To approximate arbitrary slopes
, we may alternate poles and
zeros so that the average slope of the stick diagram equals
.
For example, to achieve a slope of
(``half an
integrator''), we may start with a pole near
, where
is our lowest frequency of interest (nominally
-
Hz for
audio), which causes the slope to approach
. Then, half an octave
to the right, e.g., we can locate a zero to cancel the pole's roll-off,
pushing the slope from
back toward 0
. Continuing in this way,
we may locate a pole at each octave point
,
, with zeros interlacing at
, in order
to achieve an
average slope of
.
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Figure 2 shows the Bode plot and the corresponding
stick diagram for
poles located to give breakpoints distributed
along octaves starting with
. That is, the poles are at
, for
and
. To approximate a
power-response having slope
nepers per neper, we
place
zeros at
,
, i.e.,
shifted half an octave toward higher frequency, interlacing the poles.
We see that the response is flat for frequencies below the first
break-frequency
as before, but the gain drops by less the
dB at
due primarily to the influence of the upcoming
first zero at
. Between the pole-zero frequencies
and
, the Bode plot smoothly
interpolates the stick diagram which alternates between slopes of 0
and
dB per decade, as the poles and zeros break in alternation,
yielding an average roll-off of
dB per decade, as desired. After
the last zero breaks, the response levels off to a final slope of 0
.
Alternatively, the last zero could be omitted to have a final
dB
per decade slope, etc. Additionally, we plot the pole symbol `X' and
zero symbol `O' along the upper horizontal axis at their corresponding
break frequencies. This plot suggest that we may be able to choose
,
, and
to achieve any desired accuracy over any finite
band.
More generally, for any desired slope
, we place the
th zero on the negative-real axis of the complex
plane at
,
, where
denotes the
th pole, exponentially distributed along the
negative-real axis with spacing ratio
, starting at
.
Note that
cancels all of the poles with zeros, yielding a
constant magnitude frequency response, while
cancels all
poles except the first
, leaving a slope of
nepers per neper
(an integrator) for
. When
, the
pole-zero sequence starts out from the origin of the
plane with a
zero, thereby giving a net positive slope to the Bode magnitude plot.
In particular, at
, all of the poles are canceled by zeros,
leaving behind a single zero at
, yielding a slope of
(differentiator) for
. Between these extremes,
the poles and zeros interlace asymmetrically to approximate any
desired slope
.
As examples of other slopes, Fig.3 shows a Bode plot
analogous to Fig.2 for the case
(``half
a differentiator''), and Fig.4 shows
,
showing the resulting asymmetric pole-zero layout on a log-frequency
scale.
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To reduce the maximum error, the interlacing pole-zero pattern can be
made more dense, e.g., by placing poles every half octave, or third
octave, etc. As an example, Fig.5 shows the
improvement obtained over Fig.2 by increasing the
order from
to
.
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It is not necessary for the slope
of the spectral roll-off to
be restricted between
and
neper per neper. Any number of
poles or zeros can be used in the low-frequency region to establish
any integer part for the slope, or some number of the regularly spaced
poles (zeros) can be skipped before the partial cancellation array
of zeros (poles) begins. The subsequent interlacing poles and zeros
then only need to set the fractional value of the slope above
.
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