Appendix C: Convolution Extension
It may prove useful to extend the convolution technique of chapter 2 to cover
the case of convolution of two
combs, one of which does
not have unit spacing. Although not
required for the specific technique of spectrum computation just described, it
is a related problem and is solved in a similar manner.
Consider the general case of convolution between combs of spacing N and
N+1, where N ³ 2 . The limitation on the size of N means that the
highest frequency (N+1) is not a multiple of the lowest frequency
(N), so that the previous convolution algorithms cannot be used.
As before, an observation
frequency where the spectrum is desired to be calculated is chosen and
denoted by n = g.
The N+1 comb sits atop each d-function of
the N-comb in turn and whenever one of the d-functions of the
N+1 comb is coincident with g another term in the complex
amplitude at g is added. The d-function of the N-comb upon
which the N+1 combs then sits is denoted mN, and mN+1
denotes the d-function of the N+1 comb sitting at g.
If g is specified by
where
aN specifies the even part of
g and
b allows
the formation of odd frequencies,
a,
b integer, then it can
be shown
where k is integer ,
-¥ < k < +
¥.
These equations are the result of comparison of a dozen or so
tables such as table 2.1, the values of m
N and m
N+1 determined
by using two graduated paper strips representing each
d-function
comb. Their validity has been verified in as much as comparison of the
resulting spectrum with that as determined by Fast Fourier Transform
has always shown excellent agreement and they are used in the proof below.
The observation frequency need not be limited to those
specified by either comb as the process of convolution, it will now be shown,
always results in a final comb of
unit spacing between its
member
d-functions.
The convolution of a comb of spacing N with a comb of spacing
N+1 units, N ³ 2, results in a comb of unit spacing independent of
the integer value of N
Both mN and mN+1 specify the d-function frequency of
observation g by
|
d(n - [ mNN + mN+1(N+1) ] ) |
| (4) |
Let
n1 be one point of observation determined by
Let the location of a neighbouring frequency
n2 be
specified by
|
n2 = (mN+a)N + (mN+1+b)(N+1) |
| (6) |
where `a' and `b' are integers. This second frequency is not a point
of observation and as such the premultipliers of frequencies N and
N+1 may take any value .
The separation of
n1 and
n2 is then
The summation indices premultiplying N and N+1 range over
-¥ to +¥, so mN, mN+1, `a' and `b' also
lie in this range. It is then legitimate to set
so that the closest separation of the distinct frequencies
n1 and
n2 is
which completes the proof.
This proof may also be modified to prove the earlier statement that
convolution of an N-comb with a
unit-comb always results in
unit spaced comb, if so required.
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