with cut-off frequency f1
HPF G(z) with cut-off frequency f2
If f1 > f2 If f1 > f2
Band-pass filter is given by H(z) G(z) or h(n) * g(n) (cascaded filters)
Band reject by 1 H(z)G(z) (spectral inversion of band pass)
If f1 < f2
Band reject is give by parallel h(n) + g(n)
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Draw the cascaded/parallel filtering system for Band pass and band reject filtering
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The Moving average filter (MAF)
The moving average filter operates by averaging a number of
points from the input signal to produce each point in the
output signal.
Example: Consider a 5-point MAF, i.e., M=5
=
+=1
0
)(1
)(M
k
kixM
iy
Example: Consider a 5-point MAF, i.e., M=5
What is the impulse response?
What is the step response?
Write a pseudo-code to implement it
What is the frequency response?
The main aim of the MAF is to remove the noise while
keeping sharp transitions intactEEE6209 24
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Noise reduction performance
Consider the signal below
MATLAB commands:
X=[zeros(1,200) ones(1,100) zeros(1,200)];
Y= X+ 0.2*randn(1,500); % original signal
What is the output (recovered X) when the following three MAFs are used on Y.
M=5
M=11
M=25
Explain your observations using the step and frequency responses of the above filters.
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Frequency response of MAF
Mathematically described by the Fourier transform of the
rectangular pulse.
How to write the rectangular pulse ?
Frequency response?
)sin(
)sin(][
fM
fMfH
=
An alternative way
zero pad the impulse response to make it a length N filter, where
N is a power of 2
Take the FFT
0 to N/2 points represent the Frequency response from 0 to 0.5 of
the normalised frequency.
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The roll-off is very slow and the stopband attenuation is
ghastly. Clearly, the moving average filter cannot separate one ghastly. Clearly, the moving average filter cannot separate one
band of frequencies from another. (Good performance in the
time domain results in poor performance in the frequency
domain, and vice versa)
the moving average is an exceptionally good smoothing filter
(the action in the time domain), but an exceptionally bad low-
pass filter (the action in the frequency domain). EEE6209 29
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Multiple pass MAF
Consider M=5 MAF and involve passing the input signal
through a moving average filter two or more times.
Find out the
Filter kernal (the impulse response)
Step response
Frequency response Frequency response
For 1 pass, 2-pass and 3-pass MAF.
How is the noise removal performance for the (example in
slide 12)
For 2-pass MAF
For 3-pass MAF
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Multiple pass MAF
Consider the output after the first stage
y1 = h*x
Second stage
y2 = h*y1y2 = h*h*x
h = h*h
y2 = h*h*x
h2 = h*h
Third stage
y3 = h*y2y3 = h*h2*x
h3 = h*h2
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Multiple pass MAF
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Multiple pass MAF
Two passes are equivalent to using a triangular filter
kernel (a rectangular filter kernel convolved with itself).
After four or more passes, the equivalent filter kernel looks
like a Gaussian.
As shown in (b), multiple passes produce an "s" shaped
step response, as compared to the straight line of the step response, as compared to the straight line of the
single pass.
The frequency responses in (c) and (d) are given by the
equation in slide 14 multiplied by itself for each pass. That
is, each time domain convolution results in a multiplication
of the frequency spectra.
Why are the resulting filters from multiple passes better
than the MAF itself?EEE6209 34
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Recursive implementation of MAF
Consider a MAF with M=5
y[n] is computed as
y[n] = (x[n-2]+x[n-1]+x[n]+x[n+1]+x[n+2])/5
y[n+1] is computed as
y[n+1] = (x[n-1]+x[n]+x[n+1]+x[n+2]+x[n+3])/5
In other words,In other words,
y[n+1] = y[n]+(x[n+3]-x[n-2])/5
For any M, y[i] is computed as
y[i] = y[i-1]+(x[i+p]-x[i-q])/M
where p=(M-1)/2
q=p+1
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Filters can be summarized by
Their use
1) In time domain
E.g. ?
2) In frequency domain
E.g. ?
3) Custom
E.g. ?
Their implementation Their implementation
A) Convolution
B) Recursion
Examples for 1A, 1B, 2A, 2B, 3A, 3B ?
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Recursive implementation of MAF
The equation in the previous slide uses two sources of data to calculate each point in the output: points from the input and
previously calculated points from the output.
This is called a recursive equation, meaning that the result of one calculation is used in future calculations.
Be aware that the moving average recursive filter is Be aware that the moving average recursive filter is very different from typical recursive filters.
Is this an IIR or a FIR?
This algorithm is faster than other digital filters for several reasons. there are only two computations per point, regardless of the
length of the filter kernel.
addition and subtraction are the only math operations needed (if division by M is omitted at the individual step and performed at the end) --- Integer representation.
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Custom filters
Most filters have one of the four standard frequency
responses: low-pass, high-pass, band-pass or band-reject.
But most of the time, we have to design digital filters with
an arbitrary frequency response, tailored to the needs of
our particular application. our particular application.
Two important uses of custom filters:
Deconvolution - a way of restoring signals that have undergone an
unwanted convolution, and
Optimal filtering - the problem of separating signals with
overlapping frequency spectra. (lectures by WL)
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Deconvolution
The detected signal is the
desired signal convolved with an
unwanted kernel.
How do we recover the desired
signal back?
If the exact frequency response
of the desired signal not known
how can we get the desired signal
back? (blind deconvolution -
from WL lectures)
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