Fourier Analysis of Discrete Time Signals

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Fourier Analysis of Discrete Time Signals

• For a discrete time sequence we define two
  classes of Fourier Transforms
• the DTFT (Discrete Time FT) for sequences
  having infinite duration,
• the DFT (Discrete FT) for sequences having
  finite duration.

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The Discrete Time Fourier Transform (DTFT)
Given a sequence x(n) having infinite duration,
we define the DTFT as follows
..
..
continuous frequency
discrete time
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• Observations
• The DTFT is periodic with period
  
• The frequency is the digital frequency and
  therefore it is limited to the interval

Recall that the digital frequency is a
normalized frequency relative to the sampling
frequency, defined as
one period of
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Example
since
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Example
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Discrete Fourier Transform (DFT) Definition
(Discrete Fourier Transform) Given a finite
sequence
its Discrete Fourier Transform (DFT) is a finite
sequence
where
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Definition (Inverse Discrete Fourier Transform)
Given a sequence
its Inverse Discrete Fourier Transform (IDFT) is
a finite sequence
where
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• Observations
• The DFT and the IDFT form a transform pair.

DFT
IDFT
back to the same signal !

• The DFT is a numerical algorithm, and it can be
  computed by a digital computer.

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DFT as a Vector Operation
Let
Then
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Periodicity From the IDFT expression, notice
that the sequence x(n) can be interpreted as one
period of a periodic sequence
original sequence
periodic repetition
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This has a consequence when we define a time
shift of the sequence.
For example see what we mean with
. Start with the periodic extension
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If we look at just one period we can define the
circular shift
A
B
C
D
D
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• Properties of the DFT
• one to one with no ambiguity
• time shift
• where is a circular shift

periodic repetition
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• real sequences
• circular convolution

where both sequences must have the
same length N. Then
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Extension to General Intervals of Definition
Take the case of a sequence defined on a
different interval
How do we compute the DFT, without reinventing a
new formula?
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First see the periodic extension, which looks
like this
Then look at the period
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Example determine the DFT of the finite sequence
Then take the DFT of the vector