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Fourier Analysis of Discrete Time Signals

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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Fourier Analysis of Discrete Time Signals

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  1. 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.

  2. The Discrete Time Fourier Transform (DTFT) Given a sequence x(n) having infinite duration, we define the DTFT as follows: ….. ….. continuous frequency discrete time

  3. 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

  4. Example: since

  5. Example:

  6. DFT Discrete Fourier Transform (DFT) Definition(Discrete Fourier Transform): Given a finite sequence its Discrete Fourier Transform (DFT) is a finite sequence where

  7. IDFT Definition (Inverse Discrete Fourier Transform): Given a sequence its Inverse Discrete Fourier Transform (IDFT) is a finite sequence where

  8. 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.

  9. DFT as a Vector Operation Let Then:

  10. 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

  11. 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

  12. A B C D D If we look at just one period we can define the circular shift A B C D D

  13. Properties of the DFT: • one to one with no ambiguity; • time shift • where is a circular shift periodic repetition

  14. real sequences • circular convolution where both sequences must have the same length N. Then:

  15. 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?

  16. First see the periodic extension, which looks like this: Then look at the period

  17. Example: determine the DFT of the finite sequence Then take the DFT of the vector

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