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UNIT-II FOURIER TRANSFORM

Learn how Fourier transform analyzes aperiodic signals in terms of frequency components, properties, and applications in signal processing.

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UNIT-II FOURIER TRANSFORM

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  1. UNIT-IIFOURIER 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. Continuous Time Discrete Time Fourier Series Discrete Fourier Transform Periodic Continuous Fourier Transform Fourier Transform Aperiodic Introduction Contd.

  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 in time-domain in frequency-domain Signal Analysis and Processing （1）Time Domain Analysis: t-A （2）Frequency Domain Analysis: f-A Fourier Transform 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 nonperiodicsignals Discrete periodic signals

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

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

  28. Conclusion （1）Sampling in time domain brings periodicity in frequency domain. （2）Sampling in frequencydomain 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-periodicFourier Transform Discrete non-periodic Continuous periodicSequence’s Fourier Transform Discrete periodic Discrete periodic Discrete Fourier Series Periodic  Discrete; NonperiodicContinuous

  29. (1) Discrete (2) Finite length 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: 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

  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

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