Introduction To Radix Two Fft (Fast Fourier Transform ) Algorithm

Radix 2 FFT(Fast Fourier Transform)

As nosotros know that, the discrete Fourier transform (DFT) plays an of import role inwards many applications of digital signal processing, including linear filtering, correlation analysis, together with spectrum analysis. H5N1 major argue for its importance is the beingness of efficient algorithms for computing the DFT. The principal topic of this article is the description of computationally efficient algorithms for evaluating the DFT. Two dissimilar approaches are used for calculating FFT. One is a divide-and-conquer approach inwards which a DFT of size N, where due north is a composite number, is reduced to the computation of smaller DFTs from which the larger DFT is computed. In particular, nosotros acquaint of import computational algorithms, called fast Fourier transform (FFT) algorithms, for computing the DFT.

Let us reckon the computation of the N = 2^v point DFT yesteryear the divide-and-conquer approach. We split the N-point information sequence into 2 N/2-point information sequences f₁(n) and f₂(n), corresponding to the even-numbered together with odd-numbered samples of x(n), respectively, that is,

Thus f₁(n) and f₂(n) are obtained yesteryear decimating x(n) yesteryear a element of 2, together with thence the resulting FFT(Fast Fourier Transform) algorithm is called a decimation-in-time algorithm.

TAG: View Matlab code for Radix-2 decimation inwards Frequency

Now the N-point DFT tin survive expressed inwards price of the DFT's of the decimated sequences every bit follows:

But W_N² = W_N/2. With this substitution, the equation tin survive expressed as

where F₁(k) and F₂(k) are the N/2-point DFTs of the sequences f₁(m) and f₂(m), respectively. Since F₁(k) and F₂(k) are periodic, amongst menses N/2, nosotros have F₁(k+N/2) = F₁(k) and F₂(k+N/2) = F₂(k). In addition, the element W_N^k+N/2 = -W_N^k. Hence the equation may survive expressed as

We disclose that the straight computation of F₁(k) requires (N/2)² complex multiplications. The same applies to the computation of F₂(k). Furthermore, at that topographic point are N/2 additional complex multiplications required to compute W_N^kF₂(k). Hence the computation of X(k) requires 2(N/2)² + N/2 = N ²/2 + N/2 complex multiplications. This firstly pace results inwards a reduction of the seat out of multiplications from N ² to N ²/2 + N/2, which is most a element of 2 for due north large.

Figure TC.3.1 First pace inwards the decimation-in-time algorithm

By computing N/4-point DFTs, we would obtain the N/2-point DFTs F₁(k) and F₂(k) from the relations

The decimation of the information sequence tin survive repeated i time to a greater extent than together with i time to a greater extent than until the resulting sequences are reduced to one-point sequences. For N = 2^v, this decimation tin survive performed v = log₂N times. Thus the full seat out of complex multiplications is reduced to (N/2)log₂N. The seat out of complex additions is Nlog₂N.
For illustrative purposes, Figure TC.3.2 depicts the computation of N = 8 point DFT. We disclose that the computation is performed inwards 3 stages, get-go amongst the computations of 4 two-point DFTs, then 2 four-point DFTs, and finally, i eight-point DFT. The combination for the smaller DFTs to shape the larger DFT is illustrated inwards Figure TC.3.3 for N = 8.

Figure TC.3.2 Three stages inwards the computation of an N = 8-point DFT

Figure TC.3.3 Eight-point decimation-in-time Fast Fourier Transform algorithm

Figure TC.3.4 Basic butterfly computation inwards the decimation-in-time FFT algorithm

An of import observation is concerned amongst the gild of the input information sequence later it is decimated (v-1) times. For example, if nosotros reckon the instance where N = 8, we know that the firstly decimation yields the sequence x(0), x(2), x(4), x(6), x(1), x(3), x(5), x(7), and the minute decimation results inwards the sequence x(0), x(4), x(2), x(6), x(1), x(5), x(3), x(7). This shuffling of the input information sequence has a well-defined gild every bit tin survive ascertained from observing Figure TC.3.5, which illustrates the decimation of the eight-point sequence.

Figure TC.3.5 Shuffling of the information together with combat reversal

Decimation inwards frequency FFT algorithm

Another of import radix-2 FFT algorithm, called the decimation-in-frequency algorithm, is obtained yesteryear using the divide-and-conquer approach. To derive the algorithm, nosotros laid out yesteryear splitting the DFT formula into 2 summations, i of which involves the amount over the firstly N/2 information points together with the minute amount involves the in conclusion N/2 information points. Thus nosotros obtain

TAG: View Matlab code for Radix-2 decimation inwards Frequency

Now, permit us split (decimate) X(k) into the even- together with odd-numbered samples. Thus nosotros obtain

where nosotros accept used the fact that W_N² = W_N/2
The computational physical care for inwards a higher house tin survive repeated through the decimation of the N/2-point DFTs X(2k) and X(2k+1). The entire physical care for involves v = log₂N stages of decimation, where each phase involves N/2 butterflies of the type shown inwards Figure TC.3.7. Consequently, the computation of the N-point DFT via the decimation-in-frequency FFT requires (N/2)log₂N complex multiplications and Nlog₂N complex additions, merely every bit inwards the decimation-in-time algorithm. For illustrative purposes, the eight-point decimation-in-frequency algorithm is given inwards Figure TC.3.8.

Figure TC.3.6 First phase of the decimation-in-frequency FFT algorithm

Figure TC.3.7 Basic butterfly computation inwards the decimation-in-frequency.

Figure TC.3.8 N = 8-point decimation-in-frequency FFT algorithm

We disclose from Figure TC.3.8 that the input information x(n) occurs inwards natural order, but the output DFT occurs inwards bit-reversed order. We every bit good authorities annotation that the computations are performed inwards place. However, it is possible to reconfigure the decimation-in-frequency algorithm so that the input sequence occurs inwards bit-reversed gild spell the output DFT occurs inwards normal order. Furthermore, if nosotros abandon the requirement that the computations survive done inwards place, it is every bit good possible to accept both the input information together with the output DFT inwards normal order. That what all most Radix 2 FFT algorithm.

Reference Book: Digital Signal Processing Fourth Edition yesteryear John G. Proakis together with Dimitris G. Manolakis

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