The Basic Theory of Filtering
UNIVERSAL ENGINEERING COLLEGE
Mail: firstname.lastname@example.org , Mob: 8907305642
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What is a filter?
Simply put, it retains history of the input signal.
Take this simple lowpass filter for an example.
This simple filter is a kind of integrator, in which the capacitor integrates the charge
provided through the resistor. As the rate of change of Vout depends on the current
through the resistor, low frequencies will be filtered less than high frequencies.
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In other words:
• The output of a filter is a function not only
of the input at the present time, but also of
• That’s what a linear filter does. No more,
no less. We are sticking to linear filters
today, thank you!
• There are many ways to build filters.
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The Time and Frequency
Response of the Analog Filter
Time (impulse) Response
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How does the analog filter exhibit
• In the case shown, the capacitor is the
– It retains previous history.
– It does so by summing the history into one
value, the voltage across the capacitor.
– In such a way, a single component can have
a long memory.
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Ok, what’s the “impulse response”
Impulse response is the response of the circuit to a
(mathematical) signal of infinite height (and power),
and infinitely short duration. This “unit impulse” has
very special characteristics:
1) It contains all frequencies
2) It has energy of “1” at all frequencies.
3) It describes the behavior of the filter at all frequencies.
POINT 3 is what you need to remember. The impulse
response of a filter is a complete description of what it
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Another way to look at the impulse
• The impulse response of a system shows
how a filter captures the HISTORY of the
signal. In other words:
– The value of the impulse response at a time ‘t’
demonstrates how much of the HISTORY of
the signal is added to the output at time ‘t’
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Remember last month?
• Multiple speakers created a “comb filter”?
– Yep, that’s a filter. Different distance means
“different times in the history of the signal.
– You can plot a time response for such a filter
just like you can plot the one for the simple
RC filter above.
– We saw some of that last month.
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Getting back to the analog filter:
• The analog filter’s output can not change
instantly as that would require infinite
current in the resistor, that means that the
output depends on the HISTORY of the
• In fact, if you have an input signal, and
you integrate the product of the time-
reversed impulse response times the
signal, you get the filter’s output.
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DID YOU SAY INTEGRATE?
• Well, it can be a sum, rather than an
integral, and in fact in the digital domain, it
is a sum, rather than an integral.
• In the digital domain, rather than having all
frequencies, you have all frequencies
inside the digital passband, which will be
½ the sample rate wide.
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I’ll say more about that later.
• For now, let’s get back to that analog filter
again for a minute.
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So, what did that filter actually do?
• It stored part of the history of the signal.
As the time went on, it “forgot”,
exponentially, the contributions to its
output from previous history.
– Not all impulse responses are so simple.
– An impulse response may be positive or
– An impulse response may “ring” or not.
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How about a “Digital” filter?
The digital filter uses history explicitly.
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That digital delay
• In digital terms, a delay of one sample is
defined as a multiplication by ‘z’.
– Some texts use z-1
– Either works. The rules are the same, except
for how you interpret some things that I’ll
leave out for now.
– Remember a delay of ‘z’ is one sample. If I
say I multiplied a signal by z3
, it means I
delayed it by 3 samples. Yes, it really is that
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What is it’s time and frequency
The result is the same (below half the sampling rate)
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What about not below half the
• Inside the ‘z’ domain, there is no such thing.
– The ‘z’ domain is also called the “digital domain” by many people. This
does involve a potential confusion that I will ignore for the time being.
• Remember when we talked about the sampling theorem a long time
ago? We used an anti-aliasing filter.
• All other “frequencies” are represented by aliases below fs/2, and
you don’t let them in in the first place!
• All information in the sampled domain is contained inside that
bandwidth. All aliases/images contain exactly the SAME information
as the baseband spectrum. No more, no less.
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Why is the response the same?
• This filter also stores the history of the signal in an
exponentially decaying fashion. In the digital filter, you
can see the storage more directly, as the ‘z’ element. In
the analog filter, reactive components provide the history
in a continuous fashion.
• I picked the parameters to provide the same visible
response as long as we stay well below half the
• Recursive (IIR) filter outputs depend on both previous
outputs and input(s).
• Analog filters are mostly (but not completely) filters that
depend on both output and input.
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Back to Impulse Response
• The response of either filter to an impulse is called its
“impulse response”. A digital impulse is simpler (as the
bandwidth is finite), and consists of one ‘1’ sample but
both have the same use.
• This is the “time response” plotted in the proceeding
• The impulse response of a filter defines its memory
(history) of a signal.
• Remember. The impulse response of the filter contains
exactly all the information about a filter.
• This is, you will find out, very handy.
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Ok, what’s the big deal, JJ?
• Since the impulse response of the filter
defines its interaction with the signal, this
means that we can either use the
recursive form shown before to implement
the filter, or we can simply multiply the
time reversed impulse response by the
signal and sum (integrate) the result, to
get the filter output.
• The two operations are exactly the same.
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Let us use a 9th
order Elliptical filter
as an example:
Red: By time reverse, multiply, and sum Green: By direct filtering
(the two exactly overlap)
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• The process of multiplying the time-reversed
signal by the impulse response, and summing
(integrating) is called “convolution”.
Think of it as a way to expressly include the history of the signal in the filter output.
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I thought we multiplied transfer
S(ω) H(ω) Y(ω) = S(ω)*H(ω)
The usual way we see transfer functions expressed:
(notice this is in the ‘ω’, or frequency, domain)
What’s actually happening in the time domain:
(Note: ⊗ is used here to denote convolution.
There are other notations.)
s(t) s(t)⊗h(t) y(t)
These are two ways of saying the exact same thing! 22AJAL.A.J , 8907305642
Multiplication in the time domain is the same
as convolution in the frequency domain.
Multiplication in the FREQUENCY domain is
the same as convolution in the TIME
It works either way. If you read a DSP text,
you will see the word “duality”. This is
duality in action.
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• For typical filters, convolution is what
happens in the time domain.
• Convolution is merely another way of
expressing what happens when you filter a
• It’s the same as multiplying the signal by
the transfer function.
This relationship holds for the ‘s’ domain, the ‘z’ domain, and quite some other domains as well. 24AJAL.A.J , 8907305642
• Convolution is important because:
– Convolving in the time domain (like we just saw here)
is the same as multiplying the Fourier Transform of
the signal by the Fourier Transform of the Impulse
Response and then taking the Inverse Fourier
– This works the other way around, too, but isn’t usually
as interesting to discuss in most filtering applications
except perhaps as window functions. (There are
exceptions, for instance, “TNS”.)
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Multiplication of Transforms of
Signal and Filter
Product of Spectra
Of Product 26AJAL.A.J , 8907305642
An example of Convolution
In this plot, it’s easier to see the linear superposition in the time domain
because the two parts of the signal s(t) do not overlap. 27AJAL.A.J , 8907305642
What’s my point?
• Filtering is Convolution. Convolution is filtering.
They are the same thing expressed in different
domains. There are several ways to do filtering:
– IIR (Infinite Impulse Response) filters, like the two
shown much earlier. They are called IIR because the
filter’s impulse response continues to infinity (yes, at
infinitely small value for a stable filter). These filters,
effectively, use a topology that implements the history
inside a few (very important, sensitive) state
– FIR (Finite Impulse Response) filters, in other words,
just do the convolution using a ( potentially arbitrary)
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So, they are the same?
• Well, no. In fact, FIR filters have zeros,
and IIR filters have poles. (In reality,
nearly all IIR filters have both poles and
zeros, which is to say that they have both
an FIR and an IIR part.
• FIR and IIR filters can have quite different
properties, and usually do, they are two
different means to an end. Neither one nor
the other is always better.
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More about IIR filters
• IIR filters must be implemented using feedback
to implement the poles, in order to be truly IIR.
– IIR filters are “longer” (in terms of impulse response)
compared to the memory they directly use. (i.e. a 2nd
order filter can have 1000’s of samples of significant
energy in its impulse response.)
• The impulse response length is what can determine the
sharpness of the filter’s frequency and phase response.
– This extension places substantial requirements on the
implementation in terms of accuracy, both of
coefficients (analog or digital), and of related
processes (digital storage, multiplication, addition).
– The data is stored in a few variables, so the accuracy
required for those variables rises accordingly.
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• FIR filters are not generally as sensitive to
• FIR filters often require more computation,
because you must do a multiply-add for
each term in the impulse response
• FIR filters can be constant delay, IIR filters
can not. Sometimes this matters.
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What are the meaningful properties
of a filter?
• The amplitude response (plotted in terms
of amplitude vs. frequency)
• The phase response (plotted in terms of
phase vs. frequency)
– What does phase response mean?
• Linear phase (i.e. constant time delay)
• Minimum phase
• Non-minimum-phase Linear phase is an important
subset of this class that has all zeros.
Attention: We are talking about single filters here, not filter banks. That
is another subject, and one that places more constraints on individual filters!
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A bit more on phase response
• “Linear Phase” (constant delay)
– If a filter has a constant delay, the phase shift of the filter will be
t*w, where t is the time delay, and w the natural frequency
(2 pi f).
• This means that a delay can exhibit enormous phase shift.
• This phase shift, however, is ONLY delay.
• Non-linear delay
– This is the part of the phase shift (in and around the filter’s
passband) that is not modeled by a straight line)
• The part that does not correspond to a straight line constitutes non-
constant-time phase shift.
• Phase shift of “1 million degrees” in and of itself tells you
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Some example plots:
Notice similar frequency
(poles and zeros)
“Linear” phaseNote phase
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Properties of Impulse Responses
• DC Gain
• Fs/2 Gain
• Frequency response
• Phase Response
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• The DC gain of an impulse response is
exactly the sum of all of its non-zero
• For many applications, one wishes to set
this to one.
– This is easy. Divide the entire impulse
response by the sum of all values of the
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Gain at FS/2
• This is also easy.
• Sum all of the EVEN taps
• Sum all of the ODD taps
• The difference of the two is the gain of the
filter at FS/2
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Some useful things to know
(I won’t prove them here)
• A symmetric impulse response implies:
– The passband phase response (one or multiple passbands) will
look like a pure delay (linear phase)
• “Linear phase” phi = w*t, where omega is the natural frequency
and ‘t’ is the time delay
• An antisymmetric impulse response has some
interesting (and special) properties. They are beyond this
introductory tutorial, but are worth looking into for some
applications. Such filters will have “linear phase” in the
passband, but the intercept of such a filter at DC must be
at +_ 90 degrees, and the filter must have a zero at DC.
• An asymmetric impulse response implies:
– The passband phase response is not a pure delay.
– Practically speaking this means that the response is the sum of
a symmetric and an antisymmetric response.
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Implications of the previous page
No IIR filter can be linear
phase. If it were, it would have
to extend to infinity on both
sides, and have infinite delay.
Some IIR filters can “come
close” under some
circumstances. In such cases,
they have substantial “pre-
ringing” (as they must). A true
IIR filter with linear phase must
be “non-causal”, i.e. it must be
able to “look ahead” in time
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FIR filters are usually designed as “type 1 linear phase” meaning that they are
symmetric, with even filters having two identical center taps, and odd filters
symmetric about a single center tap. Symmetric filters with an even number of taps
must have a zero at pi and can not be highpass.
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There are other kinds of FIR filters, in particular antisymmetric even tap filters, which
have linear phase in the passband, but do not have zero phase shift at DC. Rather, they
have + or – 90 degree phase shift at DC, and must have a zero at DC. These “type 4
filters” can not be used as lowpass filters.
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A completely asymmetric FIR filter is a valid filter, and in some cases (phase
compensation, etc) may be used. Such filters are usually for special-purpose
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A comparison of 3 FIR filters
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Even vs. odd length
(we will compare 32 tap vs. 33 tap Lowpass FIR’s with identical parameters except
Zero at pi Nonzero at pi
Delay of 15.5 samples. Delay of 16 samples
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• If you need an integer delay in the filter, use an
odd-length filter. (N.B. In many applications,
where even filters are applied twice, you can use
two even filters.)
• If you need a zero at pi, use an even-length
• If you don’t want a zero at pi, you can’t use a
symmetric even-length filter. You can use an
antisymmetric even length filter if you want a
highpass filter, but then you’ll have a zero at DC.
– This means that symmetric high pass filters are of
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More useful things to know
• The longer the impulse response is at a
given level, the sharper the filter cutoff will
be to that level
– This expresses the old, familiar knowledge
that df · dt >= 1 (for a two-sided Gaussian)
– Yes, this means that if you want 1 Hz
resolution, you need a 1 second impulse
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Frequency Response vs. Length
32 tap 64 tap
The top filters have a wide transition band (.25) The bottom a .05 transition band.
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The filters vs. their responses
32 tap first
Note: The passband ripple performs in a similar fashion.
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Filter vs. response
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There are other tradeoffs possible
• IIR filters can have:
– Passband ripple only
– Stop band ripple only
– Neither passband nor stop band ripple (monotonic
– Both passband and stop band ripple
• FIR filters as usually designed can have:
– Ratio of passband ripple to stop band ripple controlled
via design parameters. The filter response is not
defined in a “transition” band.
– There are other FIR types possible, they are not that
common in most present-day uses.
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FIR Example – Passband vs.
Both filters have 32 taps and the same edge frequency and transition bandwidth
Passband weight 10, stop band .1 Passband weight .1, stopband 10
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What’s this about “windows”?
• A window is just another filter, usually a
– It is a filter that is most often used to mitigate
“edge effects” or other artifacts of truncation
– It is normally MULTIPLIED in the time
domain, therefore it CONVOLVES in the
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Several examples of windows:
These are examples of a windowed sync (brick wall) filter
All filters are length 8191
Black is rectangular window, red is Hann, green is Blackman,
blue is Hamming, cyan is Kaiser(5), magenta is Bartlett, yellow is Nutall 53AJAL.A.J , 8907305642
How are filters described?
• FIR filters usually are simply listed by
either the tap weights (individual values)
or by a function that describes the tap
weights. This is the same as providing
• IIR filters are described as sets of poles
and zeros. More on that now:
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Poles? Zeros? WHAT!?
• Poles and zeros are a way of expressing a
transfer function as two polynomials, one in the
numerator, and one in the denominator.
• For either numerator or denominator, a
polynomial can be described as
a1+ a2 * z1
+ a3* z2
… where a1, a2, a3 are the
• One can also calculate the roots of the
– The roots of the numerator are the ZEROS.
– The roots of the denominator are the POLES.
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Why poles? Why Zeros?
• A zero shows a value for the polynomial variable
that results in a ZERO output.
• A pole shows a value for the polynomial that has
an INFINTE output. (the response looks like a
• The meaning of poles and zeros in terms of
frequency changes depending on the kind of
transfer function (i.e. ‘s’ or Laplace domain, ‘z’
domain, ‘ω’ or Fourier domain, or others) but for
the commonly used domains will still be some
expression of frequency.
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A pole/zero plot for a 5th
Butterworth, using bilinear Z form
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Expressing Poles and Zeros
• In the FIR filter, the zeros are expressed by directly
providing an impulse response, corresponding to the
polynomial that results in the zeros.
• In an IIR filter, both the poles and zeros are often
factored. This leads to a variety of topologies, shown on
the next page.
• Factoring depends on the fact that any real coefficient
polynomial can be factored into real roots or complex
pairs of roots. A complex pair of roots will always have
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Direct form 2nd
These are not the only two possibilities.
One Second Order Section
(multiple sections are cascaded)
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The Direct Form
• The direct form creates a number of difficulties.
– It increases the size (in terms of bit depth) of
– It increases the depth required for accumulators
(mantissa for floating point)
• It’s not, generally speaking, very common or
useful for more than 3rd
order. Don’t do this.
– This can result in instabilities due to numerical
– You can get “limit cycles” and other disturbing
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Factoring into second-order
• There are a number of ways to make second order
– All depend on the fact that you can factor a real-valued
polynomial into second-order sections with real coefficient
– How does this relate to filtering?
• If you convolve a set of factored polynomials, you get the original
– That means that if you cascade sections with the polynomials
implemented, you MULTIPLY the transfer functions. This is the same
old duality in another form. What you’re doing is convolving things a
part at a time, and then doing more and more in cascade.
• A second-order section is easy to check for stability.
• By factoring both numerator and denominator and grouping things
correctly, you can ensure the best gain structure for a given filter.
• Your computer does this for you!
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So we factor FIR’s as well?
• Generally not. There are several reasons:
– The coefficient bit-depth growth is not nearly as
– Coefficients are not generally as large (in FIR filters
coefficients are most often considerably smaller than
– FIR’s can’t go unstable, have limit cycles, or some
other kinds of disturbing behavior.
– Of course, they require more calculation, and they
may require a wider accumulator than you expect.
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Some examples of Filter
• For an IIR 3rd
order bilinear Z Butterworth filter with a cutoff at .125 fs/2, the
0.0053 0.0159 0.0159 0.0053
• The denominator is:
1.0000 -2.2192 1.7151 -0.4535
• For a similar FIR filter, the tap values are:
0.0003 0.0034 0.0067 -0.0031 -0.0312 -0.0397 0.0349 0.1945
0.3343 0.3343 0.1945 0.0349 -0.0397 -0.0312 -0.0031 0.0067
• Notice the difference in the size of the tap weights. In this example, the tap
weights are quite moderate for an IIR denominator. Longer filters will have
often have a substantially larger range of values.
In general, this kind of tradeoff is well beyond the scope of a beginning tutorial,
but everyone must be aware of this kind of issue. As you will discover in the
next part of this tutorial, most filter design packages take care of this
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How to write a transfer function, in
the ‘z’ domain:
• The transfer function for
the third order
Butterworth is written
• In factored form, it
would look like this:
Doing factoring is one of the things Matlab, Octave, and other linear
algebra and/or filter design packages are for.
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More about poles and zeros
• We’ll show some pole/zero plots, along
with the impulse responses, frequency
responses and phase responses.
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Symmetric FIR (odd length)
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• Steve will show you Octave, a freeware
program that allows you to design both IIR
and FIR filters.
• We’ll discuss a bit, here, about designing
both kinds of filters.
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Designing IIR Filters
• First, decide what kind of filter you want:
– Butterworth (no ripple, monotonic amplitude
response, requires more poles/zeros)
– Chebychev 1 (passband ripple, monotonic
– Chebychev 2 (stopband ripple, monotonic
– Elliptical (equiripple passband, equiripple
stopband. Shortest filter for a given rejection
ratio. Has issues.)
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How to do that?
• Use the “help” function.
– help butter (for Butterworth)
– help cheby1 (for Chebychev 1)
– help cheby2 (for Chebychev 2)
– help ellip (for Cauer elliptical)
• Follow the directions. Time does not permit a full
examination of all of the calling parameters.
• All have the form
– [bb, aa]=butter(3,.125) for example.
– BB is the zero polynomial
– AA is the pole polynomial
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What does the Frequency
Response look like?
• Use “freqz(bb,aa)”
• It will give you frequency and phase
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Designing FIR filters
• Use “remez”
• This takes a bit of doing.
• Before you use “remez” you need to
– Length of the filter (a single integer)
– The points at which frequency response
– The amplitudes at each of those points.
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So you have
• len=15 (NOTE: That means a 16 tap filter, the
order is 1 less than the filter length)
• freq=[ 0 .2 .6 1] (that is a list of 4 frequencies,
corresponding to DC, .2 .6 and 1 times half the
sampling rate, whatever that is). 0 and 1 must
• amp=[1 1 0 0] (that means that the amplitude at
0 and .2 you want to be close to 1. at .6 and 1,
you want it to be zero.
• bb=remez(len,freq,amp) will give you a filter that
is optimized to be as close as possible to that
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But I care about passband ripple!
(or stopband ripple)
• w=[10 1] (this is half as long as freq and amp
vectors, both of which must be even length)
Here, the 10 means that the ERROR in the filter
design between the first two frequency points is
counted 10 times as much as the weight (1)
between the second two points.
bb=remez(len,freq,amp,w) will give you a filter
with the error weighting you specify.
• NOTE: you can not weight the error in a
transition band. By definition, there is no error in
a transition band.
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Now, frequency response
• That’s all it takes. You will see frequency and
• remez always designs symmetric filters, unless
you tell it to do something else
• “help remez” will get you as many options as you
wanted to ever know about.
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To Take Home:
• Filtering is the practice of convolving an
impulse response (the time response of
the filter) with the signal.
• FIR filters directly implement this
• IIR filters use a functional representation
that does the convolution implicitly, with
some cost in implementation issues.
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More to take home:
• Frequency (complex spectrum) response and
impulse response are duals.
• The relationship df * dt > matters in filter design
just like it does in anything else.
– If you want a sharp filter, you have a long impulse
– If you want a short impulse response, you can not
have a sharp filter cutoff.
• IIR filters do not shorten the impulse response,
they simply operate in a different fashion.
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To come after the break:
• Steve Hastings will give you a set of tips
for tools that you can get off the net to
help you design, plot, and understand
filters (and a whole lot more things)
• boB Gudgel will show you what it sounds
like to implement various filters, and offer
tips on how to do this kind of work in the
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• Ok, now what is a filter?
• And a filterBANK?
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• A filterbank is nothing but a way to
implement a set of filters, generally
strongly mathematically related, in one
• A filterbank can always be decomposed
into a set of individual filters
– This is usually a lot more work that it’s worth,
but not always.
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The famous audio filterbank:
• This would be the “MDCT”, or “Modified
Discrete Cosine Transform”.
– Annoyingly, it’s not a transform, it’s a
– It is an exact reconstruction filterbank, though,
so it does obey most of the rules of
transforms, except that it has overlap between
blocks, and remains critically sampled. A
transform either has no overlap, OR is not
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Critically Sampled – Whaaa???
• Critically sampled is a simple concept at
its heart, it means that in the filtered
domain, you have the same number of
samples that you do in the unfiltered
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The theory of filterbanks is long,
deep, and wide
• And I won’t even try to relate it in an hour.
• BUT, what you need to remember is that an
output of a filterbank is just like running some
particular kind of a filter on the signal. The
filterbank just does a lot of these at once.
• It may also:
– Downsample (i.e. critical sampling)
– Be oversampled (more values in the filtered domain
than in the input and output domains)
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What are some applications?
• For an MDCT, the obvious one is coding:
– It is critically sampled. Ergo, no extra data to
– It does a good job of frequency analysis, so
you can relate the perceptual model well, and
you can also get good signal processing gain
– It has an efficient form for calculation, very
similar to an FFT of half its length.
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Some rules about critically sampled
• If you’re not careful, very odd things happen
when you modify the filtered results.
– The best known of these is the “pre-echo” in audio
– There are also other things that can go wrong
• Why? That critical sampling means that the
filterbank creates a lot of aliasing, and then
cancels it on reconstruction. Mess with the
signal in the filtered domain, and the aliasing
does not cancel.
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How about oversampled filterbanks
• You can avoid aliasing problems, so
– You can modify the signal
– You can use it for things like equalizers
– Gain compressors work well with this kind of
• There are other applications that are far
too complicated to bring up at present.
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So - Filterbanks
• At their heart, nothing but a handy way of
implementing a whole set of filters at the
– There are more things to this than
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