READING ASSIGNMENTS
Signal Processing First
ƒ This Lecture:
ƒ Chapter 2, pp. 9-17
LECTURE #1
Sinusoids
ƒ Appendix A: Complex Numbers
ƒ Appendix B: MATLAB
ƒ Chapter 1: Introduction
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CONVERGING FIELDS
Math
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COURSE OBJECTIVE
ƒ Students will be able to:
ƒ Understand mathematical descriptions of
signal processing algorithms and express
those algorithms as computer
implementations (MATLAB)
Physics
EE
CmpE
Computer
Science
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Applications
ƒ What are your objectives?
BIO
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WHY USE DSP ?
Fourier Everywhere
ƒ Mathematical abstractions lead to
generalization and discovery of new
processing techniques
ƒ Telecommunications
ƒ Sound & Music
ƒ CDROM, Digital Video
ƒ Fourier Optics
ƒ X-ray Crystallography
ƒ Protein Structure & DNA
ƒ Computer implementations are flexible
ƒ Computerized Tomography
ƒ Nuclear Magnetic Resonance: MRI
ƒ Radioastronomy
ƒ Applications provide a physical context
ƒ Ref: Prestini, “The Evolution of Applied Harmonic Analysis”
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LECTURE OBJECTIVES
TUNING FORK EXAMPLE
ƒ Write general formula for a “sinusoidal”
waveform, or signal
ƒ From the formula, plot the sinusoid versus
time
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ƒ What’s a signal?
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:
A cos(2π ( 440)t + ϕ )
ƒ It’s a function of time, x(t)
ƒ in the mathematical sense
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TUNING FORK A-440 Waveform
T ≈ 8.15 − 5.85
= 2.3 ms
ƒ Break x(t) into its sinusoidal components
f = 1/ T
ƒ Called the FREQUENCY SPECTRUM
= 1000 / 2.3
≈ 435 Hz
ƒ More complicated signal (BAT.WAV)
ƒ Waveform x(t) is NOT a Sinusoid
ƒ Theory will tell us
ƒ x(t) is approximately a sum of sinusoids
ƒ FOURIER ANALYSIS
Time (sec)
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SPEECH EXAMPLE
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Speech Signal: BAT
DIGITIZE the WAVEFORM
ƒ Nearly Periodic in Vowel Region
ƒ x[n] is a SAMPLED SINUSOID
ƒ Period is (Approximately) T = 0.0065 sec
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ƒ A list of numbers stored in memory
ƒ Sample at 11,025 samples per second
ƒ Called the SAMPLING RATE of the A/D
ƒ Time between samples is
ƒ 1/11025 = 90.7 microsec
ƒ Output via D/A hardware (at Fsamp)
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STORING DIGITAL SOUND
SINES and COSINES
ƒ x[n] is a SAMPLED SINUSOID
ƒ Always use the COSINE FORM
ƒ A list of numbers stored in memory
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A cos(2π ( 440)t + ϕ )
CD rate is 44,100 samples per second
16-bit samples
Stereo uses 2 channels
Number of bytes for 1 minute is
ƒ Sine is a special case:
sin(ω t ) = cos(ω t − π2 )
ƒ 2 X (16/8) X 60 X 44100 = 10.584 Mbytes
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SINUSOIDAL SIGNAL
ω
ƒ Radians/sec
ƒ Hertz (cycles/sec)
ƒ AMPLITUDE
ƒ Magnitude
ω = (2π ) f
ƒ PERIOD (in sec)
T=
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1 2π
=
f
ω
ƒ PHASE
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EXAMPLE of SINUSOID
A cos(ω t + ϕ )
ƒ FREQUENCY
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ƒ Given the Formula
A
ƒ Make a plot
5 cos(0.3π t + 1.2π )
ϕ
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PLOTTING COSINE SIGNAL
from the FORMULA
PLOT COSINE SIGNAL
5 cos(0.3π t + 1.2π )
5cos(0.3π t +12
. π)
ƒ Determine period:
ƒ Formula defines A, ω, and φ
T = 2π / ω = 2π / 0.3π = 20 / 3
A=5
ƒ Determine a peak location by solving
ω = 0.3π
ϕ = 1.2π
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(ω t + ϕ ) = 0 ⇒ (0.3π t + 1.2π ) = 0
ƒ Zero crossing is T/4 before or after
ƒ Positive & Negative peaks spaced by T/2
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PLOT the SINUSOID
5 cos(0.3π t + 1.2π )
ƒ Use T=20/3 and the peak location at t=-4
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