Post on 06-Jul-2018
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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