Butterworth filter POLES lie along a circle and are spaced at equal angular distances around a circle.Select one:TRUENOT always TRUEFALSE

Step 1: Break Down the Problem

We need to determine the truth of the statement regarding the poles of a Butterworth filter and their spatial arrangement on a circle.

Step 2: Relevant Concepts

• Butterworth Filter: A type of signal processing filter designed to have a frequency response as flat as possible in the passband.
• Poles: The points in the complex plane where the transfer function of the filter is infinite. For a Butterworth filter, the poles are derived from the Butterworth polynomial.
• Location of Poles: The poles of an n-th order normalized Butterworth filter are located on a circle in the left-half of the complex plane, evenly spaced at equal angular distances.

Step 3: Analysis and Detail

For an n-th order Butterworth filter, the poles are derived using the formula:

$p_k = e^{j\left(\frac{(2k + 1)\pi}{2n} + \pi\right)} \quad \text{for } k = 0, 1, \ldots, n-1$

This representation shows that the poles are located symmetrically around a circle with a radius of 1, specifically in the left half of the complex plane for real implementations.

Step 4: Verify and Summarize

Since the poles are indeed distributed evenly around the circle at equal angular distances, we can conclude that the statement is not just sometimes true but consistently true for normalized Butterworth filters.

Final Answer

TRUE - The poles of a Butterworth filter lie along a circle and are spaced at equal angular distances around that circle.