Digital Signal Processing Digital Signal Processing

Prof. Nizamettin AYDIN

Lecture 5
naydin@yildiz.edu.tr

http://www.yildiz.edu.tr/~naydin Periodic Signals, Harmonics &

Time-
Time-Varying Sinusoids

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Problem Solving Skills LECTURE OBJECTIVES

• Math Formula • Plot & Sketches
– Sum of Cosines – S(t) versus t • Signals with HARMONIC Frequencies
– Amp, Freq, Phase – Spectrum
– Add Sinusoids with fk = kf0
N
• Recorded Signals • MATLAB x (t ) = A0 + ∑ Ak cos(2π kf 0t + ϕ k )
– Speech – Numerical
k =1
– Music – Computation
– Plotting list of FREQUENCY can change vs. TIME
– No simple formula
x(t) = cos(αt )
numbers Chirps: 2

Introduce Spectrogram Visualization (specgram.m)

(plotspec.m)

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1
 SPECTRUM for PERIODIC ?
SPECTRUM DIAGRAM

• Recall Complex Amplitude vs. Freq • Nearly Periodic in the Vowel Region
1 X k∗ 10 1 X k = ak – Period is (Approximately) T = 0.0065 sec
2 jπ / 3 − jπ / 3 2
7e 7e
4e − jπ / 2 jϕ k
4e jπ / 2
X k = Ak e
–250 –100 0 100 250
f (in Hz)

x (t ) = 10 + 14 cos(2π (100)t − π / 3)
+ 8 cos(2π (250)t + π / 2)
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PERIODIC SIGNALS
Period of Complex Exponential
• Repeat every T secs
– Definition
x (t ) = e jω t
– Example: x (t ) = x(t + T ) x (t + T ) = x (t ) ? Definition: Period is T

x (t ) = cos2 (3t ) T =? e jω ( t +T ) = e jω t e j 2π k = 1
T = π3 jωT
– Speech can be “quasi-periodic” T= 2π
3 ⇒e = 1 ⇒ ωT = 2π k
2π k  2π 
ω= =  k = ω0 k k = integer

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T  T  10

Harmonic Signal Spectrum

Define FUNDAMENTAL FREQ
N

Periodic signal can only have : f k = k f 0 x (t ) = A0 + ∑ Ak cos(2π kf 0t + ϕ k )

k =1
N
x (t ) = A0 + ∑ Ak cos(2π kf 0t + ϕ k )
k =1 f0 =
1 fk = k f0 (ω0 = 2π f 0 ) 1
jϕ k T f0 =
X k = Ak e T0

{ } f 0 = fundamental Frequency (largest)

N
x(t ) = X 0 + ∑ 1
2
X k e j 2π kf 0t + 12 X k∗e − j 2π kf0t
k =1 T0 = fundamental Period (shortest)
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2
 POP QUIZ: FUNDAMENTAL
Harmonic Signal (3 Freqs)
• Here’s another spectrum:
10
3rd 7 e jπ / 3 7e − jπ / 3
− jπ / 2
4e jπ / 2
5th
4e

–250 –100 0 100 250

f (in Hz)
What is the fundamental frequency? 10 Hz
What is the fundamental frequency?

100 Hz ? 50 Hz ?
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Harmonic Signal (3 Freqs)

IRRATIONAL SPECTRUM
T=0.1

SPECIAL RELATIONSHIP
to get a PERIODIC SIGNAL

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NON-Harmonic Signal FREQUENCY ANALYSIS

• Now, 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
NOT
PERIODIC
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3
Time-Varying FREQUENCIES Diagram SIMPLE TEST SIGNAL

• C-major SCALE: stepped frequencies

Frequency is the vertical axis

A-440
– Frequency is constant for each note

IDEAL

Time is the horizontal axis

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SPECTROGRAM SPECTROGRAM EXAMPLE

• Two Constant Frequencies: Beats

• 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 cos(2π (660)t ) sin(2π (12)t )
• Then uses the FFT (Fast Fourier Transform)

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AM Radio Signal SPECTRUM of AM (Beat)

• Same as BEAT Notes • 4 complex exponentials in AM:

cos(2π (660)t ) sin( 2π (12)t )
1
2
(e j 2π ( 660) t
+ e − j 2π ( 660)t ) (e
1
2j
j 2π (12 ) t
− e − j 2π (12 ) t ) 1
4 e jπ / 2 1
4 e − jπ / 2 1
4 e jπ / 2 1
4 e − jπ / 2

1
4j
(e j 2π ( 672) t
− e − j 2π ( 672) t − e j 2π ( 648) t + e − j 2π ( 648) t ) –672 –648 0 648 672
f (in Hz)

What is the fundamental frequency?

1 cos(2π (672)t − π2 ) + 12 cos(2π (648)t + π2 )
2 648 Hz ? 24 Hz ?
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4
 STEPPED FREQUENCIES SPECTROGRAM of C-Scale

• C-major SCALE: successive sinusoids Sinusoids ONLY

– Frequency is constant for each note

IDEAL From SPECGRAM

ANALYSIS PROGRAM

ARTIFACTS at Transitions

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Spectrogram of LAB SONG Time-Varying Frequency

• Frequency can change vs. time

Sinusoids ONLY
– Continuously, not stepped
Analysis Frame = 40ms
ARTIFACTS at Transitions • FREQUENCY MODULATION (FM)

x (t ) = cos(2π f c t + v (t ))
VOICE
• CHIRP SIGNALS
– Linear Frequency Modulation (LFM)

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New Signal: Linear FM INSTANTANEOUS FREQ

• Called Chirp Signals (LFM)
– Quadratic phase QUADRATIC • Definition
x(t ) = A cos(ψ (t ))
Derivative
x (t ) = A cos(α t 2 + 2π f 0 t + ϕ ) ⇒ ωi (t ) = dt
d ψ (t )
of the “Angle”

• For Sinusoid:
• Freq will change LINEARLY vs. time x (t ) = A cos(2π f 0t + ϕ )
– Example of Frequency Modulation (FM)
ψ (t ) = 2π f 0t + ϕ Makes sense
– Define “instantaneous frequency”
⇒ ωi ( t ) = d ψ (t )
dt
= 2π f 0
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5
 INSTANTANEOUS FREQ
of the Chirp
CHIRP SPECTROGRAM
• Chirp Signals have Quadratic phase
• Freq will change LINEARLY vs. time

x (t ) = A cos(α t 2 + β t + ϕ )
⇒ ψ (t ) = α t 2 + β t + ϕ
⇒ ωi ( t ) = d ψ (t )
dt
= 2α t + β
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CHIRP WAVEFORM OTHER CHIRPS

ψ(t) can be anything:

x (t ) = A cos(α cos( β t ) + ϕ )

⇒ ωi (t ) = dtd ψ (t ) = −αβ sin( β t )

ψ(t) could be speech or music:

– FM radio broadcast

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SINE-WAVE FREQUENCY MODULATION

(FM)

Look at CD-ROM Demos in Ch 3

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