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Spectral Lines. Spectral Lines. Gives the frequency composition of the function Amplitude, phase of sinusoidal components Could provide information not found in time signal E.g. Pitch, noise components May help distinguish between signals E.g speech/speaker recognition.

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Spectral lines1
Spectral Lines

  • Gives the frequency composition of the function

    • Amplitude, phase of sinusoidal components

  • Could provide information not found in time signal

    • E.g. Pitch, noise components

  • May help distinguish between signals

    • E.g speech/speaker recognition


Spectral lines example
Spectral Lines Example

QUESTIONS --

DC Yes____ ao = ? No_____ ao = 0

Symmetry

Even____ an = ? bn = 0

Odd____ an = 0 bn = ?

Nether even nor odd ____ an = ? bn = ?

Halfwave symmetry

Yes_____ only odd harmonics

No______ all harmonics

Discontinuities

Yes_____ proportional to1/n

No______ proportional to1/n2

Note ? means find that variable.

Comment on the general form of the Fourier Series coefficients [an and/or bn.]

X

X

X

X





The fourier transform
The Fourier Transform

  • Applied to aperiodic signals

  • Could be continuous or discrete

  • Why Transform and not series?

  • Do we want continuous or discrete?


The fourier transform1
The Fourier Transform

  • Continuous Fourier Transform:


The fourier transform2
The Fourier Transform

  • The Discrete Fourier Transform:


Famous fourier transforms
Famous Fourier Transforms

Sinusoid

Delta function


Famous fourier transforms1
Famous Fourier Transforms

Gaussian

Gaussian


Famous fourier transforms2
Famous Fourier Transforms

Sinc function

Square wave


Famous fourier transforms3
Famous Fourier Transforms

Exponential

Lorentzian


Fast fourier transform
Fast Fourier Transform

  • The Fast Fourier Transform (FFT) is a very efficient algorithm for performing a discrete Fourier transform

  • FFT principle first used by Gauss in 18??

  • FFT algorithm published by Cooley & Tukey in 1965

  • In 1969, the 2048 point analysis of a seismic trace took 13 ½ hours. Using the FFT, the same task on the same machine took 2.4 seconds!


Fft in matlab
FFT in matlab

  • Assign your time variables

    • t = [0:255];

  • Assign your function

    • y = cos(2*pi*n/10);

  • Choose the number of points for the FFT (preferably a power of two)

    • N = 2048;

  • Use the command ‘fft’ to compute the N-point FFT for your signal

    • Yf = abs(fft(y,N));

  • Use the ‘fftshift’ command to shift the zero-frequency component to center of spectrum for better visualization of your signals spectrum

    • Yf= fftshift(Yf);

  • Assign your frequency variable which is your x-axis for the spectrum

    • f = [-N/2:N/2-1]/N; - this is the normalized frequency symmetrical about f0 and about the y-axis

  • Plot the spectrum

    • plot(f, Yf)


Fft in matlab1
FFT in matlab

  • Vary the fundamental frequency and see what happens



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