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## Real outputs from complex filters

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.

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