Question

H * (Force) ^ x = ((Maxv) ^ 2 * (Radiax) ^ 2)/((Timeperiad) ^ 4) ,the value of X is (A) 1 (B) 4 (C) 3 (D) 2​

Answer 1

Answer:

Decimation in time

The radix-2 decimation-in-time algorithm rearranges the discrete Fourier transform (DFT) equation into two parts: a sum over the even-numbered discrete-time indices n=[0,2,4,…,N−2] and a sum over the odd-numbered indices n=[1,3,5,…,N−1] as in Equation:

X(k)====∑N−1n=0x(n)e−(i2πnkN)∑N2−1n=0x(2n)e−(i2π×(2n)kN)+∑N2−1n=0x(2n+1)e−(i2π(2n+1)kN)∑N2−1n=0x(2n)e−⎛⎝i2πnkN2⎞⎠+e−(i2πkN)∑N2−1n=0x(2n+1)e−⎛⎝i2πnkN2⎞⎠DFTN2[[x(0),x(2),…,x(N−2)]]+WkNDFTN2[[x(1),x(3),…,x(N−1)]]

The mathematical simplifications in Equation reveal that all DFT frequency outputs X(k) can be computed as the sum of the outputs of two length-N2 DFTs, of the even-indexed and odd-indexed discrete-time samples, respectively, where the odd-indexed short DFT is multiplied by a so-called twiddle factor term WkN=e−(i2πkN). This is called a decimation in time because the time samples are rearranged in alternating groups, and a radix-2 algorithm because there are two groups. Figure graphically illustrates this form of the DFT computation, where for convenience the frequency outputs of the length-N2 DFT of the even-indexed time samples are denoted G(k) and those of the odd-indexed samples as H(k). Because of the periodicity with N2 frequency samples of these length-N2 DFTs, G(k) and H(k) can be used to compute two of the length-N DFT frequencies, namely X(k) and X(k+N2), but with a different twiddle factor. This reuse of these short-length DFT outputs gives the FFT its computational savings.