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turney last three lectures we consider it the design
%uh infinite impulse response digital filters the two
main classes design procedures that we discussed
involved mapping continuous-time designs to discrete-time designs
that was the first class and the second class was the use about the rhythmic
design procedures
in this lecture I would like to
discuss the design a finite impulse response
digital filters and I remind you
dead as we've stressed that a number of the lectures
one of the very important aspects a finite impulse response digital filters
is the fact dad they can be implemented
to have exactly linear phase so
finite impulse response filters as opposed to infinite impulse response
filters
are useful for problems
where linear phase is an important constraint it turns out also in this is
isn't the point that will be touching on in these lectures
but it turns out also dead finite impulse response digital filters
have some important advantages over infinite impulse response filters
with regard to round of noise that is with regard to the facts a fine I
register line
now in general
the implementation of finite impulse response
filters is considerably simpler for discrete-time systems
than it is for continuous-time systems consequently it turns out
that there are not really available or a wide variety of continuous-time design
procedures for finite impulse response filters
as there were four infinite impulse response
and consequently the continuous-time to discrete-time
mapping as a design procedure is not particularly applicable
to the design are finite impulse response digital filters
so this includes for example the
impulse and Varian technique and the use of the by linear transformation
that is those techniques are not particularly useful
when we are talking about the design of finite impulse response filters
as an additional interesting site point
if we work to begin with
a finite impulse response analog a continuous-time filter
and map it to a digital filter using the by linear transformation
and in fact the resulting filter what not be
a finite impulse response filter the algorithmic procedures that we talked
about
in the previous lectures of course still do apply
to the design a finite impulse response filters
and there also are some other algorithmic procedures
that are discussed in the text and that I will touch i am just very briefly
in today's lecture now
for the design of finite impulse response
filters their are
basically three classes or
three types our love design techniques that will be discussing
and in this lecture
what I intend to do is present the three of these
primarily with an eye toward giving you an appreciation for
what the style love the design techniques
is that is to give you an overview these the
design techniques but I won't dwell on
many of the details so
in the design of finite impulse response
digital filters then we of course are talking
about unit sample response that is a finite lying
and will choose the Lang the interval over which
the unit sample responses 90 to be in the range from 0
to capital and -1 so busy transform
love the finite impulse response filter is then given by the song from zero
to capital and -1 Ave Japan seat to the minus an
which of course is a polynomial insitu the -1
and we know that for this class of systems
agency has only zeros except
at the 0 that is the polls have a gypsy
can only fall at sepals 0 RZ equals infinity
and I remind you also that for linear phase
finite impulse response filters the constraint on the unit sample response
is dead it basically is symmetrical
about its midpoint in other words as we look at the unit sample response
starting from the left hand and and starting from the right hand and
approaching toward the middle then we see the same
madams in other words this asymmetry that's required for linear phase
the three basic design procedures
that I'd like to talk about in this lecture basically introduced in the
lecture
our first the ball a design technique which is referred to as the window
method
the second design technique that is commonly referred to as
the frequency sampling method and
a third design technique which is an algorithmic
design procedure which results in
equal ripple filters which as
all comment on again when we get to this point in the lecture
turn out to be important because they have some optimality properties
so these are the three basic design techniques then
that I'd like to discuss in this lecture
well to begin with the window method
is basically
a method based on the following strategy
let's suppose bad we begin the design
all the finite impulse response filter
with a desired unit sample response H that the event
for example suppose that we wanted to design a
low-pass filter we could begin
with a desired unit sample response which was the unit sample response
overnight the low-pass filter
now as you know the unit sample responsive an idea low-pass filter
is bait basically has a sin x
over x-type envelope in particular the important characteristic is that it
extends from minus infinity
to plus infinity so in fact
it doesn't correspond to a finite impulse response filter
but we could imagine obtaining the unit sample response
all and F IR filter
by perhaps truncating
this infinitely long unit sample response
so that we retain only the values in the interval from 0 to capital N -1
or more generally we can imagine
multiplying the desired unit sample response
by a window the window being 0
outside the interval from 0 to capital and -1
so the window method basically consists beginning
with the desired unit sample response which in general will be infinitely long
or at least longer than the filter that we wanted design
we apply to that a window which is zero
outside the range 0 to capital N -1
and the result then is the unit sample response
of the finite impulse response filter
well what is the effective this in the frequency domain
we know that multiplication in the time domain
corresponds to convolution in the frequency domain
and consequently the frequency response
above the finite impulse response filter that results
is the periodic convolution
all the desired frequency response
that is the Fourier transform of the desired unit sample response
back convolved with the 48 transform
of the window W van
for example if we were talking about a rectangular window and in a rectangular
window
W van is simply constant over
the rain 02 capital N -1
be magnitude of the Fourier transform
above the window has a form roughly as i've indicated here
this is drawn for the case an equals eight and if the magnitude
a HIV to the jail maker I'm sorry a WV to the jail maker
if we were to remove the magnitude signs then
every alternate low would be flipped over in sin so in fact this is a
sign an axe over sin x her
so for the particular case a rectangular window
we would have not the I P a low-pass filter
that we began with but the idea low-pass filter frequency characteristic
convolved with the Fourier transform
above the window so that's depicted here
where we have the
idea low-pass filter
superimposed on top with it is
the 48 transform love in this case a rectangular window
and then to obtain
the resulting overall frequency response of the finite impulse response filter
that would be obtained by involving these two
together and we can see that the result if that
very roughly is to
in fact at the discontinuity be almost exact effect is
to integrate the window frequency response the
frequency response to the window and
the resulting frequency response for the finite impulse response filter
then as oppose to being the ideal characteristic that we desired
in fact has some ripples in it
generally then it the it has
some deviation in the passband it has some deviation in the stop band
and it has a transition region from passband to stop it
and that's introduced because up the convolution
up this window frequency response with
the desired or ideal frequency response
now ideally what would we like
as the frequency response above the window
well ideally what we would like of course is
we would like this function to be
an impulse because if it's an impulse then we can bother with the desired
frequency response and the result
will again be the desired frequency response
the overshoot that we have in the past and and a stop and
has arisen because all the side load structure
all the frequency response the window
and the transition region has arisen because
primarily because of the with love this central 0
so in fact if we want to keep the
deviation in the past band and stop and small
then that implies that we would like to keep the side lobe structure
all the window small the window frequency response small
and if we would like to keep the transition band narrow
that implies that we would like to keep this main low
narrow and
as I think that your intuition should agree with
by now generally an attempt to make
this function narrower in frequency
will have the general effect all making the window
longer in time so that in fact if we want
if we as we try to make the time window
W van or equivalent leave the order of the fine and impulse response filter
smaller the tendency
will be toward a widening transition band
in the frequency response to the filter
likewise as we allow the order the filter to get longer
we have more flexibility in the filter and consequently
were able to control better the transition band
and the overshoot in both the pass band and stopping
well and in particular again continuing with the example
love a rectangular window
if we look at the frequency respect to be unit sample response
for the case our love
a an ideal
low-pass filter multiplied by
a 51 point rectangular window
then the unit sample response we obtain
is indicated here actually have marked this as
the desired unit sample response it's only the disease equal to the desired
unit sample response
in the interval from zero to 50
and the desired unit sample responsive course continues to be 90 outside that
range
whereas multiplying by rectangular window truncated
to be 0 outside this range
the resulting frequency response for this particular example
i've indicated here and as oppose
to wanna the the previous
frequency response that I depicted this is shown on a logarithmic frequency
scale
there in fact is some ripple in the passband although it's not evident
on a logarithmic frequency scale but
generally you can see the effect are a transition band
and then other a side load structure
in the stop band and you can
see also that the asymptotic behavior this
is to in the vicinity a for this particular example
is in the vicinity of -20 DB
attenuation in stopping
so this then is an example
%uh the low-pass filter that could be obtained by
applying a rectangular window to the ideal
low-pass filter there are other
possible choices for Windows
in particular to other choices that I'd like to mention
are what are referred to as the Hamming window
and the Bartlett window
as opposed to the rectangular window the Bartlett window
is a triangular window which has the effect basically
up making the transition at the ends at the unit sample response
less abrupt than they are in the case of a rectangular window
the Hamming window is a
window which corresponds essentially
to a cosine Bell
sitting on a small pedestal
soul for this particular example again the
low-pass filter the idea low-pass filter windowed
we can compare the frequency response obtained
with the Bartlett window to that obtained with the rectangular window and
also
with the Hamming window so let's take a look at that
here we have the frequency response
love the Bartlett window and you can see
generally that the ripples aren't as severe
as in the stop band as they were in the case of the rectangular window
and in fact we can compare these simply
by referring back to the
re rectangular window frequency response that we had previously
so the rectangular window lies
slightly under were actually asymptotically touches
the frequency response obtained from the Bartlett window
but for the Bartlett window the rebels
in the stop and aren't as severe as
they are for the rectangular window
now to compare both these to the Hamming window
we can put on top with this
the frequency response for the
Hamming window
and in this case you can see
dat the frequency response obtained with the Hamming window
applied to a low pass filter has considerably better
stop and attenuation than either the rectangular
or the Bartlett window in particular it comes down to about
-30 or minus 32 or so DB
there is a slight price pay for this and the prices as you notice
dead the transition band is slightly wider
then it is for either up the other two windows
and in general this is a
type a trade-off bat arises again and again
in the design of actually always in the design
a vigil for analog filters in particular in using Windows
there is always this trade-off between
improved stop and attenuation and
a narrower or wider rather transition band
as the as you are willing to allow the transition band to become
wider you can generally obtain some advantage in terms of stop and
attenuation
alright so this is a
quick view into the use of Windows
in the text in fact there are quite a large number of windows that are
discussed besides the rectangular
the Bartlett and the Hamming window and
similar frequency responses
for filters designed you saying
a variety of these windows is presented in the text
the next design procedure that I would like to discuss
is the design procedure that is referred to
as the frequency sampling design procedure
and basically the idea behind that procedure
is to argue that if
we have a desired frequency response
which i've indicated by the continuous line
the desired frequency response then
we could think I've designing a digital filter
by specifying
samples all the desired frequency response
at equally spaced points along the frequency axis
now we know that for a finite impulse response digital filter
in general for any finite like sequence we can represent it
in terms of samples ovitz for a transform
that in fact is the Discrete Fourier Transform so basically
what we're saying is that we can begin with
a desired frequency response
and specify the Discrete Fourier Transform
of the finite impulse response filter by specifying
equally spaced samples all
the desired frequency response so
we have been the that the actual frequency response
will correspond to capital in samples
equally spaced in frequency all the desired frequency response
characteristic
now one could saying
that or one could acquire as to whether in fact
if we do that what we're able to implement
is the desired digital filter frequency response
in other words we know that we can implement a finite impulse response
filter
by using only
be value specified in the Discrete Fourier Transform
and here we specified a discrete Fourier transform
its unity in the passband and 80 in this top band
and one can ask well in fact is that
and ideal digital low-pass filter
well the answer to that is no and the reason is
bad it is important very important to keep track of the fact
that just because is we specify
the frequency response at equally spaced points
along the frequency axis doesn't mean
that in between those points the frequency characteristic has to be
particularly well behaved
that is the Fourier transform up the unit sample response
is a continuous function a frequency with specifying samples
aback continuous function of frequency and
their is implied interpolation
in between those samples
well in fact if we were to look at
a unit sample response whose
discrete Fourier transform magnitude
is unity in the passband and zero in the stop band
one possible choice in fact is
the unit sample response and frequency response that i've indicated here
this is the unit sample response
this is the corresponding frequency response
the Discrete Fourier Transform Ave Japan
in fact is exactly unity in the past band and zero in the stop band
but the interpolation in between those points is not unity in the passband
n0 in the stop and in fact by
simply putting the previous you graph
on top with this one
we can see that this
frequency response characteristic indeed
goes through the specified
frequency points but
it is not particularly
well constrained in between on sports
so in fact for the poor one example
if we chose this is the unit sample response corresponding to the center
frequency samples
the resulting frequency response all the fine and impulse response low-pass
filter
would be as we've indicated
with this continuous curve
well we know that there are other units sample response
responses whose discrete Fourier transforms are exactly the same
the magnitude of the discrete Fourier transform is exactly the same as this
one:
in particular we know that if we were to circularly rotate
a true then circularly shift henn
the magnitude of the Discrete Fourier Transform
would remain exactly the same so in fact
another unit sample response that would correspond to
a frequency response which likewise goes through the same
frequency samples is
the one that we have
here this unit sample response is simply the previous one
circularly shifted by approximately an over to
her and -1 over to and in this case
the frequency response characteristic is even worse
than it was in the previous example so
here we have a frequency response that has very wild
large fluctuations in the passband and even in the stop band
but still goes through the same frequency samples
that we had specify previously and
we can see that by again putting these two
on top of each other to
and you can see in this case again
dead this frequency characteristic goes through
the specified frequency samples
but it has a very wild behavior in between
fight so the frequency sampling method then
is basically a method dead consists love
specifying samples of the frequency response
and then accepting
the interpolation in between those samples
that is dictated by the form of the unit sample response
and I stress dead one has to be careful
in buffets that's applied to this or equivalent leave the circular shift
that's applied to the unit sample response says as we've seen in the last
several view graphs that very seriously affects
the interpolation in between these frequency points
furthermore an additional point has been made
and that is that it is not
even though we can implement a finite impulse response filter
using the discrete Fourier transform
specifying the Discrete Fourier Transform to be unity in the passband
and zero in the stop band does not mean
that we've implemented an ideal okay asphalt
alright well we had this of course is not a unit sample response that we would
be
particularly anxious to use I'll clearly the behavior in the past been a stop and
is very severe and in fact too severe
so we could at least consider a somewhat reasonable
the unit sample response and frequency response
that we had previously in this case
relatively well-behaved that is a triple someone in the passband in in the stop
and
but now we could inquire as to whether
there is some way or improving the passband
ripple and the stop and rebel
we know or we anticipate that there's a price that we would have to pay for that
probably the price and indeed it turns out that the price we have to pay
is the price of widening the transition
well how could we widen the transition band we could imagine beginning
with this particular design taking as
one possibility one of the stop band
constrain points and allow it
to move into the transition band so that
in fact we have a wider frequency range
they were permitting the transition from passband
to stop and to go to
and let me illustrate that
here where now
what we've done is to take
one all the frequency samples
in this top band move it up
into the transition ban and as we do that
you can see for this particular example the
the passband ripple hasn't changed too much
by in fact the stop and ripple has changed somewhat dramatically
as we allow this transition point
to move farther up into the transition band basically widening
the general passband region what we'll see is that
the passband ripple become somewhat smaller
and they're in fact is a trade-off dead is introduced between
the passband ripple and the stop and rebel and let me illustrate that
with this next
example where in this case
we have moved the transition point up
somewhat further
so that whereas in a previous case
the transition point was someplace down here we've now moved farther up
into the transition region closer to the passband
and the passband ripple has become correspondingly smaller so
this in fact is a design technique
that can be used for the design all
generally ban
frequency-selective filters where
we first the ball specify frequency samples
in fact this can be used really for the design love any
digital filter characteristic with transition regions
and then perhaps introd aP's
reducing the rebel in the passband and/or the stop and are in the various
bands
by allowing a an additional point
to be used to widen the transition band
and as we move this point up or down then
we will reduce the passband in the stop and ripple we can also of course
consider
movie not just one point into the transition region but
2.23 points or more points obviously as we do that
we will widen the transition ban further
and at the same time that offers
some additional flexibility which permits us to
reduce the passband and stop and ripple so
this then is basically the strategy
in the frequency sampling
method I love digital filter design the final method
dead I would like to touch on just very briefly
is a method which I
referred to at the beginning of this lecture as an equal ripple
design method and basically the notion here
is that we've seen
that their is a trade-off between
the transition region on the one hand and pass band and stop and ripple
on the other hand well one can inquire as to whether
in at least in some sense we could design for ourselves
optimum class filters and indeed
you can and in fact is possible to show
that the optimal strategy in terms
are minimizing pass pander stop and ripple
or in terms of minimizing a transition band is
to design the filter to have equal ripple
frequency response characteristic well
one way love setting up the optimal
condition it turns out that there are a variety of ways
and these are discussed in quite a bit more detail in the tax
but for example if
we pick a passband cutoff frequency
and its top and cut-off frequency furthermore
if we we've course are going to allow some deviation in the passband and some
deviation in the stop and
if in addition we fix the ratio
all stop band the passband ripple
and if we then inquirer as to what sort of characteristic should result
if don't the one the passband ripple or equivalent Lee
delta 2 stop and ripple is minimized
this should be dealt the two the it so if doctor one adult
to are minimize it turns out it can be shown
that that implies equal ripple
characterised fan and an equal rebel
characteristic in the stop and that is the ripples in the passband
should alternate between the upper and lower
limits and the ripples in a stop and
should alternate between the upper and lower limits in the text
there are discussed several algorithmic procedures
for actually designing equal ripple filters we won't go through those
details
in this lecture mainly I want to emphasize the point that
there are procedures that allow you to implement designs have this type
and dat designs have this type our optimum
let me just show you
one example love and optimum
equal ripple finite impulse response filter
that was designed when are computer budget was particularly
big this is an example all
a length 251 points
finite impulse response digital filter
designed using one of the algorithms that is discussed in the text
and you can see dead indeed
the ecological characteristics are evident in the passband
the this curve corresponds to the frequency characteristics
plotted in DB so the passband ripple doesn't show up
so with that scale expanded we can see the rebel
and then we have the rebel in
the stop band there should be 125 ripples
altogether and in fact we could see if that was true we could
123 but let's not do that I
obviously there's a very sharp transition band
and and what turned out to be and then
minus approximately 80 my approximately -85
the be attenuation in the stop and so
this is an example all a very high order
equal ripple optimum finite impulse response
digital filter that was designed using
one of the algorithms discussed in the text
okay well and this as i indicated
at the beginning at the lecture was intended primarily
as a very brief
overview all finite impulse response
digital filter design hopefully it
gives you some appreciation
for this style the design techniques
there are of course a large number details
some of which are discussed in the text and some of which
arent but a in any case
this then concludes our discussion are
the design of infinite impulse response and find an impulse response digital
filters
in the next lecture we will begin a new topic
specifically the topic involving the computation
up the Discrete Fourier Transform thank you
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