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

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

Triangular Wave

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

Triangular Wave

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

Convergence of Fourier Synthesis

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

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

Stepped Frequency Sinusoids

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

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

Frequency Modulation: Chirp Signals

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