EECE 301 Signals & Systems Prof. Mark Fowlerws.binghamton.edu/fowler/Fowler Personal...
Transcript of EECE 301 Signals & Systems Prof. Mark Fowlerws.binghamton.edu/fowler/Fowler Personal...
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EECE 301 Signals & SystemsProf. Mark Fowler
Note Set #19• D-T Signals: DTFT Details
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DTFT Details
dtetxX tj )()(
In rad/sec radians
Compare to CTFT:
nj
nenxX
][)(
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 be computed from the samples… we called that thing the DTFT.
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Example of Analytically Computing the DTFT
n
][nx
-3 -2 -1 1 2 3 4 5 6
With your brain, not a computer
qn
qnan
nx n
,00,
0,0][
q = 3
oscillates ][,0
explodes"" ][,1
decays ][,1
nxaIf
nxaIf
nxaIf
By definition:
n
njenxX ][)(
q
n
njq
n
njn aeea00
)(
Given this signal model, find the DTFT.
j
qj
aeaeX
11)(
1
rrrr
qqq
qn
n
1
1212
1
General Form for Geometric Sum:
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Characteristics of DTFT1.Periodicity of X()
X() is a periodic function of with period of 2
)()2( XX Recall pictures in notes of “DTFT Intro”:
Note: the CTFT does nothave this property
2. X() is complex valued (in general)
n
njenxX ][)( complex
Usually think of X() in polar form:)()()( XjeXX
magnitudephase Same
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:
)()( XX
)()( XX
(even symmetry)
(odd symmetry)
Same as CTFT
Inverse DTFT
Q: Given X() can we find the corresponding x[n]?
A: Yes!!
deXnx jn)(
21][
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
elsewhereperiodic
Xnnx,
),(2)(,1][
With a period of 2
X()2 2222
2 2 33
Main part
k
kX )2(2)(
Another way of writing this is:
Example:
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How do we derive the result? Work backwards!
deXnx jn)(
21][
de 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 them Compare and contrast them with the tableOf common CTFT’s
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DTFT of a Rectangular Pulse
otherwise
qqnnpq
,0
,,1,0,1,,,1][ as pulse T- D:Define
Subscript tells how far “left and right”
So, by DTFT definition:
q
qn
jnq eP )(
Use “Geometric Sum” Result…
2/sin
)2/1(sin1
)()1(
qeeeP j
qjjq
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 to those 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 exampleIt provides a way to use a CTFT table to find DTFT 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 elsewhere2. 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:
tBBt
sinc)(
nBBtnx
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 Xnnx
Note that: )cos(1][ 0nnx
elsewhereperiodic2π
),(2)(
Y
So… use the “mult. by sinusoid” property
1][ ny
k
kY )2(2)( From DTFT
Table
Another way of writing this:
Y()
k = 0term
k = 1term
k = 2term
k = -2term
k = -1term
2 3 4--2-3-4
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)()(21)( 00 YYX “mult. by
sinusoid” property says we shift up & down by 0
elsewhereperiodicX
2
,)()()(
00
Recall: )cos(1][ 0nnx 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
kkY )2()2()( 00
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To see this graphically:
elsewhereperiodic2π
),(2)(
Y
elsewhereperiodicX
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:
nBB sinc 2 ( ),
2 -periodic elsewhereBp