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Non-recirculating filter, second form

Sometimes we will need a variant of the filter above, shown in Figure 8.10, called the elementary non-recirculating filter, second form. Instead of multiplying the delay output by we multiply the direct signal by its complex conjugate . If

is any complex number, its complex conjugate is defined as:

Graphically this reflects points of the complex plane up and down across the real axis. The transfer function of the new filter is

This gives rise to the same frequency response as before since

Here we use the fact that , for any unit complex number , as can be verified by writing out in either polar or rectangular form.

Although the two forms of the elementary non-recirculating filter have the same frequency response, their phase responses are different; this will occasionally lead us to prefer the second form.

Figure 8.10: The elementary non-recirculating filter, second form.

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