STANFORD ARTIFlCl,4t INTELLIGENCE LABORATORY
: MEMO AIM- 208
STAN-CS-73 -379
THE HETRODYNE FILTER-AS A TOOL
FOR
ANALYSIS OF TRANSIENT WAVEFORMS
BY
JAMES ANDERSON MOORER
SUPPORTED BY
ADVANCED RESEARCH PROJECTS AGENCY
ARPA ORDER NO. 457
JULY 1973
COMPUTER SCIENCE DEPARTMENT
School of Humanities and Sciences
STANFORD UNIVERSITY
STANFORD ARTIFICIAL INTELLIGENCE LABORATORY
MEMO A I M-208
JULY 14, 1973
COMPUTER SCIENCE DEPARTMENT
REPORT CS-379
THE HETROOYNE FILTER AS A TOOL FOR ANALYSIS
OF TRANSIENT WAVEFORMS
bY
James Anderson Moorer
ABSTRACT:
A method of analysis of transient waveforms is
discussed. Its properties and limitations are
presented in the context of musical tones. The
method is shown to be useful when the risetimes
of the partials of the tone are not too short. An
extention to inharmonic partials and polyphonic
musical sound is discussed.
This research was supported in part by the Advanced Research Projects
Agency of the Office of the Secretary of Defense under Contract No. SD-183.
The views and conclusions contained in this document are those of the
author and should not be interpreted as necessarily representing the
o f f i c i a l p o l i c i e s , either expressed or implied, of the Advanced Research
Projects Agency or the U.S. Government.
Reproduced in the USA. Available from the National Technical Information
S e r v i c e , S p r i n g f i e l d , V i r g i n i a 22151
List of Figures
Figure 1
- Frequency response of a hetrodyne filter with center
frequency of 400 Hz and summation period of 10 milliseconds.
Figure 2 - Frequency response of a hetrodyne filter with center
frequency of 100 Hz and summation period of 10 milliseconds.
F i g u r e 3 - Frequency response of a hetrodyne filter with center
frequency of 300 Hz and summation period of 10 milliseconds with the
signal beginning halfway through the summation.
Figure 4 - Hetrodyne filter applied to a signal which is a sum of
sinusoids of frequencies 100, 200, and 300 Hz, with exponential attacks
of time constants 30, 20 and 10 milliseconds respectively. The graphs
are of the outputs of a hetrodyne fi lter with summation period of
10 mi I I iseconds. The center frequencies are, top to bottom, 100, 200
and 300 Hz.
F i g u r e 5 - Same data as in figure 4, but the attacks are linear rather
than exponential.
F i g u r e 6 - The magnitude and phase of the output of the hetrodyne
f i l t e r w h e n applied to a 132 Hz guitar tone. The apparent modulation
i s t h e r e s u It of beating with an inharmonic partial at 186 Hz.
F i g u r e 7 - Frequency
frequency
frequency,
is
not
response of a hetrodyne f i I ter
exactly an
integral
when the center
multiple of the summation
INTRODUCTION
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The analysis of the attack transients of vocal or musical
tones goes back as far as 1932 with Backhaus’ [I,21 tunable resonator
and drum recorder. Lute a n d C l a r k [3,4,51 used fi I tering methods to
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s e l e c t p a r t i a l t o n e s f o r a n a l y s i s a n d r e c o r d i n g . More r e c e n t l y , w i t h
the advent of computer music, analysis of musical instruments for the
purpose of simulation of timbre has been done by what wil I be called
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a “hetrodyne f i I ter”
f o r w a n t o f a b e t t e r n a m e , Beauchamp
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analysed each partial of a complex waveform by first multiplying the
waveform by a sin and cosine at
question.
the frequency of the partial
The result was then low-pass filtered, then squared and
s u m m e d . F r e e d m a n 17,8,91, and later Keeler
finite
in
summation
over
one period
[10,111
used a discrete
of the fundamental frequency in
place of Beauchamp’s low-pass filter, an effect which as we shall
show later conveniently places a zero of transmission at all harmonic
partials other than the one in question.
It
is
the
purpose
of this article to explore this method,’
report its characteristics, its limitations, its uses and some simple
extensions.
THE METHOD
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Let us define the hetrodyne fi I ter as fol lows, We begin with
a d i s c r e t e f u n c t i o n F i w h i c h r e p r e s e n t s a c o n t i n u o u s f u n c t i o n F(t)
a t d i s c r e t e i n t e r v a l s t=ih,
where h is the time between samples. h
is cal l e d t h e “ s a m p l i n g i n t e r v a l . ” T h e r e c i p r o c a l o f h i s c a l l e d t h e
“sampl ing
frequency” or the “sampling rate.” Let us define a and b
as fol lows:
&N-l
a =
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i
Fi cos(moih + eo)
1=Qf
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z
b =
,
UC
&-N-l
Fi sin(woih + 00)
t
i=a
(11
wi I I be c a l l e d t h e “ c e n t e r f r e q u e n c y ” .
Without loss of generality, we may define
TO = 80 + ah
(2)
i
and thus rewrite the sum as going from 0 to N-l.
N-l
a = >Fi eos(aoih + 'P,)
PO
N-l
b = ) Fi sin(woih + 'Oo)
i.=O
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(3)
Thi s change of
filter
variables
hides
the
t i me-pos i t i on
of
the
in the phase. We must remember that as the filter is advanced
t h r o u g h t i m e , the phase angle will increase,
and
that
any
results
which depend on this phase angle will be functions of time.
Since the summation operation is linear, we may represent the
input
waveform Fi
as a sum of sinusiods and may thus examine the
response of the fi I ter to a sinusiodal
excitation,
as
is
commonly
d o n e w i t h I inear fi 1 t e r s .
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i
N-l
a = r A cos(tih + vo)
i=O
N-l
b = x A cos(wih + Q) sin(xoih + u)o)
i=O
(4)
With the he I p o f t h e s u m m a t i o n c a l c u l u s 1121 and some trigonometric
?
identities,
we
may compute
the summation in closed form without
e r r o r a s f o I lows:
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A
a =:2N
i
1
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+
sinr(tim)Nh/2] cos[(tiw)(N-l)h/2++w]
sin[(w=wo)h/2]
sin[ w-cl,o)Nh 2] COS[(~-~O)(N-~)~/~+CD~~O]
$In[(w=wo)h/2]
(5)
b=& c sin[(dao)Nh/2] sin[(wftuo)(N-l)n/2+co+Qo
sin[(w+wo)h/2]
+
sin[( wwo)Nh/2] sin[(wwo) (N-l)h/2+Q- CDO
sin[ (up(go) h/2
Th S
S
simpl
fied somewhat by computing the sum of the squares of a and b.
not a very
useful expression
as it stands,
but
it
may
be
a2+b2= iI3
A2 ( ~~~1+ -f&gyp
sin[(&Jo)Nh/2]sin[(w-w)Nh/2]
$- sin[(&cl)o)h/2] sin[(a=wo)h/2] cos[wo(N-L)h + "']
(6)
Now if one chooses N to be such that
e
Nhwo =27&k= any integer # 0
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Then some terms in equation 6 collapse to produce
a2+b2 = A2
4N2
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2c0s[~oh - 2uo]
+ sin[ (u+dh/2] sin[ (w-ruo)h/2] )
f
t
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T h e s q u a r e -root o f t h e a b o v e e x p r e s s i o n
I
e
hetrodyne f i I ter.
will
be
termed
the
” magn i t ude” of the output of the hetrodyne filter. The arctangent of
the ratio of a to b will be called the “phase” of the output of the
Thi s process
i s s i m i l a r t o t h e d i s c r e t e F o u r i er transform,
e x c e p t t h a t o n l y o n e f r e q u e n c y i s p r o c e s s e d i n s t e a d of many. The
results of this analysis can
e
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the
output of the DFT by setting the period of the center frequency to a
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easily be generalized to represent
multiple of the sampling interval.
If one further assumes that the frequency of the input
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sinusoid is close to the center frequency, then we see that
lim
w-w0
a2+bp =A2
4N2
t,
= $2
O+N2+0
define au)= ~-u)o
W-W
sin~2,oh[(N-l)/2+~+Nsin~f&h[(N-l)/2+~]j
cos{2woh[(N-1)/2+oJ~Ncos@h[(N-1)/2+0;]?
if N > 1 then
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lim $ a tan [&d-4 (N-1)/2 + a13
,
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T h u s w e s e e t h a t i n t h e l i m i t , t h e m a g n i t u d e o f t h e o u t p u t ofthe
f i I ter becomes independant
of time and becomes a measure of the
amplitude of the input sinusoid, The phase of the output
filter remains always a function of time,
function of time,
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the
but is also a linear
its slope being determined by the difference of the
center frequency and the input frequency.
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of
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This reveals a method of determining the amplitude of the
input sinusiod and getting a better estimate of its frequency. If t h e
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center frequency of the summation is near the actual frequency of the
input
sinusoid,
the phase will be very nearly a linear function of
t inie, thus we may find the frequency deviation by fitting the phase
with a straight line and observing its slope.
The consequences of choosing N as above are significant. If
the center frequency is
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a multiple of some fundamental frequency,
then we may choose N to coincide with the period of the fundamental
and thus cancel out
frequency.
of
all the harmonic partials except the center
Figure 1 shows the log of the magnitude of the output
t h e hetr&yne
filter for a range of sinusoidal inputs.
The
,
center frequency in this plot is 400 Hz and the summation period, Nh,
i
is
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f
k
10 mi I I i seconds.
Figure 2 shows the log of the magnitude versus
frequency for a center frequency of 100 Hz and the same summation
per i od.
Notice the zeros of
t r a n s m i s s i o n at al I multiples of the
summation period except the center frequency.
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ERROR ANALYSIS
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F r e e d m a n [91 a n d K e e l e r [111 both show that this method is
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s u f f i c i e n t l y a c c u r a t e f o r t h e i r purposes even when the input
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is
not
a
perfect
Keeler does not even bother
negl igable. The above work may well seduce one as it did the author
into believing that this is a perfectly accurate method, universally
appl icable.
This is not so.
T o persue t h e m a t t e r f u r t h e r , l e t u s
t h e e q u a t i o n s s o m e w h a t . We s h a l l c o m p u t e n o t t w o
reformulate
summations but one:
&-N-l
G
= t
o?
x
Fie
iwoh+@o
(9)
i=a
The result will be a complex quantity whose real and imaginary parts
-
correspond
t o t h e a a n d b d i s c u s s e d e a r l i e r . We w i l l b e i n t e r e s t e d
in the magnitude and the phase
shall
of G. For the input waveform, Fi, we
take a complex sinusoid with exponential decay.
G
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s inusoids.
us with an elegant proof that the error in doing so is
and presents
.
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of
computing the summation at each p o i n t , b u t a t r e g u l a r i n t e r v a l s o n l y ,
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sum
signal
o!
= F-' .(a+j,)ih+e e jLI,oih+Qo
i=o
= eah[a+j(uI+wo)]+j(@+Oo)
eNh'a-+j(w'wo)l-
1
(10)
Again, we see
the magnitude is
related to the amplitude of
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the
input sinusoid and the phase drift with time
is related to the
t
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frequency
causes
differece. T h e e x p o n e n t i a l d e c a y o f t h e i n p u t s i g n a l
depending on
other harmonic partials,
cancellation of
imperfect
of the attack,
the speed
and
the deviation can be
important.
A more reveal ing
case would be to assume the signal begins
and rises exponentially to its steady-state value
at zero amplitude,
and that the signal begins
somewhere during the summation, say at
i=@.
t
jh(urtcuo)
- 1
e
_
p&+p)
e(N-p)hb+hhdl
e
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1
- 1
This is equivalent to setting N to some smaller value. As the
filter progresses through the attack,
window
wi I I
t h e e f f e c t i v e widtl h
of
the
approach N, and the response of the filter w i I I become
more representative.
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hb3+jh+~~)l
_
(11)
if N and the center frequency
see that even
Here we
are
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careful ly
chosen, the frequency response is not the same, Figure 3
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shows the
response for a
ten ter frequency
of 300 Hz, a s u m m a t i o n
interval of 10 mi I I i seconds, and P=NJZ. W e s e e
that the neighboring
not canceled out. This shows that if the signal begins
harmonics are
anywhere within the sindow, the output should not be taken to be an
of the amplitude of the partial.
accurate indication
It i s e x a c t l y
taking N to be a non-multiple of the period of the
analagous to
f u n d a m e n t a l f r e q u e n c y . F i g u r e 4 shows the output for an averaging
The input was a sum of
300 Hz.
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frequent i es 180,
e
center frequencies of 100, 200, and
of 10 mi I I iseconds and
window
200, and 300 Hz with exponential attacks of time
and 10 milliseconds respectively.
30, 20,
constants
sinusoids of unit amplitude and
The leakage
among the harmonics is apparent here.
If t h e a t t a c k i s n o t e x p o n e n t i a l , a n o t h e r f o r m o f d i s t o r t i o n
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can occur. F i g u r e 5 shows the magnitude of the filter output for the
same
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f r e q u e n c i e s a s f i g u r e 4 , b u t w i t h linear
and ten ter
input
attacks rather than exponential.
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a
sinii lar to
phase
is
caused
by
of a guitar
an
c a u s e an e f f e c t
Figure 6 shows the magnitude and
ampl i tude modulation.
of the fundamental
modulation
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i n h a r m o n i c partials can
The presence of
note at 132 Hz. The apparent
inharmonic partial
at 186 H z ,
the
frequency of a known box resonance.
Another source of error is
that of frequency quantization. N
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can
i
not in general be
the fundamental frequency and still have N be an integer. This can be
tolerated, but
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multiple
frequency.
t
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chosen such that Nh is exactly the period of
it also implies that
o f Zn/Nh r a t h e r
If
we
do
not
than
set
a
the
the center frequency must be a
multiple
center
of
the
frequency
fundamental
to
exactly
a
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t
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7
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!L.
m u l t i p l e o f Zn/Nh, w e g e t i m p e r f e c t c a n c e l l a t i o n o f a p o l e a n d a
zero. Figure 7 shows the
magnitude versus frequency of such a
case.
Note the doublet around 400 Hz, the center frequency.
Since Nh is not exactly the period of the fundamental
frequency,
the harmonic partials are
Fur thermore, most
not exactly cancelled
string instruments show
integral multiples of
out.
a deviation from perfect
the fundamental frequency. This deviation also
contributes to leakage among the harmonics. Equation 10 may be
to determine exactly how much leakage is present.
used
A SIMPLE EXTENSION
Although there
problem of
would
seem
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before analysis of
partials may
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no
good
solution
to
the
the inharmonic partials. if the attack
not too swift, it can be filtered
time of the partial in question is
out
be
the note beginning within the summation period, there is
a technique for dealing with
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to
then
the harmonic partials is done. The harmonic
be
filtered
out
to
analysis
allow
of
the
inharmonic partials.
is
The filter advocated here
a
comb
filter.
This is
described simply by the recurrence relation:
*I-l = x* - x* m
(12)
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With frequency response
peJdl = Jsin*(*h)
.
We
see
that
the
multiples of the
comb
filter
+ [co&nh)-l]*
has
a
of
transmission
at
all
base frequency l/mh. The only hazards with the comb
filter are those of transient response
i
zero
(13)
and preturbing the harmonic
p a r t i a l s . T h e t r a n s i e n t r e s p o n s e o f a c o m b f i l t e r i s e x p l i c i t . It is
identical ly
zero beyond mh
s e c o n d s . If t h i s can be tolerated, then
the f i lter may be useful,
If the frequencyy o f s o m e h a r m o n i c p a r t i a l f a l l s n e a r o n e o f
the zeros of transmission of the comb, it will be attenuated. We m a y
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prevent this by a
At those zeros of
t
resonater
is
method which appeared in
the comb that we wish to eliminate, a digital
used
configuration are
G o l d and R a d e r [13,141.
to
cancel
out
the
zero.
described in ful I detail
The details of the
in the references and
will not be repeated here.
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For analysis of an inharmonic partial, the harmonic partials
may
a l l be e l i m i n a t e d w i t h a s i n g l e
comb, subject to the limitation
that the partials may not be exact multiples of the fundamental, and
that the frequency
of the fundamental becomes quantized when the
comb length, m, is chosen.
The
comb
wi
I
I,
of
course,
attenuate
the
signal
under
a n a l y s i s b y s o m e a m o u n t . We m a y p r e d i c t t h e a t t e n u a t i o n f r o m
equation 13 and then multiply the results of the hetrodyne analysis
by the
reciprocal
of the attenuation
factor to obtain a better
estimate of the amplitudes of the partials.
CURRENT USES
The technique of combing out unwanted signals and then
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waveform is the basis for an automated
the remaining
analysing
system for
the analysis
of polyphonic music currently
When two or more instruments are playing
developed by the author.
simultaneously, a
pitch
and
duration
being
Fourier analysis is used to get an estimate of the
of
each
note.
All
notes
but
one
are
then
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eliminated by combing,
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determine the attack time of the note and to correct the estimated
pitch of the note. This
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piece.
The result
and the hetrodyne filter
is applied to
is iterated for each of the notes in
is subjected
the
to a heuristic analysis and is
eventually displayed as a musical score of the piece under analysis.
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Also under
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parameters of musical tones
trenielo,
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study by a colleague
and steady-state
is how
the physical
such as risetimes of the partials,
value contribute to the perceived timbre
o f t h e i n s t r u m e n t . The hetrodyne method as outlined above is used to
determine these parameters from digitized recordings of tones of
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actual musical instruments.
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CONCLUS ON
The hetrodyne
filter has been shown to be a useful
tones as long as its
for the analysis of the partials of musical
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I imi tat ions are observed.
quick,
perfect
It can fail
if the frequencies of
integral multiples of
sampling rate is
significant,
method
if the attack times are
the partial3
too
deviate too far from
the fundamental frequency, or if the
so low that frequency quantization effects become
It can also fail if substantial frequency modulation
(vi brato) i s present.
An extention of the method to tones with inharmonic partials
and even to multiple
use
of the comb
simultaneous notes was shown to be possible by
filter as long as
the effects of the transient
response of the comb was judged to be tolerable.
These methods are currently in use by the author and his
colleagues in the analysis of digitized musical sound,
,
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REFERENCES
1)
H.
Backhaus,
"Ober
Ge i genk I ange, "
Zei tschr i f t
f u r Techni sche
Physik, Vol 8, #ll, 1 9 2 7 , pp509-515
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2) H. Backhaus,
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L
in der
Austr i k, " Ze i tschri ft fur Technische Physik, vol 13, #l, ~~31-46,
1932
3) D a v i d A .
L
“Uber d ie Bedeutung der Ausgl eichsvorgange
Lute “Phys i ca I
Correlates of Nonpercuss i v e f l u s ical
Instrument Tones, " PhD thesi s, MIT, 1963
4) D a v i d Lute, flelvil l e C l a r k ,
'Physical Correlates of Brass
Instrument Tones," J. Acoust, SOL Am, vol 42, ~1232, 1 9 6 7 .
5) D a v i d Lute,
Nonpercussive
flelvi I le Clark,
'Duration of Attack Transients of
O r c h e s t r a l I n s t r u m e n t s , " J . A u d i o E n g . Sot., vol 13,
#3, ~194, 1 9 6 5 .
i
I
t
6) James W. Beauchamp, "A Computer System for Time-Variant Harmonic
Analysis
and Synthesi s o f M u s i c a l T o n e s , " i n
"flusic by Computers, "
J. W. Beauchamp and He i nz Von Foerster eds, John Wiley and Sons,
i
L
t
. York,
7) M . D .
Freedman, "Analysis of
M u s i c a l I n s t r u m e n t T o n e s , ' J,
A c o u s t . SOL Am vol 41, ~793, 1 9 6 7
t
i
New
A
8) M . D .
Freedman, "A Method For Analysing Musical Tones," J Audio
E n g . Sot., V o l 1 6 , #4, ~419, O c t . 1 9 6 8
., .
“ A Techni q u e f o r Ana Iysis o f M u s i c a l I n s t r u m e n t
9) M.D. Freedman,
IL
L
Tones, ” PhD d i s s e r t a t i o n , U n i v e r s i t y o f I I I inoi s, Urbana. 1965
‘The Attack Transients of Some Organ Pipes,” IEEE
10) J.S. K e e ler,
Tran.
and Electroacoustics,
on Aud io
V o l A U - 2 0 , #5,
pp378-391,
December 1372
t
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I
11) J . S . K e e l e r , ‘ P i e c e w i s e - P e r i o d i c A n a l y s i s o f A l m o s t - P e r i o d i c
Sounds
and
Musical
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