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EECE 301Signals & SystemsProf. Mark Fowler
Note Set #14
• C-T Signals: Circuits with Non-Periodic Sources
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Recall: Convolution Property (The Most Important FT Property!!!)
( ) ( ) ( ) ( ) ( ) ( ) y t x h t d Y X H
Here… we will explore the real-world use of the right side of this result!
Linear
Circuit
k
t jk
k ect x0)( 0( ) jk t
k
k
y t d e
Input Output
Input’s FS
Coefficients
Output’s FS
Coefficients
( )k o k d H k c
Recall: For Periodic Signal…
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Unlike for the FS case it is not easy to use these ideas numerically to find
the actual y(t)… Rather, we usually use these ideas to help us “visualize”what we need in a circuit design.
Linear
Circuit j t e
Input Output
( ) j t H e
Linear
Circuit( ) ( ) j t x t X e d
Input Output
Input’s FT Output’s FT
( ) ( ) ( )Y H X
( ) ( ) ( ) j t y t H X e d
Input x( t) is a linear combo of sinusoids… the output is a linear combo:
Recall the definition of the frequency response:
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So we have as a big picture view:
( ) H ( )
( )
x t
X
1( ) ( ) y t Y F
( ) ( ) ( )Y X H
( ) ( ) ( )
( ) ( ) ( )
Y X H
Y X H
So…
So…in general we see that the system frequency response re-shapes the inputFT’s magnitude and phase.
System can:
-emphasize some frequencies-de-emphasize other frequencies
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Example Application of Time Shift Property: Room acoustics.
Practical Questions: Why do some rooms sound bad? Why can you fix this byusing a “graphic equalizer” to “boost” some frequencies and “cut” others?
c je X Y 1)()(This is the FT of what you hear…
It gives an equation that shows how thereflection affects what you hear!!!!
c j
e X X
ct xt xct xt xY
)()(
)()()()()( F F F Use linearity and time shift to get the FT at your ear:
0c
10
Delayed signal
Attenuated signal
So… You hear:
)()()( ct xt xt y
instead of just x(t )
speaker ear
)(t x
)( ct x
Reflecting Surface
Very simple case of a single reflection:
H ( ) of the room!
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c je X Y 1)()(
)( H
changes
shape of
)( H
)( X
The big
picture!
The room changes how much of each frequency you hear…
)sin()cos(11)( c jce H jc
Let’s look closer at | H ( )| to see what it does… Using Euler’s formulagives Rectangular Form
)(sin)(cos)cos(21)(sin))cos(1( 2222222 ccccc
22
ImRe magExpand 1st squared term
2
Use Trig ID
)cos(2)1(|)(|
2c H
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)cos(2)1()()(2
c X Y
Effect of the room… what does it look like asa function of frequency?? The cosine term
makes it wiggle up and down… and the valueof c controls how fast it wiggles up and down
Speed of sound in air 340 m/s
Typical difference in distance 0.167msec5.0
m/s340
m167.0mc
What is a typical value for delay c???
Spacing = 2 kHz
“Dip-to-Dip”
“Peak-to-Peak”
Spacing = 1/c Hz c controls spacing between dips/peaks
controls depth/height of dips/peaks
The next 3 slides explore these effects
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8/13Longer delay causes closer spacing… so more dips/peaks over audio range!
Attenuation: = 0.2 Delay: c = 0.5 ms (Spacing = 1/0.5e-3 = 2 kHz)
FT magnitude at
the speaker
(a made-up
spectrum… but
kind of like audio)
|H ( )|… the effect
of the room
FT magnitude at
your ear… room
gives slight boosts
and cuts at closely
spaced locations
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9/13Stronger reflection causes bigger boosts/cuts!!
Attenuation: = 0.8 Delay: c = 0.5 ms (Spacing = 1/0.5e-3 = 2 kHz)
FT magnitude at
the speaker
|H ( )|… the effect
of the room
FT magnitude at
your ear… room
gives large boosts
and cuts at closely
spaced locations
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Attenuation: = 0.2 Delay: c = 0.1 ms (Spacing = 1/0.1e-3 = 10 kHz)
Shorter delay causes wider spacing… so fewer dips/peaks over audio range!
FT magnitude at
the speaker
|H ( )|… the effect
of the room
FT magnitude at
your ear… room
gives small boosts
and cuts at widely
spaced locations
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Matlab Code to create
the previous plots
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Room boosts and cuts various frequencies… So, fix it using an “equalizer”
amp( )eq H )(t x
Equalizer
c j
eq e H X Y
1)()()(Then:
Recall: Peaks and dips
2( ) ( ) ( )eq X X H
)()()(
)()()( 2
X H H
X H Y
eqroom
room
Want this whole thing to be = 1 so )()( X Y
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Equalizer’s | H eq( )| should peak at frequencies
where the room’s | H room( )| dips and vice versa
Room
Equalizer
Room
&
Equalizer
Image from musiciansfriend.com