1. Lecture 10: Fourier Analysis of Periodic Signals
Signals and Systems
Lecture 10:
Fourier Analysis of Periodic Signals

2. Today's lecture Fourier Analysis of Triangular Wave
Convergence of Fourier Synthesis
Frequency Modulation

3. Triangular Wave

4. Triangular Wave
ak = (e -jkπ – 1)/ k2 π2

5. Triangular Wave

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

7. Convergence of Fourier Synthesis

8. General Waveforms Waveforms can be synthesized by the equation
x(t) = A0 + ∑Ak cos(2πfkt +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

9. 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 ψ(t) = ω0t + and dψ(t)/dt = ω0
where ψ(t) denotes the time varying angle function

10. Stepped Frequency Sinusoids

11. Frequency Modulation
We can create a signal with quadratic angle function by defining
ψ(t) = 2πμt2 + 2πf0t + 
instantaneous frequency = slope of the angle function
ωi = dψ(t)/dt
fi(t) = 1/2 π dψ(t)/dt
fi(t) = 2μt + f0

12. Example 3.8: Synthesize a Chirp Formula
Synthesize a frequency sweep from f1 = 220 Hz to f2 = 2320 Hz over a 3-second time interval.
fi(t) = (f2 - f1)t / T2 + f1
ψi(t) = ∫ ωi(u) du t

13. Frequency Modulation: Chirp Signals

14. Assignment #2 End Chapter Problems P- 3.8 P- 3.10 P- 3.12 P- 3.14
Due on Tuesday 3rd March 2009