circuits with non-periodic sources

8/17/2019 Circuits With Non-Periodic Sources

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EECE 301Signals & SystemsProf. Mark Fowler

Note Set #14

• C-T Signals: Circuits with Non-Periodic Sources


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2/13

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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4/13

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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5/13

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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6/13

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


8/13

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


9/13

9/13Stronger reflection causes bigger boosts/cuts!!


FT magnitude at

the speaker


of the room

FT magnitude at

your ear… room

gives large boosts

and cuts at closely

spaced locations


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10/13


Shorter delay causes wider spacing… so fewer dips/peaks over audio range!

FT magnitude at

the speaker


of the room

FT magnitude at

your ear… room

gives small boosts

and cuts at widely

spaced locations


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11/13

Matlab Code to create

the previous plots


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12/13

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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13/13

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

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