In-Place Computation | Signals and Systems - Electronics and Communication Engineering (ECE) PDF Download

This efficient use of memory is important for designing fast hardware to calculate the FFT. The term in-place computation is used to describe this memory usage.

Decimation in Time Sequence

In this structure, we represent all the points in binary format i.e. in 0 and 1. Then, we reverse those structures. The sequence we get after that is known as bit reversal sequence. This is also known as decimation in time sequence. In-place computation of an eight-point DFT is shown in a tabular format as shown below −

Decimation in Frequency Sequence

Apart from time sequence, an N-point sequence can also be represented in frequency. Let us take a four-point sequence to understand it better.

Let the sequence be   

We will group two points into one group, initially. Mathematically, this sequence can be written as;

Now let us make one group of sequence number 0 to 3 and another group of sequence 4 to 7. Now, mathematically this can be shown as;

Let us replace n by r, where r = 0, 1 , 2….(N/2-1). Mathematically,

We take the first four points (x[0], x[1], x[2], x[3]) initially, and try to represent them mathematically as follows −

We can further break it into two more parts, which means instead of breaking them as 4-point sequence, we can break them into 2-point sequence.