Lect4 Discrete Fourier Transform (DFT) and Fast Fourier Transform (FFT) - PowerPoint Presentation

Slide1

Lect4

Discrete Fourier Transform (DFT) and Fast Fourier Transform (FFT) Slide2

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. Slide3

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 Slide4

4.3 Properties of the DFT

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) 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

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) Delta Function

𝐹𝐷

[𝛿(𝑛𝑇)]=1 …. (4.9) Slide5

4) Convolution

(a) Time Convolution

(b) Frequency Convolution Slide6

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. Slide7
Slide8

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 Slide9

In terms of these sequence the N-point DFT of x[n] is Slide10

a) The butterfly which is the basic computational element of the FFT algorithm.

b) A simplified butterfly, with only one complex multiplication. Slide11

4.6 Data Shuffling and Bit Reversal Slide12

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.