dtft details

8/17/2019 Dtft Details

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EECE 301

Signals & Systems

Prof. Mark Fowler

Note Set #19

• D-T Signals: DTFT Details


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DTFT Details

dt et x X t j )()(

In rad/sec radians

Compare to CTFT:

n j

n en x X

][)(DT frequency

in rad/sample

rad = (rad/sample) “sample”

Define the DTFT:

Very similar structure… so we should expect

similar properties!!!

What we saw: That the conceptual CTFT inside the DAC can also becomputed from the samples… we called that thing the DTFT.


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Example of Analytically Computing the DTFT

n

][n x

-3 -2 -1 1 2 3 4 5 6

With your brain,not a computer

qn

qna

n

n x n

,0

0,

0,0

][

q = 3

oscillates ][,0

explodes"" ][,1

decays ][,1

n xa If

n xa If

n xa If

By definition:

n

n jen x X ][)(

q

n

n jq

n

n jn aeea00

)(

Given this signal model, find the DTFT.

j

q j

ae

ae X

1

1)(

1

r

r r r

qqq

qn

n

1

1212

1

General Form for

Geometric Sum:


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Characteristics of DTFT

1.Periodicity of X ()

X () is a periodic function of with period of 2

)()2( X X Recall pictures in notes of “DTFT Intro”:

Note: the CTFT does not

have this property

2. X () is complex valued (in general)

n

n jen x X ][)( complex

Usually think of X () in polar form:

)()()( X je X X

magnitude

phaseSame

as

CTFT

|X ()| is periodic with period 2

X () is periodic with period 2


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3. Symmetry

If x[n] is real-valued, then:

)()( X X

)()( X X

(even symmetry)

(odd symmetry)

Same as CTFT

Inverse DTFT

Q: Given X () can we find the corresponding x[n]?A: Yes!!

d e X n x

jn)(2

1][

We can integrate instead over

any interval of length 2

…because the

DTFT is periodic

with period 2


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Generalized DTFT

Periodic D-T signals have DTFT’s that contain delta functions

elsewhere periodic

X nn x,

),(2)(,1][

With a period of 2

X () 2 2 2 2 2

2 2 3 3Main part

k

k X )2(2)(

Another way of writing this is:

Example:


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How do we derive the result? Work backwards!

d e X n x

jn)(2

1][

d e jn)(2

21

0 jne

1

Sifting property


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Transform Pairs: Like for the CTFT, there is a table of common pairs (See Web)

Be familiar with themCompare and contrast them with the table

Of common CTFT’s


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DTFT of a Rectangular Pulse

otherwise

qqnn pq

,0

,,1,0,1,,,1][ as pulse T- D:Define

Subscript tells howfar “left and right”

So, by DTFT definition:

q

qn

jn

q eP )(

Use “Geometric

Sum” Result…

2/sin

)2/1(sin

1)(

)1(

q

e

eeP

j

q j jq

q

See book for

details


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Properties of the DTFT (See table provided)

Like for the CTFT, there are many properties for the DTFT. Most are identical tothose for the CTFT!!

But Note: “Summation Property” replaces Integration

There is no “Differentiation Property”

Compare and contrast these with the table of CTFT properties

Most important ones:

-Time shift

-Multiplication by sinusoid… Three “flavors”-Convolution in the time domain

-Parseval’s Theorem


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This one has no equivalent on

CTFT Properties Table…See next example It provides a way to use a CTFT table to findDTFT pairs… here is an example

Comparing Properties of DTFT & CTFT


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Example: Finding a DTFT pair from a CTFT pair

2--2

X ()

Say we are given this DTFT and want to invert it…

The four steps for using “Relationship to Inverse CTFT” property are:

1. Truncate the DTFT X() to the - to range and set it to zero elsewhere

2. Then treat the resulting function as a function of

… call this

(

)

B-B

2--2

() = X () p2 ()

B-B

() = X () p2 ()


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4. Get the x[n] by replacing t by n in (t )

From CTFT table:

t

B Bt

sinc)(

n

B Bt n x

nt

sinc)(][

3. Find the inverse CTFT of () from a CTFT table, call it (t )


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Example of DTFT of sinusoid

?)()cos(][ 0 X nn x

Note that: )cos(1][ 0nn x

elsewhere periodic2π

),(2)(

Y

So… use the “mult. by sinusoid”

property

1][

n y

k

k Y )2(2)(

From DTFT

Table

Another way of writing this:

Y ()

k = 0

term

k = 1

term

k = 2

term

k = -2

term

k = -1

term

2 3 4--2-3-4


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)()(2

1)( 00 Y Y X “mult. by

sinusoid”

property sayswe shift up &

down by 0

elsewhere periodic X

2

,)()()(

00

Recall: )cos(1][ 0nn x so we can use the “mult. by sinusoid” result

Using the second form for Y () gives:

Or…using the first form for Y () gives:

k

k k Y )2()2()( 00


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To see this graphically:

elsewhere periodic2π

),(2)(

Y

elsewhere periodic

X

2

,)()()(

00

0-0

Red comes from Up-shifted Y ()

Blue comes from Down-shifted Y ()

X ()

2 3 4--2-3-4

Y ()

2 3 4--2-3-4

0


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Comment on Some DTFT Forms on the Table

The last four entries on the DTFT Pairs Table are:

• Note that each of them has a summation… where the summation just

adds in terms that are shifted by 2• Note that because of this shift, only the k = 0 term lies between – and • Thus… we could more simply state these by writing only the k = 0 term

and stating that the result is 2-periodic elsewhere… like this:

n B B

sinc2 ( ),

2 -periodic elsewhere

B p

dtft details

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