1. UNIT-II FOURIER TRANSFORM

2. Introduction
We have seen that periodic signals can be represented with the Fourier series
Can aperiodic signals be analyzed in terms of frequency components?
Yes, and the Fourier transform provides the tool for this analysis

3. Introduction Contd. Time Discrete Fourier Periodic Series Transform
Continuous
Time
Periodic
Aperiodic

4. Fourier Transform in the General Case
Given a signal x(t), its Fourier transform is defined as
A signal x(t) is said to have a Fourier transform in the ordinary sense if the above integral converges

5. Existence of Fourier Transform
The integral does converge if
the signal x(t) is “well-behaved”
and x(t) is absolutely integrable, namely,
Note: well behaved means that the signal has a finite number of discontinuities, maxima, and minima within any finite time interval

6. Example: The DC or Constant Signal
Consider the signal
Clearly x(t) does not satisfy the first requirement since
Therefore, the constant signal does not have a Fourier transform in the ordinary sense

7. Rectangular Form of the Fourier Transform
Consider
Since in general is a complex function, by using Euler’s formula

8. Fourier Transform of Real-Valued Signals
If x(t) is real-valued, it is
Moreover
whence
Hermitian symmetry

9. Inverse Fourier Transform
Given a signal x(t) with Fourier transform , x(t) can be recomputed from by applying the inverse Fourier transform given by
Transform pair

10. Properties of the Fourier Transform
Linearity:
Left or Right Shift in Time:
Time Scaling:

11. Properties of the Fourier Transform
Time Reversal:
Multiplication by a Power of t:
Multiplication by a Complex Exponential:

12. Properties of the Fourier Transform
Multiplication by a Sinusoid (Modulation):
Differentiation in the Time Domain:

13. Properties of the Fourier Transform
Integration in the Time Domain:
Convolution in the Time Domain:
Multiplication in the Time Domain:

14. Properties of the Fourier Transform
Parseval’s Theorem:
Duality: if

15. Generalized Fourier Transform
Fourier transform of
Applying the duality property
generalized Fourier transform of the constant signal

16. Generalized Fourier Transform of Sinusoidal Signals

17. Fourier Transform of Periodic Signals
Let x(t) be a periodic signal with period T; as such, it can be represented with its Fourier transform
Since , it is

18. Fourier Transform of the Unit-Step Function
Since
using the integration property, it is

19. Sampling Sampling is a continuous to discrete-time conversion.
Most common sampling is periodic
If a function x(t) contains no frequencies higher than B Hz, it is completely determined by giving its ordinates at a series of points spaced 1/(2B) seconds apart.

20. Types of Sampling

21. Sampling Theorem
Statement: A continuous time signal can be represented in its samples and can be recovered back when sampling frequency fs is greater than or equal to the twice the highest frequency component of message signal. i. e.
fs≥2fm.

22. Nyquist sampling rate for low-pass and band pass signals

23. Aliasing Effect
Aliasing is a phenomenon where the high frequency components of the sampled signal interfere with each otherbecause of inadequate sampling fs  < 2fm .
The overlapped region in case of under sampling represents aliasing effect, which can be removed by
considering fs >2fm
By using anti aliasing filters.

24. Fourier Transform Signal Analysis and Processing
（1）Time Domain Analysis: t-A
（2）Frequency Domain Analysis: f-A
Fourier
Transform
这里是否应该加入对频谱分析的一点解释呢？
in time-domain
in frequency-domain
In some situation, signal’s frequency spectrum can represent its characteristics more clearly.

25. ？ Fourier Transform Signal Analysis and Processing:
（1）Time Domain Analysis
（2）Frequency Domain Analysis
Fourier Transform is a bridge from time domain to frequency domain.
Characteristic: continuous—discrete, periodic—nonperiodic .
Type:
Continuous periodic signals ？
Continuous nonperiodic signals
Discrete nonperiodic signals
Discrete periodic signals

26. Continuous non-periodic function’s Fourier Transform
Time-domain
Frequency-domain
Conclusion : Continuous non-periodic function—
Non-periodic continuous function

27. Discrete-time non-periodic sequence’s Fourier Transform
Time-domain
Frequency-domain
Conclusion : Discrete non-periodic function—
Continuous-time periodic function

28. Periodic  Discrete; NonperiodicContinuous
Conclusion
（1）Sampling in time domain brings periodicity in frequency domain.
（2）Sampling in frequency domain brings periodicity in time domain.
（3）Relationship between frequency domain and time domain
Time domain Frequency domain Transform
Continuous periodic Discrete non-periodic Fourier series
Continuous non-periodic Continuous non-periodic Fourier Transform
Discrete non-periodic Continuous periodic Sequence’s Fourier
Transform
Discrete periodic Discrete periodic Discrete Fourier Series
Periodic  Discrete; NonperiodicContinuous

29. Basic idea of Discrete Fourier Transform
In practical application, signal processed by computer has two main characteristics:
(1) Discrete
(2) Finite length
Similarly, signal’s frequency must also have two main characteristics:
(1) Discrete
(2) Finite length
But nonperiodic sequence’s Fourier Transform is a continuous function of , and it
is a periodic function in  with a period 2. So it is not suitable to solve practical
digital signal processing.
Idea:
Expand finite-length sequence to periodic sequence, compute its Discrete Fourier Series, so that we can get the discrete spectrum in frequency domain.

30. Discrete Fourier Transform-DFT
Periodic sequence and its DFS

31. Discrete Fourier Transform-DFT
Periodic sequence is infinite length.
but only N sequence values contain information.
Periodic sequence  finite length sequence.
Relationship between these sequences?
Infinite  Finite
Periodic  Nonperiodic
周期序列和有限长序列存在本质上的联系

32. Discrete Fourier Transform-DFT
Relationship between periodic sequence and finite-length sequence
Periodic sequence can be seen as periodically copies of finite-length sequence.
Finite-length sequence can be seen as extracting one period from periodic sequence.
Finite-duration Sequence
Periodic Sequence
Main period

33. Properties of DTFT

34. Properties of DTFT

35. Convolution Property

36. Multiplication Property