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Fourier series. The frequency domain. It is sometimes preferable to work in the frequency domain rather than time Some mathematical operations are easier in the frequency domain the human ear works on frequencies Working in this domain really means the x axis is f and not t.

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Presentation Transcript
the frequency domain
The frequency domain
  • It is sometimes preferable to work in the frequency domain rather than time
    • Some mathematical operations are easier in the frequency domain
    • the human ear works on frequencies
  • Working in this domain really means the x axis is f and not t.
  • We need a method to convert time domain functions into frequency domain functions
fourier series3
Fourier Series
  • A sinusoid can be represented by
  • A sinusoid can be expressed as a sum of a sine and cosine at the same frequency but possibly different magnitudes independently of phase
slide4
Adding sinusoids with freq F results in a sinusoid with frequency F

x(t)=sin(4πt) + 0.6cos(4πt)

slide6
Any periodic waveform can be represented as an infinite sum of sine and cosine waves regardless of phase (as shown in the previous slide). This is the Fourier Series f(t)
  • Or more succinctly written as:
  • a0=const, an and bn are the amplitudes of the individual harmonics making up the periodic waveform
discrete fourier transform dft
Discrete Fourier Transform DFT
  • The data we will use is sampled and an infinite number of samples is impractical
  • Works with non continuous non periodic functions
  • N time domain samples transform to N complex DFT values in the frequency domain
periodic function generation
Periodic function generation
  • Remember that you created a sine wave digitally using:
slide9
The DFT is:
  • Where F is effectively a row matrix of size N
  • h is the harmonic
  • n is the time domain sample number
  • x(n) is the magnitude of the nth sample
  • N is the total number of samples
slide10
As each Fourier coefficient F(h) is complex so its magnitude and phase (with respect to the fundamental) need to be calculated:
example
Example
  • Consider 4 samples of a waveform from the time domain (from an a to d converter) {1,0,0,1}
slide12
Show that x(2)=0 and x(3)=1-j
  • So the DFT of a time vector {1,0,0,1}is a vector {2,1+j,0,1-j}
  • These coefficients are frequency independent as no account for frequency has been taken. We know that the second coefficient F(1) will represent the fundamental and the next one F(2) will be the next harmonic etc. If we need to find the fundamental frequency, we need to specify the coefficient in terms of sample frequency Fs where
power phase diagram
Power phase diagram
  • Often wish to represent Fourier spectrum diagrammatically
  • Power is magnitude squared
  • Phase is angle
  • Line up power graph with phase graph
  • Plot actual frequencies is sample rate known
  • Usually only plot samples from 0 to N/2
slide14
From example x = {1,0,0,1}giving Fourier coefficients of {2,1+j,0,1-j}
  • If Fs=1000Hz, F1=1000/N=250, F2=500, F3=750
  • Polar form of power and phase:
slide15

M

4

2

F

0

250

1k

500

750

90

45

0

-45

-90