EE360: SIGNALS AND SYSTEMS - UNLV
Transcript of EE360: SIGNALS AND SYSTEMS - UNLV
BIG IDEA: TRANSFORM ANALYSIS
Make use of properties of LTI systems to simplify analysis
Represent signals as a linear combination of basic signals with two properties
Simple response: easy to characterize LTI system response to basic signal
Representation power: the set of basic signals can be use to construct a broad/useful class of signals
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When plucking a string, length is divided into integer divisions or harmonics
Frequency of each harmonic is an integer multiple of a โfundamental frequencyโ
Also known as the normal modes
Any string deflection could be built out of a linear combination of โmodesโ
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NORMAL MODES OF VIBRATING STRING
When plucking a string, length is divided into integer divisions or harmonics
Frequency of each harmonic is an integer multiple of a โfundamental frequencyโ
Also known as the normal modes
Any string deflection could be built out of a linear combination of โmodesโ
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NORMAL MODES OF VIBRATING STRING
Caution: turn your sound down
https://youtu.be/BSIw5SgUirg
Fourier argued that periodic signals (like the single period from a plucked string) were actually useful
Represent complex periodic signals
Examples of basic periodic signals
Sinusoid: ๐ฅ ๐ก = ๐๐๐ ๐0๐ก
Complex exponential: ๐ฅ ๐ก = ๐๐๐0t
Fundamental frequency: ๐0
Fundamental period: ๐ =2๐
๐0
Harmonically related period signals form family
Integer multiple of fundamental frequency
๐๐ ๐ก = ๐๐๐๐0๐ก for ๐ = 0,ยฑ1,ยฑ2,โฆ
Fourier Series is a way to represent a periodic signal as a linear combination of harmonics
๐ฅ ๐ก = ฯ๐=โโโ ๐๐๐
๐๐๐0๐ก
๐๐ coefficient gives the contribution of a harmonic (periodic signal of ๐times frequency)
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FOURIER SERIES 1 SLIDE OVERVIEW
SAWTOOTH EXAMPLE
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signalHarmonics: height given by coefficient
Animation showing approximation as more harmonics added
๐1
๐2
๐3๐4โฆ
SQUARE WAVE EXAMPLE
Better approximation of square wave with more coefficients
Aligned approximations
Animation of FS
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1
2
3
4
Note: ๐(๐) ~ ๐๐
#๐๐
coef
fici
ents
ARBITRARY EXAMPLES
Interactive examples [flash (dated)][html]
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TRANSFORM ANALYSIS OBJECTIVE
Need family of signals ๐ฅ๐ ๐ก that have 1) simple response and 2) represent a broad (useful) class of signals
1. Family of signals Simple response โ every signal in family pass through LTI system with scale change
2. โAnyโ signal can be represented as a linear combination of signals in the family
Results in an output generated by input ๐ฅ(๐ก)
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๐ฅ๐(๐ก) โถ ๐๐๐ฅ๐(๐ก)
๐ฅ ๐ก =
๐=โโ
โ
๐๐๐ฅ๐(๐ก)
๐ฅ ๐ก โถ
๐=โโ
โ
๐๐๐๐๐ฅ๐(๐ก)
IMPULSE AS BASIC SIGNAL
Previously (Ch2), we used shifted and scaled deltas
๐ฟ ๐ก โ ๐ก0 โน ๐ฅ ๐ก = โซ ๐ฅ ๐ ๐ฟ ๐ก โ ๐ ๐๐ โถ ๐ฆ ๐ก = โซ ๐ฅ ๐ โ ๐ก โ ๐ ๐๐
Thanks to Jean Baptiste Joseph Fourier in the early 1800s we got Fourier analysis
Consider signal family of complex exponentials
๐ฅ ๐ก = ๐๐ ๐ก or ๐ฅ ๐ = ๐ง๐, ๐ , ๐ง โ โ
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Using the convolution
๐๐ ๐ก โถ๐ป ๐ ๐๐ ๐ก
๐ง๐ โถ๐ป ๐ง ๐ง๐
Notice the eigenvalue ๐ป ๐ depends on the value of โ(๐ก)and ๐
Transfer function of LTI system
Laplace transform of impulse response
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COMPLEX EXPONENTIAL AS EIGENSIGNAL
TRANSFORM OBJECTIVE
Simple response
๐ฅ ๐ก = ๐๐ ๐ก โถ ๐ฆ ๐ก = ๐ป ๐ ๐ฅ ๐ก
Useful representation?
๐ฅ ๐ก = ฯ๐๐๐๐ ๐๐ก โถ ๐ฆ ๐ก = ฯ๐๐๐ป ๐ ๐ ๐๐ ๐๐ก
Input linear combination of complex exponentials leads to output linear combination of complex exponentials
Fourier suggested limiting to subclass of period complex exponentials ๐๐๐๐0๐ก , ๐ โ โค,๐0 โ โ
๐ฅ ๐ก = ฯ๐๐๐๐๐๐0๐ก โถ ๐ฆ ๐ก = ฯ๐๐๐ป ๐๐๐0 ๐๐ ๐๐ก
Periodic input leads to periodic output.
๐ป ๐๐ = ๐ป ๐ ศ๐ =๐๐ is the frequency response of the system
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CTFS TRANSFORM PAIR
Suppose ๐ฅ(๐ก) can be expressed as a linear combination of harmonic complex exponentials
๐ฅ ๐ก = ฯ๐=โโโ ๐๐๐
๐๐๐0๐ก synthesis equation
Then the FS coefficients {๐๐} can be found as
๐๐ =1
๐โซ๐ ๐ฅ(๐ก) ๐โ๐๐๐0๐ก๐๐ก analysis equation
๐0 - fundamental frequency
๐ = 2๐/๐0 - fundamental period
๐๐ known as FS coefficients or spectral coefficients
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CTFS PROOF
While we can prove this, it is not well suited for slides.
Key observation from proof: Complex exponentials are orthogonal
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Each of the harmonic exponentials are orthogonal to each other and span the space of periodic signals
The projection of ๐ฅ(๐ก) onto a particular harmonic (๐๐) gives the contribution of that complex exponential to building ๐ฅ ๐ก
๐๐ is how much of each harmonic is required to construct the periodic signal ๐ฅ(๐ก)
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VECTOR SPACE OF PERIODIC SIGNALS
Periodic signals, ๐0
๐ฅ(๐ก)
๐๐0๐ก = 1๐0
๐๐(โ๐0)๐ก
๐๐๐0๐ก
๐๐2๐0๐ก
๐๐๐๐0๐ก
๐โ1
๐1
๐2
๐๐
HARMONICS
๐ = ยฑ1 โ fundamental component (first harmonic)
Frequency ๐0, period ๐ = 2๐/๐0
๐ = ยฑ2 โ second harmonic
Frequency ๐2 = 2๐0, period ๐2 = ๐/2 (half period)
โฆ
๐ = ยฑ๐ โ Nth harmonic
Frequency ๐๐ = ๐๐0, period ๐๐ = ๐/๐ (1/N period)
๐ = 0 โ ๐0 =1
๐โซ๐ ๐ฅ ๐ก ๐๐ก, DC, constant component, average
over a single period
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HOW TO FIND FS REPRESENTATION
Will use important examples to demonstrate common techniques
Sinusoidal signals โ Eulerโs relationship
Direct FS integral evaluation
FS properties table and transform pairs
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๐ฅ ๐ก = 1 +1
2cos 2๐๐ก + sin 3๐๐ก
First find the period
Constant 1 has arbitrary period
cos 2๐๐ก has period ๐1 = 1
sin 3๐๐ก has period ๐2 = 2/3
๐ = 2, ๐0 = 2๐/๐ = ๐
Rewrite ๐ฅ ๐ก using Eulerโs and read off ๐๐coefficients by inspection
๐ฅ ๐ก = 1 +1
4๐๐2๐0๐ก + ๐โ๐2๐0๐ก +
1
2๐๐๐3๐0๐ก โ ๐โ๐3๐0๐ก
Read off coeff. directly
๐0 = 1
๐1 = ๐โ1 = 0
๐2 = ๐โ2 = 1/4
๐3 = 1/2๐, ๐โ3 = โ1/2๐
๐๐ = 0, else
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SINUSOIDAL SIGNAL
Important signal/function in DSP and communication
sinc ๐ฅ =sin ๐๐ฅ
๐๐ฅnormalized
sinc ๐ฅ =sin ๐ฅ
๐ฅunnormalized
Modulated sine function
Amplitude follows 1/x
Must use LโHopitalโs rule to get ๐ฅ = 0 time
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SINC FUNCTION
Consider different โduty cycleโ for the rectangle wave
๐ = 4๐1 50% (square wave)
๐ = 8๐1 25%
๐ = 16๐1 12.5%
Note all plots are still a sincshape
Difference is how the sync is sampled
Longer in time (larger T) smaller spacing in frequency more samples between zero crossings
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RECTANGLE WAVE COEFFICIENTS
Special case of rectangle wave with ๐ = 4๐1
One sample between zero-crossing
๐๐ = แ1/2 ๐ = 0
sin(๐๐/2)
๐๐๐๐๐ ๐
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SQUARE WAVE
๐ฅ ๐ก = ฯ๐=โโโ ๐ฟ(๐ก โ ๐๐)
Using FS integral
Notice only one impulse in the interval
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PERIODIC IMPULSE TRAIN
PROPERTIES OF CTFS
Since these are very similar between CT and DT, will save until after DT
Properties are used to avoid direct evaluation of FS integral
Be sure to bookmark properties in Table 3.1 on page 206
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DTFS VS CTFS DIFFERENCES
While quite similar to the CT case,
DTFS is a finite series, ๐๐ , k < K
Does not have convergence issues
Good News: motivation and intuition from CT applies for DT case
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DTFS TRANSFORM PAIR
Consider the discrete time periodic signal ๐ฅ ๐ = ๐ฅ ๐ + ๐
๐ฅ ๐ = ฯ๐=<๐> ๐๐๐๐๐๐0๐ synthesis equation
๐๐ =1
๐ฯ๐=<๐> ๐ฅ ๐ ๐โ๐๐๐0๐ analysis equation
๐ โ fundamental period (smallest value such that periodicity constraint holds)
๐0 = 2๐/๐ โ fundamental frequency
ฯ๐=<๐> indicates summation over a period (๐ samples)
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DTFS REMARKS
DTFS representation is a finite sum, so there is always pointwise convergence
FS coefficients are periodic with period N
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DTFS PROOF
Proof for the DTFS pair is similar to the CT case
Relies on orthogonality of harmonically related DT period complex exponentials
Will not show in class
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HOW TO FIND DTFS REPRESENTATION
Like CTFS, will use important examples to demonstrate common techniques
Sinusoidal signals โ Eulerโs relationship
Direct FS summation evaluation โ periodic rectangular wave and impulse train
FS properties table and transform pairs
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๐ฅ[๐] = 1 +1
2cos
2๐
๐๐ + sin
4๐
๐๐
First find the period
Rewrite ๐ฅ[๐] using Eulerโs and read off ๐๐ coefficients by inspection
Shortcut here
๐0 = 1, ๐ยฑ1 =1
4, ๐2 = ๐โ2
โ =1
2๐
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SINUSOIDAL SIGNAL
๐ฅ(๐ก) = cos๐0๐ก
๐๐ = แ1/2 ๐ = ยฑ10 ๐๐๐ ๐
๐ฅ ๐ = cos๐0๐
๐๐ = แ1/2 ๐ = ยฑ10 ๐๐๐ ๐
Over a single period must specify period with period N
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SINUSOIDAL COMPARISON
Consider different โduty cycleโ for the rectangle wave
50% (square wave)
25%
12.5%
Note all plots are still a sincshaped, but periodic
Difference is how the sync is sampled
Longer in time (larger N) smaller spacing in frequency more samples between zero crossings
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RECTANGLE WAVE COEFFICIENTS
๐ฅ[๐] = ฯ๐=โโโ ๐ฟ[๐ โ ๐๐]
Using FS integral
Notice only one impulse in the interval
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PERIODIC IMPULSE TRAIN
PROPERTIES OF FOURIER SERIES
See Table 3.1 pg. 206 (CT) and Table 3.2 pg. 221 (DT)
In the following slides, suppose:
Most times, will only show proof for one of CT or DT
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CT
๐ด๐ฅ ๐ก + ๐ต๐ฆ ๐ก โท ๐ด๐๐ + ๐ต๐๐
DT
๐ด๐ฅ ๐ + ๐ต๐ฆ ๐ โท ๐ด๐๐ + ๐ต๐๐
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LINEARITY
CT
๐ฅ(๐ก โ ๐ก0) โท ๐๐๐โ๐๐๐0๐ก0
Proof
Let ๐ฆ ๐ก = ๐ฅ(๐ก โ ๐ก0)
DT
๐ฅ[๐ โ ๐0] โท ๐๐๐โ๐๐๐0๐0
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TIME-SHIFT
CT
๐๐๐๐0๐ก๐ฅ ๐ก โท ๐๐โ๐
DT
๐๐๐๐0๐๐ฅ ๐ โท ๐๐โ๐
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FREQUENCY SHIFT
Note: Similar relationship with Time Shift (dualilty). Multiplication by exponential in time is a shift in frequency. Shift in time is a multiplication by exponential in frequency.
CT
โซ๐ ๐ฅ ๐ ๐ฆ ๐ก โ ๐ ๐๐ โท ๐๐๐๐๐
DT
ฯ๐=<๐> ๐ฅ ๐ ๐ฆ[๐ โ ๐] โท ๐๐๐๐๐
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PERIODIC CONVOLUTION
CT
๐ฅ ๐ก ๐ฆ(๐ก) โท ฯ๐=โโโ ๐๐๐๐โ๐ = ๐๐ โ ๐๐
DT
๐ฅ ๐ ๐ฆ[๐] โท ฯ๐=<๐>๐๐๐๐โ๐ = ๐๐ โ ๐๐
Convolution over a single period (DT FS is periodic)
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MULTIPLICATION
Note: Similar relationship with Convolution (dualilty). Convolution in time results in multiplication in frequency domain. Multiplication in time results in convolution in frequency domain.
CT
1
๐โซ๐ ๐ฅ ๐ก 2๐๐ก = ฯ๐=โโ
โ ๐๐2
DT
1
๐ฯ๐=<๐> ๐ฅ ๐ 2 = ฯ๐=<๐> ๐๐
2
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PARSEVALโS RELATION
Note: Total average power in a periodic signal equals the sum of the average power in all its harmonic components
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Tเถฑ๐
๐๐๐๐๐๐0๐ก
2๐๐ก =
1
๐เถฑ๐
๐๐2๐๐ก = ๐๐
2
Average power in the ๐th harmonic
CT
๐ฅ(๐ผ๐ก) โท ๐๐
๐ผ > 0
Periodic with period ๐/๐ผ
DT
๐ฅ(๐) ๐ = แ๐ฅ[๐/๐] ๐ multiple of ๐
0 ๐๐๐ ๐
Periodic with period ๐๐
๐ฅ(๐)[๐] โท1
๐๐๐
Periodic with period ๐๐
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TIME SCALING
Note: Not all properties are exactly the same. Must be careful due to constraints on periodicity for DT signal.
EIGENSIGNAL REMINDER
๐ฅ ๐ก = ๐๐ ๐ก โท ๐ฆ ๐ก = ๐ป ๐ ๐๐ ๐ก ๐ฅ ๐ = ๐ง๐ โท ๐ฆ ๐ = ๐ป ๐ง ๐ง๐
๐ป ๐ = โซโโโโ ๐ก ๐โ๐ ๐ก๐๐ก ๐ป ๐ง = ฯ๐=โโ
โ โ ๐ ๐งโ๐
๐ป ๐ ,๐ป ๐ง known as system function (๐ , ๐ง โ โ)
For Fourier Analysis (e.g. FS)
Let ๐ = ๐๐ and ๐ง = ๐๐๐
Frequency response (system response to particular input frequency)
๐ป ๐๐ = ๐ป ๐ ศ๐ =๐๐ = โซโโโโ ๐ก ๐โ๐๐๐ก๐๐ก
๐ป ๐๐๐ = ๐ป ๐ง ศ๐ง=๐๐๐ = ฯ๐=โโโ โ ๐ ๐โ๐๐๐
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FOURIER SERIES AND LTI SYSTEMS I
Consider now a FS representation of a periodic signals
๐ฅ ๐ก = ฯ๐ ๐๐๐๐๐๐0๐ก
โ ๐ฆ ๐ก = ฯ๐ ๐๐๐ป ๐๐๐0 ๐๐๐๐0๐ก
Due to superposition (LTI system)
Each harmonic in results in harmonic out with eigenvalue
๐ฆ(๐ก) periodic with same fundamental frequency as ๐ฅ(๐ก) โ ๐0
๐ =2๐
๐0- fundamental period
FS coefficients for ๐ฆ ๐ก
๐๐ = ๐๐๐ป(๐๐๐0)
๐๐ is the FS coefficient ๐๐ multiplied/affected by frequency response at ๐๐0
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FOURIER SERIES AND LTI SYSTEMS III
System block diagram
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๐โ๐๐โ๐๐๐0๐ก ๐ป(๐๐)
๐ฅ ๐ก =
๐ฆ(๐ก)๐0๐๐(0)๐0๐ก ๐ป(๐๐)
๐๐๐๐๐๐0๐ก ๐ป(๐๐)
โฎ
โฎ
๐โ๐๐ป(โ๐๐๐0)๐โ๐๐๐0๐ก
๐๐๐ป(๐๐๐0)๐โ๐๐๐0๐ก
๐0๐ป(โ๐0)
DTFS AND LTI SYSTEMS
๐ฅ ๐ = ฯ๐=<๐>๐๐๐๐๐2๐/๐๐ โ
๐ฆ ๐ =
๐=<๐>
๐๐๐ป(๐๐2๐๐๐)๐๐๐2๐/๐๐
Same idea as in the continuous case
Each harmonic is modified by the Frequency Response at the harmonic frequency
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LTI system with
โ ๐ = ๐ผ๐๐ข ๐ ,โ1 < ๐ผ < 1
Find FS of ๐ฆ[๐] given input
๐ฅ ๐ = cos2๐๐
๐
Find FS representation of ๐ฅ[๐]
๐0 = 2๐/๐
๐ฅ ๐ =1
2๐๐2๐/๐๐ +
1
2๐โ๐2๐/๐๐
๐๐ = เต1
2๐ = ยฑ1,ยฑ ๐ + 1 ,โฆ
0 else
Find frequency response
๐ป ๐๐๐ = ฯ๐ โ ๐ ๐โ๐๐๐
๐ป ๐๐๐ = ฯ๐๐ผ๐๐ข[๐]๐โ๐๐๐
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EXAMPLE 1
๐ป ๐๐ =
๐=0
โ
๐ผ๐๐โ๐๐๐
๐ป ๐๐ =
๐=0
โ
๐ผ๐โ๐๐๐
Let ๐ฝ = ๐ผ๐โ๐๐
๐ป ๐๐ =1
1 โ ๐ฝ
๐ป ๐๐ =1
1 โ ๐ผ๐โ๐๐
EXAMPLE 1 II
Use FS LTI relationship to find output
๐ฆ ๐ = ฯ๐=<๐> ๐๐๐ป(๐๐๐๐0) ๐๐๐๐0๐
๐ฆ ๐ =1
2๐ป ๐๐1
2๐
๐๐ ๐๐1
2๐
๐๐ +
1
2๐ป ๐โ๐1
2๐
๐๐ ๐โ๐1
2๐
๐๐
๐ฆ ๐ =1
2
1
1โ๐ผ๐โ๐๐2๐/๐๐๐
2๐
๐๐ +
1
2
1
1โ๐ผ๐๐๐2๐/๐๐โ๐
2๐
๐๐
Output FS coefficients
๐๐ = เต1
2
1
1โ๐ผ๐โ๐๐2๐/๐๐ = ยฑ1
0 ๐๐๐ ๐
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Periodic with period ๐
๐ฅ(๐ก) has fundamental period ๐and FS ๐๐
Sometimes direct calculation of ๐๐ is difficult, at times easier to calculate transformation
๐๐ โ ๐ ๐ก =๐๐ฅ ๐ก
๐๐ก
Find ๐๐ in terms of ๐๐ and ๐, given
โซ๐2๐๐ฅ ๐ก ๐๐ก = 2
๐0 =1
๐โซ๐ ๐ฅ ๐ก ๐โ๐ 0 ๐0๐ก๐๐ก =
1
๐โซ๐ ๐ฅ ๐ก ๐๐ก โ
2
๐
From Table 3.1 pg 206
๐๐ โ ๐๐2๐
๐๐๐ โ ๐๐ =
๐๐
๐๐2๐/๐
๐๐ = แ2/๐ ๐ = 0๐๐
๐๐2๐/๐๐ โ 0
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EXAMPLE PROBLEM 3.7
EXAMPLE PROBLEM 3.7 II
Find FS of periodic sawtooth wave
Take derivative of sawtooth
Results in sum of rectangular waves
FS coefficients of rectangular waves from Table 3.2 to get ๐๐ โ ๐(๐ก)
Then use previous result to find ๐๐ โ ๐ฅ(๐ก)
See examples 3.6, 3.7 for similar treatment
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FILTERING
Important process in many applications
The goal is to change the relative amplitudes of frequency components in a signal
In EE480: DSP you can learn how to design a filter with desired properties/specifications
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LTI FILTERS
Frequency-shaping filters โ general LTI systems
Frequency-selective filters โ pass some frequencies and eliminate others
Common examples include low-pass (LP), high-pass (HP), bandpass (BP), and bandstop (BS) [notch]
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MOTIVATION: AUDIO EQUALIZER
Basic equalizer gives user ability to adjust sound from to match taste โ e.g. bass (low freq) and treble (high freq)
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Log-log plot to show larger range of frequencies and response
dB = 20 log10 ๐ป ๐๐
Magnitude response matches are intuition Boost low and high frequencies
but attenuate mid frequencies
EXAMPLE: DERIVATIVE FILTER
๐ฆ ๐ก =๐
๐๐ก๐ฅ ๐ก โท ๐ป ๐๐ = ๐๐
High-pass filter used for โedgeโ detection
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EXAMPLE: AVERAGE FILTER
๐ฆ ๐ =1
2๐ฅ ๐ + ๐ฅ ๐ โ 1
Low-pass filter used for smoothing
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Continuous Case
๐ฅ ๐ก = ฯ๐ ๐๐๐๐๐๐0๐ก
๐๐ =1
๐โซ๐ ๐ฅ ๐ก ๐โ๐๐๐0๐ก๐๐ก
Fundamental frequency ๐0
Fundamental period ๐ =2๐
๐0
Discrete Case
๐ฅ[๐] = ฯ๐=<๐>๐๐๐๐๐๐0๐
๐๐ =1
๐ฯ๐=<๐> ๐ฅ ๐ ๐โ๐๐๐0๐
Fundamental frequency ๐0
Fundamental period ๐ =2๐
๐0
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FOURIER SERIES SUMMARY