[PPT] - Information Transmission, Chapter 2, Sinusoidal functions & the PowerPoint Presentation

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Information Transmission, Chapter 2, Sinusoidal functions & the Fourier transform

OVE EFORS ELECTRICAL AND INFORMATION TECHNOLOGY

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Learning outcomes

T

h e s t u d e n t s h

u

l d

– u

n d e r s t a n d h

w

s i n u s

i

d a l i n p u t s t

L

T I s y s t e m s g e n e r a t e s i n u s

i

d a l

u

t p u t s ( s i n u s

i

d a l s b e i n g e i g e n f u n c t i

n

s

f

L T I s y s t e m s ) ,

– b

e a b l e t

c

a l c u l a t e t h e f r e q u e n c y f u n c t i

n

/ t r a n s f e r f u n c t i

n
f

a n L T I s y s t e m ,

– u

n d e r s t a n d a n d b e a b l e t

c

a l c u l a t e t h e F

u

r i e r t r a n s f

r

m

f

a t i m e s i g n a l , u s i n g a n i n t e g r a l ,

– u

n d e r s t a n d h

w

t h e F

u

r i e r t r a n s f

r

m r e l a t e s t

t

h e f r e q u e n c y c

n

t e n t ( s p e c t r u m )

f

a s i g n a l ,

– b

e a b l e t

u

s e F

u

r i e r t r a n s f

r

m p r

p

e r t i e s a n d F

u

r i e r t r a n s f

r

m p a i r s l i s t e d i n t h e f

r

m u l a c

l

l e c t i

n

t

q

u i c k l y fi n d F

u

r i e r t r a n s f

r

m s ,

– u

n d e r s t a n d t h e r e l a t i

n

s h i p b e t w e e n c

n

v

l

u t i

n

( i n t i m e ) a n d m u l t i p l i c a t i

n
f

F

u

r i e r t r a n s f

r

m s ( i n f r e q u e n c y ) , a n d h

w

i t c a n b e u s e d t

s

i m p l i f y a n a l y s i s

f

L T I s y s t e m s .

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Wh e r e a r e w e i n t h e B I G P I C T U R E ?

Electronics for analog input and output, including sampling and reconstruction. Models of transmission and storage media. Lecture relates to pages 43–58 in textbook.

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On the importance of being sinusoidal

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Two notations for frequency: Hertz radians per second

F r e q u e n c y n

t

a t i

n

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S

m

e t r i g

n
m

e t r i c i d e n t i t i e s

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S

m

e t r i g

n
m

e t r i c i d e n t i t i e s

8b

Typo in the book

n page 45, Table 2.2.

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E u l e r ’ s f

r

m u l a

In school we all learned about complex numbers and in particular about Euler's remarkable formula for the complex exponential where is the real part is the imaginary part

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A c

m

p l e x i n p u t s i g n a l s p l i t i n t

i

t s r e a l a n d i m a g i n a r y p a r t

IF IMPULSE RESPONSE IS REAL

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Complex sinusoidal input to an LTI system

In the previous lecture we learned that the output from an LTI system with impulse response h(t) is calculated as the convolution Using as input, we get The same sinusoidal as output … but multiplied by a complex number, which typically depends on the frequency of the sinusoidal.

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Tie t r a n s f e r f u n c t i

n

is called the frequency function or the transfer function for the LTI system with impulse response h(t).

Remember the two notations for frequency:

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P h a s e a n d a m p l i t u d e f u n c t i

n

s

The frequency function is in general a complex function of the frequency: where is called the amplitude function and is called the phase function.

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F i n a l l y …

For a linear, time-invariant system with a (bi-infinite) sinusoidal input, we always

btain a (bi-infinite) sinusoidal output!

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The Fourier transform

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Tie t r a n s f e r f u n c t i

n

is the frequency function (transfer function) for the LTI system with impulse response h(t). The frequency function H(f0) specifies how the amplitude and phase of the sinusoidal input of frequency f0 are changed by the LTI system.

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F r e q u e n c y c

n

t e n t

f

a p u l s e ?

Which frequencies does a pulse contain?

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Tie F

u

r i e r t r a n s f

r

m

There is a mathematical way of solving this problem, namely using the Fourier transform of the signal x(t) given by the formula This function is in general complex: where is called the (amplitude) spectrum of x(t) and its phase angle.

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S p e c t r u m

f

a c

n

s i n e

Consider now the sinusoidal signal where Which frequencies does it contain? In order to answer this fundamental question we use Euler's formula as

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S p e c t r u m

f

a c

n

s i n e

Hence we have a Fourier transform pair

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F r e q u e n c y c

n

t e n t

f

a p u l s e ?

Which frequencies does a pulse contain?

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P r

p

e r t i e s

f

t h e F

u

r i e r t r a n s f

r

m

1. Linearity
2. Inverse
3. Translation (time shifting)
4. Modulation (frequency shifting)

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P r

p

e r t i e s

f

t h e F

u

r i e r t r a n s f

r

m

5. Time scaling
6. Differentiation in the time domain
7. Integration in the time domain
8. Duality

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P r

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t h e F

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r

m

9. Conjugate functions
10. Convolution in the time domain
11. Multiplication in the time domain
12. Parseval's formulas

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F

u

r i e r t r a n s f

r

m

f

a c

n

v

l

u t i

n

Since the output y(t) of an LTI system is the convolution of its input x(t) and impulse response h(t) it follows from Property 10 (Convolution in the time domain) that the Fourier transform of its output Y(f) is simply the product of the Fourier transform of its input X(f) and its frequency function H(f), that is,

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S

m

e u s e f u l F

u

r i e r t r a n s f

r

m p a i r s

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S

m

e u s e f u l F

u

r i e r t r a n s f

r

m p a i r s

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S

m

e u s e f u l F

u

r i e r t r a n s f

r

m p a i r s

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S

m

e u s e f u l F

u

r i e r t r a n s f

r

m p a i r s

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S

m

e u s e f u l F

u

r i e r t r a n s f

r

m p a i r s

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E x a m p l e

What is the spectrum of a modulated rect signal?

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S u m m a r y

S

i n u s

i

d s – r e a l a n d c

m

p l e x

F

r e q u e n c y a n d a n g u l a r f r e q u e n c y

S

i n u s

i

d s a r e ” e i g e n f u n c t i

n

s ”

f

L T I s y s t e m s ( w i t h c

m

p l e x s i n u s

i

d

n

i n p u t , a s i n u s

i

d w i t h t h e s a m e f r e q u e n c y

n

t h e

u

t p u t , m u l t i p l i e d b y a c

m

p l e x n u m b e r )

T

r a n s f e r f u n c t i

n
f

a n L T I s y s t e m a n d i t s p h a s e a n d a m p l i t u d e f u n c t i

n

s

T

h e F

u

r i e r t r a n s f

r

m ( ” d e r i v e d ” f r

m

t h e t r a n s f e r f u n c t i

n
f

a n L T I s y s t e m )

– F

r e q u e n c y c

n

t e n t

f

s i g n a l s

– F

u

r i e r t r a n s f

r

m p r

p

e r t i e s

– F

r

u r i e r t r a n s f

r

m p a i r s

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