Signal Processing First Lecture 5 Periodic Signals, Harmonics & Time-Varying Sinusoids

READING ASSIGNMENTS Fourier Series ANALYSIS This Lecture: Chapter 3, Sections 3-2 and 3-3 Chapter 3, Sections 3-7 and 3-8 Next Lecture: Fourier Series ANALYSIS Sections 3-4, 3-5 and 3-6

COMPLEX AMPLITUDE General Sinusoid Signal Complex AMPLITUDE or Phasor = X Then, any Sinusoid Cosine Signal = REAL PART of Xejwt

Summary: GENERAL FORM Each sinusoid is decomposed into two rotating phasor pairs, one positive frequency and another with negative.

INVERSE Euler’s Formula Solve for cosine (or sine)

SPECTRUM Interpretation Cosine = sum of 2 complex exponentials: t j A e 7 2 ) cos( - + = One has a positive frequency The other has a negative freq. Amplitude of each is half as big Spectrums of a signal are the representation of complex amplitudes or phasors at the relevant sinusoidal frequencies.

GRAPHICAL SPECTRUM EXAMPLE of SINE w 7 -7 AMPLITUDE, PHASE & FREQUENCY are shown Note that the complex amplitudes are complex conjugate.

SPECTRUM of SINE Sine = sum of 2 complex exponentials: Positive freq. has phase = -0.5p Negative freq. has phase = +0.5p

GRAPHICAL SPECTRUM EXAMPLE of SINE w 7 -7 AMPLITUDE, PHASE & FREQUENCY are shown Note that the complex amplitudes are complex conjugate.

FREQUENCY DIAGRAM Plot Complex Amplitude vs. Freq 100 250 –100 –250 f (in Hz) X(t) = 10 + 7e-j/3 ej2(100)t + 7ej/3 e-j2(100)t + 4ej/2 ej2(250)t + 4e-j/2 e-j2(250)t = 10 + 14 cos(200t - /3) + 8 cos(500t + /2) = 10 + 14sin(200t-/3+/2) + 8sin(500t+/2+/2) (Two-sided spectrum, representation in frequency-domain)

LECTURE OBJECTIVES Signals with HARMONIC Frequencies Add Sinusoids with fk = kf0 FREQUENCY can change vs. TIME Chirps: Introduce Spectrogram Visualization (specgram.m) (plotspec.m)

3.2 Beat Notes Beat note: audio effect created by multiplying two sinusoids having different frequencies => warble, AM Spectrum of a product (5Hz and 0.5Hz), Spectrum components are at +11, -11, +9, -9 corresponding to +-5.5Hz and +-4.5Hz. (neither of original 5Hz and 0.5Hz): a product of two sinusoids with a big difference becomes a sum of two sinusoids with a little difference. A product of two cosine signal will simply have no phase shift.

3.2 Beat Note Waveform Beat notes are produced by adding two sinusoids with nearly identical frequencies. f1=fc-f, f2=fc+f; deviation frequency is assumed to be much smaller than center frequency

EX 3-3: Suppose fc = 200 and f = 20 Hz, so Fig. 3-3 Envelope is obtained by drawing 2cos(2 (20)t) and - 2cos(2(20)t) Signal is ‘beating.’ (fade in and out) Note that f1=+-220Hz and f2=+-180Hz

Decreasing f to 9 Hz causes the envelope of the 200Hz tone changes much more slowly. Fig 3-4 Note that f1=+-209Hz and f2=+-191Hz. When two frequencies are closer, the beating phenomenon becomes less and slow. When they are the same, the two are “in tune”.

3-2.3 Amplitude Modulation Carrier signal fc: carrier frequency >> frequency of v(t):voice When v(t) = 5 + 4cos(40t) and fc = 200 Hz, then (AM is always positive, DC >amp.of cos) Fig 3-5 Amplitude=1

Amplitude Modulation The spectrum is, Fig. 3-7 These subset is a shifted version of the two-sided spectrum of v(t).

SPECTROGRAM EXAMPLE Two Constant Frequencies: Beats

3.3 Periodic Waveforms Nearly Periodic in the Vowel Region Period is (Approximately) T = 0.0065 sec

PERIODIC SIGNALS Repeat every T secs Definition Example: Fundamental period: the smallest interval Speech can be “quasi-periodic”

Harmonic Signal Spectrum Any periodic signal can be synthesized by adding two or more cosine waves that have harmonically related frequencies. f0: fundamental frequency fk : kth harmonic of f0

Define FUNDAMENTAL FREQ Any periodic signal: Great common divisor Such that, f0=gcd{fk}: If the signal is sum of 1.2, 2, and 6 Hz, then f0=0.4 Hz. Fundamental frequency is the largest f0 such that, The fundamental frequency may not be in the spectrum of the signal. (kth harmonic frequency) 1/f0 The fundamental period is defined as the shortest possible period such that x(t+T0) = x(t) For a signal v(k) with a specific k, v(t+1/f0) = v(t). But, fundamental period is kf0 as v(t+1/kf0) = v(t). see ex. 3.3

Harmonic Signal (3 Freqs) 3rd 5th What is the fundamental frequency? 10 Hz

Periodic Waveform

3-3.1 Synthetic Vowel Table 3-1: ak for vowel ‘ah’ with a-k=ak* k fk (Hz) ak Mag Phase 1 100 2 200 386+j 6101 6113 1.508 3 300 4 400 -4433+j 14024 14708 1.877 5 500 24000-j 4498 24418 -0.185 6 600 : 15 1500 16 1600 828-j 6760 6811 -1.449 17 1700 2362+j 0 2362 Note that the f0 = 100 Hz.

Fig. 3-8 Spectrum of the Signal Negative frequencies has the same magnitude with negative phase angles.

K=2, f0=200, T=5m Sum of K=2, 4 f0=200, T=5m Sum of K=2, 4, 5 f0=100, T=10m Sum of all f0=100, T=10m

POP QUIZ: FUNDAMENTAL Here’s another spectrum: 100 Hz ? 50 Hz ? 100 250 –100 –250 f (in Hz) What is the fundamental frequency? Fundamental frequency is the largest f0 such that f0 = k f0. Or, f0 = gcd {fk} 100 Hz ? 50 Hz ?

3.3-2 Nonperiodic Signal SPECIAL RELATIONSHIP to get a PERIODIC SIGNAL 1.The frequencies are not integer multiples of a common fundamental frequency. 2. Slight shifts of frequency make a dramatic difference in the time waveform.

Harmonic Signal (3 Freqs) T=0.1

NON-Harmonic Signal NOT PERIODIC

3.7 Time-Varying FREQUENCIES Diagram Spectrogram: a time-frequency spectrum A-440 440Hz Frequency is the vertical axis axis

SIMPLE TEST SIGNAL C-major SCALE: stepped frequencies Frequency is constant for each note IDEAL : 563Hz : 440Hz : 262Hz

3.8 Time-Varying Frequency Frequency can change vs. time Continuously, not stepped FREQUENCY MODULATION (FM): Demo FM Frequency variation produced by the time-varying angle function CHIRP SIGNALS Linear Frequency Modulation (LFM): frequency changes linearly from some low value to a high one. Linear Frequency Modulation (LFM) VOICE

New Signal: Linear FM Called Chirp Signals (LFM) Quadratic phase Freq will change LINEARLY vs. time Example of Frequency Modulation (FM) Define “instantaneous frequency” QUADRATIC

INSTANTANEOUS FREQ Definition For Sinusoid: Derivative of the “Angle” (rad/sec) Makes sense

INSTANTANEOUS FREQ of the Chirp Chirp Signals have Quadratic phase Freq will change LINEARLY vs. time Why the derivatives are instaneous frequencies? => Fig.3-26

Fig. 3.26

CHIRP SPECTROGRAM

CHIRP WAVEFORM

Thank You !