EE 313 Linear Systems and Signals Fall 2018

EE 313 Linear Systems and Signals Fall 2018 Fourier Series Prof. Brian L. Evans Dept. of Electrical and Computer Engineering The University of Texas at Austin Textbook: Mc. Clellan, Schafer & Yoder, Signal Processing First, 2003 Lecture 3 http: //www. ece. utexas. edu/~bevans/courses/signals

Periodic Signals – SPFirst Sec. 3 -2 Beat Notes • Occurs when multiplying two sinusoidal signals Audio: singing when holding a note, some musical instruments Communications: amplitude modulation (AM radio, Wi-Fi) • Example: Spectrum Rewrite x(t) as sum of complex sinusoids f 3 -2

Periodic Signals – SPFirst Sec. 3 -2 Beat Notes • Example: Equal to: f 0 = 10; f 1 = 1000; fs = 8000; Ts = 1/fs; t = 0 : Ts : 3; x 0 = cos(2*pi*f 0*t); x 1 = sin(2*pi*f 1*t); x = x 0. * x 1; sound(x, fs); % plot one period T 0 = 1/f 0; n = T 0 / Ts; plot( t(1: n), x(1: n) ); f 2 = 1010; f 3 = 990; fs = 8000; Ts = 1/fs; t = 0 : Ts : 3; x 2 = cos(2*pi*f 2*t - pi/2); x 3 = cos(2*pi*f 3*t - pi/2); x = 0. 5*x 2 + 0. 5*x 3; sound(x, fs); % plot one period f 0 = gcd(f 2, f 3); T 0 = 1/f 0; n = T 0 / Ts; plot( t(1: n), x(1: n) ); 3 -3

Periodic Signals – SPFirst Sec. 3 -3 Periodic Waveforms • A signal has period T if x(t + T) = x(t) for all t Also periodic with periods 2 T, 3 T, etc. , and –T, – 2 T. . . Smallest positive period T 0 is called the fundamental period Fundamental frequency f 0 is computed as 1 / T 0 • Synthesize periodic signals Add two or more cosine waves with harmonic frequencies • Finding fundamental frequency Largest f 0 such that fk = k f 0 , i. e. f 0 = gcd{ fk } Consider notes A 440 Hz, E 660 Hz and F♯ 740 Hz. f 0 = __ 3 -4

Periodic Signals – SPFirst Sec. 3 -3. 1 Notation and Speech Example • Alternate periodic signal synthesis formula Expand cosines into sum of two exponentials (Euler’s formula) Explicitly represents positive and negative frequencies: where • Synthesis of “ah” vowel from SPFirst (page 45) http: //dspfirst. gatech. edu/chapters/03 spect/demos/vowel Matlab command: vowel Comment out lines that start with capture and imwrite Combines components at 200, 400, 500, 1600 and 1700 Hz 3 -5

Periodic Signals – SPFirst Sec. 3 -3. 1 Speech Example Revisited t Magnitude of Spectrum Time domain • A recording of the “ah” vowel being spoken “ah” recording f t Magnitude of Spectrum Time domain • How close did synthesized vowel sound like “ah”? synthesized sound f 3 -6

Periodic Signals – SPFirst Sec. 3 -4 Fourier Series • Any periodic signal can be synthesized With a sum of harmonically related sinusoids • Mathematical theory realized by Fourier series T 0 fundamental period f 0 fundamental frequency (f 0 = 1 / T 0) kth complex exponential in summation has frequency fk = k f 0 • Special case Conjugate symmetric amplitudes: Leads to real-valued x(t): 3 -7

Periodic Signals – SPFirst Sec. 3 -4 Fourier Series • Analysis: start with x(t) and compute { ak } Integrate x(t) over fundamental period T 0 Calculation of a 0 simplifies to average value of x(t) Can often “eyeball” a 0 without performing integration Example #1: With x(t) = cos(2 p f 0 t), what is a 0 ? Example #2: With x(t) = cos 2(2 p f 1 t), what is a 0 ? • Synthesis: start with { ak } and compute x(t) General case: 3 -8

Periodic Signals – SPFirst Sec. 3 -5 Spectrum of the Fourier Series • Find Fourier series coefficients for x(t) = cos 3(3 pt) Approach #1: Fourier analysis formulas Plot x(t) to find T 0 = 3 s / 4. 5 = 2/3 s and “eyeball” a 0 = 0 and then t Approach #2: Expand into complex exponentials Spectrum • Resulting spectrum w 0 = gcd(3 p, 9 p) = 3 p f 0 = 1. 5 Hz f -3 f 0 -f 0 3 -9

Periodic Signals – SPFirst Sec. 3 -6. 1 Fourier Analysis of a Square Wave • Periodic square wave with 50% duty cycle Defined for one period as x(t) 1 t • Fourier coefficients 1. a 0 = ½ because x(t) is 1 half the time and 0 half the time 2. Then, For k ≠ 0 3 -10

Periodic Signals – SPFirst Sec. 3 -6. 2 Spectrum for a Square Wave • Fourier coefficients Independent of T 0 • Example T 0 = 0. 04 s f 0 = 25 Hz 11 x(t) 0. 01. 02 0. 04 t © 2003 -2016, JH Mc. Clellan & RW Schafer 3 -11

Periodic Signals – SPFirst Sec. 3 -6. 3 Fourier Synthesis of a Square Wave • Synthesis using up to the 7 th harmonic demo © 2003 -2016, JH Mc. Clellan & RW Schafer 3 -12

Periodic Signals – SPFirst Sec. 3 -6 Spectrum & Fourier Series © 2003 -2016, JH Mc. Clellan & RW Schafer 3 -13

Periodic Signals – SPFirst Sec. 3 -6 Fourier Synthesis vs. Analysis • Fourier Synthesis Given (fk, Ak, fk) values, create x(t) Implementing synthesis formula is somewhat straightforward Achieving high perceptual quality in x(t) is difficult, e. g. synthesized speech or music • Fourier Analysis Very difficult task, esp. for physical signals Given x(t), extract (fk, Ak, fk) values. How many? Using (fk, Ak, fk), how close is model to signal? Need mathematical algorithms for computer 3 -14