Example 2: Rational Approximation

Instead of raising to the non-integer power , suppose we express as and compute the frequency response as ? Perhaps working with integer powers and integer root extractions will be better behaved.

Generalizing Fig.B.28 of the previous section (page ), the wrapped phase response of , for any integer , consists of N negatively sloped linear segments (like those in Fig.B.28), with one segment always centered about , and with each segment traversing the vertical range to from left to right. The slope of each segment is radians per radian.

Now, when we take the th root of to get , we see that the phase of is divided by . Thus, the phase response of will still have linear segments, but now traversing the vertical range to . In our present example, we take the square root of to get to obtain the phase response shown in Fig.B.25b on page . This phase response is what is computed by the matlab expression angle(sqrt(exp(-j*9*omega*T))). As can be seen in the plot, there are now two bandpass regions in which the sign is inverted by the phase jumps of radians. This is even worse than the result of Example 1.

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written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.