1. S. Mandayam/ ECOMMS/ECE Dept./Rowan University Signals & Systems & Music ECE.09.402 Spring 2010 Shreekanth Mandayam ECE Department Rowan University March 8, 2010

2. S. Mandayam/ ECOMMS/ECE Dept./Rowan UniversityPlan Sinusoidal signals Fundamental Harmonics Non-sinusoidal signals Square wave Fourier Series for non-sinusoidal signals Why is this important?

3. S. Mandayam/ ECOMMS/ECE Dept./Rowan University Sinusoidal Signal t s(t) A T 0 = 1/f 0 >> A=1; f=400; >> t = 0:1/8000:1; >> s = A*cos(2*pi*f*t); >> sound (s); Instrument Demo: Fundamental and harmonics

4. S. Mandayam/ ECOMMS/ECE Dept./Rowan University Non-Sinusoidal Signal e.g. A Square Wave A -A w(t) t >> A=1; f=400; >> t = 0:1/8000:1; >> w = A*square(2*pi*f*t); >> sound (w); Instrument Demo: Listen to square wave T 0 = 1/f 0

5. S. Mandayam/ ECOMMS/ECE Dept./Rowan University Fourier Series Any periodic non-sinusoidal signal Infinite sum of sines and cosines at harmonic frequencies Fourier Series

6. S. Mandayam/ ECOMMS/ECE Dept./Rowan University Fourier Series Fourier Series Applet: http://www.gac.edu/~huber/fourier/ Any periodic non-sinusoidal signal Infinite sum of sines and cosines at harmonic frequencies Fourier Series

7. S. Mandayam/ ECOMMS/ECE Dept./Rowan University Why is this important? “Linear” System f1f1 f1f1 f2f2 f2f2 f3f3 f3f3 Sinusoid signal in Sinusoid signal out No “new” frequencies! Electric circuits Mechanical systems (musical instrument)

8. S. Mandayam/ ECOMMS/ECE Dept./Rowan UniversitySummary