1. cmpe 362- Signal Processing Instructor: Fatih Alagöz Assistant: Yekta Sait Can

2. Example 1: Voltage on a capacitor as a function of time.
What is Signal?
Signal is the variation of a physical phenomenon / quantity with respect to one or more independent variable
A signal is a function.
Example 1: Voltage on a capacitor as a function of time.
RC circuit + -

3. What is Signal?
Example 2: Two different vocoder signals of “Otuz yedi derece” male/female, emotion, background noise, any aspects of vocoder functionallity Have you had something spicy, salty, sour just before you say it; function of everything !!!

4. What is Signal? M T W F S Index
Example 3 : Closing value of the stock exchange index as a function of days
Example 4:Image as a function of x-y coordinates (e.g X 256 pixel image) M T W F S
Index
Fig. Stock exchange

5. What is Signal Processing
Process signal(s) for solving many scientific/ engineering/ theoretical purposes
The process is includes calculus, Differential equations, Difference equations, Transform theory, Linear time-invariant system theory, System identification and classification, Time-frequency analysis, Spectral estimation, Vector spaces and Linear algebra, Functional analysis, statistical signals and stochastic processes, Detection theory, Estimation theory, Optimization, Numerical methods, Time series, Data mining, etc
“The IEEE Transactions on Signal Processing covers novel theory, algorithms, performance analyses and applications of techniques for the processing, understanding, learning, retrieval, mining, and extraction of information from signals. The term "signal" includes, among others, audio, video, speech, image, communication, geophysical, sonar, radar, medical and musical signals. Examples of topics of interest include, but are not limited to, information processing and the theory and application of filtering, coding, transmitting, estimating, detecting, analyzing, recognizing, synthesizing, recording, and reproducing signals” REFERENCE: IEEE Transactions on Signal Processing Journal

6. Signal Processing everywhere
SYSTEM (PROCESS)
x(n) or x(t)
INPUT SIGNAL
y(n) or y(t)
OUTPUT
SIGNAL
Audio, speech, image, video
Ranging from nanoscale to deepspace communications
Sonar, radar, geosensing
Array processing,
Control systems (all industry)
Seismology,
Meteorology,
Finance,
Health, etc..
IT IS GOOD TO KNOW THIS COURSE

7. I am currently working on gender classification problem for chicken hatchery eggs
DİŞİ
FTIR (Fourier Transform Infrared Spectrometer) yöntemi ile allantoik sıvının kimyasal bileşenine (östrojen seviyesi gibi) duyarlı frekans bandındaki (kırmızı bant) ışığın soğurulma oranına dayalı olarak yumurtadaki östrojen miktarı yumurtaya zarar vermeden (uzaktan) saptanabilir.
ERKEK

8. The course in a single slide

9. ALL ABOUT THREE DOMAINS
Z-TRANSFORM-DOMAIN: poles & zeros
POLYNOMIALS: H(z)
Use H(z) to get
Freq. Response
FREQ-DOMAIN
TIME-DOMAIN
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10. Lets start from the beginning...
Signals…

11. Signal Processing First
LECTURE #1
Sinusoids

12. READING ASSIGNMENTS This Lecture: Appendix A: Complex Numbers
Chapter 2, Sects. 2-1 to 2-5
Appendix A: Complex Numbers
Appendix B: MATLAB
Chapter 1: Introduction
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13. LECTURE OBJECTIVES
Write general formula for a “sinusoidal” waveform, or signal
From the formula, plot the sinusoid versus time
What’s a signal?
It’s a function of time, x(t)
in the mathematical sense
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14. TUNING FORK EXAMPLE CD-ROM demo “A” is at 440 Hertz (Hz)
Waveform is a SINUSOIDAL SIGNAL
Computer plot looks like a sine wave
This should be the mathematical formula:
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15. SINUSOID AMPLITUDE EXAMPLES
1-)0.25*sin(2π*300t)
2-)0.50*sin(2π*300t)
3-)1.00*sin(2π*300t)
Different Amplitude Sound Examples
Amplitude = Asin(2π*300t)
Amplitude = 5*Asin(2π*300t)
Amplitude =10*A sin(2π*300t)
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16. SINUSOID FREQUENCY EXAMPLES
1-)1.00sin(2π*300t)
2-)1.00*sin(2π*600t)
3-)1.00*sin(2π*1200t)
Different Amplitude Sound Examples
Amplitude = 1.00sin(2π*300t)
Amplitude = 1.00sin(2π*600t)
Amplitude = 1.00sin(2π*1200t)
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17. SINUSOID PHASE EXAMPLES
1-)1.00sin(2π*300t)
2-)1.00*sin(2π*300t+ π/2)
3-)1.00*sin(2π*300t+ π)
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18. HUMAN VOICE WITH DIFFERENT SAMPLING FREQUENCIES
With the original sampling frequency Fs
When sampled with Fs/2
When sampled with 2*Fs
When sampled with 5*Fs
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19. TIME VS FREQUENCY REPRESENTATION OF ONE CLAP RECORD(SPECTROGRAM)
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20. TIME VS FREQUENCY REPRESENTATION OF TWO CLAPS RECORD (SPECTROGRAM)
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21. TIME VS FREQUENCY REPRESENTATION OF CLAP + SNAP RECORD (SPECTROGRAM) (FIRST CLAP THEN FINGER SNAP)
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22. TUNING FORK A-440 Waveform
Time (sec)
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23. SPEECH EXAMPLE More complicated signal (BAT.WAV)
Waveform x(t) is NOT a Sinusoid
Theory will tell us
x(t) is approximately a sum of sinusoids
FOURIER ANALYSIS
Break x(t) into its sinusoidal components
Called the FREQUENCY SPECTRUM
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24. Speech Signal: BAT Nearly Periodic in Vowel Region
Period is (Approximately) T = sec
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25. DIGITIZE the WAVEFORM x[n] is a SAMPLED SINUSOID
A list of numbers stored in memory
Sample at 11,025 samples per second
Called the SAMPLING RATE of the A/D
Time between samples is
1/11025 = 90.7 microsec
Output via D/A hardware (at Fsamp)
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26. STORING DIGITAL SOUND x[n] is a SAMPLED SINUSOID
A list of numbers stored in memory
CD rate is 44,100 samples per second
16-bit samples
Stereo uses 2 channels
Number of bytes for 1 minute is
2 X (16/8) X 60 X = Mbytes
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27. SINUSOIDAL SIGNAL FREQUENCY AMPLITUDE PERIOD (in sec) PHASE
Radians/sec
Hertz (cycles/sec)
PERIOD (in sec)
AMPLITUDE
Magnitude
PHASE
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28. EXAMPLE of SINUSOID Given the Formula Make a plot 9/12/2018
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29. PLOT COSINE SIGNAL
Formula defines A, w, and f
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30. PLOTTING COSINE SIGNAL from the FORMULA
Determine period:
Determine a peak location by solving
Zero crossing is T/4 before or after
Positive & Negative peaks spaced by T/2
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31. PLOT the SINUSOID Use T=20/3 and the peak location at t=-4 9/12/2018
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32. PLOTTING COSINE SIGNAL from the FORMULA
Determine period:
Determine a peak location by solving
Peak at t=-4
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33. © 2003, JH McClellan & RW Schafer
ANSWER for the PLOT
Use T=20/3 and the peak location at t=-4
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34. © 2003, JH McClellan & RW Schafer
TIME-SHIFT
In a mathematical formula we can replace t with t-tm
Then the t=0 point moves to t=tm
Peak value of cos(w(t-tm)) is now at t=tm
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35. TIME-SHIFTED SINUSOID
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36. PHASE <--> TIME-SHIFT
Equate the formulas:
and we obtain:
or,
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37. © 2003, JH McClellan & RW Schafer
SINUSOID from a PLOT
Measure the period, T
Between peaks or zero crossings
Compute frequency: w = 2p/T
Measure time of a peak: tm
Compute phase: f = -w tm
Measure height of positive peak: A
3 steps
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38. © 2003, JH McClellan & RW Schafer
(A, w, f) from a PLOT
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39. SINE DRILL (MATLAB GUI)
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40. © 2003, JH McClellan & RW Schafer
PHASE
The cosine signal is periodic
Period is 2p
Thus adding any multiple of 2p leaves x(t) unchanged
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41. © 2003, JH McClellan & RW Schafer
COMPLEX NUMBERS
To solve: z2 = -1
z = j
Math and Physics use z = i
Complex number: z = x + j y y z
Cartesian
coordinate
system x
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42. © 2003, JH McClellan & RW Schafer
PLOT COMPLEX NUMBERS
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43. COMPLEX ADDITION = VECTOR Addition
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44. © 2003, JH McClellan & RW Schafer
*** POLAR FORM ***
Vector Form
Length =1
Angle = q
Common Values
j has angle of 0.5p
-1 has angle of p
- j has angle of 1.5p
also, angle of -j could be -0.5p = 1.5p -2p
because the PHASE is AMBIGUOUS
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45. POLAR <--> RECTANGULAR
Relate (x,y) to (r,q)
Most calculators do
Polar-Rectangular
Need a notation for POLAR FORM
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46. © 2003, JH McClellan & RW Schafer
Euler’s FORMULA
Complex Exponential
Real part is cosine
Imaginary part is sine
Magnitude is one
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47. © 2003, JH McClellan & RW Schafer
COMPLEX EXPONENTIAL
Interpret this as a Rotating Vector
q = wt
Angle changes vs. time
ex: w=20p rad/s
Rotates 0.2p in 0.01 secs
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48. © 2003, JH McClellan & RW Schafer
cos = REAL PART
Real Part of Euler’s
General Sinusoid
So,
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49. © 2003, JH McClellan & RW Schafer
REAL PART EXAMPLE
Evaluate:
Answer:
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50. © 2003, JH McClellan & RW Schafer
COMPLEX AMPLITUDE
General Sinusoid
Complex AMPLITUDE = X
Then, any Sinusoid = REAL PART of Xejwt
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51. Z DRILL (Complex Arith)
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52. © 2003, JH McClellan & RW Schafer
AVOID Trigonometry
Algebra, even complex, is EASIER !!!
Can you recall cos(q1+q2) ?
Use: real part of ej(q1+q2) = cos(q1+q2)
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53. © 2003, JH McClellan & RW Schafer
Euler’s FORMULA
Complex Exponential
Real part is cosine
Imaginary part is sine
Magnitude is one
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54. Real & Imaginary Part Plots
PHASE DIFFERENCE = p/2
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55. © 2003, JH McClellan & RW Schafer
COMPLEX EXPONENTIAL
Interpret this as a Rotating Vector
q = wt
Angle changes vs. time
ex: w=20p rad/s
Rotates 0.2p in 0.01 secs
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56. © 2003, JH McClellan & RW Schafer
Rotating Phasor
See Demo on CD-ROM
Chapter 2
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57. © 2003, JH McClellan & RW Schafer
Cos = REAL PART
Real Part of Euler’s
General Sinusoid
So,
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58. © 2003, JH McClellan & RW Schafer
COMPLEX AMPLITUDE
General Sinusoid
Sinusoid = REAL PART of (Aejf)ejwt
Complex AMPLITUDE = X
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59. © 2003, JH McClellan & RW Schafer
POP QUIZ: Complex Amp
Find the COMPLEX AMPLITUDE for:
Use EULER’s FORMULA:
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60. © 2003, JH McClellan & RW Schafer
WANT to ADD SINUSOIDS
ALL SINUSOIDS have SAME FREQUENCY
HOW to GET {Amp,Phase} of RESULT ?
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61. © 2003, JH McClellan & RW Schafer
ADD SINUSOIDS
Sum Sinusoid has SAME Frequency
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62. © 2003, JH McClellan & RW Schafer
PHASOR ADDITION RULE
Get the new complex amplitude by complex addition
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63. © 2003, JH McClellan & RW Schafer
Phasor Addition Proof
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64. POP QUIZ: Add Sinusoids
ADD THESE 2 SINUSOIDS:
COMPLEX ADDITION:
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65. © 2003, JH McClellan & RW Schafer
POP QUIZ (answer)
COMPLEX ADDITION:
CONVERT back to cosine form:
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66. © 2003, JH McClellan & RW Schafer
ADD SINUSOIDS EXAMPLE
tm1
tm2
tm3
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67. Convert Time-Shift to Phase
Measure peak times:
tm1= , tm2= , tm3=
Convert to phase (T=0.1)
f1=-wtm1 = -2p(tm1 /T) = 70p/180,
f2= 200p/180
Amplitudes
A1=1.7, A2=1.9, A3=1.532
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68. © 2003, JH McClellan & RW Schafer
Phasor Add: Numerical
Convert Polar to Cartesian
X1 = j1.597
X2 = j0.6498
sum =
X3 = j0.9476
Convert back to Polar
X3 = at angle p/180
This is the sum
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69. © 2003, JH McClellan & RW Schafer
ADD SINUSOIDS X1
VECTOR
(PHASOR)
ADD X3
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