


 Basic operators and adjoints  

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Convolution is the computational equivalent of
ordinary linear differential operators (with constant coefficients).
Applications are vast,
and end effects are important.
Another choice of data handling at ends
is that zero data not be assumed
beyond the interval where the data is given.
Careful handling of ends is important in data in which the crosscorrelation changes with time.
Then it is sometimes handled as constant in shorttime windows.
Care must be taken that zero signal values not be presumed
off the ends of those shorttime windows;
otherwise,
the many ends of the many short segments
can overwhelm the results.
In equations (4) and (5),
the top two equations explicitly assume the input data vanishes
before the interval on which it is given, and likewise at the bottom.
Abandoning the top two and bottom two equations in equation (5)
we get:

(8) 
The adjoint is

(9) 
The difference between equations (9)
and (7) is that here,
the adjoint crosscorrelates a fixed portion of output
across a variable portion of input;
whereas, with (7)
the adjoint crosscorrelates a fixed portion of input
across a variable portion of output.
In practice, we typically allocate equal space for input and output.
Because the output is shorter than the input,
it could slide around in its allocated space;
therefore, its location is specified by an additional parameter called its lag.
user/gee/icaf1.c
for( b=0; b < nb; b++) {
for( y = SF_MAX(lag,b+1); y <= ny; y++) { x = y  b  1;
if( adj) bb[b] += yy[ylag] * xx[x];
else yy[ylag] += bb[b] * xx[x];
}
}

The value of lag always used in this book is lag=1.
For lag=1 the module icaf1
implements not equation (8)
but equation (10):

(10) 
It may seem a little odd to put the required zeros at the beginning
of the output, but filters are generally designed so the
strongest coefficient is the first, namely bb(1), so
the alignment of input and output in equation (10)
is the most common one.
The
end effects
of the convolution modules are summarized
in Figure 2.
conv
Figure 2.
Example of convolution endeffects.
From top to bottom:
input;
filter;
output of tcai1();
output of icaf1() also with (lag=1).







 Basic operators and adjoints  

Next: Zero padding is the
Up: FAMILIAR OPERATORS
Previous: Transient convolution
20140927