1. Fourier Transform Periodicity of Fourier series
Limiting behaviour of Fourier series
Limiting form of Fourier series
Fourier transform pairs
Existence of Fourier transform
„Signal Theory” Zdzisław Papir

2. Periodicity of Fourier series
sawtooth signal
time t
period T = 1
„Signal Theory” Zdzisław Papir

3. Limiting behaviour of Fourier series
„Signal Theory” Zdzisław Papir
x(t)
time t
-T/2
xT(t)
Periodic extension of a signal window xT(t)
through Fourier series
+T/2

4. Limiting behaviour of Fourier series
„Signal Theory” Zdzisław Papir

5. Limiting behaviour of Fourier series
„Signal Theory” Zdzisław Papir

6. Limiting behaviour of Fourier series
„Signal Theory” Zdzisław Papir

7. Limiting behaviour of Fourier series
amplitude of the 1st spectrum line of an exponential puls
Fourier series window
„Signal Theory” Zdzisław Papir

8. Limiting behaviour of Fourier series
Fourier series window T
amplitude spectrum – exponential pulse
„Signal Theory” Zdzisław Papir

9. Limiting behaviour of Fourier series
amplitude spectrum – exponential pulse
Fourier series window 3T
„Signal Theory” Zdzisław Papir

10. Limiting behaviour of Fourier series
amplitude spectrum – exponential pulse
Fourier series window 10T
„Signal Theory” Zdzisław Papir

11. Limiting behaviour of Fourier series
amplitude spectrum – exponential pulse
Fourier series window 100T
„Signal Theory” Zdzisław Papir

12. Limiting behaviour of Fourier series
Squeezing Fourier series coefficients in FREQUENCY:
Squeezing Fourier series coefficients in AMPLITUDE:
„Signal Theory” Zdzisław Papir

13. Riemann integral a b x
f(x)
„Signal Theory” Zdzisław Papir

14. Limiting form of Fourier series
Fourier series coefficients:
FORWARD FOURIER TRANSFORM:
„Signal Theory” Zdzisław Papir

15. Limiting form of Fourier series
INVERSE FOURIER TRANSFORM:
„Signal Theory” Zdzisław Papir

16. Fourier Integral Theorem
„Signal Theory” Zdzisław Papir

17. Fourier integral theorem
Fourier transform
Fourier integral theorem
Forward
Fourier transform
„Signal Theory” Zdzisław Papir

18. Inverse Fourier transform
„Signal Theory” Zdzisław Papir

19. Fourier transform pairs
FORWARD
INVERSE
TRANSFORM PAIRS
„Signal Theory” Zdzisław Papir

20. Fourier transform pairs
FORWARD FOURIER TRANSFORM:
„Signal Theory” Zdzisław Papir

21. Fourier transform pairs
time t
FOURIER TRANSFORM:
„Signal Theory” Zdzisław Papir

22. Fourier transform pairs1
-T/2
T/2
frequency f
FOURIER TRANSFORM:
„Signal Theory” Zdzisław Papir

23. Fourier transform pairs
frequency f
FOURIER TRANSFORM:
„Signal Theory” Zdzisław Papir

24. Existence of Fourier transform
Dirichlet conditions are necessary for Fourier transform existence.
Signal x(t) must have only a finite number of maxima and minima, as well as a finite number of discontinuities over the entire range [–, + ].
Signal x(t) is also allowed to be unbounded provided that it is absolutely integrable:
„Signal Theory” Zdzisław Papir

25. Summary
Fourier series is a spectral decomposition of periodic signal or produces a periodic extension of signal window.
Fourier transform is a tool for spectral decomposition of nonperiodic signals.
Fourier transform is a limiting case of Fourier series with signal window being extended up to infinity.
Dirichlet conditions are necessary for Fourier transform existence.
In engineering applications it is commonly assumed that signals of limited energy are Fourier transformable.
„Signal Theory”  Zdzisław Papir