1. Signals & Systems Lecture 17: Chapter 3 Spectrum Representation

2. 2 Today's lecture  Error signal  Convergence of Fourier synthesis  Frequency modulation  Chirp formula

3. 3 Convergence of Fourier Synthesis  Error Signal:  Worst-case error:

4. 4 Convergence of Fourier Synthesis

5. 5 General Waveforms  Waveforms can be synthesized by the equation x(t) = A 0 + ∑A k cos(2πf k t +  k )  These waveforms maybe –constants –cosine signals ( periodic) –complicated-looking signals (not periodic)  So far we have dealt with signals whose amplitudes, phases and frequencies do not change with time

6. 6 Sinusoid with Changing Frequency

7. 7 Frequency Modulation  Most real-world signals exhibit frequency change over time e.g. music.  Frequency of a signal may change linearly with time which sounds like a siren or chirp  Chirp signal: Signal whose frequency changes linearly with time from some low value to high value  Let x(t) =Acos(ω 0 t +  ) =Acos ψ(t) ψ(t) = ω 0 t +  and dψ(t)/dt = ω 0 where ψ(t) denotes the time varying angle function

8. 8 Frequency Modulation  We can create a signal with quadratic angle function by defining ψ(t) = 2πμt 2 + 2πf 0 t +   instantaneous frequency = slope of the angle function ω i = dψ(t)/dt f i (t) = 1/2 π dψ(t)/dt f i (t) = 2μt + f 0

9. 9 Example 3.8: Synthesize a Chirp Formula Synthesize a frequency sweep from f 1 = 220 Hz to f 2 = 2320 Hz over a 3-second time interval. f i (t) = (f 2 - f 1 )t / T 2 + f 1 ψ i (t) = ∫ ω i (u) du 0 t

10. 10 Frequency Modulation: Chirp Signals