5.2 Phase Response

We wish to recognize features of the original signal from the coefficients produced by transformations involving QFs, so it is necessary to keep track of which portion of the sequence contributes energy to the filtered sequence.

Suppose that F is a finitely supported filter with filter sequence f( n). For any sequence u ? ? ², if Fu( n) is large at some index n ? Z, then we can conclude that u( k) is large near the index k = 2 n. Likewise, if F*u( n) is large, then there must be significant energy in u( k) near k = n/2. We can quantify this assertion of nearness using the support of f, or more generally by computing the position of f and its uncertainty computed with Equations 1.57 and 1.60. When the support of f is large, the position method gives a more precise notion of where the analyzed function is concentrated.

Consider what happens when f( n) is concentrated near n = 2 T:

(5.25)

Since f( j + 2 T) is concentrated about j = 0, we can conclude by our previous reasoning that if Fu( n) is large, then u( k) is large when k ? 2 n - 2 T. Similarly,

(5.26)

Since f(2 j - n +...