The Continuous - Time Fourier Transform (CTFT)

1. The Continuous - Time Fourier Transform (CTFT)

2. Extending the CTFS • The CTFS is a good analysis tool for systems with periodic excitation but the CTFS cannot represent an aperiodic signal for all time • The continuous-time Fourier transform (CTFT) can represent an aperiodic signal for all time

3. Objective • To generalize the Fourier series to include aperiodic signals by defining the Fourier transform • To establish which type of signals can or cannot be described by a Fourier transform • To derive and demonstrate the properties of the Fourier transform

4. CTFS-to-CTFT Transition . . . . . .

5. CTFS-to-CTFT Transition Below are plots of the magnitude of X[k] for 50% and 10% duty cycles. As the period increases the sinc function widens and its magnitude falls. As the period approaches infinity, the CTFS harmonic function becomes an infinitely-wide sinc function with zero amplitude.

6. CTFS-to-CTFT Transition 10% duty cycle 50% duty cycle

7. CTFS-to-CTFT Transition

8. Definition of the CTFT (fform) Forward Inverse Commonly-used notation

9. Definition of the CTFT (ω form) Forward Inverse Commonly-used notation

10. Convergence and the Generalized Fourier Transform This integral does not converge so, strictly speaking, the CTFT does not exist.

11. Convergence and the Generalized Fourier Transform (cont…) But consider a similar function, Its CTFT integral does converge.

12. Convergence and the Generalized Fourier Transform (cont…) Now let s approach zero.

13. Convergence and the Generalized Fourier Transform (cont…) By a similar process it can be shown that and These CTFT’s which involve impulses are called generalized Fourier transforms (probably because the impulse is a generalized function).

14. Negative Frequency This signal is obviously a sinusoid. How is it described mathematically? It could be described by But it could also be described by

15. Negative Frequency (cont…) x(t) could also be described by or and probably in a few other different-looking ways. So who is to say whether the frequency is positive or negative? For the purposes of signal analysis, it does not matter.

16. CTFT Properties Linearity

17. CTFT Properties (cont…) Time Shifting Frequency Shifting

18. CTFT Properties (cont…) Time Scaling Frequency Scaling

19. The “Uncertainty” Principle The time and frequency scaling properties indicate that if a signal is expanded in one domain it is compressed in the other domain. This is called the “uncertainty principle” of Fourier analysis.

20. CTFT Properties (cont…) Transform of a Conjugate Multiplication- Convolution Duality

21. CTFT Properties (cont…) In the frequency domain, the cascade connection multiplies the transfer functions instead of convolving the impulse responses.

22. CTFT Properties (cont…) Time Differentiation Modulation Transforms of Periodic Signals

23. CTFT Properties (cont…) Parseval’s Theorem Even though an energy signal and its CTFT may look quite different, they do have something in common. They have the same total signal energy.

24. CTFT Properties (cont…) Integral Definition of an Impulse Duality

25. CTFT Properties (cont…) Total area under a time or frequency-domain signal can be found by evaluating its CTFT or inverse CTFT with an argument of zero Total-Area Integral

26. CTFT Properties (cont…) Integration