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

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

8. DFT IDFT Observations: The DFT and the IDFT form a transform pair.
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:

11. 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

12. 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

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

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

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

16. 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?

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

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