Fft Analysis

Frequency Analysis – Fast Fourier Transform (FFT)
Dr Michael Sek

FREQUENCY ANALYSIS FAST FOURIER TRANSFORM, FREQUENCY SPECTRUM
FOURIER SERIES
In the exercises of module "FUNCTIONS IN MATLAB – STRUCTURED PROGRAMMING" you have developed a function that can generate a cosinusoidal (harmonic) signal {x} with the given amplitude xo , frequency f in Hz and the initial phase φo in degrees for thegiven time vector {t}, according to the equation x (t ) = x o cos(2πft + ϕ o ) . In this Module you will use this function to generate signals that comprise a known blend of various harmonics to practice their decomposition into the frequency components with the Fourier Transform spectrum.
xo=1 f = 1Hz φo=-90o xo=1/3 f = 3Hz φo=-90o xo=1/5 f = 5Hz φo=-90o

Notice that a sinusoid is the same ascosinusoid with the initial phase of -90o. The effect of summing more and more terms of a series of sinusoids in the time span of one second by applying the following pattern of amplitude: 1, 1/3, 1/5, 1/7, 1/9 etc and the corresponding pattern of frequency: 1, 3, 5, 7, 9 etc. The summations of the first 2,4 and 6 terms for the two cases are illustrated are illustrated in Figure 1 and Figure 2.Two cases are shown: • Figure 1 - the initial phase is -90o for all cosinusoidal terms, i.e. signals start from 0 and are sinusoids. The combination produces a square wave as seen below for the sum of 100 components.

xo=1/7 f = 7Hz φo=-90o

xo=1/9 f = 9Hz φo=-90o

xo=1/11 f = 11Hz φo=-90o

1+2

1 to 4

1 to 6

Figure 1 First six frequency components of a square wave

xo=1 f = 1Hzφo-random

xo=1/3 f = 3Hz φo-random

xo=1/5 f = 5Hz φo-random

xo=1/7 f = 7Hz φo-random

xo=1/9 f = 9Hz φo-random

xo=1/11 f = 11Hz φo-random

1+2

1 to 4

• 1 to 6

Figure 2 First six frequency components of a random signal Figure 1 and Figure 2 are illustrations of the Fourier Theorem and the Fourier Series.

Figure 2 - the initial phase of each cosinusoid varies and isallocated from the uniform random distribution between –π and +π radians (i.e. -180o and 180o), i.e. signals start randomly. The combination produces a random signal.

FOURIER SERIES A periodic function can be represented as a sum of infinite number of (co-)sinusoidal components at equally spaced frequencies with the interval of 1/T, where T is the period of the function, the so called Fourier Series.1

Frequency Analysis – Fast Fourier Transform (FFT)
Dr Michael Sek

FREQUENCY SPECTRUM
When harmonic components of a signal are known, the signal can be presented in a different way that highlights its frequency content rather than its time domain content. Introducing the third axis of frequency perpendicular to the amplitude-time plane the harmonic components can be plotted in the planethat corresponds to their frequencies. For example, a random signal similar to the one in Figure 2 can be presented as shown in Figure 3 in such a way that both time and frequency details are visible.

Figure 3 Time-Frequency-Amplitude representation of a random signal The graph in Figure 3 can be rotated in such a way that the time axis is perpendicular to the observer. This frequency domainview when the time axis is no longer visible and only positive values of magnitude of each sine are plotted, is called the magnitude spectrum. Figure 4 depicts the magnitude spectrum of the signals shown in Figure 1, Figure 2 or Figure 3. Components of this spectrum appear as lines to reflect the fact that they are planes in which the cosines are placed. A spectrum however can be plotted showing onlythe end points of each line connected together. For a spectrum with a large number of points the gaps between the lines are barely visible and the spectrum appears continuous. Irrespectively, the components of a spectrum are traditionally called the lines.

Figure 4 Magnitude spectrum Magnitude spectrum is not sufficient to fully define the signal. The phase spectrum of each frequency...