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Time and Frequency RepresentationThe most common representation of signals and waveforms is in the

time domainMost signal analysis techniques only work in the frequency domainThis can be a difficult concept when first introduced to itThe frequency domain is just another way of representing a signalFist 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 ……

0 0.5 1 1.5 2 2.5

-1

-0.5

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0.5

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itude

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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

01

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Fourier SeriesAny periodic function f(t), with period T, may be represented by an

infinite series of the form:

where the coefficients are calculated from:

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Fourier SeriesProvides 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

0 0.5 1 1.5 2 2.5 3 3.50

0.2

0.4

0.6

0.8

1

1.2

1.4

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Complex Fourier SeriesUsing 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

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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……..

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Discrete Fourier Transform (DFT)Example: Find the DFT of the sequence {1, 0, 0, 1}

Solution: { 2, 1+j, 0, 1-j }

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Computational Complexity of the DFTConsider 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

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Computational Complexity of the DFTFor an N-point DFT - N2 multiplications and N(N-1)

additions

Therefore for a 1024-point DFT (N=1024)Multiplications: N2 = 1048576Additions: N(N-1) = 1047552

Clearly some means of reducing these numbers is desirable

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Computational Complexity of the DFT

where

X(k) x(0) x(1) x(2) x(3) x(4) x(5) x(6) x(7)

0 0 0 0 0 0 0 0 0

1 0 π/4 π/2 3π/4 π 5π/4 3π/2 7π/4

2 0 π/2 π 3π/2 2π 5π/2 3π 7π/2

3 0 3π/4 3π/2 9π/4 3π 15π/4 9π/2 21π/4

4 0 π 2π 3π 4π 5π 6π 7π

5 0 5π/4 5π/2 15π/4 5π 25π/4 15π/2 35π/4

6 0 3π/2 3π 9π/2 6π 15π/2 9π 21π/2

7 0 7π/4 7π/2 21π/4 7π 35π/4 21π/2 49π/4

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Computational Savings

FFT Algorithmic Development

Multiplies Adds Multiplies Adds2 4 2 1 24 16 12 4 88 64 56 12 2416 256 240 32 6432 1024 992 80 16064 4096 4032 192 384128 16384 16256 448 896256 65536 65280 1024 2048512 262144 261632 2304 46081024 1048576 1047552 5120 102402048 4194304 4192256 11264 225284096 16777216 16773120 24576 491528192 67108864 67100672 53248 106496

DFT FFTN