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In most applications, we start with a real-valued signal to filter and we need
a real-valued output, but in general, a compound filter with a transfer
function as above will give a complex-valued output. However, we can
construct filters with non-real-valued coefficients which nonetheless give
real-valued outputs, so that the analysis that we carry out using complex
numbers can be used to predict, explain, and control real-valued output
signals. We do this by pairing each elementary filter (with coefficient
or ) with another having as its coefficient the complex conjugate
or .
For example, putting two non-recirculating filters, with coefficients and
, in series gives a transfer function equal to:
which has the property that:
Now if we put any real-valued sinusoid:
we get out:
which, by inspection, is another real sinusoid.
Here we're using two properties of complex conjugates. First, you can
add and multiply them at will:
and second, anything plus its complex conjugate is real, and is in fact
twice its real part:
This result for two conjugate filters extends to any compound filter; in
general, we always get a real-valued output from a real-valued input if we
arrange that each coefficient and in the compound filter is
either real-valued, or else appears in a pair with its complex conjugate.
Next: Two recirculating filters for
Up: Elementary filters
Previous: Compound filters
Contents
Index
Miller Puckette
2006-09-24