1. Continuous Time Signals All signals in nature are in continuous time

2. From Discrete Time to Continuous Time A continuous time signals can be viewed as the limit of a discrete time signal with sampling interval

3. From Discrete Time FT (DTFT) … We saw the DTFT of a discrete time signal Substitute and obtain:

4. … to Continuous Time FT Now take the limit so that discrete time -> cont. time Then we obtain the Fourier Transform sampling freq -> infinity sum -> integral

5. Fourier Transform We want to represent a signal in terms of its frequency components. Define: Fourier Transform (FT)

6. Example of a Fourier Transform Take a Rectangular Pulse

7. Example of a Fourier Transform

8. Properties of the FT: 1. Symmetry If the signal is real, then its FT is symmetric as since Example: just verify the previous example

9. Symmetry of the FT Magnitude has “even” symmetry Phase has “odd” symmetry

10. Properties of the FT: 2. Time Shift since In other words a time shift affects the phase, not the magnitude

11. Bandwidth of a Baseband Signal A Baseband Signal has all frequency components at the low frequencies, around F=0 Hz; Bandwidth: the frequency interval where most of the frequency components are.

12. What does it mean? If you take the signal at two different times and with then since

13. For Example: zoom samples spaced by less than 0.1msec are fairly close to each other

14. Computation of the Fourier Transform Whatever we do, physical signals are in continuous time and, as we have seen, they are described by the FT; The FT can be computed in one of two ways: 1.Analytical: if we have an expression of the signal (like in the example); 2.Numerical: by approximation using the Fast Fourier Transform (FFT).

15. Fourier Transform and FFT Consider a signal of a finite duration with Bandwidth. Then we can approximate, by simple arguments, where (say at least an order of magnitude smaller)

16. Fourier Transform and FFT Using the FFT: Take an even integer. Then compute the N point FFT of the sampled data, padded with zeros: Assign the frequencies: positive frequencies negative frequencies

17. Example Take a sinusoid with frequency and length Let the sampling frequency be

18. Example X=fft(x, N); F=(-N/2:N/2 -1)*Fs/N; plot(F,fftshift(20*log10(abs(X))))

19. Example (Zoom in at the Peak) Max at F=10kHz Sidelobes due to finite data length

20. Complex Signals All signals in nature are real. There is not such as a thing as “complex” signal. However in many cases we are interested in processing and transmitting “pairs” of signals. We can analyze them “as if” they were just one complex signal: Real Signals Complex Signal

21. Amplitude Modulation: Real Signal You want to transmit a signal over a medium (air, water, space, cable…). You need to “modulate it” by a carrier frequency:

22. Amplitude Modulation: Complex Signal However most of the times the signal we modulate is Complex Notice now that the modulated signal is real and it contains both signals a(t) and b(t).

23. FT of Modulated Signal See the different steps:

24. FT of Modulated Signal Put things together: Usually