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Transcript of Digital Signal Processing
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Prof. Nizamettin AYDIN
http://www.yildiz.edu.tr/~naydin
Digital Signal Processing
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Lecture 5
Periodic Signals, Harmonics & Periodic Signals, Harmonics & Time-Varying SinusoidsTime-Varying Sinusoids
Digital Signal Processing
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READING ASSIGNMENTS
• This Lecture:– Chapter 3, Sections 3-2 and 3-3– Chapter 3, Sections 3-7 and 3-8
• Next Lecture:
– Fourier Series ANALYSISFourier Series ANALYSIS– Sections 3-4, 3-5 and 3-6
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Problem Solving Skills
• Math Formula– Sum of Cosines
– Amp, Freq, Phase
• Recorded Signals– Speech
– Music
– No simple formula
• Plot & Sketches– S(t) versus t
– Spectrum
• MATLAB– Numerical
– Computation
– Plotting list of numbers
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LECTURE OBJECTIVES
• Signals with HARMONICHARMONIC Frequencies– Add Sinusoids with fk = kf0
FREQUENCY can change vs. TIMEChirps:
Introduce Spectrogram Visualization (specgram.m) (plotspec.m)
x(t) cos(t2 )
N
kkk tkfAAtx
100 )2cos()(
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SPECTRUM DIAGRAM
• Recall Complex Amplitude vs. Freq
kk aX 21
0 100 250–100–250f (in Hz)
3/7 je 3/7 je2/4 je 2/4 je
10
)2/)250(2cos(8)3/)100(2cos(1410)(
tttx
kjkk eAX
kX2
1
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SPECTRUM for PERIODIC ?
• Nearly Periodic in the Vowel Region– Period is (Approximately) T = 0.0065 sec
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PERIODIC SIGNALS
• Repeat every T secs– Definition
– Example:
– Speech can be “quasi-periodic”
)()( Ttxtx
)3(cos)( 2 ttx ?T
3T
32T
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Period of Complex Exponential
Definition: Period is T
k = integer
tjTtj ee )(
?)()()(
txTtxetx tj
12 kje
kTe Tj 21
kkTT
k0
22
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N
k
tkfjk
tkfjk
jkk
N
kkk
eXeXXtx
eAX
tkfAAtx
k
1
2212
21
0
100
00)(
)2cos()(
Harmonic Signal Spectrum
0:haveonly can signal Periodic fkfk
Tf
10
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Define FUNDAMENTAL FREQ
00
1
Tf
(shortest) Periodlfundamenta
(largest) Frequencylfundamenta
)2(
)2cos()(
0
0
000
100
T
f
ffkf
tkfAAtx
k
N
kkk
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What is the fundamental frequency?
Harmonic Signal (3 Freqs)
3rd5th
10 Hz
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POP QUIZ: FUNDAMENTAL
• Here’s another spectrum:
What is the fundamental frequency?
100 Hz ? 50 Hz ?
0 100 250–100–250f (in Hz)
3/7 je 3/7 je2/4 je 2/4 je
10
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SPECIAL RELATIONSHIPto get a PERIODIC SIGNAL
IRRATIONAL SPECTRUM
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Harmonic Signal (3 Freqs)
T=0.1
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NON-Harmonic Signal
NOTPERIODIC
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FREQUENCY ANALYSIS
• Now, a much HARDER problemNow, a much HARDER problem• Given a recording of a song, have the
computer write the music
• Can a machine extract frequencies?– Yes, if we COMPUTE the spectrum for x(t)
• During short intervals
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Time-Varying FREQUENCIES DiagramF
req
uen
cy i
s th
e ve
rtic
al a
xis
Time is the horizontal axis
A-440
26Ekim2k11
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SIMPLE TEST SIGNAL
• C-major SCALE: stepped frequencies– Frequency is constant for each note
IDEAL
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SPECTROGRAM
• SPECTROGRAM Tool– MATLAB function is specgram.m– SP-First has plotspec.m & spectgr.m
• ANALYSIS program– Takes x(t) as input &
– Produces spectrum values Xk
– Breaks x(t) into SHORT TIME SEGMENTS • Then uses the FFT (Fast Fourier Transform)
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SPECTROGRAM EXAMPLE
• Two Constant Frequencies: Beats
))12(2sin())660(2cos( tt
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tjtjj
tjtj eeee )12(2)12(221)660(2)660(2
21
AM Radio Signal
• Same as BEAT Notes
))12(2sin())660(2cos( tt
))648(2cos())672(2cos( 221
221 tt
tjtjtjtjj eeee )648(2)648(2)672(2)672(2
41
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SPECTRUM of AM (Beat)
• 4 complex exponentials in AM:
What is the fundamental frequency?
648 Hz ? 24 Hz ?
0 648 672f (in Hz)
–672 –648
2/41 je
2/41 je2/
41 je2/
41 je
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STEPPED FREQUENCIES
• C-major SCALE: successive sinusoids– Frequency is constant for each note
IDEAL
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SPECTROGRAM of C-Scale
ARTIFACTS at Transitions
Sinusoids ONLY
From SPECGRAMANALYSIS PROGRAM
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Spectrogram of LAB SONG
ARTIFACTS at Transitions
Sinusoids ONLY
Analysis Frame = 40ms
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Time-Varying Frequency
• Frequency can change vs. time– Continuously, not stepped
• FREQUENCY MODULATION (FM)FREQUENCY MODULATION (FM)
• CHIRP SIGNALS– Linear Frequency Modulation (LFM)
))(2cos()( tvtftx c VOICE
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)2cos()( 02 tftAtx
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
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INSTANTANEOUS FREQ
• Definition
• For Sinusoid:
Derivativeof the “Angle”)()(
))(cos()(
tt
tAtx
dtd
i
Makes sense
0
0
0
2)()(
2)()2cos()(
ftt
tfttfAtx
dtd
i
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INSTANTANEOUS FREQof the Chirp
• Chirp Signals have Quadratic phase
• Freq will change LINEARLY vs. time
ttt
ttAtx2
2
)(
)cos()(
tttdtd
i 2)()(
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CHIRP SPECTROGRAM
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CHIRP WAVEFORM
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OTHER CHIRPS
(t) can be anything:
(t) could be speech or music:– FM radio broadcast
))cos(cos()( tAtx
)sin()()( ttt dtd
i
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SINE-WAVE FREQUENCY MODULATION (FM)