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

2. 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

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

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

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

6. 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.

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

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

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

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

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

12. 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.

13. 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

14. 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

15. 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”.

16. 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

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

18. SPECTROGRAM EXAMPLE
Two Constant Frequencies: Beats

19. 3.3 Periodic Waveforms Nearly Periodic in the Vowel Region
Period is (Approximately) T = sec

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

21. 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

22. 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

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

24. Periodic Waveform

25. 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.

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

27. 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

28. 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 ?

29. 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.

30. Harmonic Signal (3 Freqs)
T=0.1

31. NON-Harmonic Signal
NOT
PERIODIC

32. 3.7 Time-Varying FREQUENCIES Diagram
Spectrogram: a time-frequency spectrum
A-440
440Hz
Frequency is the vertical axis
axis

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

34. 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

35. 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

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

37. 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

38. Fig. 3.26

39. CHIRP SPECTROGRAM

40. CHIRP WAVEFORM

41. Thank You !