Lect4 Discrete Fourier Transform (DFT) and Fast Fourier Transform (FFT)

4.1 DFT In practice the Fourier components of data are obtained by digital computation rather than by analog processing. The analog values have to be sampled at regular intervals and the sample values are converted to a digital binary representation by using ADC.

4.2 Inverse Discrete Fourier Transform IDFT IDFT is used to carry out discrete transformation from the frequency to the time domain. IDFT is given, IDFT is defined by

4.3 Properties of the DFT 1) Symmetry 𝑅𝑒[𝑋(𝑁−𝑘)]=𝑅𝑒[𝑋(𝑘)] … (4.6) 𝐼𝑚[𝑋(𝑁−𝑘)]=−𝐼𝑚[𝑋(𝑘)] … (4.7) 2) Perceval's theorem: The normalized energy in the signal is given by either of the expressions 3) Delta Function 𝐹𝐷[𝛿(𝑛𝑇)]=1 …. (4.9)

4) Convolution (a) Time Convolution (b) Frequency Convolution

4.4 Computational complexity of the DFT For an 8 point DFT the expansion for X(kΩ) becomes Eq(4.15) contains eight terms on the right hand side. Each term consists (8) complex multiplications and seven complex addition to be calculated. For 1024 point DFT required (1024)2 complex multiplication and 1024×1023 addition. - Thus amount computation involved may be reduced if we note that there is amount of redundancy in computation of eq(4.15) due to the rotation factor.

4.5 Decimation in Time FFT The decimation in time FFT algorithm is based on splitting (decimating) x[n] into smaller sequence and finding X(k) from the DFT's of these decimated sequences. Let x[n] be a sequence of length N=2𝑃(i.e Radix -2) and suppose that x[n] is split (decimated) into two subsequence each of length (N/2) as shown in fig (4.1), the first sequence, is found from the even index terms 𝑔[𝑛]=𝑥[2𝑛] 𝑛=0.1.2.….𝑁/2−1 an the second sequence, h[n] is formed from, the odd index ℎ[𝑛]=𝑥[2𝑛+1] 𝑛=0.1.2.….𝑁/2−1

In terms of these sequence the N-point DFT of x[n] is

a) The butterfly which is the basic computational element of the FFT algorithm. b) A simplified butterfly, with only one complex multiplication.

4.6 Data Shuffling and Bit Reversal

4.7 Decimation in Frequency FFT Another class of FFT algorithms may be derived by decimating the output sequence X(k) into smaller and smaller subsequences.