Polyphase filters - A model for teaching the art of discovery in DSP

Abstract

By its very nature DSP is a mathematically heavy topic and to fully understand it students need to understand the mathematical developments underlying DSP topics. However, relying solely on mathematical developments often clouds the true nature of the foundation of a result. It is likely that students who master the mathematics may still not truly grasp the key ideas of a topic. Furthermore, teaching DSP topics by merely "going through the mathematics" deprives students of learning the art of discovery that will make them good researchers. This paper uses the topic of polyphase decimation and interpolation to illustrate how it is possible to maintain rigor yet teach using less mathematical approaches that show students how researchers think when developing new ideas.

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POLYPHASE FILTERS – A MODEL FOR TEACHING THE ART OF DISCOVERY IN DSP
Mark L. Fowler
Dept. of Electrical and Computer Engineering
State University of New York at Binghamton
mfowler@binghamton.edu
ABSTRACT
By its very nature DSP is a mathematically heavy topic and
to fully understand it students need to understand the
mathematical developments underlying DSP topics.
However, relying solely on mathematical developments
often clouds the true nature of the foundation of a result. It
is likely that students who master the mathematics may still
not truly grasp the key ideas of a topic. Furthermore,
teaching DSP topics by merely “going through the
mathematics” deprives students of learning the art of
discovery that will make them good researchers. This paper
uses the topic of polyphase decimation and interpolation to
illustrate how it is possible to maintain rigor yet teach using
less mathematical approaches that show students how
researchers think when developing new ideas.
Index Terms— DSP education, discovery, polyphase
filters, decimation and interpolation
1. INTRODUCTION
Teaching DSP necessarily requires heavy use of
mathematics – the nature of the material requires
mathematics to precisely specify the methods and firmly
establish their characteristics
Furthermore, mathematics enables efficient descriptions that
help authors provide precise details despite length
constraints imposed by book and journal editors. Thus, it is
not surprising that DSP educators rely heavily on using
mathematics when teaching DSP – and perhaps too often
our use of mathematics, although precise, fails to convey the
true essence of the topics. Nonetheless, our students
persevere and learn the material – but likely in the process
fail to learn the art of discovery needed to become creative
DSP researchers. DSP educators should strive to present
material in ways that simultaneously demonstrate the art of
discovery in DSP and present the mathematical results and
insights the students need to learn.
The back cover of one of Richard Hamming’s books
[1] states that the book is “…intended to instill in the reader
a style of thinking that will enhance his ability to function as
a problem solver of complex technical issues…” The book
and performance.
describes DSP-related developments and “…relates how
those discoveries came about, and most importantly,
provides analysis about the thought processes and reasoning
that took place…” [1]. Another example in the literature
that shows the thought process that lies behind something
that is otherwise developed mathematically is a description
of how the Hough transform was invented [2]; in fact that
paper is part of the ongoing “DSP History Column” in the
IEEE Signal Processing Magazine, which has focused at
least partly on how DSP results were discovered. Likewise,
here we take a specific DSP idea (i.e., polyphase filters) that
ordinarily is presented solely in terms of mathematical
developments and instead develop it from a more intuitive
point of view. This approach allows students to see an
example of the thought process used in DSP research –
namely that innovation (even in math-heavy areas) more
often flows from non-mathematical visualization/intuition
than from blindly manipulating equations. Furthermore,
this particular development of polyphase filters provides
students with a clearer view of exactly what polyphase
filters are and how they work – then going through the
mathematical development will make more sense.
For conciseness in this paper we will focus only on FIR
polyphase filters for decimation and interpolation by an
integer factor. This is sufficient to show discovery in action
and to show that polyphase filters are essentially a re-
configuration that allows convenient computing. We close
the paper with some general comments about teaching
discovery techniques in DSP classes.
2. STANDARD PRESENTATION OF POLYPHASE
General DSP books (e.g., [3],[4]) and multirate DSP books
(e.g., [5]-[7]) generally develop decimation polyphase filters
as follows. First, the idea of filter-then-decimate is
introduced. Namely, the signal to be decimated, x[n], is first
filtered by a filter with impulse response h[n] to give the
intermediate signal v[n] given by
?
[ ] v n[ ] [x i h n i]
i
??
(1)
and then that filtered signal is decimated by an integer factor
M to give the lower rate signal y[n] as
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[ ] y n[][ ] [ x i h nM]
i
v nMi
???
?
, (2)
which is simply (1) with n replaced by nM, to enact the
decimation. In such developments it is then pointed out that
although this structure accomplishes the desired goal it is
computationally inefficient.
Up to here the development is intuitive and instructive.
However, at this point the polyphase structure is then
developed one of two ways, neither of which provides much
insight or understanding – even when fully understood. The
first way is a time-domain development that uses a non-
obvious re-indexing of the summation in (2) given by
,with ,0,1,, 1,i i M
?
mimM
?
?????
??
(3)
which, after some mathematical manipulation of double
summations, leads to the desired polyphase filter structure.
The second way is a z-domain development that starts by
first demonstrating that the filter transfer function can be re-
organized into a sum of the polyphase component transfer
functions, as given by
1
( )
m
?
with
( )[]
m
n
completed by multiplying this form of H(z) by X(z),
applying the z-domain result for decimation, and finally
exploiting the noble identities (also called the multirate
identities) to move the decimation to the front of the
resulting system.
Both of these developments lead to the same polyphase
structure but neither gives much insight into what is really
going on nor what the resulting structure really does. This
leaves students with a shallow understanding of the real
essence of polyphase filters and likely unprepared to apply
them in any new developments of their own. Furthermore, it
gives the student no insight into how the development was
conceived or discovered. In the first development, the
introduction of the re-indexing in (3) seems to “come out of
nowhere” and leaves many fundamental questions
unanswered. Why is this re-indexing used? How did those
who first developed these ideas ever come up with such an
idea? In the second development, the expansion in (4) also
seems to “come out of nowhere” and the need to apply the
noble identities further shrouds what is really going on.
Furthermore, the application of the noble identities leads to
(imprecise) statements that “the decimation is now being
done before the filtering,” which seems contradictory to
what they learned first (i.e., filter-then-decimate, as in (2)).
Each approach hides from students what is really going
on and gives them the impression that to become DSP
innovators they must learn to pull such confounding steps
“out of thin air.” Perhaps even worse, the resulting
structure is touted as an efficient implementation, yet the
development gives no insight into where that efficiency
0
(),
M
mM
m
H zzP z
?
?
??
(4)
n
P zh nM m z?
??
?
. The development is then
comes from (except perhaps that the parallel paths that
result admit the potential for parallel implementation, but
that is only part of the story). Alternatively, it is desirable to
find a development that (i) flows transparently from (2) to
the final polyphase structure, (ii) clearly shows where the
efficiency in the structure really comes from, and (iii)
illustrates how researchers think conceptually/intuitively to
arrive at precise mathematically results. Point (iii),
obviously, applies to topics beyond the specific coverage of
polyphase filters and should be done more in DSP textbooks
and class notes. By showing how (iii) can be accomplished
for the topic of polyphase filters, hopefully this paper will
motivate ways to address point (iii) for other topics. Some
discussion of this is given in the last section of the paper.
3. DISCOVERY OF POLYPHASE FILTERS
When DSP researchers seek to discover new ideas it is
common to use visual methods rather than just using
mathematics. This is clear when glancing at researchers’
office whiteboards. For the topic at hand we can start by
going back to the inefficient method shown in (2) and
visualizing what it says; Figure 1 shows such a
visualization. Along the top are the samples of the signal x
indexed by i as in (1) and (2). Under that are the various
flipped and shifted versions of the impulse response
(labeled along the left side with the appropriate h[n – i]); the
arrows on the right indicate the act of assigning the results
of multiplying that row’s shifted-flipped impulse response
by the x[i] above and summing to create the corresponding
v[n]. Note that all the possible shifts are shown for n = 6 to
11. That is, the figure even shows the shifts in (1) that are
ultimately thrown away in (2), which are the ones that have
their corresponding v[n] crossed out.
The key insight here (and one that is obvious) is that for
efficiency one should not compute the samples that are to be
thrown away by the decimation. And that is all that is
needed to discover the fundamental insight needed to derive
polyphase filters! The top part of Figure 2 retains only
those outputs that are kept by the decimation; to aid in
visualization we have used color-coded symbols for the
samples of the impulse response rather than the math
symbols used in Figure 1. Now notice that we could
implement this simply by revising our view of convolution:
rather than shifting the flipped impulse response ahead by 1
prior to computing each output we now shift it ahead by M
each time. This is an efficient way to compute the filter and
decimate; but, it is not desirable from an implementation
point of view: we can’t use standard software code for
convolution nor could we use standard hardware blocks for
convolution, because they use single-step shifting for
convolution. Furthermore, this shift-by-M convolution view
is not easily applicable to the implementation of recursive
filters for decimation.
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h[7 – i]
[7]v
x[i]
M = 3
x[0] x[1] x[2] x[3] x[4] x[5] x[6] x[7] x[8]
h[3]h[2] h[1]h[0]h[4]h[5]
h[3]h[2] h[1]h[0]h[4]h[5]
h[3]h[2] h[1]h[0]h[4]h[5]
h[3]h[2] h[1]h[0]h[4]
h[5]
h[3]h[2] h[1]h[0]h[4]h[5]
h[3]h[2] h[1]h[0]h[4]h[5]
x[9]x[10] x[11]
h[8 – i]
[8]v
h[9 – i]
[9]v
h[10 – i]
[10] v
h[11 – i]
[11]v
[6]v
h[6 – i]
[2]y
[3]y
Figure 1: Inefficient form of filter-then-decimate.
Illustrated for the case of decimation factor M = 3.
x[i]
i = 01234567891011
x[0] x[1] x[2] x[3] x[4] x[5] x[6] x[7] x[8] x[9] x[10] x[11]
h[6 – i]
[2]y
x[12]
12
h[9 – i]
[3]
[4]y
y
h[12 – i]
Original Filter…
h[0] h[1] h[2] h[3] h[4] h[5]
Figure 2: Efficient structure from Figure 1 obtained by
eliminating outputs that are discarded by decimation.
But this viewpoint makes it clear that there is a definite
structure in Figure 2 that can be exploited. Looking
carefully (what good researchers do when exploring to find
new results) we see that the purple symbols (trapezoid and
square) only interact with signal samples that have an index
that is a multiple of 3; Thus we see that the purple symbols
taken together as a two-point impulse response are
convolved in a standard way (i.e., requiring shifts of 1
sample) with the sequence of input samples having an index
that is a multiple of 3. Similar relationships can be observed
for the yellow symbols (circle and cross) and the red
symbols (heart and triangle). This view is highlighted by the
rounded rectangular shapes. It leads to the polyphase
structure shown in Figure 3, where the input signal gets
“dealt” into M sub-signals and the filter impulse response
gets split into M sub-filters – and most importantly each
sub-signal is input to each sub-filter for standard
convolution (i.e., requiring only single-step shifting). Thus,
this structure can be easily implemented using standard
software/hardware building blocks and furthermore the sub-
structures have potential to be extended to more general
cases.
Thus, without the use of any precise-yet-obfuscating
mathematics it is possible to precisely develop the basis of
polyphase filters. The important thing here is that the
development shows that the result is not a complicated
mathematical development but rather just a re-grouping of
the structure of the inefficient filter-then-decimate
viewpoint shown in Figure 1 into the efficient polyphase
form shown in Figure 3.
Once this simple re-structuring provides the key idea
behind polyphase decimation for the FIR case it is possible
to (i) derive the result using the time-domain re-indexing
approach and the z-domain polyphase decomposition/noble
identities approach and (ii) extend the idea to IIR polyphase
ideas. Doing such derivations after seeing this development
will make it more obvious to the student why such a re-
indexing or filter decomposition is used.
Original Filter…
…gets split into M subfilters:
… x[3] x[6] x[9] x[12] …
… x[1] x[4] x[7] x[10] …
… x[2] x[5] x[8] x[11] …
?
…
[2]y
[3]y
h[0] h[3]
h[0] h[1] h[2] h[3] h[4] h[5]
h[1] h[4]
h[2] h[6]
…
Figure 3: The resulting polyphase decimation structure
obtained by re-interpreting the structure in Figure 2.
Of course it is possible to develop polyphase
interpolation structures using similar approaches. There the
insight comes from the fact that the effect of the inserted
zeros on the convolution can be exploited by re-grouping
the structure. Figure 4 shows the signal [ ]
input signal with inserted zeros, as it is filtered using the
impulse response h[n]. The tall dashed rectangles show the
multiplications that need not be performed. The solid-filled
symbols of the impulse response need to be multiplied and
the large solid-lined rounded rectangles collect those into
groups that all operate on the same subset of input samples.
x i
?
, which is the
i = 012345678910 11
ˆ
x n
Interpolation: [ ][ ] [
x i h n
??
]
i
i
??
h[6 – i]
h[7 – i]
h[8 – i]
h[9 – i]
h[10 – i]
L = 3
x[0]x[1]x[2]x[3]00000000
ˆ[6]
x
ˆ[7]
x
ˆ[8]
x
ˆ[9]
x
ˆ[10]
x
h[10 – i]
ˆ[11]
x
[ ]x i
?
Figure 4: Inefficient form of zero-stuff-then-filter form
of interpolation.
The result of eliminating from Figure 4 the operations
that involve only the inserted zeros is shown in Figure 5,
where it is clear that the original signal (i.e., without
inserted zeros) gets filtered with M different sub-filters to
create M successive output samples and then get shifted
ahead to the next non-zero sample. Figure 5 is precisely the
structure of FIR polyphase interpolation, and again we see
the same sub-filter structure (the polyphase filters) come
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into play – although in a slightly different way. It now
makes sense to derive the results mathematically and it will
be obvious why the mathematical steps are taken.
L= 3
Filter…
… gets split into L = 3 subfilters:
h[0] h[1] h[2] h[3] h[4] h[5]
h[0] h[3]
h[1] h[4]
h[2] h[5]
x[0] x[1] x[2] x[3] …
] 6 [
) 3(?
x
ˆ
] 7 [
) 3(?
x
ˆ
) 3(?
x
…
ˆ
] 8 [
] 9 [
) 3(?
x
ˆ
]10[
ˆ
x
) 3(?
ˆ
x] 11[
) 3(?
……
…
…
…
Figure 5: The resulting polyphase interpolation
structure obtained from Figure 4.
4. TEACHING DISCOVERY IN DSP
In order to extend to other DSP topics the ideas of teaching
discovery that have been illustrated above, we need to
identify what characteristics in the above descriptions are
general approaches that can be carried over. Looking back
at how the discovery of polyphase filters was made in
Section 3 we see that there were three main components:
1.Visualize the problem
o
Block Diagrams
o
Sketches of simple plots of functions, etc.
2. Start with small examples
3.Look for patterns and familiar structures
Equation (2) was visualized for a small example using block
diagrams and signal sketches. Then the familiar pattern of
individual convolutions was discovered.
Although not present in the polyphase filter discovery
given above, there are other components that can be useful
for discovery. A few of them are:
4.Start with simplified assumptions then remove
them
o
For example, in developing uniform DFT
filter banks first assume that the filter
length is the same as the number of
channels [8]
5.Use analogies to relate new ideas to topics already
covered
o
For example, in developing the idea of I
& Q signals it is useful to first explore the
idea using the familiar concept of phasors
[8]
6.Divide and conquer
o
For example, deriving the FFT algorithm
is often done using this approach.
Here are some other DSP topics that can be used as vehicles
for teaching the art of discovery.
Uniform DFT Filter Banks [8]: This uses components
1 – 4 by visualizing a small example under the simplified
assumption that the filter length equals the number of
channels and that there is no decimation. Once this scenario
is understood through visualization, the mathematical view
is extracted and it is possible (through continued
visualization) to successively remove assumptions to
discover the uniform DFT filter bank with decimation and
filter length not equal to the number of channels.
Bandpass Sampling [8]: This uses components 1 – 4 by
visualizing a small example under the simplified assumption
that the upper band edge is an integer multiple of the
signal’s bandwidth. Then, as understanding is gained, the
discovery progresses to the general case of an upper band
edge that is not an integer multiple of the bandwidth.
DFT & FT Results: An example is the proof of
1
2/
0
0,
N
2,3,
0
?
,1
,
N
?
jkn N
?
n
N
k
e
k
?
?
?
?
?
?
??
?
(5)
The proof is commonly available in DSP books (e.g., [4])
but, again, students may wonder how one discovers such
results (it is easy to prove, but may be elusive to discover).
However, visualizing this as the sum of vectors in the
complex plane helps with the discovery: when k = 0 all N
vectors in the sum line up on the positive real axis and
therefore sum to N, but for the other values of k you get N
vectors that are uniformly spaced in angle and thus sum to
0. Visualization together with looking for patterns in small
examples leads to this discovery.
REFERENCES
[1] R.W. Hamming, The Art of Doing Science and Engineering:
Learning to Learn. Gordon and Breach, Amsterdam, 1997.
[2] P.E. Hart, “How the Hough Transform Was Invented,” IEEE
Signal Processing Magazine, DSP History Column, pp. 11 –
22, Nov. 2009.
[3] J.G. Proakis and D.G. Manolakis, Digital Signal Processing:
Principles, Algorithms, and Applications, 4th Ed., Pearson
Prentice Hall, Upper Saddle River, NJ, 2007.
[4] B. Porat, A Course in Digital Signal Processing, John Wiley
and Sons, New York, 1997.
[5] P.P. Vaidyanathan, Multirate Systems and Filter Banks,
Prentice-Hall, Englewood Cliffs, NJ, 1993.
[6] N.J. Fliege, Multirate Digital Signal Processing: Multirate
Systems, Filter Banks, Wavelets, John Wiley and Sons, West
Sussex, England, 1994.
[7] f.j. harris, Multirate Signal Processing for Communication
Systems, Prentice Hall, Upper Saddle River, NJ, 2004.
[8] M.L. Fowler, Lecture Notes for EECE521, Binghamton
University, available at
http://www.ws.binghamton.edu/fowler/fowler%20personal%2
0page/EE521.htm
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