Time and Frequency Representation The most common representation of signals and waveforms is in the time domain Most signal analysis techniques only work in the frequency domain This can be a difficult concept when first introduced to it The frequency domain is just another way of representing a signal Fist consider a simple sinusoid The time-amplitude axes on which the sinusoid is shown define the time plane. If an extra axis is added to represent frequency then the sinusoid would illustrated as ……

Time and Frequency Representation The frequency-amplitude axes define the frequency plane in the same way as the time-amplitude axes defines the time plane The frequency-plane is orthogonal to the time-plane and intersect with it a line on the amplitude axis. The actual sinusoid can be considered to be existing some distance along the frequency domain

Fourier Series enableservice('automationserver',true) Any periodic function f(t), with period T, may be represented by an infinite series of the form: where the coefficients are calculated from:

Fourier Series Provides a means of expanding a function into its major sine / cosine or complex exponential components These individual terms represent various frequency components which make up the original waveform Example: Square wave

Complex Fourier Series Using Eulers formula to derive the complex expressions for , and substituting these into the Fourier series it can be shown that the complex form of the Fourier series is: where

Discrete Fourier Transform (DFT) The Fourier transform provides the means of transforming a signal in the time domain into one defined in the frequency domain. The DFT is given by: DFTExpanded.m DFT.m Example: Find the DFT of the sequence {1, 0, 0, 1} Solution……..

Discrete Fourier Transform (DFT) Example: Find the DFT of the sequence {1, 0, 0, 1} Solution: { 2, 1+j, 0, 1-j }

Computational Complexity of the DFT Consider an 8-point DFT Letting Each term consists of a multiplication of an exponential term by another term which is either real or complex. Each of the product terms are added together. There are also eight harmonic components (k = 0, … ,7) Therefore for an 8-point DFT there are 82 = 64 multiplications and 8 x 7 additions . For an N-point DFT - N2 multiplications and N(N-1) additions

Computational Complexity of the DFT For an N-point DFT - N2 multiplications and N(N-1) additions Therefore for a 1024-point DFT (N=1024) Multiplications: N2 = 1048576 Additions: N(N-1) = 1047552 Clearly some means of reducing these numbers is desirable

Computational Complexity of the DFT where X(k) x(0) x(1) x(2) x(3) x(4) x(5) x(6) x(7) 1 π/4 π/2 3π/4 π 5π/4 3π/2 7π/4 2 2π 5π/2 3π 7π/2 3 9π/4 15π/4 9π/2 21π/4 4 4π 5π 6π 7π 5 25π/4 15π/2 35π/4 6 9π 21π/2 7 49π/4

FFT Algorithmic Development Computational Savings