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BME 310 Biomedical Computing -J.Schesser

54

Sampling and Aliasing

Lecture #6

Chapter 4


55

Homework

• Exercises:– 4.1 – 4.5

• Problems:– 4.1, 4.2, 4.3,

– 4.4, Use Matlab to plot the signal in part a.

– 4.5, 4.8, 4.11, 4.19


56

Homework

• Problems:– 4.1

1

( ) 10cos(880 )

[ ] ( ) 10cos(880 )

440 ; 0.0001; 10,000 /

10000Number of samples per cycle 22.7

440

[ ] ( ) 10cos(0.0880 )

cos( 2 ) cos( )

0.0880 2 2.088

2.088 / 2.088

s s

s s

s

s

x t t

x n x nT nT

f Hz T f s s

x n x nT n

n

T

1

/ 0.0001 20880

20880 / 2 10440

10000Number of samples per cycle .957

10440

f Hz


57

Homework


ffHzf

nnnx

ssf

ffHzf

nnnx

ssf

ffHzf

nnnx

ssf

fnnx

tttx

s

s

s

s

s

s

s

2;ˆ D;OVERSAMPLE ;5.5

)2/733.0cos(7)2/15/11cos(7][

/15

2 ;ˆLED; UNDERSAMP;5.5

)2/2.2cos(7)2/5/11cos(7][

/5

2;ˆ LED; UNDERSAMP;5.5

)2/1.1cos(7)2/10/11cos(7][

/10

)2//11cos(7][

)2/11cos(7)11sin(7)(

><--

<>--

<>--

-

-


58

Homework


Hzfl

flf

Hzfl

Hzfl

flf

flf

ssf

nnx

s

s

s

s

5100;1

/2).32(

and

6900;1

900;0

/2;)2.3(

)2.3(2

are aliases

/6000

)3/3.0cos(2.2][

´-

´´

-


59

Homework• Problem 4-4

4

4 4

4

) ( ) [10 cos(2 (2,000) )]cos(2 (10 ) )

10cos(2 (10 ) ) cos(2 (2,000) )cos(2 (10 ) )

1 110cos(2 (10 ) ) cos(2 (10,000 2,000) ) cos(2 (10,000 2,000) )

2 21 1

10cos(2 (10,000) ) cos(2 (8,000) ) cos(2 (122 2

a x t t t

t t t

t t t

t t

-

2 (10,000) 2 (10,000) 2 (8,000) 2 (8,000) 2 (12,000) 2 (12,000)

2 (10,000) 2 (10,000) 2 (8,000) 2 (8,000) 2 (12,000) 2

,000) )

1 110( ) ( ) ( )

2 2 2 2 25 5 0.25 0.25 0.25 0.25

j t j t j t j t j t j t

j t j t j t j t j t j

t

e e e e e e

e e e e e e

- - -

- - -

(12,000)

2 (8,000) 2 (8,000) 2 (10,000) 2 (10,000) 2 (12,000) 2 (12,000)

1 2 3 1 2 3

4 4

5 5

6 6

0.25 0.25 5 5 0.25 0.25

) 2000 ; 6; 1/ 2000 .0005 5 sec

0

0.25

5

t


o

e e e e e e

b LCD Hz N T

a a a a a a a

a a

a a

a a

- - -

- - -

-

-

-

max

0.25

) 12,000 ; 24,000sb f Hz f Hz

10,000

12,0008,000-10,000

-12,000-8,000

5

0.250.25

5

0.250.25


60


4

4 4

4

) ( ) [10 cos(2 (2,000) )]cos(2 (10 ) )

10cos(2 (10 ) ) cos(2 (2,000) )cos(2 (10 ) )

1 110cos(2 (10 ) ) cos(2 (10,000 2,000) ) cos(2 (10,000 2,000) )

2 21 1

10cos(2 (10,000) ) cos(2 (8,000) ) cos(2 (122 2

a x t t t

t t t

t t t

t t

-

2 (10,000) 2 (10,000) 2 (8,000) 2 (8,000) 2 (12,000) 2 (12,000)

2 (10,000) 2 (10,000) 2 (8,000) 2 (8,000) 2 (12,000) 2

,000) )

1 110( ) ( ) ( )

2 2 2 2 25 5 0.25 0.25 0.25 0.25


j t j t j t j t j t j

t

e e e e e e

e e e e e e

- - -

- - -

(12,000)

2 (8,000) 2 (8,000) 2 (10,000) 2 (10,000) 2 (12,000) 2 (12,000)

1 2 3 1 2 3

4 4

5 5

6 6

0.25 0.25 5 5 0.25 0.25

) 2000 ; 6; 1/ 2000 .0005 5 sec

0

0.25

5

t


o

e e e e e e

b LCD Hz N T

a a a a a a a

a a

a a

a a

- - -

- - -

-

-

-

max

0.25

) 12,000 ; 24,000sb f Hz f Hz

10,000

12,0008,000-10,000

-12,000-8,000

5

0.250.25

5

0.250.25


61


HzfHzfb

aa

aa

aa

aaaaaaa

TNHzLCDb

eeeeee

eeeeee

eeeeee

ttt

ttt

ttt

tttxa

s

o

tjtjtjtjtjtj

tjtjtjtjtjtj

tjtjtjtjtjtj

000,24;000,12)

25.0

5

25.0

0

sec50005.2000/1;6;2000)

25.025.05525.025.0

25.025.025.025.055

)2

(2

1)

2(

2

1)

2(10

))000,12(2cos(2

1))000,8(2cos(

2

1))000,10(2cos(10

))000,2000,10(2cos(2

1))000,2000,10(2cos(

2

1))10(2cos(10

))10(2cos())000,2(2cos())10(2cos(10

))10(2cos()])000,2(2cos(10[)()

max

66

55

44

321321

)000,12(2)000,12(2)000,10(2)000,10(2)000,8(2)000,8(2

)000,12(2)000,12(2)000,8(2)000,8(2)000,10(2)000,10(2

)000,12(2)000,12(2)000,8(2)000,8(2)000,10(2)000,10(2

4

44

4

-

-

-

-

---

---

---

---

10,000

12,0008,000-10,000

-12,000-8,000

5

0.250.25

5

0.250.25


62

Matlab Code used in problem 4.4

%problem 4.4fc=10^4;fi=2*10^3;Period=1/fi;Points=100;Timestart=0;Timedelta=Period/Points;Timeend=4*Period;time = (Timestart:Timedelta:Timeend);max=length(time);for i=1:max

y(i)=(10+cos(2*pi*fi*time(i)))*cos(2*pi*fc*time(i));endenv1=cos(2*pi*fi*time)+10;env2=-10-cos(2*pi*fi*time);plot(time,y,'r',time,env1,'b',time,env2,'b');title('Amplitude Modulation Plot');xlabel('Seconds');axis([ Timestart Timeend -1.5*10 +1.5*10]);

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

x 10-3

-15

-10

-5

0

5

10

15Amplitude Modulation Plot

Seconds


63

Matlab Code used in problem 4.4%problem 4.4

clear allfc=10^4;fi=2*10^3;ratio=fc/fi-1;fs=100*fc;ts=1/fs;cycles=5;DC=5;mod=.5*DC;time=(0:ts:cycles/fi);maxtime=length(time);for i=1:maxtime

x(i)=(DC+mod*cos(2*pi*fi*time(i)))*cos(2*pi*fc*time(i));endenv1=DC+mod*cos(2*pi*fi*time);env2=-DC-mod*cos(2*pi*fi*time);subplot(2,1,1)plot(time,x,'b',time,env1,'r',time,env2,'r');title('Time Domain');xlabel('Time (Seconds)')freqs=(0:fs/maxtime:fs-fs/maxtime);subplot(2,1,2)y=fft(x)/maxtime;stem(freqs,abs(y));title('Frequency Domain');xlabel('Frequency (Hz)')axis([(fc-ratio*fi) (fc+ratio*fi) 0 1.1*max(abs(y))])

0 0.5 1 1.5 2 2.5

x 10-3

-10

-5

0

5

10Time Domain

Time (Seconds)

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

x 104

0

1

2

Frequency Domain

Frequency (Hz)


64

Homework


)7/400cos(10)(

200;2.ˆ

/2;.2

.22

are aliases

/2000

)7/2.0cos(10][ )

-

´

-

ttx

Hzf

ff

ff

ssf

nnxb

s

s

s

edUndersampl

)7/1800cos(10)(

900;2.2ˆ;1

/2).22(

and

dOversample

)7/200cos(10)(

100;2.ˆ;0

/2;)2.2(

)2.2(2

are aliases

/1000

)7/2.0cos(10][

-

´-

-

´´

-

ttx

Hzfl

flf

ttx

Hzfl

flf

flf

ssf

nnx

s

s

s

s


65

Homework


50 50 700 700

50 700 50 700 700 50 700 50

750 650 650 750

750 650 650

) ( ) cos(50 )sin(700 )

( ) ( )( )2 2

( )4

( )4

4 4

j t j t j t j t

j t j t j t j t j t j t j t j t

j t j t j t j t

j t j t j

a x t t t

e e e ex t

j

e e e e e e e e

j

e e e e

j

e e e

j j

- -

- - - -

- -

-

-

- -

- -

-

750

3 3

2 2 2 2750 650 650 750

2 2 2 2750 650 650 750

2 2 2 2750 750 650 650

4 4

4 4 4 4

4 4 4 4

4 4 4 4

t j t

j j j j

j t j t j t j t

j j j j

j t j t j t j t

j j j j

j t j t j t j t

e

j j

e e e ee e e e

e e e ee e e e

e e e ee e e e

-

- - - -

- -

- -

- -

- -

- -

-


66

Homework


(750 ) (750 ) (650 ) (650 )2 2 2 2

750 752 2

OR

( ) cos(50 )sin(700 ) cos(50 ) cos(700 )2

1[cos(750 ) cos(650 )]

2 2 21

[cos(750 ) cos(650 )]2 2 2

1[ ]

2 2 2

1[

4

j t j t j t j t

j jj t j

x t t t t t

t t

t t

e e e e

e e e e

- - - - - -

- -

-

- -

- -

0 650 6502 2 ]

750) Maximum frequency is 375

2Nyquist rate is mimium sampling rate = 2 375 750 /

j jt j t j te e e e

b

s s

- -

´


67

Homework

• Problems: 4.11 Short Method( ) [3 sin( )]cos(13 / 2)

( ) [3 cos( / 2)]cos(13 / 2)

3cos(13 / 2) cos( / 2) cos(13 / 2)

1 13cos(13 / 2) cos(12 ) cos(14 )

2 21 1

cos(12 ) 3cos(13 / 2) cos(14 )2 2

Highest frequenc

x t t t

x t t t

t t t

t t t

t t t

-

-

y 14 , 28s >

6 6.5 7

3/2e jππ/2

1/4 e -jp 1/4 e j0


68

Homework• Problems: 4.11 Long Method

28,14frequency Highest

)12cos(2

1)14cos(

2

1)2/13cos(3

)2/12sin(2

1)2/14sin(

2

1)2/13cos(3

]4

)()([]

2[3

]4

[]2

[3

]4

[]2

[3

]2

][2

[]2

[3

]2

][2

3[

)2/13cos()]sin(3[)(

)2/12()2/12()2/14()2/14()2/13()2/13(

)2/14()2/12()2/12()2/14()2/13()2/13(

)2/13()2/13()2/13()2/13()2/13()2/13(

)2/13()2/13()2/13()2/13(

)2/13()2/13(0

>

-

-

---

--

--

-

-

---

---

-----

---

--

s

tjtjtjtjtjtj

tjtjtjtjtjtj

tjtjtjtjtjtjtjtjtjtj

tjtjtjtjtjtj

tjtjtjtjj

ttt

ttt

j

eeeeee

j

eeeeee

j

eeeeeeeeee

ee

j

eeee

ee

j

eee

tttx

6 6.5 7

3/2e jπ/2

1/4 e -jπ 1/4 e j0


69

Homework


0

0.5

1

1.5

2

2.5

3

3.5

0 1 2 3 4 5

0 2.42711 2.90022 2.98163 2.66034 1.9798

17

23696.

)9816.2(2

9002.26603.2cos)

][2

]1[]1[(cos

]1[]1[cos][2

])1[]1[(]1[]1[cos][2cos][2

]1[]1[cos][2

cos2][

]1[

][

]1[

][]1[

][]1[

][][][

]}[{}{][

: trick thisuseor plot thefrom guesscan You

11

)1(

1)1(

jj

jjj

j

j

-

---

-

-

-

ÂÂ

--

-

----

nx

nxnx

nxnxnx

nynyjnxnxnjynx

nznznz

eenz

nz

nz

nz

enzeeAeeAenz

enzeeAeeeAeeAenz

njynxeAenz

nzeeAeenx

o

o

oo

o

ojj

jjnjjnjj

jjnjjjnjjnjj

njj

njj

oo

oooo

oooooo

o

o

)5/17/2cos(3][

3809.

4271.2cos/]0[

5/62831.361242.

93247.19492.1tan

)tan(361242.93247.19492.14271.2

9002.2

)tan()sin()cos(cos

)sin()sin()cos()cos(

cos

)cos(

]0[

]1[

)cos(]1[;cos]0[

1

j

j

j

jj

jjjj

jj

-

---

-

-

--

-

nnx

xA

x

x

AxAx

oooo

o

o

+rphzrun - new jersey institute of technologyjoelsd/fundamentals/homework/bme... · 2019. 11....

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