... Lakshmanan¹

lakshman@nssl.noaa.gov

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... axis. ²

This means that the filter is zero everywhere except in an ellipse-shaped region, where it has a constant non-zero value.

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... pixel ³

In the radial direction; in the azimuthal direction, resolution varies with radar range

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... non-linear. ⁴

A system is linear only if the sum of the responses to two inputs $x$ and $y$ is the same as the response to $x+y$. It is easy to see that the presence of the $\bigvee$ operation in the equation makes it non-linear. If we denote by $x_k$ the results of $fil$ using $E_k$, the response of the equation to a pixel $x$ is $\bigvee(x_1, x_2, \ldots, x_n)$. One can verify that $\bigvee(x_1, x_2, \ldots, x_n) + \bigvee(y_1, y_2, \ldots, y_n)$ is not the same as $\bigvee(x_1+y_1, x_2 + y_2, \ldots, x_n+y_n)$. In this discussion, it has been assumed that the operation $fil$ is linear, i.e. the result of filtering $x+y$ with $E_k$ is $x_k + y_k$. We show in the next paragraph that this is, indeed, the case.

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... filter. ⁵

If there were no quantization effect, then $N_{k}$ would translate to the area of the ellipse and would not be changed by rotation. However, there is a quantization effect and one must keep track of the $N_k$'s

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... numbers. ⁶

Although Radix-2 algorithms are usually the ones that are taught in most college courses, the underlying principle of divide-and-conquer is true for any number that can be expressed as a multiple of small primes. Typical FFT implementations support sequence lengths that are a multiple of 2, 3 and 5. For example, a sequence of length 515 has to be padded to a length of 540 before its DFT can be computed efficiently. The data padding when using a FFT implementation based on small primes requires far less padding than would be required by a purely Radix-2 algorithm (which would need padding to a length of 1024 for a sequence of length 515).

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.... ⁷

$(q*( N*2*p )^2)/((3q+2)*N^2*log_2(N))$ is approximately $(4qp^2)/(3q*log_2(N))$ if the number of filters, $q$, is large enough.

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